STEREOCOMPARATOR OPERATOR TRAINING MANUAL (PRELIMINARY SUBMITTAL) U.S. GOVERNMENT (Sanitized)
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STEREOCOMPARATOR
OPERATOR TRAINING MANUAL
(Preliminary Submittal)
U. S. Government
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This document is presented for review by the customer
in the form of a Preliminary Operator Training Manual
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I.
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VECTORS,
1.1
TENSORS,
Vectors
TABLE OF CONTENTS
Page
Number
and MATRICES
1
1
1.1.1
Need for Coordinate Systems
1
1.1.2
Properties of Vector Addition
1
1.1.3
Use of Vector Components
2
1.1.4
Arbitrariness of Coordinate Systems
3
1 . 1 .5
Examples of "Directed Strokes" Which
Are Not Vectors
4
1.1.6
Basic Importance of 'Components' in
Defining Vectors
5
1 .1 .7
Vectors Defined by Components
6
1 .1.8
Two Vectors Drawn from Practical Applications
8
1.1.9
Definition of' Contravariant and Covariant
Vectors
10
1.1.1 0
Use of Special Index Letters to Denote
Coordinate Systems
11
1.1.11
Short HandiNotation for Partial Derivatives
12
1.1.12
Omission of Special Sign for Summation
12
1.1.13
Summary of Conventions for Notation
13
1.1.14
Inverse Transforms
14
1.1.15
Scalar Product of Two Vectors
14
1.1.16
Alternate Derivation of Inverse Transforms
1 6
1.2
Tensors
18
1.2.1
Generalization of Vectors
18
1.2.2
General Definition of Tenslors
19
1.2.3
Addition of Tensors
20
1.2.4
Multiplication of Tensors
20
1.2.5
Contraction of Tensors
21
1.2.6
Inner Product of Tensors
21
1.2.7
Some Special Types of Tensors
22
1.2.8
The Metric Tensor
24
1.2.9
Formula for the Magnitude of A Vector
26
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Page
Number
1.2.10 Example of Use of the Magnitude Formula
27
1.2.11 Determinants and the Tensor Densities
of Levi Civita
29
1.2.12 Vector Product of Two Vectors
30
1.2.13 Two Types of Relations Which Can Be
Written for Tensors
31
1.2.14 Translation of Tensor Expressions into
FORTRAN
33
1.3
Matrices
36
1.3.1 Use of Matrices to Exhibit Components
of A Tensor
36
1.3.2 Linear Combination of Matrices
37
1.3.3 Multiplication of Matrices
38
1.3.4 Transposition of Matrices
39
1.3.5 Inverse of A Matrix
40
1.4
Cartesian Coordinate Systems
41
II.
OPTICAL
IMAGING AND AERIAL PHOTOGRAPHS
44
2.1
Gaussian Optics
44
2.2
Projective Optics
46
2.2.1 The General Equation of Optical Imaging
47
2.2.2 Invariance of the Optical Imaging Equation
SO
2.2.3 Use of Separate Coordinate Systems for
Object Space and Image Space
51
2.3
Application to Aerial Photography
52
2.3.1 Rotation of Image
53
2.3.2 Application to Frame, Strip, and Pan
Types of Photography
53
2.3.2.1 Frame Type Photography
SS
2.3.2.2 Strip Type Photography
56
2.3.2.3 Panoramic Type Photography
57
2,4
Taylor Series Expansion of the Projective Equations
59
2.4.1 Evaluation of the Partial Derivatives
61
2.4.2 Time (t) Treated as A Function of the
Ground Coordinates
62
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Number
2.4.2.1 Frame Photography 62
2.4.2.2 Strip Photography 62
2.4.2.3 Panoramic Photography 63
2.4.3 Generalized Formula for the Partial
Derivatives 64
2.4.4 Invariance of the Generalized Formula 66
2.4.5 Importance of Moving Coordinate Systems 67
III. COMPUTATION OF GROUND COORDINATES 69
3.1 Approximate Ground Computations Based on
Only One Photograph 70
3.1.1 Use of A Tangent Plane 70
3.1.2 Tangent Plane to A Spherical Earth 72
3.1.3 Plane Tangent at the Nadir 80
3.1.4 Treatment of the Photograph Coordinates
for the Different Types of Photography 81
3.1.4.1 Frame Type Photography 82
3.1.4.2 Strip Type Photography 83
3.1.4.3 Panoramic Type Photography 85
3.2 Stereoscopic Triangulation of Ground Points 86
IV. AUTOMATIC STAGE TRACKING IN THE OPERATION OF
THE STEREOCOMPARATOR 90
4.1 Automatic Without Correlator Tracking Mode 92
4.1.1 Operations Performed 92
4.1.2 Computation of Slave Stage Coordinates 93
4.1.3 Computation of Slave Photo Point Time
of Exposure 95
4.1.3.1 Frame Type Photos 96
4.1.3.2 Strip Type Photos 96
4.1.3.3 Panoramic Type Photos 97
4.1.4 Modified Computations for RealTime
Stage Control 98
4.2 Automatic With Correlator Tracking Mode 104
4.3 Digital Integration of the Control Commands 107
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V.
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Page
Number
OF DISTORTIONS FOR STEREO VIEWING
109
5.1
Conditions for Stereo Viewing
109
5.2
Elements of the Stereocomparator Optical System
111
5.3
Matrix Representation of the Functions of the
Optical Elements
114
5.3.1 Zoom Lens
117
5.3.2 Anamorphic Lens
117
5.3.3 Image Rotator
118
5.4
Equivalent Frame Images
119
5.5
Solution of Equations
122
5.6
Automatic Without Correlator Control of Optics
124
5.7
Automatic With Correlator Control of Optics
129
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OPERATOR TRAINING MANUAL
I. VECTORS, TENSORS, and MATRICES
1.1 Vectors*
1.1.1 Need for Coordinate Systems
Vectors have often been defined as entities which have
both magnitude and direction. In exploring the concept direction, one
finds that it cannot be defined except in relation to reference points or
reference directions. Any particular statement or rule which establishes
a given set of reference directions is said to thus establish a coordinate
system. Thus the above definition of vectors is made mathematically
meaningful by relating the magnitude and direction to some particular
coordinate system. In any established space, however, a given set of
reference points may be used to define a large number of different co
ordinate systems, any one of which may be suitable for expressing the
desired vector. Thus a complete definition of a vector must provide a
scheme for expressing the vector with respect to any desired coordinate
system.
1.1.2 Properties of Vector Addition
Vector analysis includes a rule for vector addition, and
this rule is purposely designed so that vector addition is both "commuta
tive" and "associative." This means that if two vectors A,and B are added
then the resultant is the same whether the vector addition is performed as
A +B (vector addition)
or as
B Link (vector addition)
*This discussion is not intended as an introduction to vector analysis.
It is written for readers who are already, at least somewhat, familiar
with most of the ideas presented.
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Furthermore if three vectors A, B, and C are added, then the resultant
Is the same whether any of the following schemes is employed:
A + B + C,
(A + B) + C,
A ?(B + C),
A ?(c + B),
(C + A) + B, etc.
1.1.3 Use of Vector Components
In implementing the ideas presented in the foregoing two
paragraphs one arrives at the notion of resolving vectors into other
vectors such that the latter may be vectorially added to produce the
original vectors as resultants. Thus assume that a coordinate system
x, y, z is to be used for the vector addition of two vectors A and B.
The vector A is first resolved into the three component vectors Ax,
A , and Az such that vectorial addition of these component vectors
yields the vector A as resultant. Similarly the vector B is resolved
into component vectorsB B' and Bz which vectorially added produce
x' y
the vector B as resultant. The above stated properties of vectorial
addition imply that the vector sum of the vectors A and B must be the
same as the vector sum of resultants obtained by vectorially adding
corresponding components. The rule of vector addition is also such
that the vector sum of two vectors which have the same direction is
a vector with this common direction whose magnitude is the algebraic
sum of the two magnitudes. Thus the vector sum of the vectors A and
815 the same as the vector sum of the three component vectors
A + B, A + B, and A+ B
x x y y z
z
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where each component is obtained by algebraic addition of magnitudes.
Thus one way of implementing vector addition is by resolving the
vectors into components with respect to some established coordinate
system and defining the vector sum as that vector whose components
are the algebraic sums of corresponding components of the vectors
being added. This turns out to be a practical scheme for computing
vector sums.
1.1.4 Arbitrariness of Coordinate Systems
From all of the foregoing it follows that there is considerable
arbitrariness in selecting a coordinate system but that any vector sum
must be, in some sense, independent of which particular coordinate
system is chosen. Thus if two coordinate systems x, y, z and u, v, w
are equally suitable, then the vector sum of A and B is the vector whose
components in the two coordinate systems are
Ax +Bx ,Ay +By ,A +B
z z
and
Au + Bu, Av + Bv, Aw + Bw
respectively. Vector addition of these two sets of component vectors
must therefore yield resultants which are equal in magniture and equiva
lent in direction. The concepts "magnitude" and "direction" must then
be so related to any set of components as to have equal (or at least
equivalent) values regardless of which suitable coordinate system is
being employed.
If the "suitable" coordinate systems were to be limited
to only those which are called Cartesian coordinate systems (i.e.,
systems with 3 mutually perpendicular axes with fixed directions) then
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magnitude might be defined by the Pythagorean rule:
= 1V2+V21V
x
where V is any vector (including those obtained by adding other
vectors). It is desirable, however, to include more general co
ordinate systems as being "suitable" (for example, spherical
coordinates, which may consist of the distance from earth's center,
longitude, and geocentric latitude, respectively). Thus the defini
tion of "magnitude" must be a generalized version of the Pythagorean
relation which is suitable for the most general coordinate system to
be employed (a correspondifig discussion of "direction" will be
omitted).
1.1.5 Examples of "Directed Strokes" Which Are Not Vectors
Before continuing, some examples will be given of entities
which have magnitude and direction but which do not conform to the
mathematical relations which are customarily employed in vector
analysis. (The point is that, in actual usage, the term "vector" is
properly used to designate only a certain subclass of all possible
entities which have magnitude and direction). Suppose there are three
noncoplanar vectors A, B, and C which are so established as to have
their tails all conjoined in one point. Then one may draw three directed
strokes to represent the "vector angles" between the three pairs of
vectors.
Thus the directed stroke a has a direction perpendicular to
both of the vectors A and B and a magnitude equal to the angle between
these two vectors (this angle being in the plane of the two vectors).
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Similarly the directed stroke 13, has a direction perpendicular to the
two vectors B and C and a magnitude equal to the angle between
these two vectors. Finally the directed stroke y has a direction
perpendicular to the two vectors A and C and a magnitude equal to
the angle between these two vectors. With these definitions it may
be demonstrated that the directed stroke y is, in general, not equal
to the vector sum of the directed strokes a and p. Thus it is not.
strictly proper to call the directed strokes a, p and y by the name
"vectors."
Somewhat similarly, directed strokes which join various
points in space behave like "true" vectors only if the allowable
coordinate systems are restricted to those in the same class as
Cartesian coordinate systems. As was noted previously, this is
not, in general, a desirable restriction.
1.1.6 Basic Importance of 'Components' in Defining Vectors
Evidently then it is not entirely satisfactory to begin a
deductive discussion of vectors by defining them as entities having
magnitude and direction (although the concept of the directed stroke
is, nevertheless, extremely useful in qualitative visualization of
vectors). Present day general discussions of vector analysis are
often built around the calculus of partial derivatives which may, at
first, seem very far afield. This, however, turns out to be a very
satisfying procedure. Most practical computations of vector relations
operate by manipulating the components of the vectors  hence the
latter are of fundamental importance. It turns out then, that, if needed,
the direction and magnitude can be obtained from the components.
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1.1.7 Vectors Defined by Components
Thus a vector is, first of all, an entity which has as many
components as there are dimensions in the space being considered
(i.e., 3 components in 3 dimensional space or N components in N
dimensional space, where N is a positive integer). In some computa
tions nothing more is said about the vectors being used, but usually
the vectors can be assumed to have magnitudes (and directions) even
though it may not be of interest to compute these properties. Usual
computing practice is to use one or more coordinate systems and to
state vector components relative to these coordinate systems. If all
the computations are carried out in only one coordinate system then
it is not necessary to evaluate the components relative to any other
coordinate system. Vector theory assumes, however, that any one
coordinate system defines a special case and that (at least poten
tially) there are a large number of other equally good coordinate
systems. Thus there must be a set of rules whereby, if the com
ponents of a vector are known in any one coordinate system, they
can be evaluated in any other "suitable" coordinate system. The
various relations which are used for operating on and/or combining
vectors must then be equally valid in all "suitable" coordinate
systems. This concept is known as "invariance" (sometimes
"covariance") and is fundamental in much of present day mathematics.
The answer to the question "What constitutes a "suitable" coordinate
system?" has a long history of evolution, but presently it includes
about any coordinate system whieh can be imagined.
Suppose then that there is an x, y, z coordinate sy,stem
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(defined somehow) and that relative to this a u, v, w coordinate
system can be computed by the (quite general) functional relations
u = f1 (x, y, z)
v = f2 (x, y, z)
w = 13 (x, y, z):
Suppose, furthermore, that these functions can be solved to yield
the inverse relations
x = g1 (u, v, w)
y = g2 (u, v, w)
z = g3 (u, v, w).
The first set of functions will be assumed to have all of the nine first
order partial derivatives:
au af1 au 8f1 au 811
ax ax' ay ay' az az'
ay 812 av af2 av 8f2
ax ax' ay= ay' az = az'
aw af3 aw af3 aw af3
ax ax' ay= ay' az= az '
Likewise the second set of functions will be assumed to have all of
the nine first order partial derivatives
a x a x
Su ay'
(Sometimes higher order partial derivatives are also assumed, but
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these need not be considered here.) Since the second set of functions
are inverse to the first set (i.e., substituting the values of x, y, and
z given by the second set of functions into tht first set of functions
will result in identities in u, v, and w), the two sets of partial
derivatives must satisfy the nine relations
L
ax au ay au az au
Du ax + au _Di + au az =
ax Dv ay ay az Dv
au ax + au + au az = 0
ax aw ay aw az aw '
etc.
Evidently there is serious need for some short hand notation in
representing these relations.
Hence let
x. = (x, y, z);
(i = 1,
2,
3)
and
(u, v, w);
(i = 1,
2,
3)
Then the above relations may be written in the form:
3 Du. ax.
1 3
ax. Du
= ) k
fl if i = k
10 if i j k
Similarly, a set of inverse relations must also hold:
3 ax.
u
= I 3
jl if i = k
 10 if i k
(In both cases the relations are true for all combinations of i and k
independently taking values 1, 2, or 3.)
1.1.8 Two Vectors Drawn From Practical Applications
In vector analysis two vectors which occur quite frequently
are the differential displacement
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dr = (dx, dy, dz)
= (dxi); (i = 1, 2, 3)
and the gradient of a scalar field
Lt\
= laX ay aZ
(ax.); (1= 1, 2, 3)
Note that these vectors are stated by defining their components relative
to the x1 . coordinate system. The corresponding components relative
to the u, coordinate system are then (by the rules of differential
calculus):
and
3 au,
i
du. = 27, dx.?'
1 3
i = 1 ax,.1
(i = 1, 2, 3)
3 a 4) ax,
= E     ' ? (i = 1, 2, 3) .
au, ax. au.
1 j = 1 1
These two transformation laws have similar form (linear) but differ in
that one uses the partial derivatives
au.
1
ax.
as coefficients of the vector components relative to the xi system whereas
the other uses the inverse partial derivatives
ax
au.
1
as coefficients of the vector components relative to the xi coordinate
system.
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1.1.9 Definition of Contravariant and Covariant Vectors
Using the above two vectors as models we, more generally,
define a vector A as having contravariant components
(Az, Ay, Az) = Axi; = 1, 2, 3)
if the appropriate transformation law (which gives the components
relative to .the u1 . coordinate system) is
3 au.
Au. =  ;?c A_ (i = 1, 2, 3).
1 j = 1 j i`j
Similarly we define a vector B as having covariant components
(B , B , B ) = B ; (i = 1, 2, 3)
x y z x.
if the appropriate transformation law is
3 ax.
B =e (i = 1, 2, 3).
ui j = 1 Oui xj
Thus our calculus leads to two basic types of vector: the
contravariant and the covariant type.* For convenience, a notation is
commonly used which marks any particular components as to which type
they are. This notation uses superscripts (which should not be con
fused with exponents) for contravariant components and it uses sub
scripts for covariant components. Since the differential displacement
dr takes contravariant components the latter are represented by
dxi (i = 1, 2, 3) rather than by dx1*
, Correspondingly the coordinates
themselves are therefore represented by xi and ui (i = 1, 2, 3) rather
than by xi and ui as was done previously.** Thus the two transforma
tion laws may be written in the forms:
*In some discussions it is assumed that the vector entity exists above
and beyond its components and that it may be represented by components
of either type. Here, components of one type (only) will be used for
some vectors and components of the other type (only) will be used for
other vectors.
**Note that the change in notation does not imply any change whatsoever
in meaning.
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and
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: 3
1, au
A = . A; (contravariant)
j = 1 axJ
3
axi
B. = B.; (covariant
1 j = 1 au' )
1.1.10 Use of Special Index Letters to Denote Coordinate Systems
While it is somewhat traditional to use the letters i, j,
and k for indices (subscripts and superscripts) there is really no
reason for not using other letters (or even more general symbols)
as well, if some purpose is served by doing so. In succeeding
chapters there will be occasion to use.a number of different
coordinate systems and it will avoid ambiguities if a notation is
used which identifies the coordinate system which any particular
set of components is relative to. Hence a unique set of letters
will be used as indices for each different coordinate system. For
now, only two different coordinate systems will be distinguished,
but these two are to be thought of as generalized representations
of all pairs of coordinate systems which may be introduced later.
The letters a, b, c, ? ? ? will be used as indices for one
representative coordinate system, and the letters m, n, p, ? ? ? will
be used as indices for the other representative coordinate system
(note that any index is assumed to represent any positive integer
from 1 to N, where N is the number of dimensions in the space being
considered). With this convention the two sets of coordinates are
represented by:
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and
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xa x, y, z;
a = 1, 2, 3
xm= u, v, w; m = 1, 2, 3.
At times xa will be replaced by xb or xc (to avoid ambiguities which
would otherwise occur). This does not imply any change in meaning.
Similarly xm will sometimes be replaced by xn or xP, but no different
meaning is intended.
written as:
and
With this notation the two transformation laws may be
3
Am = ax Aa; (m = 1, 2, 3)
a = 1 axa
3 ?a
ax
B = 2;
B; (m = 1, 2, 3) .
a=1 axm a
1.1.11 Short Hand Notation for Partial Derivatives
am a
ax
Because the two sets of partial derivatives ? and
ax a axm
occur so frequently it will be convenient to represent them by the short
hand symbols Xam and Xam respectively. Then the transformations become:
and
3 m a
Am = )c A
3
a
Bm = Xm Ba
a = 1
1.1.12 Omission of Special Sign for Summation
3
Finally it may be observed that the summation symbol E is
a = 1
redundent, since the index "a" occurs in such a way that it alone may
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be considered as signifying the same thing which was heretofore
signified by the combined presence of the summation symbol and
an index used in this particular manner. Thus the transformation
formulae will be written in the form:
and
m m a
A =Xa A
Bm =a Xm Ba
which mean precisely the same things as though the summation
3
symbol 1, were written immediately after the = sign in each
a= 1
case. Note that there is also an implication that the formulae hold
for m = 1, 2, 3, but explicit statement of this fact is customarily
omitted.*
1.1.13 Summary of Conventions for Notation
To recapitulate; the short hand notation
m m a
A =Xa A
means precisely the same thing as
Likewise;
u. N au. x.
j
A
A ; (i = 1, 2, "SN).
2' a
=
B = Xa B
m ma
means precisely the same thing as
N ax.
B = 2 B ' ? (i = 1, 2, :"N),
ui j = 1 1
*In N dimensional space the implied range of all indices is 1 to N.
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where the integer N is the number of dimensions in the space being
considered.
1.1.14 Inverse Transforms
The two transformation laws may be solved to yield the
corresponding inverse transformations
and
Aa = Xa Am
Ba = Xa Bm.
Evidently these are entirely analagous to the'first forms.
1.1.15 Scalar Product of Two Vectors
Returning to the two vectors, differential displacement,
and gradient, (which will now be represented by dxa and
axa
respectively) we note that the dot (scalar) product of these may
be written in the form:
dxa
a
ax
This expression illustrates the general convention that an index which
appears both contravariantly and covariantly in the same term signifies
summation (just as though the summation symbol 21.; preceded the term).
1
Now substitute the (inverse) transformation laws for both vectors
and
act' =m ack
a a m
ax ax
a
dxa = Xm dxm.
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Then
al) dxa = 1,n1 .a q5 m ixa axril
i n '
axa
ax
where n has the same meaning as m but runs through its range of values
independently of in. This may also be written (without in the least
changing its meaning)
a _ap c dxn.
dxa Xm =
a a n
ax ax
As was stated earlier (in different notation),
m aii if m = n
XXn = 10 if m n.
Hence:
m a aq, dxn =m dxn
Xa Xn
ax ax
act) n
(=>; m
m = 1 n = 1 n ax'
dx )
=
ax
where
m = {1 if m = n
n 0 if m * n.
(f)
Thus the expression a dxa , in the xa coordinate system, transforms
axa
into the expression adxm, in the xm coordinate system. This
ax
duplication of form before and after transformation is an example of
invariance, and in this case it allows us to identify the result with
the differential c14.
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In precisely the same way we represent the dot (scalar)
product of the vectors Aa and Ba by
Aa Ba = (Xa Am) (X: Bn)
m
a n m
= X X A B
man
n m
5mA n
B= Am Bm.
Again the result is an invariant form.
1.1.16 Alternate Derivation of Inverse Transforms
By applying a somewhat similar procedure to the
transformation laws we can obtain the inverse transformation p in
straight forward fashion. Thus
Likewise:
a a
A = X Am
m
m a m a n m n
Xa A = Xa Xn A = b A
n
=Am.
Ba = Xa Bm
a a n
Xm Ba = Xm Xa Bn = bmBn
= Bm.
At this point it would perhaps be logical to introduce the
general definition of the nmagpitude" of a vector. This topic will be
more easily discussed, however, when the reader is familiar with
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ati
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the subject matter in the next section. Similarly the operation "cross
product" can be discussed more meaningfully at a later point.*
*If any part of the discussion to this point is not clear to the reader,
then the reader should substitute enough of the longer notation in
the various expressions to satisfy himself that all of the results given
really do follow logically from the definitions and from the principles
of differential calculus. The reader should not be deceived by the
brevity of the notation. The underlying meaning of these expressions
is fundamental to all that follows.
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1.2 Tensors*
1.2.1 Generalization of Vectors
Tensors are a straightforward generalization of vectors.
Whereas vectors have N components (in Ndimensional space), tensors
may have N2 or N3 or, in general, N  where M is any nonnegative
integer  components. M is then called the "order" (sometimes "rank")
of the tensor. Thus vectors are first order tensors and scalars are
zero order tensors. Like vectors, tensors may have either contravariant
or covariant components. Tensors, of order two or greater may, however,
also have mixed (i.e., partly contravariant and partly covariant) com
ponents.
A simple example of a second order tensor is the algebraic
product of two vectors. Since each vector has N components then the
product has N2 components. For example, let the vectors A and B both
have contravariant components. Then their transformation laws are:
and
Am m a
a
A = X A
m m a
B = Xa B .
Multiplying these together (and changing the indices so as to avoid
confusion):
Am Bn = (Xm Aa) (Xn B b)
a
m n
= Xa Xb Aa Bb.**
+*This discussion is intended to discuss only such aspects of tensor
analysis as are considered pertinent to the material which follows.
Standard texts are available which cover a number of topics not treated here.
**The reader should now be able to recognize that this expression means
the same thing as
Am Bn 1Xxbn Aa Bb;
m=1, ? ' ?N; = 1, ? ? ?N.
= am
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This is an example of a tensor of contravariant order 2. If the two
vectors are both covariant then the product takes the form
A B = (Xa A) (Xb B)
mn ma nb
a b
A
= Xm nabb'
and this is an example of a tensor of covariant order 2. Finally the
mixed tensor
Am
B = (Xm Aa) (Xbn Bb)
n a
= Xm Xb Aa Bb
a n
has contravariant order 1 and covariant order 1.
1.2.2 General Definition of Tensors
Thus a tensor of contravariant order p and covariant order q
has its components relative to the first representative coordinate system
represented as:
al, a2, ? ? ? a
b b2' ? ? ? b
Its components relative to the second representative coordinate system
are then:
ml' m2' ? mp ml bl m2 b2? ?
al nl a2 n2 ? ? ? a
=X X X X ? T
n2, ? ? ? n b1 ? ? ? b
Thus the number of components in both coordinate systems is
The notation on the right side of the transformation formula, of course,
means independent summation on each?of the indices al' a2' ? ? ? a ,
b b2' ? ? ? bq each of which appears both contravariantly and
'
covariantly in the same term. The tensor property resides in the fact
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that when the components are defined relative to some one coordinate
system then corresponding components relative to any other coordinate
system arc given by a linear transformation law (such as the one shown)
involving the partial derivatives Xam and the inverse partial derivatives
Xa as coefficients.
1.2.3 Addition of Tensors
The following discussion is given using third order tensors
but it could equally well be given for tensors of any order, the equations
would then differ only in having the required number of indices and of
ab
partial derivatives. Let Aab and Bc be two tensors of contravariant
order 2 and covariant order 1. Then their transformation laws are
and
mn
= X A
m n c ab
A a Xb Xp c
Bmn = Xm Xn XC Bab
a bp c ?
Adding these together then gives
Amn Bmn m xc Aabi ivm xn xe Bab)
1 a 'lb p 1"a b p
m n c ab ab
= Xa Xb Xpc + Bc ).
Thus the sum of two tensors of equal order is a tensor of the same
order as both of them.
1.2.4 Multiplication of Tensors
Likewise multiplying the above two transformations:
Amn ty
Bqr m q r f de
P s 11'a ' tx x
1D d e Xs Bf )
mncqr f abde
= Xa Xb Xp Xd Xe Xs Ac Bf .
vara
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Thus the product of two tensors is a tensor whose contravariant order
is the sum of the contravariant orders, and whose covariant order is
the sum of the covariant orders, of the two tensors being multiplied. #
1.2.5 Contraction of Tensors
Mixed tensors are also subject to an operation known as
"contraction." This operation consists of simultaneously equating
and summing on one of the contravariant and one of the covariant
indices. Thus consider the transformation formula
mn m n c ab
Ap = Xa Xb Xp Ac
Now equate and sum on the indices n and p, then:
A =X Xmn m n X A c ab
n
a bn c
m c ab
= Xa bb Ac
m ab
= Xa Ab .
Thus the result is a tensor with its order reduced by two. The above
tensor can similarly be contracted by equating and summing on the
indices m and p:
Amn m n c ab
m = Xa Xb X A
m c
n c ab
= Xb ba Ac
n ,ab
= Xb a .
This result is also a tensor with order reduced by two but it is not,
in general, equal to the one above.
1.2.6 ?Inner Product of Tensors
Two tensors may, of course, be multiplied and then
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immediately contracted on a contravariant index of one tensor and a
covariant index of the other tensor (assuming that the two tensors
have such indices). This combination of operations is sometimes
known as the "inner product" of the two tensors. If the two tensors
are both vectors (one contravariant and the other covariant) then the
result is the familiar "dot product" of the two vectors. Most of the
applications of tensors in the following material will involve either
this dot product of two vectors or else the inner product of a second
order tensor by a vector; the latter producing a vector.
1.2.7 Some Special Types of Tensors
Evidently if the components of a tensor relative to some
one coordinate system are all zero then the components relative to
any other coordinate system are likewise all zero. Evidently, also,
if the components of two tensors are respectively equal in one co
ordinate system then they are also equal in all other coordinate systems:
i.e. , if
then
Amn = Xm Xn Xc Aab
a bp c
n c ab
= Xma Xb Xp Bc
Bmn.
Tensors of contravariant order 2 or of covariant order 2, cannot, in
general, have their two indices interchanged (without altering the
value of the tensor). That is, in general,
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and
There is, however, a special class of tensor  known as symmetric 
in which
Aab = Aba
(Note  that when this is true
mn m n ab
A = Xa Xb A
m n ba
= X X A
a b
n m
= X X Aba
ba
n m ab
= Xa Xb A
= Anm .)
Likewise the covariant symmetric tensor is such that
Aab = Aba
(hence A =mn A ).
nm
There is also a special class of tensor  known as skew
symmetric  such that
Aab = Aba; (hence Amn = Anm)
or
Ab = Ab; (hence Amn = Anm)
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1.2.8 The Metric Tensor
An important example of a symmetric tensor is the socalled
"metric" tensor. This may have either contravariant or covariant
components and is usually represented by the base letter g. Thus:
and
inn m n ab
g = Xa Xb g
m n ba
= Xa Xb g
nm
g
a b
gmn =X m Xn gab
a b
=X m Xn gba
= gnm?
This metric tensor, besides being symmetric, has the important property
that its components relative to any Cartesian coordinate system are
components of the unit tensor. Thus let a and b be indices for a
Cartesian coordinate system. Then:
and
Hence
and
ab ab {1 if a = b
g = =
0 if a # b
= l if a = b
= 5
'ab ab 0 if a * b
mn m n ab
g = Xa Xb b
Xm Xn)
a a
a = 1
a b
gmn =X m Xn ab
a
Xm Xa)
a=l n
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Assume (again) that m and n are indices for a general
coordinate system but that a and b are indices for a Cartesian
coordinate system. Then:
Likewise:
np (xa
=
'mn rn n a
txn vp cd1
c
a _hiX n p
= Xm An a Xd b bcd
ab
= Xa XP 6b 6 cd
m d c ab
=
a p d
Xm Xd 6a
=
a p
Xm Xa
= op .
mn txm x n cab\ fxc xd 6
a b n p cd'
mncdab
= Xa Xb Xn Xp 6 6 cd
= xm xd bc bab
a p b cd
= Xm Xd 6a
a p d
=ma
a p
=
Thus the contravariant and covariant components of the metric tensor
are reciprocal to one another. The symbolsab,ab' and 6a are all
forms of what is known as the "Kronecker delta." All three forms have
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the values 1 if a = b and 0 if a 4 b. The corresponding matrices are
often called the unit matrix or the identity matrix. In the general co
ordinate system b mn and bmn do not occur, since they are replaced by
gmn and gmn. The particular Kronecker delta b m is, however, valid
a m b m
even in the most general coordinates (since b b Xa Xn = Xa Xn = bn)?
1.2.9 Formula for the Magnitude of A Vector
The metric tensor is used to give a general definition of
the magnitude of a vector. Thus if the components of a contravariant
vector are
Am = Xm Aa
a
then the magnitude of this vector is defined as
mn Am An
= 1(xy, xbn gab) (xcrn
Ac)
= .48 Xm n c d
mc ndgabA A
16 a 6 b Ac Ad
c d
1gab Aa Ab .
Likewise for a covariant vector;
a
Bm = Xm Ba,
the magnitude is defined as
/ n
gmv BmBn
m n gab) c ,d
VO"Ca )(1) (Xm bc) n ?di
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xc xn xd gab
a m b Bc Bd
joc od _ab B B
a b c d
, 'gab Ba Bb.
In both cases the magnitude is thus an invariant form. Hence if
a and b are indices for a Cartesian coordinate system then the two
magnitudes are
and
1/6ab Aa Ab = .1Z,N (A )
a 2
1
VoabBa _
Bb
a= 1
2
(Ba)
a= 1 
where the 2's are both exponents. Thus the magnitude of a vector is
defined so as to be a generalization of the Pythagorean theorem.
1.2.10 Example of Use of the Magnitude Formula
As an example of the above, consider a vector relative
to a spherical coordinate system. Let the spherical coordinates
be r  the distance from the origin, 0  the colatitude, and 4)  the
longitude. Let x, y, and z be Cartesian coordinates related to
r, 0, and 0. by the relations
x = r sin 0 cos (I)
y = r sin 0 sin.cp
z = r cos 0.
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From these we calculate the nine partial derivatives:
Ox
? = sin 0 cos 4)
ar
ax
? = r cos 0 cos 4
30
ax
= r sin 0 sin .1)
a4)
ay _
 sin 0 sin 4)
ar
raY = r cos 0 sin 4)
ao
= r sin 0 cos 4)
a(1)
az
a?r cos 0
az
? = r sin 0
ao
az 0
For convenience these may be listed in matrix form:
4=
ein 0 cos 4)
sin 0 sin 4)
cos 0
Using these:
g = 8 Xa Xb
mn ab m n
r cos 0 cos 4)
r cos 0 sin 4)
r sin 0
3
(= >,
a
=
1
Xa Xa)
m n
0
0
r2
0
0
r2
sin2 0_
r sin 0 sin 4)?
r sin 0 cos .4)
0
Let the vector under consideration be the velocity vector  with
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components relative to the spherical coordinates:
dxm fdr dO
dt dt ' dt
Hence the magnitude squared of this vector is
dxm dxn dr 2 2 dO 2 2 2 Ell 2
= () + r + r sin 0 (dt
mn dt dt dt dt
?
1.2.11 Determinants and the Tensor Densities of Levi Civita
Since the metric tensor gmn is defined to be nonsingular
it has a determinant. This determinant is customarily designated by
the letter g for one coordinate system or by g' for another coordinate
system. Since
then
g = Xa Ab g
ran m n ab
2
g'= kard g,
where g is the determinant of gab, I Xa
III
is the determinant of Xa and,
g' is the determinant of gmn. A little consideration shows that this
determinant is unity for a Cartesian coordinant system. This trans
formation law is commonly said to define a scalar density of weight 2.
Corresponding to this there are entities known as tensor densities 
but the general definition will not be given here. Important examples
of tensor densities, however, are those known as the tensor densities
of Levi Civita. These may be generally defined for N dimensional
space but only 3 dimensional examples will be given here. In three
dimensional space the tensor densities of Levi Civita are represented
by the symbols (for the two representative coordinate systems):
abc mnp
E cabc' , and E mnp
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They are defined, in each case, to be skew symmetric in any pair
of indices, to take the value 0 if any two indices are equal, and to
take the value ? 1 if all three indices are unequal, i.e., the value + 1
if the indices are an even permutation of the numbers 1, 2, 3 and the
value 1 if the indices are an odd permutation of the numbers 1, 2, 3.
With these definitions it may be demonstrated that
and
abc Xa m
Xn XID = I 1 mnp
b c a
Xa
Cabo m
xb xc _ I xa
n p I m
`mnp
Where the symbol 1 Xam I means the determinant of Xam and V means
the determinant of X.
.
Multiplying the second expression by la gives
E xa xbA a
" abc m n p Arg." IX
m mnp
= cmnp?
, Hence the product ,rg eabc is a tensor with covariant order 3 (in 3
dimensional space). Similarly (1Arg) Eabc may be shown to be a
tensor with contravariant order 3 (in 3 dimensional space).
1.2.12 Vector Product of Two Vectors
The tensor densities of Levi Civita are used to define the
vector (cross) product of two vectors. Thus, for the vectors Aa and
Bb:
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cabc Ab BC 1/67 (Xb Am) (XC Bn )
abc m n
d r b
= dbc X
XC AmBn
cm
_X dX p 1? ?bA LI
?m ,n
 p a E A
bc m
n
_ xProi db cmn
Xp Xm Xn A B
Edbc
=X vrg
a
a
Xm
Am Bn
pmn
= XP gc AM. Bn
a pmn
Hence the cross product of two contravariant vectors is a covariant
vector. Similarly, for the vectors Aa and Ba:
1 abc 1 Eabc
c A B = (x An ) (Xg Bp)
Ab c
Vg
a 1 dbc n p
=d ?r EXXA B
b c np
Vg
,a fri 1 dbc
 A A E Xn XP A B
m d r bcnp
= Xa1 cdbc Xm Xn XP A B
m d bcnp
=a 1 Emnp A B
np .
Xm
The cross product of two covariant vectors is thus seen to be a
contravariant vector.
1.2.13 Two Types of Relations Which Can be Written for Tensors
From all of the foregoing it should be evident that two
types of expressions may be written which involve tensors (i.e.,
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tensor components). One type of expression shows the transformation
of the tensor components relative to one (representative) coordinate
system into corresponding components relative to some other (repre
sentative) coordinate system. This transformation formula shows,
explicitly, the contravariant and covariant orders of the components.
The other type of expression may be said to define new tensors in
terms of existing tensors. Thus the sum and product of two existing
tensors are each of them newly defined tensors. Evidently the trans
formation formula may always be used to test whether any particular
combination of existing tensors does, in fact, define a new tensor.
Expressions which define new tensors in terms of existing
tensors are said to be invariant (or covariant), since the same form
may always be identified before and after the transformation. While
several methods have been given for forming new tensors, there are
other methods which have not been discussed. Some of these get
quite involved and are better left to the standard texts on the subject.
One might expect, for example, that a new tensor may be formed by
differentiating an existing tensor. Hence starting with the trans
formation formula
Tmn _ xm xn xc Tab
a bp c
and differentiating both sides with respect to `x , we obtain
aTmn
aTmn
Xq
8x axq
aTab
axa n c ab
axb c ab m n axp lab
mnc c
= Xa Xb Xp d
d Xb X T Xm X T +X X
p d
c a cp a b d c ?
Bx ax ax
ax
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The result is seen to be a tensor only if the partial derivatives
Xm and Xa are independent of xd, which is not generally the
a
case. The way in which this gets fixed up is shown in standard
text books on tensor analysis, but will not be discussed here.
1.2.14 Translation of Tensor Expressions into FORTRAN
It is of, interest to show how tensor relations may be trans
lated into the computer programming language FORTRAN. Thus con
sider the tensor equation
m m+ m bc
Xa = Ya Uab V W .
This may be translated into the following FORTRAN routine* (assuming
that the space of interest has 7 dimensions):
INTEGER M, A, B, C
DIMENSION XMA (7,7), YMA (7,7), UMAB (7, 7, 7)
DIMENSION VBC (7, 7), WC (7), TB (7)
DO 10 B = 1, 7
TB (B) = 0.0
DO 10 C = 1, 7
10 TB (B) = TB (B) + VBC (B, C) *WC (C)
DO
20
M = 1,
7
DO
20
A = 1,
7
XMA (M, A) = YMA (M, A)
DO 20 B = 1, 7
20 XMA (M, A) = XMA (M, A) + UMAB (M: A, B) *TB (B)
*Other FORTRAN routines are also possible; the one given here is
thought to be more efficient than some others.
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Fig. 1 shows a flow chart of this routine. Although this is only one
example, and general rules have not been formulated, this example
is thought to be sufficiently typical that readers who are familiar
with FORTRAN programming will be able to thus program tensor
expressions practically by inspection.
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K2 M + M *(A  1)
XMA (K2) YMA (K2,)
7
+M *A *(B_1)
in m
XMA (K2) XMA (K2) + UMAB (K3) * TB (B)
END
Figure 1. Flow Chart for example tensor equation.
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1.3 Matrices
1.3.1 Use of Matrices to Exhibit Components of A Tensor
Although the availability of digital computers has greatly
reduced the requirements for pencil and paper type computations, it
is still sometimes desirable to be able to set down on paper the
components of tensors relative to some coordinate system. Matrices
provide an organized way of doing this. Corresponding to some of
the tensor operations there are also rules for manipulating matrices.
The latter are, in fact, sufficiently complete as to permit matrix
solution of many problems without any use at all of tensor notation.
For those who are proficient with tensor notation, however, the latter
often brings to light useful relatibns ( simplifications) which would
probably not be discovered if the less explicit matrix notation were
being used exclusively.
Although there are rules for dealing with third (and even
higher) order matrices, practically all matrix computations are limited
to operations with only first and second order matrices. Hence only
these will be discussed here. First order matrices (representing
components of vectors) are linear arrays (either rows or columns).of
numbers. Second order matrices are rectangular (often square) arrays
of numbers (which hence have bothrows and columns).
The vector Va may be represented by either a row or a
column of components. Similarly the vector Va may also be repre
sented by either a row or a column. In other words, there is no fixed
correspondence between contravariant and covariant vectors on the one
hand and row vectors and column vectors on the other hand.
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Similarly Tab, Tab' and Ta may all be represented by
rectangular arrays of components. Thus, in general, a matrix does
not show explicitly whether it represents covariant, contravariant, or
mixed components. Likewise a matrix does not normally carry an
explicit indication of what coordinate system its components are
relative to. As a further source of ambiguity, either the first or
second (upper or lower) index of the tensor may correspond to rows
(or columns) of the matrix.
1.3.2 Linear Combination of Matrices
Matrix addition or subtraction means simply addition or
subtraction of the elements (components) of the matrices. Likewise
multiplication of a matrix by a scalar means multiplication of each
element by the scalar. Thus a linear combination of matrices may be
represented as follows:
ci
?All A12 ? ? ? Aln
A21 A22 ? ? ? A2n
Aml A21 ? ? ? Amn
+ p
73 11 B12 ? ? ? B
B21
? ? ?
B22
? ? ? B
fl
Bml Bm2 ? ? ? B
mn
(a p Bid (a Al 2 + 1312) ? ? (a Al n Bln)
(a A21 + B21) (a A22 522) ? A2n 132n)
? ? ?
(a Aml +13 Bml) (a. Am2 + 13 B m2) ? ? ? (a. Amn + 13 Bmn)
where a and 13 are scalars and All' Al2 ? ? ? Bmn are the elements of
two m x n matrices; the result is, of course, also amxn matrix.
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The above matrix combination might, for example, represent any of
the three tensor combinations:
ab ab
aA +0B
a Ab + Bab
a e +0Ba
B. ?
1 . 3. 3 Multiplication of Matrices
The socalled "product" of two matrices corresponds to the
inner product (i.e., general product combined with contraction) of two
second order tensors. Matrix multiplication is defined so as to be
noncommutative; that is, (pre) multiplying the first matrix by the
second is not the same as (pre) multiplying the second matrix by the
first. The operation is defined so as to apply to rectangular matrices
generally (i.e., nonsquare as well as square matrices). It requires,
however, that the number of columns in the matrix which is to be the
prefactor must be equal to the number of rows in the matrix which is
to be the postfactor. The result then has the same number of rows as
the prefactor and the same number of columns as the postfactor.
The rule for multiplying two matrices may be symbolized
as follows:
1. Let A. .represent the elements of the first matrix, with i
corresponding to rows and j corresponding to columns.
2. Let Bik represent the elements of the second matrix, with j
corresponding to rows and k corresponding to columns.
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3. The product Cik is then given by the rule
= '2; A.. B. .
Cik
.= 1
Obviously, N is the number of columns in the first matrix and also
the number of rows in the second matrix. The similarity to the
following tensor inner products is obvious:
Aab Bbc
A?
Ab Bb
Ab Bbc.
1.3.4 Transposition of Matrices
Matrices are also subject to the operation of transposing
(i.e., interchanging) their rows and columns. If a particular matrix
is represented by the symbol M (imagine bold face type) then the
result of interchanging its rows and columns (called its transpose)
is represented by Mt (again imagine bold face type). If the matrices
A and B are both square and are equal in number of rows (and columns)
then four (generally different) products can be formed by making use
of their transposes:
AB, AT B, ABT, AT BT
In addition the following products may also be represented:
BA, Br A, BAT, BT AT.
It may be shown, however, that each of these is the transpose of one
of the first four products. That is, it is generally true that if
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C = AB
then
CTBT.
The four matrix products correspond to, for example, the following
four tensor inner products:
ae
A Bed,
, Aa Bce, Aeb B.
1.3.5 Inverse of A Matrix
If a particular matrix is square (i.e., it has an equal
number of rows and columns) then one may compute the determinant
of its elements. If this determinant is not equal to zero, the matrix
is said to be nonsingular, and a matrix may be found such that if
multiplied by the original matrix (either pre or post multiplication)
then the result is the unit matrix. (The unit matrix has all of its
main diagonal elements equal to one and all other elements equal to
zero.) In such a case the second matrix is called the inverse of the
first matrix. Text books on matrix manipulations give rules for com
puting the inverse of a matrix, but these will not be stated here.
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1.4 Cartesian Coordinate Systems
The preceding discussion has been written in terms of
general coordinate systems in order to bring out the principle of
invariance. According to this principle any proper combination of
tensors has the same form irrespective of the coordinate system.
Consequently it is not necessary to actually carry out a transforma
tion of tensor forms when transforming from one coordinate system to
another. One merely duplicates the form existing in one coordinate
system  but substitutes the components of each tensor relative to
the new coordinates.
In actual calculations, one uses Cartesian coordinates 
almost always. This does not make the various tensor relations take
a significantly simpler form, but it limits the numerical values that
the components of the various tensors may have. For example, the
a
partial derivatives Xa and Xm become constants (with respect to the
coordinates  not necessarily with respect to other parameters, such
as time) for any particular pair of Cartesian coordinate systems. In
addition, as has been stated, the metric tensors become the
Kronecker deltas with respect to any set of Cartesian coordinates.
Hence the determinant g takes the value one when only Cartesian
coordinate system's are used. Consequently this determinant is often
omitted from the various tensor equations  but doing so makes the
equations no longer invariant with respect to general coordinate trans
formations. The equations are then said to be invariant with respect
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to transformations among Cartesian coordinate systems (only). This
will be done in much of what follows.
It can be demonstrated that all of the possible transforma
tions between any two different sets of Cartesian coordinates are
equivalent to combinations of parallel translations and rotations of
the coordinate axes. For most purposes, the parallel translations
are of trivial importance, and transformations among various Cartesian
coordinate systems are commonly referred to as "coordinate rotations."
The partial derivatives Xam and )(1,1 (which then have unity determinants)
are hence often referred to as "direction cosines" (of the various
coordinate axes in one system with respect to those in the other).
Accordingly the notations Cam and Cran (instead of Xam and Xam) will be
used to represent transformations among Cartesian coordinate systems.
As a result, in what follows it will generally be true that
and these determinants, also, will often be omitted. It is also a fact
that when two reciprocal sets of direction cosines Cam and Cam are
represented by their respective matrices (without indices) these two
matrices are transposes of one?another. This fact is used in the
computer program, but is otherwise of little consequence.
When the permissible coordinate systems are limited to
only those of Cartesian type, certain entities may be treated like
tensors which are not tensors under more general transformations.
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One example is the partial derivatives of the components of a tensor
with respect to the coordinates. These derivatives were considered
near the end of section 1.2 and found to define a tensor only when
the partial derivatives are independent of the coordinates  exactly
the situation when only Cartesian coordinates are considered. Another
example is the incremental (more than just differential) displacement:
xm xm =C m (xa xa).
a
These quantities may now be treated like components of a vector
since the magnitude squared is
(x x xm\ fxn_ )xx)
mn
= 6 mn [Cam (xa_ x0a) [Cbn kb_ xbo)]
6
Cm Cn (xa xa) (xb xb)
rnn a b
_ ix a_ xal txb_ xb\
ab
i.e., invariant with respect to coordinate rotations. "Vectors" of this
type will be used extensively in what follows.
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It. OPTICAL IMAGING AND AERIAL PHOTOGRAPHS
2.1 Gaussian Optics
Most optical lenses consist of a series of interfaces
between transparent media of different refractive index, which are
all, as nearly as practical, spherical surfaces with their centers
lying on a single straight line. The common line of centers is
called the optical axis and the various spherical surfaces are
incomplete spheres (usually less than hemispheres), usually with
circular boundaries also centered on the optical axis. In some
what rare instances lenses include one or more interfaces which
are intentionally ground so as to depart from a spherical surface by
a small, but definite, amount.? These surfaces are, nevertheless,
rotationally symmetrical (as nearly as practical) about the optical
axis.
Socalled "Gaussian optics" consists of a body of
mathematical analyses of the refraction occurring at a series of
such centered spherical interfaces between various optical media,
which preserves only the degree of approximation obtained as the
various "rays" considered, approach parallelism to the optical axis
and also approach only an infinitesimal displacement from the optical
axis. Thus Gaussian optics might be considered as a "zero order"
approximation to the true analysis. This approximate analysis
results, however, in what might be called a theory of "ideal"
imaging, hence it is of prime importance. Real lenses are built so
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as to have actual imaging properties substantially like the ideal
imaging of Gaussian optics. Deviations from this ideal imaging
are referred to as "aberations, " and these aberations are made as
small as is consistent with the intended price of any particular
lens.
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2.2 Projective Optics
Projective optics is a branch of geometrical optics that
states a formal deductive theory of ideal imaging which corresponds
closely to that of Gaussian optics. The basic assumptions of pro
jective optics will here be used to derive the "general equation of
optical imaging" which is the basis for the optical analysis used in
subsequent chapters. This socalled "general" equation is, as
stated above, an idealization of the real optical situation which
neglects aberations completely.
Projective optics considers that any optical system which
has cylindrical symmetry about the optical axis may be treated as
having the fundamental elements: (1) two principal planes, which
are normal to the optical axis, (2) two focal points, which are on the
optical axis, and (3) two nodal points, which are on the optical axis.
These elements are illustrated in Figure 2, which also shows three
rays diverging from a typical object point 0 and converging on the
corresponding image point 0'. These three rays are particular cases
of three classes of rays. One class enters the first principal plane
in a direction parallel to the optical axis; all such rays continue
parallel to the optical axis until they intersect the second principal
plane and then become "refracted" by just the right angle so they
hence pass through the second focal point (F'). Rays of the second
class pass through the first focal point (F) and continue until they
intersect the first principal plane, at which point they become
"refracted" so as to henceforth be parallel to the optical axis. The
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third class consists of rays which pass through the first nodal point
(N) without having been previously refracted. All such rays emerge
from the second nodal point (N') in a direction which is parallel to
the direction in which they entered the first nodal point. Thus the
ray N'O' is parallel to the ray ON.
2.2.1 The General Equation of Optical Imaging
The above stated principles (of projective optics) and
Figure 2 may be used to derive the relations between a typical object
point (0) and the corresponding image point (0'). For this purpose
assume an arbitrary Cartesian coordinate system (moving or stationary).
a
Let Xa and X1 be the coordinates of 0 and N respectively. Likewise
a
let xa and xI be the coordinates of 0' and N' respectively. So long as
only Cartesian type coordinate systems are considered, the displace
ments
X1  Xa and x? x1 may be treated as components of two vectors
which from the discussion above are known to be parallel to each other.
(These two vectors are shown in Figure 2 as U and u respectively.)
Parallel vectors have corresponding component which are respectively
a
proportional. Hence: xa  x1 must be equal to some scalar quantity
times Xa  Xa.
1
Figure 2 shows two pairs of similar triangles. One pair of
similar triangles has a common junction at F and has a pair of corre
sponding legs lying along the optical axis. The other pair of similar
triangles has a common junction at r' and also has a pair of corre
sponding legs lying along the optical axis. The lengths of the first
pair of corresponding legs are seen to be (p.U f') and f respectively,
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Fig. 2. Relations between an object point 0 and the image point 0'
formed by a lens with principal planes P and P I. Drawn for a positive
lens, with f and f both taken positive. For negative lenses f and f
are both negative.
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where p is a unit vector parallel to the optical axis. The lengths of
the second pair of corresponding legs are likewise seen to be f' and
(p.0  f) respectively. Consequently the value of the scalar multi
plier, mentioned at the end of the previous paragraph must be:
_ p. u  f
p . U  ff ' ?
Now let ka be the components of the unit vector p. Then
and
p.U= a (Xa  Xa)
1
p.0 =X u xa).
a 1
Using the ratio from the first pair of similar triangles, the desired
relation between the two ve6tors U and u is thus
a a
f (Xa  Xa)
1
x  x
1
b 1
)
1 (1)
)13 (Xb 4) ?
This will be referred to as the optical imaging equation. Multiplying
both sides of this equation by ka gives:
Hence:
f k (Xa  X?)
(xa 4.) _ a
a
)`13 (Xb XI1))
p.0  f = ka (xa  xa)  f
1
f (xa , _a,
_ a "1)
+ kb (Xb  Xb)
1
f [I1 + (xa  Xa)]
f. (xi) 
I
+ kb (Xb  XII)) p.U
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Thus the ratios derived separately from the two pairs of similar
triangles are indeed equal to each other.
2.2.2 Invariance of the Optical Imaging Equation
That the optical imaging equation has been set up so as
to be invariant to coordinate rotations, may be checked as follows:
(1.)
Let Ca be the direction cosines of the (three dimen
sional) coordinate rotation, and let Cam be the reciprocal
direction cosines.
(2.) The three vectors then have their components relative
to the xm coordinates given by
and
mm maa
x  = Ca (x 
a
m = ma
(3.) The above relations may be inverted to give:
and
a
xa  x1 = Ca (xm  xm1)
'
aa amm
X  X1 = Cm (X  X1 )'
X = Cm X
a a m
(4.) The imaging equation is therefore
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f Ca (Xn Xn)
a n n 1
Cn (x  x ) 
I (Cbm X)C ()CP  xPi)
f Ca (Xn  Xn)
1
+ Cm Cb X m (XP  e)
b p 1
(5.) Multiplying both sides by Cam and using
m a
Ca Cn = b n
then gives
(and Cm Cb = bm)
b p p
I (xm 
m m
xx1 
f'+ txn ?v'i n '
n 1'
which has the desired form.
(2)
2.2.3 Use of Separate Coordinate Systems for Object Space
and Image Space
The preceding discussion assumed that the same coordinate
system was used for both the object space and the image space
(these two spaces may be considered as separated by the principal
planes). In practice, it is often desired to use one coordinate
system for the object space and a different coordinate system for
the image space. Evidently the imaging equation then takes the form
f ixa va?
xm
x (3)
1
+ Xb (Xb  Xb)
1
This equation is invariant to transformations among Cartesian
coordinate systems both (separately) for the object space and for
the image space. Cam is, of course, the set of direction cosines of
the image space coordinate system with respect to the object space
coordinate system.
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2.3 Application to Aerial Photography
In an aerial camera the object point (0) is normally some
point on the surface of the earth and the corresponding image
point (0') is in the emulsion of a photographic film. Since the
photographic film lies in a two dimensional surface (i.e., a
plane surface for frame and strip cameras, a cylindrical surface
for panoramic cameras) the object points which strictly satisfy
the imaging equation also lie in a two dimensional surface.
Aerial cameras, however, normally have sufficient depth of
focus to satisfactorily image a substantial volume of space.
Thus, in general, the imaging equation is only approximately
satisfied with respect to camera focus. Specifically, the
imaging equation correctly gives phoeograph coordinates as a
function of ground coordinates but cannot be solved directly
to give ground coordinates as a function of photograph co
ordinates. The most which can be inferred about ground
coordinates from a single aerial photograph is their projective
directions. Some additional information (or assumption) is
necessary to determine the points at which rays projected
from the photograph intersect the ground. Since the flying
height is normally very much greater than the camera focal
length, it is usual to neglect the rear focal length (f') which
appears in the denominator and to write the imaging equation as:
f Cam (Xa  Xa)
m m
X x
1 kb
1
This approximate form will be used from here on.
(4)
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2 .3.1 Rotation of Image
From Figure 2 it may be seen that the image in a camera
is rotated 1800 about the optical axis as compared to the apparent scene
viewed directly. If, however, the photographic negative is viewed
through its film base there is no mirror type reversal unless the camera
contains one or more (i.e., an odd number of) mirrors, in addition to its
lens. Correspondingly, then, a dup positive would normally be viewed
emulsion side up in order not to show a mirror type reversal. By rotating
the photograph coordinate system 180? about the optical axis, with
respebt to the ground coordinate system, one may make photograph
coordinates take substantially the same algebraic signs as corresponding
ground coordinates.*
2.3.2 Application to Frame, Strip, and Pan Types of Photography
Equation (4) is to be interpreted ?for three different types
of photography: frame, strip, and panoramic. For frame and strip type
the photographic film is exposed while confined in a geometric plane at
a fixed normal distance from the camera lens. Assume that the camera
motion is such that the direction normal to the film does not appreciably?
change direction during the exposure time for the photograph. Then the
lens normal kb must also be constant in direction during the exposure
time. Let the photograph coordinate system be oriented with its x3 (z) ?
axis normal to the film. Then X = C3 and equation (4) becomes:
*The common practice of regarding a positive as a negative projected
through the point of perspective and having a negative focal length
introduces a negative unit matrix which is not formally correct but
which gets cancelled out in the usual photogrammetric computations.
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m
m m (xa a
f Ca  x )
1
x x
1 3' _hi b
Cb  Xi)
(5)
This is the basic equation for frame and strip type photographs.
Xa and x1 are constant for frame type but time functions for strip
1
type. Ca (including C3') are constant for frame type and, for
small regions at least, of strip type photographs.
For panoramic photographs the film is maintained in the
form of half of a circular cylinder during exposure. The lens
(and a slit) are rotated to produce a sweeping exposure. Hence
the lens normal kb must be expressed in terms of the camera
sweep angle ai (=wt). Let the photograph coordinate system be
oriented so the x axis is parallel to the cylinder axis and the
x3I axis is normal to the tangent plane at tIe "top" of the half
cylinder. Let al be zero at the x3' axis and increasing positively
in the direction toward the x2' axis. Then
km = Cm kb = (0' sin a cos a1)
Hence (using (4))
m m 2' 2'
'
k m (x  x1) = (x  x1 ) sin a1 + (x3  x1 3') cos a1 = f (6)
Evidently (6) will be true if
and
x2'  x2' = f sin a1 (7)
1
x3'  x3' = f cos al (8)
1
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Thus xrn  xIfl
1 are rectangular coordinates of the circular cylinder
relative to a coordinate system in which the film is stationary.
Combining (7) and (8) with (4):
with
m
m m a
Ca (Xa  XI)
x  x1 = f cos a1 3' b b
Cb (X  Xi)
C2 iva
al = tan 1 a v,  Xad
C3'b
b (X 41)
(9)
(10)
Equations (9) and (10) are basic for a panoramic type photograph.
a m
XI' x1' and a1 are time functions depending on the camera motion,
the mechanism, and the lens sweep mechanism.
Equations (5) and (9), (10) are basically correct but are of
little value until the various time functions are evaluated.
Evaluation of these time functions is dependent on the particular
cameras and on the flight pattern. The following examples .are
cursory and are based on descriptions of particular cameras which
have been published in the literature and assume uniform flight
velocity.
2.3.2.1 Frame type photography
The film is maintained in a plane and exposed
simultaneously over the whole photograph. Image motion is
neglected  since the exposure time is brief.
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f1 Cm (Xa  Xa)
a 1
x  x1 
3 b b
Cb (X  X1)
X1 = m1 0 (constant)
a a
X1 = X10 (consfant)
m m1' ' 2'
. . x  xi = (x  x10, x  x10, f1)
2. 3.2,2 Strip type photography
The film is maintained in a flat plane but is
exposed by moving it past a narrow slit which results in a
sweeping exposure. The exposure time for any small area is
brief but the time interval required to sweep the whole photo
graph is appreciable. There is a fixed angle (31 associated
with the slitlens scanning operation. This angle is here taken
positive for backward looking, or negative for forward looking,
slits. Let the photograph coordinate system be oriented with
its x3' axis normal to the film plane and its x1' axis parallel,
but opposite, to the direction of lensslit motion relative to the
film
Assume that the time functions are linear in t.
m m
fl Cm (Xa  Xa)
a 1
x x 
1 3'
Cb (Xb  Xb)
1
a
X1 0 + Va t
m,, m
1
vx1 x0 + 51' t
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m m 2' 2'
x  x1 = (fi tan 131, x  x10, fi)
t. .
x  x10  tarif31
(Cali  Ca3I tan 131) (Xa  Xa3.0)
(Cbi  C133 tan 131) Vb
where Vb is the camera ground speed vector, and v is the velocity
(here taken as a negative number).
2.3.2.3 Panoramic type photography
The film is maintained in half of a circular cylinder
and exposed by a slit which scans around the cylinder. The
exposure time for any small area is brief, but the total scanning
time is appreciable. There is a scanning angle al whIch increases
at a rate approximately uniform in time, and is here taken positive
in the direction from the x3' toward the x2' axes. Let the photo
graph coordinate system be oriented with its x1 axis parallel to
the axis of the cylinder and with al = 0 at the x3' axis. Assume
that the lens motion for
is in the negative x1' direction.
a
Assume that X1 is a linear function of time but that the
velocity is proportional to cos ai .
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STAT
STAT
STAT
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m m cna' (xa  X7)
x x1 =f cos a
1 C7 (X/3  Xi))
1
1 Ca (X  X1)
a = tan
1 C3I (Xi? 
al
a a
X1 = X1 0 + Va t
m1 x1 m , M m??  x ? sin al 0 1
For measurement, the panoramic photograph is laid out flat.
The linear coordinate in the scan direction is then
 Y10 = fl
11 11
but there is no change in the coordinate x  xi  parallel to
the cylinder axis. Quantitative use of a panoramic photograph
is usually for calculation of derived quantities (ground co
ordinates, or an equivalent image) as a function of measured
photograph coordinates. Hence the following substitutions
will usually be made:
m m 1' 1'
VM Y  Y10 sin Y  Y10 Y Y10
x  x = (x  x  sin
, f
1 10 1 f1, cos )
f1 f1 1
These follow quite easily from the preceding equations but they
are discussed further in section 3.1.4.3.
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STAT 2.4
Series Expansion of the Projective Equations
Equations (5), (9), and (10) may be collected and written
in the following generalized format:
cam (xa xa1)
m m
x =x + f cos a ,
C3 (Xb  Xb)
1
with
; 0;
frame photos
strip photos
21 a
1 Ca (Xa  X)
1
tan,
; pan photos.
C3(Xb  Xb)
1
(12)
These equations are not, however, entirely explicit since
a
a, Ca , and Xi are, in general, functions of time (t). These
functions of time are not usually available in closed form but can
often be approximated by the first few terms of their
expansions. Hence it is of interest to examine the
expansion of equation (11).
To expand (11) as a
series
series
series it is necessary to find
the various orders of partial derivatives of equations (11) and
(12) with respect to Xa. In doing so it will be convenient to
?
represent the partial derivatives taken two ways. The expressions
axa m axm
' n '
ax ax1
axm axran
a
ax axi
? ? ?
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STAT
STAT
STAT
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will represent partial derivatives taken by treating x1 ' a, Ca ,
a
and X1 like independent variables. On the other hand the
expressions
X, rn ,m v
Ab Abc
? ? ?
IT1
will represent partial derivatives taken by treating x1 ' a, Cm
a '
a
and X1 as functions of t (time) which is in turn treated as a function
of Xa.
Hence:
rn
m axm ax dxn1 axm da axm dCn axm dXc
1 at
xa  + (
?) a
axa ax1 dt aa dt a Cn dt 8x1 dt ax
aXm aXm dxn aXm da aXm dCn aXm dXc
a a Xm 1 a a c a 1  ? + (? ? + ? ? + +
ab c
axb axn dt @a dt acn dt DXI dt
1 c
n 2 n 2 c
axam d2 xi. aXm d2a axam d Cc DXm a d X1 at
d
? + + 4   )
dxl
n 2 a ,da , a 2 c 2 ?la dt ? t dCn dt2 dX1 dt ax
dt c
a a a
dt dt dt
aXab
Xabc+ ? ? ? etc.
ax
The
series is then
(13)
(14)
STAT
mm1 m (15)
m alm a b
x =xo j Xa AX +? X AX AXAXa AXII) AXc + ? ? ?
2 ab 3. ac
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where x111 are the photograph coordinates of the center" of the
region over which (15) is useful. If the expansion is desired for
m
m a a
a certain ground region then x1 ' a, Ca , X , and X must all be
1
determined for a point near the center of this ground region and
substitut6d into (11) and (12), to give xnol, and into (15). Thus for
any particular ground region, over which (15) is considered valid,
m a a
x1 ' a, Ca , X , and X1 are all treated as constants. Hence over
this ground region, xo are constants and AXa are the independent
variables which correspond to motion over the region.
2.4.1 Evaluation of thie Partial Derivatives
From (11)
axm
axa
The various partial derivatives are obtained as follows.
f cos a
m Cm b 3 Cb (Xb  X1) Ca
j 7 a
3 b .
Cb (Xb Xi) [Cc(X  Xi)]2
(Cm C31C3' Cm) (Xb  Xb)
a 7b a b 1
= f cos a
[c (XC  Xci)]2
ax
n
ax1
cm txa .v.a\
ax
 f sin a a, Al/
act 3
Cb (Xb  Xia)
1
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i
6111 (XC  Xc) Crn (Xa  Xa) 63 (XC  Xc)
a xill
 f cos a i n i 1 a
I 1 n 1
a C11
c [C3 (Xd  Xd]
d 1 [Cd3 (Xd  4)]2
(6i" c 3'  ' 6 3'a a
C ) (X X1)(XC  Xci)
n a . a n
= f cos a.
ax DX
ic3 fxd _ ,d,12
d
dxn da dCn dXc
1
The total derivatives ? ?
dt ' dt ' dtc, and dt
camera motion along the flight path and
are taken from the
(19)
(20)
sweep STAT
motions in the camera, and are often treated as constants though,
in general, their values vary from one region to another.
2%4.2 Time (t) Treated as A Function of the Ground Coordinates
at
The partial derivatives ? are different for each
axa
different type of photography and are as follows:
2.4.2.1 Frame Photography
simultaneously.
Hence:
at
axa
All points in the photograph are exposed
0
2.4.2.2 Strip Photography
1
Since xI  x1 = f tan p where 1E1 is a con
stant angle, characteristic of each particular strip camera (zero for
systems currently of interest), equation (11) form = 11 becomes
tan 13 
1 a
Ca (Xa  X1)
C3I (XbXb)
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(21)
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Multiplying through by the denominator of the right side and
differentiating with respect to Xa gives
f 3
= tan iCb
1
dXb at dC at
b
dt dt
?ax axa (xb  4)
dxb at dC at
b b
a ?X"`"
dt a dt
ax ax
at
Solving (22) for ? then gives
a
ax
form
. .
1 3
Ca  Ca tan 13
at = ,
1 3
axa dxb idc1 b dCb
ii 3. 1 b
(C  C tan d)
b b . ? dt ?dt dt tan (3)(Xb Xi)
2.4.2.3 Panoramic Photography
The last part of (12) may be written in the
2 a a 3 a a
C (X  x) = ca (X  x) tan a.
1 1
Differentiating (24) with respect to Xa gives
dXb at dC2 at
 ?
b a dt a dt a
ax ax
dx at dcb3 at b h
= (6a 71t? ?) Tt? ? (x  xi)] tan a
axa axa
b da at
sec 2 a
+ C 3 (Xru 1 ) EIT ?axa
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(2)
(23)
(24)
(25)
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Equation (25) may be solved for ata. Equation (24) is
ax
then used to substitute for tan a and for sec2 = 1 + tan2 a. The
result is
(c2 c3  c3 c2 ) (xb  xbt
at _ a b a b
axa
2' 31 31 21 d, 1
dXc
(Cc Cd  C C ) .(x  X1) dt
c d
2 3
(4. dGc 3' dCc 2'
c ..d d
461 dt Cd dt Cd ) (XC  X1) (x  Xi) +
9,{[C21(Xc  xc112 + [C31 (xc _ x)]2}ddta
1 c
Air
Evidently (16) through (20) and (23) or (26) must be
substituted in (13) before applying (14), and similarly for higher
order derivatives. Fortunately, however, only terms of (15)
through the first order will be needed in what follows. Note that
(15) gives the rectangular coordinates, hence for panoramic
photographs it must be supplemented by the following formula for
the photograph coordinate in the scan direction:
21 2'
_1 x  xi 0
 Y10 = f tan 3,
3' ?
x  0
2.4.3 Generalized Formula for the Partial Derivatives
Equations (13), (14), (22), and (25) are written for
coordinate systems which are stationary with respect to the
ground (i.e., the object photographed). For such coordinate
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systems
and
dXa
 0
dt
d2Xa
 O.
(zit2
Using (28), equation (13) can have the term
axm cixc at
axc dt axa
(28)
(29)
added to its right side without upsetting the equality. If this is done,
then (13) becomes
dxm at
m axm
Xa
axa dt axa
Similarly (14) can have
M C M 2 c
axm
dX axm
d X
axc + at
( dXc dt
b dt 2
ax
adt
added to its right side, and it then becomes
axm cpcm at
xm  a + a
ab
ax dt aXb ?
Similarly (22) and (25) are respectively equivalent to
1, 3,
aa [ (Cc  Cc tan (3) (xc  xc)
ax
d 3'
d?t [(Cc cc tan 13) (XC  Xc1)] ata  0
ax
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(30)
(31)
(32)
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and
a [(C2  C3 tan a) ()CC  Xcn
a c 1
ax
at
+ dt 211 [(C2  C3 tan a) (XC  Xc) = 0.
c c 1 a
ax
Hence (2 3) may be written
1,
 C3,
tan
at Ca a f3
a
aX d r11 31
crt [(ub  cb tan (3) (xl?  4)1
and (2 6) may be replaced by
2 3
at Ca C tan a
a
ax d 3' b b
Ert
[(Cd  cb tan a) (X  X1)]
Note that (24) cannot be substituted into (35) until after the
differentiation in the denominator has been performed.
(33)
(34)
(35)
2.4.4 Invariance of the Generalized Formula
Examination of (30) through (35) shows that all but (31)
have forms which are invariant with respect to translation and/or
rotation of the ground coordinate system. Equation (31) would also
be invariant if the derivatives were covariant derivatives; the latter
are not discussed in this paper, however. Thus these equations are
valid for all Cartesian coordinate systems, whether moving or stationary
with respect to the ground. The fact that these equations have invariant
forms allows Xam ' X m ab ? ? ? to be treated as tensors. Hence if Xa is
treated as a tensor, then (15) also is valid for moving, as well as for
stationary, coordinate systems.*
*Note that the photograph coordinate system remains unchanged for the
transformations of the ground coordinate system considered here.
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2.4.5 Importance of Moving Coordinate Systems
Moving coordinate systems are of, at least theoretical,
interest since the photographs are taken by moving cameras. The
usual object in reducing the data contained in aerial photographs
is to obtain a series of ground coordinates with respect to a
coordinate system fixed on the ground. Nevertheless the data
available are those in the photographs, and the equations of motion
for the camera. Thus, at least in principle, the data reduction is
equivalent to transforming ground coordinates from a system fixed
on the camera to a system fixed on the ground.
Evidently, then, one may distinguish three types of
coordinate systems which may be used in computing the various
terms in (30), (31), and similar equations for the higher order
derivatives. One type is that which is fixed on the ground. For
this type, it is usually the case that
dXa a
a0
dt
dX,
0, and cFt?` # 0.
dt 0' dt
A second type is that which is fixed on the camera. For this type,
usually
dCm dXa dXa
1
dta  0, # 0, and
dt
Finally, there is the general coordinate system type, which is
moving with respect to the ground but yet not fixed on the camera.
In this case, generally
dCm dXa dXa
a 1
dt 0, ?dt 0, and 71t t 0.
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m a a
Similarly for the higher order derivatives of Ca ' X , and Xl.
Furthermore, in all these ground coordinate system types
and
dx1
dt 4 0 except for frame type photographs,
dC1
r u for panoramic type photographs.
dt
As an example of a somewhat similar situation one
may note that the vector net effective velocity of the camera
with respect to a particular ground point is given by the formula
a d m b _ h
[C (X  X)].
m dt b 1
That this formula is reasonable may be seen by noting first that
it is invariant to transformations among Cartesian coordinate
systems, and second that for a coordinate system fixed on the
camera (36) gives
/la dXa
VR
dt
Thus (36) may be used to compute the camera to ground relative
effective velocity in any of the three coordinate system types
listed above.
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(36)
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'III. COMPUTATION OF GROUND COORDINATES
The optical imaging equation (3) can be solved for Xa, giving:
,Ca m m
f m (x  x1 )
?
f \ (xn  xn)
n 1
If, however, (37) is applied to an aerial photograph, in an attempt
to compute ground coordinates (Xa), then it is usually found that the
magnitude of the denominator [f  Xn (xn  xnin is so much less than
the focal length f that the errors in the determination of f cause an
unacceptable degree of orr?r in the values computed for xa. Hence
(37)
(37) is not a practical formula for computing ground coordinates from
the coordinates in an aerial photograph. This situation may be
likened to the reverse side of the fact that an aerial photograph has
a practical depth of focus which is somewhat larger than the range
of relief in the terrain photographed. Therefore it is not practically
meaningful to solve for the (ground) surface which is theoretically
in "perfect" focus, as would be done by computing with equation (37).
It is for a somewhat similar reason that the projective equations
((4), or(5), or (11) and (12)) cannot be completely solved for xa. In a
praciical sense the most which can be strictly inferred about ground
points from a single aerial photograph (and the corresponding camera
parameters) is their projective directions. Some additional assumption,
or other information, is necessary to determine where the projected
directions intersect the earth's surface. A common way out of this is
to assume some approximate geometrical shape (a plane or a sphere
or a spheroid) for the earth's surface. Such an assumption allows one
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to write a constraint equation which may be used in conjunction with
the projective equations to solve for approximate ground coordinates.
Another possibility, of particular interest here, is to use two over
lapping aerial photographs taken with different camera locations.
In this case rays may be projected from corresponding coordinates in
both photos so as to triangulate ground coordinates. Both these schemes
will be discussed in the following sections.
3.1 Approximate Ground Computations Based
on Only One Photograph
3.1.1 Use of A Tangent Plane
Let it be assumed that the ground coordinates
over some local region may be adequately approximated as lying in a
geometrical plane, whose parameters are known (or may be assumed
with sufficient accuracy). The necessary parameters are the direction
cosines p.a of a normal to the plane and the normal distance Di from the
plane to the camera station. It is assumed that the camera parameters
a
Ca and X1 are known (i.e., known functions of time). Hence the
quantities
may be computed. Multiplying these quantities by both sides of
equation (11) gives
m a 1
p.m xm = p.m x1 + f cos a
C 3 (Xb  XI))
1
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(38)
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In (38), the quantity
.1 cm ixa . txa val
rn a '1' '11
may be identified as the negative of the normal distance from the
plane to the camera station. That is,
[la (Xa = DI ?
Substituting this in (38) and rearranging somewhat, then gives
m m
(x
f cos a   m 1)
?
C133 ()(13  x1) 1
(40) may be substituted in (11) and the result solved for Xa?
D
a a i a m m
X = X1 n% Cm (x x )?
, n
11n kx  x1 )
Thus equation (41) may be used to compute coordinates of points
lying in a plane surface whose normal has the direction cosines
a and whose normal distance from the camera location (coordinates
a
X1) is  D1? Obviously the extent to which the points in this plane
surface approximate actual ground coordinates depends on the
(39)
(40) .
(41)
accuracy with which and D1 represent real conditions of the local
ground region. If three or more ground control points (i.e., ground
points with known coordinates whose images can be recognizdd in the
aerial photograph being examined) are available, then the "best
average" plane may be passed through these points. Otherwise it's
usually necessary to compute a level surface tangent to some point
of the geometrical model (plane, or sphere, or spheroid) assumed for
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the earth. The latter procedure will be discussed here  using a
spherical model of the earth.
3.1.2 Tangent Plane to A Spherical Earth
In equation (91) it is assumed that, relative to
some established ground coordinate system, the camera parameters
Ca and Xa are known (as functions of time) and that, relative to the
m 1
photograph coordinate system, the lens coordinates xI are known
(as functions of time  determined by the
pan sweep STAT
mechanism). Thus the coordinates xm of any particular photo point
determine a projective ray frdm the known camera location (4). The
problem to be solved in this section is to determine p.a and DI for a
plane surface which is tangent to the earth (assumed spherical)
at the point at which this projective ray intersects the plane. Thus
the tangent plane is made a level datum plane at the point of the
earth's model which is intersected by the particular projective ray,
and the coordinates of the intersection point are actually approximate
ground coordinates.
The solution given here for the problem stated
above is one by successive approximations  suitable for iterative
computation by a digital computer. Figure 3 illustrates the first two
steps of the method. In Figure 3 the camera is represented as having
a
ground coordinates X1 and flying height H over a spherical earth with
its center at coordinates Xa' P1 represents a plane tangent to the
0
earth at the ground nadir, and Xa (1) are the coordinates of the point
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a
X3 (Center of Earth)
R
(2)
Earth's Surface
(Radius of Earth)
Figure 3
First two steps of iterative solution for level datum plane
t.anjent at point intersected by a protective ray.
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at which this plane is intersected by the projective ray from the photo
point with coordinates xm. Now an earth's radius vector is passed
through the point X8 (1); i (2) are the direction cosines of this radius
vector which has, in fact, the vertical direction through the point Xa (1).
P2 is a plane tangent to the earth at the point intersected by the vertical
through Xa (1). Thus ?a (2) are the direction cosines of the normal to
the level tangent plane P2. D1 (2) is then the normal distance from
this plane to the camera station, and Xa (2) are the coordinates of the
point at which this plane is intersected by the projective ray from xm.
The third step in the solution (not shown in Figure 3)
consists of passing a plane P3 tangent to the earth at the point verti
cally below Xa (2) and finding the intersection Xa (3) of P3 by the
projective ray from xm. Evidently the process may then be iterated
until a plane PN is tangent to the earth at a point as close as desired
to the intersection Xa (N) of this plane by the projected ray from xm.
Thus Xa (N) is a satisfactory approximation of the intersection of the
projective ray from xm with the surface of the spherical model of the
earth.
The actual computations are as follows:
xa  xal = Ca (xm  xm1)
1.
2. R + = 15b (Xa X((1)) (XII)  Xbo)
a 1
3. (x13  Xb)/(R + H)
" ab 1 0
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4. i =
5. D1
a
6. X
7. RI
8.tia
9. If
2003/01/28
1
(i)= (i)]
. a
0.) =?
I
: CIARDP78605171A0006000300010
[X7  Xao]  R
Di (i) (xa  xa1)
? ? ? (e.g., (41))
['lb (i)1 b b1
x 
b
(i) = ab
(i + l)= bab
R1 (i)  R
[xa (i)  xa01 [ (i)  xb0]
[Xb (i)  X0 ]/R1 (1)
the allowable error threshold:
increase i by 1 and repeat steps 5 through 9; otherwise
the computation is complete.
The following FORTRAN subroutine was written to test
the computation scheme given above (BETA is the pitch angle; roll
and yaw are each taken as zero):
SUBROUTINE GNDCUR
INTEGER A, B, M
REAL XAO (3), XA1(3). MUA(3), MITM(3), X.A(3), CAM(3,3)
REAL DLTAAB(3,3), XM(3), X1 MXO (3), MUDOTX
10 FORMAT (3E15.5)
11 FORMAT (/3F15.5)
READ (1,10) H, BETA, R
READ (1,10) XM
DO 12 A=1,3
DO 12 B=1,3
IF (A EQ .B) DLTAAB (A, B) = 1.0
IF (A.NE.B) DLTAAB (A, B) = 0.0
12 CONTINUE
C=COS (BETA/57.2958)
S=SIN(BETA/57.2958)
CAM(1, I) = C
CAM(1,2) = 0.0
CAM (1,3) = S
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CAM (2, 1) = 0.0
CAM (2, 2) = 1.0
CAM(2, 3) = 0.0
CAM (3, 1) = S
CAM(3, 2) = 0.0
CAM (3, 3) = C
DO 13 A = 1,2
)(AO (A) = 0.0
13 XA1(A) = 0.0
XA0(3) = R
XA1 (3) = H
RPLSH = 0.0
D015 A  1, 3
X1MX0(A) = )(Al (A)  XAO(A)
15 MUA(A) = U.0
DO 2 0 A = 1, 3
DO 16 B = 1,3
16 MUA(A) = MUA(A) + DLTAAB(A, B)*XlIVIXO(B)
20 RPLSH = RPLSH + M1JA(A)*X1MX0(A)
RPLSH = SQRT(RPLSH)
1D021 A= 1,3
21 MUA(A) = MUA(A)/RPLSH
30 D1 R
DO 31 A = 1,3
31 DI = D1 + MUA(A)*X1MX0(A)
D035 M = 1, 3
MUM(M) = 0.0
D035 A = 1, 3
35 MUM (M) = MUM (M) + CAM(A, M) * MUA(A)
D041 A = 1, 3
XA(A) = 0.0
MUDOTX = 0.0
D040 M= 1,3
XA(A) = XA(A)  CAM (A, M)*XM(M)
40 MUDOTX = M UDOTX+M UM (M)*XM (M)
41 XA(A) XA(A)*D1/MUDOTX+Xl MXO (A)
R1 = 0.0
DO 46 A = 1,3
MUA(A) = 0.0
DO 45 B = 1, 3
45 MUA(A) = MUA(A)HDLTAAB (A, B)*XA(B)
46 R1 = R1 +MUA(A)*XA(A)
R1 = SQRT(R1)
DO 5 0 A = 1,3
MUA(A) = MUA(A)/R1
50 XA(A) = XA(A)+XAO (A)
WRITE (1, 11) MUA
WRITE (1, 11) xA
X = R1 /R
Y = ABS(1 .0X)
IF (Y10.0**(6)) 55, 30, 30
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55 RETURN
END
$0 END OF JOB
It may be seen that this FORTRAN program does not exactly follow
all of the steps of the computational scheme given above, but is
equivalent in all respects. The program was compiled and run on
STAT a
computer with the allowable error threshold set
at 106 (see the IF statement just before statement 55). Conver
gence occurred in from one to three passes (only one case required
three passes  all others were less) over the iterative loop. Table I
shows some of the results obtained
Table I
Some Results Obtained with Subroutine GNDCUR
H= 15240.0
13= 15.0
R= 6378388.0
111171,
(x x 1 pi = 0.1
0.0
1.0
p.a (2) = 0.00090
0.00000
1.00000
X' (l) = 5761.92969
0.00000
0.00000
H= 15240.0
13= 15.0
R = 6378388.0
X rn x1 /f 0.2
0.0
1.0
kia(2) 0.00118
0.00000
1.00000
Xa(1) = 7535.35742
0.00009
0.0000.0
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= 15.0 R= 6378388.0
( In
x111xT)/f= 0.5
0.0
1.0
[I (2) = 0.00212
0.00000
1.00000
Xa(1) = 13514.08203
0.00000
0.00000
11,1(3) = 0.00212
0.00000
1.00000
Xa (2) = 13527.11133
0.00000
15.00000
H =15240.0
0
=
15.0
R= 6378388.0
(xmxT) = 1.0
0.0
1.0
Ila(2) = 0.00414
0.00000
0.99999
Xa(1) = 26396.42578
0.00000
0.00000
(3) = 0.00415
0.00000
0.99999
Xa (2) = 26491.28125
0.00000
55.00000
II = 15190.0
0
=
15.0
R= 6378438.0
(xmx) = 0.1
0.0
1.0
Fla(2) = 0.00090
0.00000
1.00000
X8(1) = 5743.02539
0.00000
0.00000
H = 15190.0
=
15.0
R= 6378438.0
(xmxmi ) = 0.2
0.0
1.0
= 0.00118
0.00000
1.00000
Xa(1) = 7510.63477
0.00000
0.00000
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11 = 1 5 1 9 0 . 0
(x111x1111)/f =
pa (2) =
Xa(1)
tia (3) =
Xa(2)
0.5
0.00211
13469.74609
0.00211
13482.60352
H = 1 5 1 9 0
Ocalxr171)/f=
fia (2) =
Xa (1) =
=
Xa(2)=
1.0
0.00412
26309.82422
0.00414
26406.91406
H = 15240.0
0.1
0.00115
7337.97852
H = 15240.0
(>cmx111:1)/f =
1La (2) =
Xa(1) =
=
Xa(2) =
1.0
0.00512
32682.25391
0.00515
32863.65625
0 = 15.0
0 . 0
0.00000
0.00000
0.00000
0.00000
0 = 15.0
0.0
0.00000
0.00000
0.00000
0.00000
13 =20.0
0.0
0.00000
0.00000
0 = 20.0
0.0
0.00000
0.00000
0.00000
0.00000
R = 6378438.0
1.0
1.00000
0.00000
1.00000
15.00000
R=6378438.0 .
1 . 0
0.99999
0.00000
0.99999
57.00000
R= 6378388.0
1.0
1.00000
0.00000
R= 6378388.0
1.0
0.99999
0.00000
0.99999
85.00000
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II = 30480.0
13
=
15.0
R= 637838/3.0
(x m .... x lin ) A _ 0 . 1
0 . 0
1.0
it a(2) = 0.00181
0.00000
1.00000
Xa(1) = 11523.48437
0.00000
1.00000
= 0.00181
0.00000
1.00000
Xa(2) = 11527.97070
0.00000
11.00000
H = 30480.0
13
=
15.0
R= 6378388.0
(xmxT)/f= 1.0
0.0
1.0
ria(2)= 0.00828
0.00000
0.99997
Xa(1) = 52791.14062
0.00000
1.00000
Pa(3) = 0.00834
0.00000
0.99997
Xa(2) = 53174.14844
0.00000
221.00000
3.1.3 Plane Tangent at the Nadir
From Figure 3 and from the computational scheme given
above, it may be seen that the first pass over the iterative loop corre
sponds to the plane P1 which is tangent at the ground nadir.* The
FORTRAN subroutine may be seen to be based on a ground coordinate
system with its origin at the nadir and its Z axis vertical. Thus in
Table I the X1 = X and X2= Y results are in the plane PI and the X3 = Z
*Since the camera is moving, the term ground nadir is possibly
ambiguous; here it is used to mean a fixed ground point which is
vertically below the camera lens at some defined instant of time.
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results are perpendicular to PI ? If only the coordinates in P1 are
considered then Table I shows that the values obtained on the first
pass are within a fraction of a percent of the final values  for the
cases computed. Thus it is sometimes sufficiently accurate to com
pute approximate (horizontal) ground coordinates by treating the
earth's surface as a level plane through the nadir.
For a level plane through the nadir and with the
ground coordinate system having its X3 axis vertical equation (41)
3
may be put in a more familiar form. In this case DI = X1  X3 and
= (0, 0, I). Hence
and
C _a , 3 _3
= u ,
m ma m
Ca
m (x111 Xa
= X +  xI )
a in . 1
3 n n
Cn (x  x1)
It is common practice to write (42) with X3  X3 set equal to (IIh)
1
(42)
where H is the flying height and h is the ground elevation at the nadir 
both with respect to some datum (as mean sea level). Table I includes
cases which illustrate the effect of varying the ground elevation by
50 meters.
3.1.4 Treatment of the Photograph Coordinates for the
Different Types of Photography
Equations (41) and (42) give the ground coordinates
as functions of the rectangular components of the displacement of the
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photograph point from the instantaneous position of I he camera lens
(x111  x1111). The relations of this displacement vector to actual photo
graph measurements are different for the three different types of
photography.
3.1 .4 .1 Frame Type Photography
In analyzing frame photographs it is usual
practice to neglect the finite time required for exposure, and to con
sider the entire photograph to be exposed in the same instant of time
(thus also neglecting the scanning time for the focal plane shutter  if
one is used). Hence both the position of the camera with respect to
the ground and the position of the lens with respect to the film are the
a
same for all points in the photograph. In other words x, Ca , and X1
are treated like constants (i.e., constant at the particular values they
have at the instant of exposure). Thus the displacement vector
(x111  x1m ) is related to the actual photograph measurements simply by
translation and/or rotation of the measurement coordinate system into
the photograph coordinate system. Both of these coordinate systems
are usually taken with the x3 (z) axis normal to the photograph  hence
3
(x3  x1 ) = f (the camera foca, length), and the coordinate transforma
tion (if any) is effectively a two dimensional transformation. Thus the
displacement vector may be stated as:
m  x1 = Cn (x a
a.
x 
(1 x10)
where Cm, and x10 are constants with x3  x3 = f, and x are the
a. 1 0
photograph measurements. The lens coordinates x10 are usually
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determined from fiducial marks on the photograph.
In discussions from this point on the dis
tinction between the measurement (i.e., comparator stage) coordinate
system and the photograph coordinate system will be neglected unless
there is particular reason to discuss it, and photograph coordinates
will be treated as though they were measured directly. Hence (43)
will be stated simply as:
M M M M
X  X1 = X  10 = (X 1 1 10' x2  x21 0' f)
(44)
3.1 .4.2 Strip Type Photography
Strip type photographs are exposed by having
a narrow slit, with its long dimension extending across the width of
the film and perpendicular to the edges, scan in a direction parallel
to the edges of the film which are approximately parallel to the
direction of flight. The scanning speed is coordinated with the flight
speed so the image on the film is as nearly stationary as is practical.
Thus the projected ray from a strip camera lies in a plane which bears
a fixed angular relation to the camera and which includes the scanning
slit. The angular relation is stated as the tilt of the plane with respect
to the normal to the film, which is here called 0. This tilt angle (fi) is
normally fixed, for any particular strip camera, by the camera design
and is zero for systems currently of interest.
Throughout this and following discussions the
following conventions will be assumed for examining (i.e., measuring)
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processed photographs:
a. For cameras which do not produce a mirror type reversal:
negatives viewed emulsion side down (away from observer) and
positives viewed emulsion side up (toward observer).
b. For cameras which do produce a mirror type reversal:
negatives viewed emulsion side up and positives viewed emulsion
side down.
In all cases the photograph coordinate system
(right handed Cartesian) for strip type photos will be assumed to have
the x axis approximately in the direction of flight (i.e., the direction
of the camera motion with respect to the ground), the y axis parallel
to the long dimensidn of the scanning slit, and the z axis upward
normal to the film plane. Analysis shows that, under these assumptions,
the displacement vector for strip photos has the following components
(to be substituted in (41) or (42):
xm  x1 = (1 tan 13, x2
 x2 f
10'
(45)
Since (45) does not include the x photograph
coordinate, it is not sufficient to merely substitute (45) in formulae
for ground coordinates. The effect of the x photo coordinate is to
enable determination of the time at which a particular point of the
photo was exposed. This value of time must then be used to determine
a
the corresponding instantaneous values of X1 and Ca (which are time
functions). Strip photos usually have time tics along the edge to
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facilitate determination of the actual time at which any particular
point was exposed. Thus (41) and (42) are applied in a slightly
different fashion for strip photos than for frame type photos.
3.1.4.3 Panoramic Type Photography
Panoramic cameras expose the film by a slit
which scans around half of a circular cylinder to which the film is
clamped. The axis of the cylinder is approximately parallel to the
flight direction, and the scan is from high oblique on one side to
vertical to high oblique on the other side. The displacement vector,
with rectangular components (x111  x1 ) is from the lens (on the cylinder
axis) to the cylindrical surface of the film emulsion. The photo co
ordinate system (right hand Cartesian) will here be taken with the x1
axis parallel to the axis of the cylinder and approximately in the
flight direction, the x2 axis approximately horizontal and to the left,
and the x3 axis upward normal to an imaginary plane which is tangent
to the "top" of the cylinder. The scanning angle a (= cot) is positive
left handed about the x1 axis, and zero at the x3 axis. The
velocity is assumed proportional to cos a.
When laid out flat, for ,measurement, a
panoramic photograph exhibits its x1 axis unchanged but has the
original x2' and x3 axes combined into one  which is here called
Y  Yio?
Thus (see section 2.3.2):
2 2
x x1
y y10 =fa=ftan1(3 , ). (46)
3
x x1
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Putting the above stated definitions together and solving for the
rectangular components then leads to
1' yyio y y10
m m YYi0
x  xl  x sin f sin f cos ) (47)
10 (4.) f ' f '
where V is the maximum rate
conventions used here).
(a negative number in the
Equation (47) is to be substituted in formulae
for ground coordinates, such as (41) and (42), when these are applied
to panoramic photographs. As with strip photography it is also nec
essary to evaluate the time functions Ca and Xa for the particular
1
time at which any given photo point was exposed. For pan photos
this time is given (at least approximately) by
a Y  Y10
t
I
(48) is therefore to be substituted in the time functions assumed for
(48)
Ca and X'a whenever (47) is used in a formula for ground coordinates.
1
3.2 Stereoscopic Triangulation of Ground Points
In any aerial photograph, at the instant of exposure,
each photo point lies on a ray from some particular ground point
through the camera lens ("point of perspective"). (This ray is here
assumed straight  since atmospheric refraction is being neglected.)
If two photographs are taken with different camera locations but
including coverage of the same ground region then corresponding
points (as they are Called here) in the two photos, together with their
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common ground point, define a plane which includes the instantaneous
positions of the two points of perspective. The line segment joining
the instantaneous points of perspective will henceforth be referred to
as the "airbase" (i.e., the particular airbase associated with a partic
ular pair of corresponding points). nonce two overlapping photographs,
together with data stating the positions and orientations of their
respective cameras, may be used to corupute ground coordinates by
triangulation.
Equation (42) was written as giving ground coordinates
when certain assumptions are justified. Under somewhat more general
conditions, however, (42) defines a projective ray from the instan
taneous camera position. Thus, if two photographs are located and
oriented as they were originally exposed then projective rays from
corresponding points must intersect at the common ground point. An
equation equivalent to (42) but stated for the other photograph (with
photo coordinate system axes designated by xr, xs, ? ? ?) is
a r
Cr (x x)
3
= (X
S 3  X3)
2 s
Cs (x x2)
Equating (42) and (49) gives
m a
C (x  xi ) 3 C (xr xr,)
a in x3 X2).
3
X + (X X3 ) Xa + r
1 3 n n 1 2 3 s s
Cn (x x1) Cs (x x2)
*Note that in this sense (42) is valid even if the X3 axis is not
restricted to have the vertical direction.
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(49)
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Multiplying by the product of the two denominators then gives
a 3 3 a
in in r r 3 a 3 in m r r 3
(C  C C ) (x  x1 ) (x  x2) X C C (x  x1) (x x2) X1
m m r in r
3 am m r r3 3 3 in in r r a a
C C (x  x1) (x x2) X2 C C (x xi ) (x  x2) (X2  Xi). (51)
in r r
Now set the index a  1 and solve (51) for X3:
3 3 1 1 3 1 3 1 3 3 m m r r
3 [Cm Cr (X2X1)  (CinCr X2 Cm Cr X1)1 (x Xl ) (x  x2)
1 3 3 1 n n s
(Cn Cs  Cn Cs) (x  x1 ) (x x2
Hence:
C3 [C3 (X1 X1)  C1 (X3 X3) (xrn xin) (xr xr )
x3 3 mr 2 1 r 2 1 1 2

X1 1 3 3 Inn ss
(CnCs  Cn Cs)(x x1) (x x2)
Combining (42) and (53) results in
a 3 1 1 1 3 3 m in r r
(X2  Xi) I (x  xi )(x  x2)
1 3 3 1 n n s . s
(Cn Cs  Cn Cs) (x  x1 ) (x  x2 )
3 3
Similarly, using (52) to find X  X2 and combining the result with
(49) results in
Ca [C3 (Xi  X1) C1  X3).1 (x111x1 (xrxr)
r m 2 1 1 . 2
1 3 3 1 nnrr
(C C C C ) (x x1) (x x2)
Ft s n s
a
Evidently (54) gives ground coordinates based on XI' the coordinates
of the first camera station, and (55) gives ground coordinates based
on Xa ' the coordinates of the second camera station. Both (54) and
2
(55) involve the orientations of both camera stations, the flight base,
and the photo coordinates of corresponding points in the two photo
graphs. If there were no errors, in measurement or in computation,
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(52)
(53)
(54)
(55)
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and if there were no approximations involved, then (54) and (55)
would be expected to give the same values for ground coordinates.
Since there generally arc errors (54) and (55) may be expected to
give slightly different values and presumably the averages of the
two are the "best" values for ground coordinate's.
Equations (54) and (55) involve the displacement vectors
m  x1) and (xr  xr'
) Substitutions for these vectors are to be
2
made according to the method stated in section 3.1.4. Section 3.1.4
specifically states the method for the first photograph but can easily
be modified to state it also for the second photograph. All that's
necessary is to replace the various indices for the first photo with
corresponding indices for the second photo. The details will not be
given here.
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IV. AUTOMATIC STAGE TRACKING IN THE OPERATION
OF THE STEREOCOMPARATOR
The preceding chapter brings out the fact that for accurate
computation of ground coordinates from aerial photographs it's at
least desirable and perhaps actually necessary to use two over
lapping photographs taken with different camera locations. Reduction
of the data in these overlapping photographs involves measuring the
photo coordinates of pairs of points in the two photos which corre
spond, in each case, to the same ground point. To facilitate this
operation the Steroocomparator is designed to automatically adjust
its measuring stages and its various optical elements so as to enable
viewing selected regions of the two photos in stereo. The method by
which the two stages are maintained on approximately corresponding
points will be discussed in this chapter. The method for controlling
the optical system will be given later.
Before details of the stage tracking are given, two other matters
will be stated. One is that the primary functions of the Stereocomparator
involve measurement of the coordinates of corresponding points and
output of the resulting digital information, but not further reduction of
this information. Computations ( for example, ground coordinates)
based on the digital output of the Stereocomparator are performed by
a computer which is external to the Stereocomparator (not by the con
trol colLputer which is part of the Stereocomparator). The other matter
has to do with the precision of automatic stage tracking. This is
intended to be sufficient for comfortable stereo viewing bust not
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necessarily sufficient for automatic digitization of corresponding
points. That is, it is expected that the operator will, in general,
perform the final setting of the stages to precisely corresponding
points  after the automatic tracking has set them approximately on
the desired points.
A joystick and two trackballs are provided for use by the
operator in directing the two stages to desired points. Pushbuttons
are available which permit selection of various modes wherein the
joystick or either trackball controls either stage or both stages
together. Probably the most frequently selected mode will be one
in which the joystick controls both stages together, and each track
ball Controls one stage independently. In this mode the operator
directs both stages to various selected areas with the joystick and
performs final setting on precisely corresponding points with the two
trackballs. Under these conditions the Stereocomparator, unless
inhibited by the operator having selected a nontracking mode,
performs automatic stage tracking when directed by the joystick and
temporarily discontinues automatic stage tracking'when directed by
either trackball. No alteration of the pushbutton setting is required
to produce switching in or out of automatic tracking, since such
switching is automatically performed, as required, by deflection of
the joystick or either trackball. lithe joystick is standing at its
neutral position and neither trackball is being rotated then both
stages remain stationary as last directed by the operator.
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The Stereocomparator automatic tracking system can be described
as a master slave type of control system. That is, the operator, in
effect, directs one stage (the master) lo a desired point, and the
Stereocomparator electronics direct the other (i.e., slave) stage to
the corresponding point (approximately). To the operator, however, it
appears that both stages move simultaneously to corresponding points
since the time delay in servoing the slave stage to the master is so
small as to be unnoticeable. Somewhat similarly, the optics control
is by a master slave type of system, but this will be discussed later.
Two tracking modes are available for selection by the operator, which
function  one with and one without  an optoelectronic correlation
system. These will be described separately.
4 .1 Automatic Without Correlator Tracking Mode
4 .1 .1 Operations Performed
Automatic tracking in this mode is initiated by
the operator directing the stages successively to three different points
and manually establishing a stereo model at each of these three points.
In each case the operator depresses the REORIENT button after having
established a stereo model.
Depressing the REORIENT button causes the
internal control computer
to read the stage coordinates
for both stages, transform these into the respective photo coordinate
systems and substitute in equation (54), thus obtaining X, Y, Z co
ordinates for?one model point. When three model points have been thus
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obtained the control computer, in effect, passes a geometric plane
through these three points. That is, it computes the direction cosines
for the normal to such a plane and then computes the normal distance
from the plane to the first camera station. A plane thus determined
will hereafter be referred to as a "tracking plane."
Once a tracking plane has been established,
then tracking motion (i.e., stage motion such that the floating dot
appears to remain in the tracking plane) may be directed by the joystick.
Motion of the floating dot out of the tracking plane (for example, to the
top of a building if the tracking plane corresponds to the ground level)
may be directed by the trackballs.
4.1.2 Computation of Slave Stage Coordinates
The omputer controls stage tracking
by periodically reading the coordinates of the master stage and com
puting corresponding coordinates for the slave stage. For this purpose
equation (41) is used to compute model coordinates in the tracking
plane. Equations (11) and (12) are then used to compute the slave
stage coordinates. Equation (11) for the second photograph is written
as:
a a
r (X X2) 
r r a
x x2 2 f cos '2 3" Cb _x)
b (X
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equation for Xd 
Equation (41) is used to obtain the following
DI Ca (x"  x")  (Xa  Xd)
, n n m 1 2 1
x1)
11
D1 IIIn
a a 1(x  x)
in
X1) m
, n [cm ? DI 1
L pc 
n
Hence (56) becomes
Cra Yam (xIn  xlin)
xr = xr + f2 cos a2
2 3 b n
Cb Yn (x  x1)
wherein
a a I'M a a
Y rn C rn + ?D (X2  X1)*
1
The angle a2 is given by
0
It
tan1 C2a Yam (x x1)
 xi1)
11
(xn  xn1)
b n
frame type slave photo
strip type slave photo
pan type slave photo
Equations (57), (58), and (59) give slave
photo coordinates as functions of master photo coordinates,
utilizing the parameters p.m and D1 of the tracking plane and the
camera station parameters for both cameras. Since the latter are
functions of t1 and t2' the times at which the corresponding master
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and slave points were exposed, it is necessary to determine these
times. The time ti for the master photo is determined from the master
photo coordinates as described in section 3.1.4. This method cannot
be used for the slave photo, however, since the object at this point
is to compute slave photo coordinates  hence the latter must be
treated as unknown until after t2 has been determined.
4.1.3 Computation of Slave Photo Point Time of Exposure
The method of solving for the time of exposure of
the unknown slave photo point is somewhat similar to that given in
section 2.4.2. Each camera type imposes a constraint on its scanning
and/or exposure which is peculiar to its design. This constraint may
be combined with the flight equations to write a function of time which
can be equated to zero. The resulting equation may then be solved for
the particular value of time for which it is satisfied.
An iterative scheme, known as the Newton
Raphson method, is used to solve the equation. This method assumes
that a first approximation is known for the solution and enables com
puting a correction to this  thus yielding a second approximation.
The method is then successively repeated to give third, fourth, etc.,
approximations as far as is desired. Thus the equation
f (t) = 0
may have its left side approximated by the first two terms of its
series expansion:
f (to) + (to) (t  to) = 0.
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Solving this for t:
t = t
0 f' (t 0) '
f (to)
(60)
Hence if f(t) and its first derivative f (t) are stated explicitly and
t is an approximate solution of some order then t, as given by
0
(60), is an approximate solution of order one greater. Thus (60)
may be used iteratively to determine a value of t which makes f(t)
as close as desired to zero. Application of this method to the different
types of photography is described below.
4.1.3.1 Frame Type Photos
The idealized frame camera constrains
all of its points to be exposed simultaneously and instantaneously.
The time of exposure must be stated as part of the auxiliary input
data, and no additional computation is necessary.
4.1.3.2 Strip Type Photos
The strip camera scanning slit causes
a constraint which was stated as equation (21). Reexpressing (21)
for the slave photo and arranging the result in a slightly different'
form gives
[Ca  C3a tan 132[ (Xa ) = 0 (61)
Equation (61) is a function of time L2 since Cr and Xa are functions
a 2
of time  stated by the equations of flight. Hence (61) may be solved
by the method of equation (60). That is
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..,
1 3 a. a
( Ca  Ca tan (3) (X  X2)
t = tn n II
Tt [ (Cb  CI) tan 132) (Xb  x2)]
(62)
wherein the second term on the right is evaluated at a particular
approximation to for the solution of (61), and t is hence the next
(better) approximation. Thus (62) is used iteratively until successive
values of t cease to show significant change from one to the next.
This can only occur when the numerator of the second term is suffi
ciently near zero. Hence (61) is adequately satisfied by the final
value of t.
4.1.3.3 Panoramic Type Photos
The last statement of (59) may be
used as a constraint to solve for the slave photo time of exposure.
3 a a
Time t2 enters (59) by Ca (i.e., C2 and Ca ) and Ym (X2). In this
a
case the formal scheme of (60) can be somewhat simplified, since it
is assumed that
where (.02
(12 = w2 (t2 t20)
is a constant (stated in the input data) and t20 is the value
of time at which the scan angle a2 is zero. Thus if t is an approxi
mation of some order for the value of time which satisfies (59) then
the next better approximation t2 is given by
2" am m\
1 Ca Ym (x  x1 /
t2 = t + ?1 tan
20 (.,023 b n n ?
Cb Yn (x  x1)
wherein Ca and Y am are evaluated at the time t. The value t2 is then
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used to reevaluate Cr and Ya and (63) is used again  to obtain
a
the next better approximation. Thus (63) may be used iteratively
until successive values of t2 cease to show significant change from
one to the next. The final value of t2 is hence the time of exposure of
a point of the slave photo corresponding to the master photo point
whose coordinates are xm (as substituted in (63)). This value of
time is then substituted in (57) and in (27)* to give the predicted slave
photo coordinates.
4.1.4 Modified Computations for RealTime Stage Control
The Stereocomparator design specifications require
that, for smooth stage motion, the control computer must output stage
drive commands at the rate of 120 per second. Each such drive com
mand consists of incremental values of x and y (either of which may be
zero) for each stage. The two stages attempt to follow each incremental
drive command, but the frequency response of the stage servo systems
is much below 120 hertz. Hence the incremental (step) nature of the
computer outputs is very much smoothed out in the actual stage motion.
In order to satisfy the requirement for 120 cycles
of input/output transfer per second the control computer program is ?
separated into two parts  called the real time and the nonreal time
programs respectively. The real time program includes the I/0 transfers
and performs only the minimum amount of computing which is necessary 
*That is (27) rewritten for the slave photograph.
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according to the scheme which is described below. The nonreal
time program does the bulk of the computing and feeds results to the
real time program. Since the nonreal time program requires con
siderably more than (1/120) second for complete execution, it is
interrupted 120 times a second and transfers control to the real time
program. The latter requires much less than (1/120) second for one
complete cycle and, after executing one cycle, returns control to the
nonreal time program at the location from which the latter was
interrupted. The nonreal time program is, executed repetitively 
requiring about 1 or 2 seconds for each cycle of execution.
As a means of separating the computations into
those which must be performed in real time and those which may be
done in nonreal time a number of the formulae are expanded as
series. This enables computing the coefficients (parameters) in non
real time and evaluating the series each time the real time program
cycles. The frequency of cyling the real time program ensures that
the variations between any two successive cycles are small; hence
only the zero and first order terms are retained for each series.
The aerial photographs which are measured by
the Stereocomparator, generally speaking, have appreciable amounts
of distortion due to tilt and motion* of the camera during exposure.
To enable viewing two distorted pictures in stereo the Stereocomparator
optical system introduces automatic compensation for the distortion
*Thel kechanism minimizes multiple exposure but still permits
geometric distortion.
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within any particular field of view. The means for doing this will be
described later, but the point here is that the distortion correcting
features of the optical system also affect the way the images in the
eyepieces appear to move when the stages are moved. Hence the
control computer is programmed so that when the operator deflects the
joystick or one of the trackballs in a certain direction and by a certain
amount then the stages are commanded to move in such a direction and
at such a rate that the images in the eyepieces move in the directions
and by the amounts directed by the control which was deflected.
Motion of the images may be represented by
the two dimensional vector Axj, which will apply to either image or
to both images simultaneously as is appropriate in any particular
discussion. Motion of the master and slave stages will be represented
by the two dimensional vectors Axm and Axr respectively. Thus the
operator deflects the joystick or one of the trackballs in a direction
and by an amount corresponding to the vector Axj (with x and y but
not z components). The computer must determine the vectors Axm
and Axr and output corresponding commands to the two stages.
Evidently for masterslave stage tracking
Ax3 corresponds to deflection of the joystick and the above mentioned
vectors must be related by the expressions
and
m m j
Ax = X Ax
Axr Xr Axm
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(64)
(65)
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where Xl.'1 and Xr are 2 x 2 matrices to be determined by the computer.
The matrix Xr.11 is the reciprocal of the master optics transformation
matrix which will be described later. 'The matrix XI is called the
tracking matrix anci is derived, basically, from equation (57). Thus
the computer operates to satisfy (64) so as to move the master stage
as required to produce motion of the master image as directed by the
operator. Simultaneously the computer operates to satisfy (65) so as
to move the slave stage as required to maintain the two photos with .
corresponding points at (or at least close to) the respective optical
axes. Hence the floating dot appears to move as directed by the
operator and to remain in the tracking plane.
For nontracking motion of the stages (64) still
applies to the master stage but control of the slave stage must be
according to the expression
r r
= j
xAx X. A
where Xr. is the reciprocal of the slave optics transformation matrix.
In this case Axi for one image corresponds to deflection of one
trackball and A for the other image corresponds to deflection of the
other trackball. Discussion of (64) and (66) will be deferred until
later. At this point computation of Xrm, which appears in (65), will
be discussed.
The
series expansion of equation (57)
(66)
may be derived by a method similar to that given in section 2.4. The.
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result is
with
xr xo + Xr 1 r  X AX Axn + ? ? ?
2 inn
,r ax dxr at
x
ax n1
dt Dx111
axr jr
Xr m
, etc.
11111 Dxn
dt Dxn
(67)
(68)
(69)
at
and determined from the appropriate constraint equation. Only
Dx11)
the first two terms on the right side of (67) will be retained in the
computations, however. Equation (57)  hence also (67)  is basically
a three dimensional relation between two photographs both of which are
two dimensional. Hence if both photograph coordinate systems are
oriented with their Z axes normal to the respective photographs then
the Z axis components in (67) may be ignored. Thus, neglecting
higher order terms, (67) may be written as
A xr = xr  xr = Xr Axril
0
and the 2 x 2 (x, y) portion of Xrm is seen to be the same as the
tracking matrix in (65).
From (57), (58), and (59) it may be seen
dxr
that the explicit formula for the derivatives ? is quite complex.
dt
Consequently the nonreal time program does not use (68) to compute
the tracking matrix (as would be the theoretically correct procedure).
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Instead it uses an approximate method based directly on (57).
Equation (57) may be written symbolically:
r r m
x f (x ).
For any increment Axm, (70) may be written in the form
xr + Ar fr (x +
i.e., Axr= fr (xm +Axrn)  fr (xm).
(70)
(71)
Thus (57), in the form of (71), may be used to compute two values
of Axr corresponding to two arbitrarily chosen values of Axm. These
may then be substituted in (65) and the resulting equations solved
for Xr . If the values chosen for Axm are smaller than the diameter
rn
of the largest portion of a photograph over which Xr may be con
sidered constant and not so small as to cause excessive truncation
errors in computing (71) then the resulting values for Xrm should be
approximately correct. As tracking proceeds the values of xm and xr
will change and Xrm will have to be recomputed  hence the nonreal
time program runs repetitively. For accuracy and convenience the
nonreal time, program computes with floating point arithmetic.
Thus the computing which is left for the
real time program  with regal ci to stage tracking  is to evaluate
(64) and (65) or (66)?using values of )(71, Xr , and Xr, supplied by
the nonreal time program. Only a few multiplications and additions
are needed, and, for maximum speed, these are performed in fixed
point arithmetic. The values of Axi are obtained by reading the
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latest position of the joystick and the trackballs. The computed
values of xm and xr are transferred to the master and slave stage
servo systems, and the stages move accordingly.
In general, it may be expected that stage
tracking, as described above, will have a limited range over which
it may proceed without introducing appreciable noncorrespondence.
Thus when operating in the automatic without correlator mode it may
be expected that new tracking planes will have to be established
from time to time. The three point procedure described earlier may,
of course, be repeated whenever desired. However another procedure
is also available if a tracking plane has been previously established.
This consists of manually establishing a stereo model on some one
point and depressing the REORIENT button, and then immediately
using the joystick to reestablish the tracking mode. Under these
conditions the computer will compute a new tracking plane through
the desired point and parallel to the previously established tracking
plane.
4.2 Automatic With Correlator Tracking Mode
The opto electronic correlator consists of two image
disector type TV camera tubes and electronics for controlling the
beam deflections, correlating the video signals, and computing 6
analog output signals. The image clisector tubes are mounted so as
to "see" essentially the same views as the operator's two eyes.
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The 6 output signals are derived from the video correlation in such
a way as to represent approximate measures of 6 respects in which
the two images differ from one another. These 6 respects are called:
x displacement, y displacement, x scale factor, y scale factor,
x skew factor, and y skew factor. Thus two of the output signals
(x and y displacement) may be used to aid stage tracking. The other
four output signals are used to aid in setting the optical system  as
will be described later.
Supplementary to the prime functions described above,
the optoelectronic correlator (called the "Image Analysis System"
or IAS) also outputs a digital signal showing whether it can or cannot
satisfactorily correlate the video from the images which it is seeing
at any particular time. Some minimum degree of detail, contrast,
and brightness are needed for satisfactory correlation, but also it is
necessary that the two images be within the "pullin" range in each
of the 6 respects. Whenever the IAS is called for by the operator's
mode selection but is nevertheless unable to satisfactorily correlate
then it outputs a digital signal which operates an indicator light on
the Stereocomparator control console and tells the control computer
to disregard the IAS output lines.
In initiating operation in the Automatic With Correlator
mode the operator manually establishes a stereo model at some one
pair of corresponding points. This may be done somewhat approxi
mately since the IAS will automatically take hold as soon as the two
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images are brought within the pullin range in all 6 respects. Once
the IAS has taken hold then the operator may direct tracking or
nontracking operation by way of the joystick or trackballs as was
described for the Automatic Without Correlator mode.
Computer control with the aid of the IAS differs in two
respects from the scheme which was described in section 4.1. In
the first place the tracking plane is established by the computer with
the aid of the correlator, without any need for manual setting on three
points. For this purpose a plane tangent to the "spherical" earth is
established by the method of section 3.1.2 and parallel shifted by an
automatic one point reorient so as to permit best correlation. In the
second place when the slave stage vector is computed by the equiva
lent of equation (65) then the IAS x, y displacement outputs are used
to compute a correction term. Equations (64) and (66) are, however,
used just as before.
The IAS images are essentially the same as those seen by
the operator  hence the IAS ,coordinate system is logically the same
as that used for the images. In other words the IAS x, y displacement
signals may be treated as components of the two dimensional vector
. Consequently for stage tracking with the aid of the IAS, equation
(65) is replaced by
Axr = Xr Axm Xr Axi
c
with Xr and Xr the. same as defined in section 4.1.4. Thus the
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(72)
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correlator signals are applied as corrections to the slave stage only,
rather than being divided between both stages. This is done because
it results in simpler stability conditions for the servo systems.
4.3 Digital Integration of the .Control Commands
Although equations (64), (65) or (72), and (66) are used,
as has been described, to compute the required increments of stage
motion, the actual outputs to the stages sometimes differ from these
increments. This is done since there may be time lags in the actual
stage motion as compared to that which is commanded.
When operation is initiated in either of the automatic
modes, the computer reads and stores the x, y coordinates of both
stages. Henceforth each time the computer obtains values for Axm
and Axr (i.e., 120 times a second) it adds these increments to the
stored values which it is maintaining for commanded stage coordinates.
This is equivalent to digital integration of the computed increments,
and the initially read stage cooedinates are the initial values of the
integrals thus obtained. In each cycle of the real time program the
computer compares these integrated increments with actual stage
coordinates at the time of that particular cycle. The vector differences
are the values actually output to the stage servo system. Thus the
stages are continually being commanded to move by the amounts which
their actual positions differ from where the computer has determined
they ought to be. Consequently time lags occurring during periods
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when the stage commands call for acceleration tend to be offset by
overshoots during periods of deceleration.
Servo theory tells us that for zero steady state positional
error a servo open loop transfer function should have a pole at the
complex frequency origin. This is the effect which is achieved by
the digital integration described above. Zero positional error is
particularly relevant for the correction signals derived from the IAS
since it's desired that the two stages approach precisely corresponding
points. Zero positional error is also relevant for trackball control
since it's desirable that the operator be able to direct precise stage
positioning. In both these cases the temporary time lags are incon
sequential, since they do not result in net position errors. All this
is accomplished by having the computer perform digital integration of
the computed position increments.
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V. COMPENSATION OF DISTORTIONS FOR STEREO VIEWING
Figure 4 illustrates the importance of the points of perspective
for stereo viewing. Pi and P3 represent two frame photographs both
having their point of perspective at coordinates X. Similarly P2 and
P4 represent two frame photographs with their point of perspective at
X. A and B are two ground ;points and al, a2, a3, a4 are the respective
photograph points corresponding to A. Likewise b b2' b b4 are
photograph points corresponding to B. From the figure it's evident that
photographs P1 and 13 contain the same information (though possibly
in different geometric form). Likewise photographs P2 and P4 contain
the same information. In general, however, the photographs P2 and P4
do not have the same information content as P1 and P3. In what follows,
P1 and P3 will be referred to as "equivalent" photographs, and likewise
for P2 and P4' Evidently, then, a pair of photographs which are equiva
lent to one another cannot be viewed in stereo. In other words, a
necessary (but not sufficient) condition that two photographs be suitable
for stereo viewing is that they be taken with different points of per
spective.
5,1 Conditions for Stereo Viewing
Figure 4 is drawn to suggest that Pi and P2 are vertical
photographs whereas P3 and P4 are tilted photographs. If vertical frame
photographs are regarded as undistorted standards, then tilted photo
graphs may be said to contain tilt distortion, and strip and pan photo
graphs also contain motion distortion. Generally speaking, these
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Figure 4.
Photographs P1 and P2 illustrate stereo viewing. Photographs
P3 and P4 are represented as "equivalent" to P1 and P2 respectively.
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100
.41
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distortions tend to interfere with stereo viewing, but some degree of
distortion can be tolerated without undue discomfort.
Experience seems to demonstrate that frame type photo
graph pairs with 'oblique tilt (not convergence) can be viewed about as
comfortably as vertical photographs. Convergence type tilt,on the other
hand, seems to make stereo viewing quite uncomfortable. Hence the
criterion which will be used here for correcting distortions will be to
produce images as nearly as practical like oblique or vertical (not
convergent) frame photographs. Furthermore, the obliquity angles for
the two images of a stereo pair will be made equal. Discussion of the
details of this scheme will, however; be deferred until after the func
tioning of the optical system has been outlined.
5.2 Elements of the Stereocomparator Optical System
Figure 5 shows the main projection elements of onehalf
of the Stereocomparator optical system (the other half is analogous to
a mirror image of Figure 5). The system is designed to provide continu
ously variable magnification over two ranges: 10X to 100X and 20X to
200X (by way of the zoom lens and two interchangeable objtctive
lenses), It also provides continuously adjustable allomorphic stretch
over the range 1/1 to 2/1  with a continuously variable angle for the
stretch axis. Continuously variable rotation of the magnified and
anamorphosed image is also provided. In other words, each half of the
main projection optical system is continuously adjustable through 4
degrees of freedom: magnification, anamorphic stretch ratio, anamorphic
 
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Fixed
Mirror ?70
Reticle
Anamorphic Lens
Zoom Lens
Z
Beamsplitter
Image Diseetor
and PhotomUltiplier
Tubes
/ \
Imago
Rotator
Beamsplitter
Collimating Objectives
Film plane (mounted
on measuring stage)
Illumination
Figure 5
Partial diagram of onehalf of Stereocomparator Optical System
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0 0 
Eyepieces
(with switching
optics)
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stretch angle, and image angle. In addition the eyepieces can be
switched 4 ways: (1) normal stereo, (2) crossed stereo, (3) binocular
viewing of the left photograph, and (4) binocular viewing of the right
photograph.
Figure 5 shows a beamsplitter just above the objective
lens  to allow superposing a reticle image on the image of the photo
graph (i.e., appearing to lie in the plane of the photograph). The
reticle optics include a zoom lens and an anamorphic lens which are
servoed to the corresponding elem ents in the main projection path so
as to minimize changes in size or shape of the reticle when the photo
graphic image is zoomed and/or anamorphosed. The reticle optics also
include yet another zoom lens which allows the operator to adjust the
reticle size over a 4 to I range. Thus the reticle serves as a fixed
pointer (marking the optical axis), to which various points of the photo
graph may be brought as directed by the operator  in order to measure

the coordinates of the selected points.
Considering both halves of the optical system (which
are like mirror images of one another), it's evident that two aerial
photographs taken of the same ground region, but with different points
of perspective, may be viewed (and measured) simultaneously. The
4 degrees of freedom available in adjusting each half of the optical
system are generally sufficient to compensate geometric distortions
in the photographs so the operator is able to form a stereo perception
of the two images. Simultaneously with forming a stereo model of the
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ground region covered by the two photographs, the operator tends to
perceive the two reticles (one in each eyepiece) fused into a single
dot which appears to lie in, or float over, the stereo model. By
adjusting the two trackballs the operator may set the floating dot so
it appears to lie precisely on the surface of the stereo model. Under
these conditions the two photographs have precisely corresponding
points brought to the respective optical axes. Depressing one of the
record buttons then produces measurement and digital output (either to
a card punch or to a link to an external computer) of the two sets of
xy coordinates for the corresponding points. Such digitization of
corresponding points is the prime function of the Stereocomparator.
Automatic adjustment of the optical system so as to enable stereo
viewing is a secondary function which greatly aids the operator in
selecting the points he wishes to have digitized.
5 . 3 Matrix Representation of the Functions of the Optical Elements
Photographs placed on the Stereocomparator's two
measuring stages are, of course, constrained to planes perpendicular
to the respective optical axes. The two images of these photographs
formed by the respective optical systems may also be thought of as
lying in planes perpendicular to the optical axes  at the top or eyepiece
ends. Hence it is convenient to orient the photograph and image co
ordinate systems with their respective z axes parallel to the optical
axes. As in chapter IV the master and slave photograph coordinate
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axes are designated byxm and xr respectively.* The coordinate axes
for both images are designated xi, and the x axes are taken in the plane
of, and normal to, the two optical axes  i.e., in the direction of the
operator's eye base. That it is reasonable to use parallel axes for the
two images follows from the criteria for stereo viewing  as will be
shown eventually.
Thus the master image may be represented as a (two
dimensional) function of the master photograph
xi =11 (x m).
Similarly the slave image may be represented as a (two dimensional)
function of the slave photograph
The
= fi (xr)
2
series expansions of these equations may then be written:
Ax = Xi Axm +1 Xj Axm Axn +... (75)
2 mn
(73)
(74)
for the master image, and
1
Ax) = X) Axr + Xi Axr ,Ax +
2 rs
for the slave image.
(76)
Evidently (75) and (76) must represent the optical trans
formations (from photograph to image in each case) produced respectively
*Although there are, in general, two dimensional coordinate trans
formations between photograph coordinates and measuring stage
coordinates, these transformations are ignored as too trivial to
require discussion.
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by the master and slave optical systems. In section 5.2 the optical
systems were stated to each have four functional parameter7, (magnifi
cation, anamorphic stretch ratio, anamorphic stretch angle, and
image rotation angle). These parameters may be used to compute the?
components of the 2 x 2 matrices Xi and Xj which appear in (75) and
(76)  as will be shown. From this it follows that the 2 x 2 matrices
Xj and Xj must be substantially independent of the x, y coordinates
about the optical axes, and hence, that the higher order derivatives
Xi ? ? ? and Xi ? ? ? may be taken as zero. The 2 x 2 matrices Xi and
mn rs
Xi will be referred to as the master and slave optics transformation
matrices.
Figure 5 shows that each optical system has three
elements in tandem  the zoom lens, the anamorphic lens, and the
image rotator. These elements all operate in collimated light. fiance
they may be thought of as se,parated by planes corresponding to inter
mediate images. The input (plane) to the zoom lens is the photograph
itself. The output (plane) from the zoom lens is also the input (plane)
to the anamorphic lens, whose output, in turn, is input to the image
rotator. Finally the output from the image rotator becomes the actual
optical image (by way of a decollimating lens and the eyepieces).
Thus it is proper to represent the function of each optical element by
a 2 x 2 matrix, and the matrix product of all the constituent matrices
is the optical transformation matrix for the whole system. The matrices
for the individual elements are as follows (for the master optical system,
which is typical of both optical systems):
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5,3.1Zoom Lens
Since the output from the zoom lens is a
magnified replica of the input, the transformation matrix must be
the identity matrix multiply by a scalar whose value is the magni
fication.
(77)
5.3.2 Anamorphic Lens
The anamorphic lens may be regarded as
having a major axis and a perpendicular minor axis  of magnification.
The magnification in the direction of the minor axis will be taken as
unity  hence the magnification in the direction of the major axis is
equal to the anamorphic stretch ratio. The anamorphic angle will be
defined as the angle of the major axis with respect to the x axis of
the optical system. The transformation matrix is hence the product
of three matrices:
XX =
a sin 02 cos 02
cos 02 sin 0
2
a
0
cos 02 sin 021
Tsin 02 cos 02
(78)
wherein a is the anamorphic stretch ratio and 02 is the anamorphic
angle. Evidently the third matrix in (78) rotates the image plane
axes so the rotated x axis is parallel to the major axis of the
anamorph. The second matrix in (78) then gives the magnifications
along the rotated x and y axes. Finally the first matrix on the right
side of (78) rotates the axes back to the original orientation. Per
forming the matrix multiplications in (78) gives
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X =
2
(a cos202 +sin 02)
(al)sin 02 cos 02
(al)sin 02 cos 02
(a sin2 02 + cos2 02)
Thus XX is a symmetric matrix which is the identity matrix if the
a
anamorphic ratio is unity.
(79)
S.3.3 Image Rotator
The transformation matrix for the image
rotator is essentially a standard two dimensional rotation matrix.
However, it represents rotation of vectors (in the image) rather than
rotation of axes. Consequently it is the transpose of the corresponding
axes rotation matrix,
xj =
cos 01  sin 0
1
sin 01 cos 0
wherein 01 is called the image rotation angle. Note, however, that
the image rotator acts on an image which is, in general, modified by
the anamorphic lens. The combined effect is such that this name
for 01 should not be taken too literally.
(80)
Multiplying together the three component matrices then
gives the overall master optics transformation matrix:
Xj = X XX Xa
Similarly the slave optics transformation matrix is the product of'
three component matrices:
p r ?
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(81)
(82)
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The forms of these component matrices are, of course, the same as
those given for the corresponding master component matrices
with values for the slave optical elements substituted.
 but
The right sides of equations (81) and (82) are each
functions of four variables (magnification, etc.). If the latttr are
regarded as unknowns then (81) and (82) may be solved for them as
functions of the components of the matrices on the left sides of (81)
and (82). These solutions will be indicated later, but first a method
will be shown for determining values for the components of the matrices
Xi andXi so that these components become the known variables in
r'
(81) and (82).
5.4 Equivalent Frame Images
The concept of an equivalent frame image will be used
to determine a method for computing the optics transformation matrices
Xj and X. Referring back to Figure 4, P3 is considered to be equiva
lent to the frame photograph P1 since it has the same ,point of perspective.
Strip and panoramic photographs do not, however, have the same point of
perspective for all of their points. Hence for these types of photographs
there is no strictly equivalent frame type. Nevertheless, the concept
of equivalence will be used, possibly loosely, to include frame images
of small extent which are considered approximately equivalent to small
regions of real photographs  by reason of having nearly the same point
of perspective.
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By reasoning similar to that leading to equation (57)
the following equation may be written for a frame type image related
to the master photograph, using the parameters of a tracking plane.
with
J ?a m m
xjx0+ f
Ca Ym (X  X1 )
= j
I 3 n n
C Y (X 
b n
a a m a
Y = C + (X0  X
m m DI 1
Equation (83) is expanded as a
eries which applies over the
(83)
(84)
region of the photograph which is, at any particular time, wilhin the
field of view of the master optical system. This series may be written:
with
m
Axj = Xm x +! X Axm Axn + ? ? ?
2 mn
j
? ax J dx at
xm
axm + dt 3xm
ax dx' at
m m
x m ,etc.
mn
3xn
dt 3x
Thus (85) has the same form as (75), and Xjrn, as computed by (86),
is seen to be the master optics transformation.
(85)
(86)
(87)
The following considerations apply in taking the
derivatives called for in (86): (1) To make the image as nearly like
a
a frame type as possible, x10, Ca, and Xi 0 are treated like constants.
a
(2) To make the image as nearly equivalent as possible Xi0 is set
equal to the instantaneous value of X7 which corresponds to the
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photograph point which is at the optical axis. (3) The tracking plane
a m a
parameters 11m and Di are treated as constants. (4) Cm, xi , and Xi
are, as usual, treated as functions of time which is itself a function of
the coordinates of the point under observation. Wherever the dis
placement vector (xm  xm) appears, it is treated as in section 3.1.4 
1
after differentiation has been performed. Equations (44), (45) and (48)
at
may be used to obtain
8x
The formulas for the image equivalent to the slave
photograph have the same forms as those given above but with appro
priate changes in indices. Thus:
va fxr _
xj xj f 1'21
20
Cb Ybs (xs  x2s)
with
wherein
and
a a a a
Y = C + ? (X0 
r r D2 2 L
a a m
r=Cr ?a= Cr Ca p.m
+ a (X2a xal ) ?
_ 8xj dxj at
Xr 
axr dt oxr
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(88)
(89)
(90)
(91)
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The symbol xi is used for the coordinates in both the
master equivalent image and the slave equivalent image. This means,
not that these two different sets of coordinates can be equated to each
other, but that they are computed for parallel sets of axes. Parallel
sets of axes are necessary in order that the images produced by the
two optical systems will appear to have their normals parallel to each
other and perpendicular to the eyebase (which is taken as the x axis
direction). These conditions on the respective normals are implied in
the criteria for stereo viewing, given in section 5.1.
5.5 Solution of Equations
In section 5.3 it was shown that the basic performance
of the Stereocomparator optical system could be expressed by the optics
transformation matrices Xim and X. In section 5.4 a method was given
?' for computing the components of these matrices in terms of the photo
? graph geometry and the criteria for stereo viewing. In this section
equations (81) and (82) will be solved for the unknowns in their right
sides, i.e., magnification, anamorphic stretch ratio, etc.
notation:
It will be convenient to use the following short hand
sl ,?= sin 0
1
Cl = cos 01
= sin 02
s2
c2 = cos 02
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Then (79) may be written
a c22 +S22
(a1) s2
(a1) s2 c2
a s22 +C22
a_
rca+1 4_ a1
2 2 cos 02) 2
2 a1
sin 2 02
271 sin 2 02 (a+1 a1
2
2 cos 2 02)
,Hence (81) may be written
Xi = M
a+1 a1
2 c + cos (01 + 2 02)
1, 2
a+1
a1
L 2 s1 + 7 sin (01 + 2 02)
(92)
a+1 a1
7 s1 + 2? sin (01 +2 02
a+1 a1
1 2 cos (01 +2 02)
Now represent the components of Xi as follows:
Xm =
A B
C D
(94)
Thus (93) and (94) are 4 equations with A, B, C, D known (by section
4
5.4) and M, a, 01 and 02 treated as the unknowns. From them may be
derived the following 4 equations:
A+ D = M (a+1)
C  B = M (a+1)
A  D = M (a1) cos (01 + 2 02)
C + B = M (a1) sin (01 + 2 02)
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(95)
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Equations (95) may be solved, and the results are:
2 .tir , r
A2 +D +C2 +B2 + L(AD)2 + iC+B)2 ,H.(A+D)2+ (CB)2,
a  2 (ADCB)
M
a
?IAD  CB
0 = tan 1 CB
1 A+D
 C+B
01 + 2 02 = tan1
A D
wherein the positive sign (only) has been chosen in front of the
radical; corresponding to the physical fact that a 5 1 Equations
(96)(99) apply (separately) both to the master and to the slave
optical systems. For convenience they may have the subscript 1
appended to all ?tariables when applied to the master optics, and
the subscript 2 appended to all variables when applied to the slave
optics. It may be shown that the determinent (ADC[3), when its
elements are computed by equation (86) for an aerial photograph,
does not vanish for any point in the photograph which corresponds
to a ground point lying below the horizon.
(96)
(97)
(98)
(99)
? 5.6 Automatic Without Correlato,r Control of Optics
As was stated in section 4.1.4 the control computer
program may be separated into a real time part and a nonreal time
part. Evidently, to control the optics, the real time part must perform
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operations equivalent to evaluating equations (96)  (99) for both
the master and the slave sides. In order that this may be done
rapidly (96)  (99) are therefore expanded as
series, with
terms of order higher than first being neglected. Hence the real
time program must evaluate the following expressions:
_ (aaaa as aa ac 4.3a 81)1/4,4.18a 8A 4.
?aA ax ' as ax ' ac ax ' aD 8x/'"" ?aA 8y ? ?
.?? (am aA am aA
taA a?x + ???)Ax + ( ?+ ???) Ay
aA ay
A 02 = ? ? ?
Ay (100)
(101)
(102)
(103)
In (100)  (103) the various parenthized expressions, which may be
labeled
aa aa 8M aM aoi ..?
ax' ay' ax ay ax '
are computed in the non real time program, and the values are whence
supplied to the real time program. Thus the real time program performs
only a few computations ?(in fixed point arithmetic) to determine the
required increments of adjustment for the optical elements. As in
stage control (see section 4.3), these incremental adjustments are
digitally integrated before being output to the optics. The reasons for
Integrating the optical control signals are substantially the same as
those given for the stage servos.
The optical servos differ from the stage servos, how
ever, in that the position feedback signals are analog rather than
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digital (i.e., they are generated by analog potentiometers rather
than by digital counters). These analog position signals are con
verted into digital form so that the computer can keep track of the
optical settings. The A/D converters, however, turn out to have a
resolution which is too low for the requirements of the optical servo
systems. Consequently the analog position signals are used
directly (rather than the digital conversion of them) in the servo
position feedback loops. Correspondingly the computer output
signals are the actual integrated values rather than the differences
between these and the optical position signals. These digital out
puts are converted to analog and the differences between them and
the position feedback signals are taken by analog circuitry as part
of the optical servo systems. In other words, by subtracting the
position feedback signals from the computer position commands
with analog circuitry rather than.digitally in the computer, the
position feedback loops bypass having them, in effect, converted
to digital and then back to analog form.
The non real time program uses (94), (86) and the indicated
partial derivatives of (96) through (99) to evaluate the parenthesized
expressions in (100)  (103). Although it would be proper to use (87)
aA
for the partial derivatives a?x, etc., an approximate method is used
instead. This consists of evaluating (86) for several nearby points
and dividing out the incremental differences in much the same way as
was described for (70) and (71).
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A study of the partial derivatives of (96) and
(97) shows that they include, among others, the expressions
A  D
and
)(A  D)2 + (C + B)2
C + B
(104)
(105)
i(AD)2 + (C + B)2
which become indeterminent when
AD=C+B=0.* (106)
Furthermore, (99) itself, becomes indeterminent under these conditions.
It may be seen, however, that under these conditions the right side of
(96) becomes equal to one. Hence the value of 02 becomes inconse
quential. The computer program contains a test for the condition
 D)2 + (C + B)2 < E
AD CB
where E is some sufficiently small positive number (say about 106).
(107)
This test is applied repetitively and whenever it suddently becomes
true a is set to 1 and the value oi 02 is frozen at the last value which
was assigned to it. The freeze continues until (107) is found to be no
longer true, at which point use of (96) and (99) is resumed for evaluating
a and 02' In other words, the indetermency condition corresponds to
there being no need for anamorphic correction, and a logical switch
causes the anamorph to be set to unity when this is true.
*The corresponding conditions A+D=CB=0 cannot occur
since AD CB > O.
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The preceding paragraphs describe the method by which
the computer adjusts the two optical systems so as to compensate for
changing distortions as the stages move to various parts of the two
photographs. During this automatic process, the operator has no manual
control of the optics except that he may direct the scale factor to which
the computer is to bring the two equivalent images. He does this by
turning either magnification control in the direction for either increasing
or decreasing scale factor as he desires. The two magnification controls
operate interchangeably (when in either AUTOMATIC mode) to drive both
zoom lenses at rates proportional to their respective magnification
settings. The computer program then resets the otherwise arbitrary scale
factor (f) in equations (83) and (88) to correspond to the new setting of
the two zoom lenses.
The OPTICS INDEPENDENT button provides the operator
with means for directing the computer to discontinue automatic tracking
of the optics. When this button is selected the operator may manually
set the various optical elements as he desires. Having done so, the
operator may either reset the OPTICS INDEPENDENT button, or he may
select the REORIENT button. In the former case the computer will cancel
the manual adjustments and will resume automatic tracking from the
settings which existed prior to the OPITCS INDEPENDENT operation.
If the REORIENT button is selected (without having reset OPTICS
INDEPENDENT) then the computer will resume automatic tracking from
the newly established settings. In other words, in this case the com
puter takes the new settings as "initial" conditions for the various
digital integrations which it is performing.
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5.7 Automatic With Correlator Control of Optics
Equations (64), (65) and (66) may be combined to give
the slave image as a function of the master image:
= Xj Xr Xm Axk
r m k '
Equation (108) may be interpreted as follows: Axk represents a (two
(108)
dimensional) vector in the master image, Xk transforms this vector into
a corresponding vector in the master photograph, Xrm further transforms
it into a vector in the slave photograph, and Xri finally transforms it into
a vector in the slave image  all in two dimensions. The matrix Xrm has
been labeled the tracking matrix and is used as the basis for keeping
the two photographs on corresponding points. In fact, however, it is
a two dimensional scaling matrix of the slave photograph relative to the
master photograph, for corresponding small regions. Hence it may also
be used as is done in equation (108). In previous sections of this
manual, methods have been described for computing values for the
three matrices in (108) such that, in general, the two images should
match as required for stereo viewing. The IAS (Image Analysis System)
makes an emperical comparison of the two images and provides feed
back signals representing differences (to the first order) between them.
These differences are used to modify the settings of the slave optical
system so as to improve the first order match between the two images,
i.e., to reduce the differences themselves.
It may be seen that the conditions for optimum stereo
viewing require that the two images be identical with one another except
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for differences higher than first order (in other words, the xparallax
required for perception of stereo is a difference of higher order than
first). This means that under optimum conditions the matrix product
in equation (108) should' equal the identity matrix. As was stated in
Section 4.2, the IAS outputs 6 analog signals  2 for stage tracking
and 4 for optics tracking. The latter are equivalent to the components
of the matrix which is reciprocal to the product matrix in (108). Hence
the matrix of the 4 IAS analog output signals  x and y scale factors
and x and y skew factors  should approach the identity matrix as the
optics adjustments approach those required for optimum stereo viewing.
Let the matrix of the IAS optical output signals be
represented by
Cc
Then the matrix
(Ac  1)
Cc
(Do 1)
(109)
(110)
may be interpreted as an error matrix representing the extent to which
the optics adjustments fail to be optimum for stereo viewing. Hence
(11 0) may be applied to the optics servo systems as a correction matrix
which will drive the optics as required to make (1 09) approach the
identity matrix.
As are other control signals, the IAS error signals are
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treated incrementally and digitally integrated. The corrections are
applied to the slave optical system only. Hence the slave optics
transformation matrix is multiplied by (110) to produce the modified
slave optics matrix:
k kjkik
j  6) Xr = Xj Xr  Xr
The four elements in ?(111) are next treated like a 4 dimensional vector
11)
AAa = (AA, AB, AC, AD) (112)
This 4vector is multiplied by the four 4vectors
? aa ann 8012 ? 2 2 12
,and
aAa aAa aAa aAa,
to produce the increments
Aa2' AM2' A?12' A?22
which are added to the results of (100)  (1 03) for the slave side. In
this way the optics adjustments computed by (100)  (103) are corrected
for the signal from the IAS. Otherwise optics control is as was
described under 5.6.
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