FINAL REPORT ELECTRONIC RECTIFIER STUDY

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP74B00752R000100160001-8
• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 139
• Document Creation Date: December 22, 2016
• Document Release Date: December 9, 2010
• Sequence Number: 1
• Publication Date: May 22, 1959
• Content Type: REPORT

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ENGINEERING & OPTICAL DIVISION \'/IJ1/.a./ THE PERKIN-ELMER CORPORATION
ENGINEERING REPORT NO. 5435
Final Report
Electronic Rectifier Study
COPY NO. DATE: May 22, 1959
STAT
CHIEF ENGINEER 6
P
DIRECTOR OF ENGIN
STAT
NUMBER OF PAGES 132 + vi
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
Page No.
SECTION I
INTRODUCTION
II
GENERAL DISCUSSION OF PROBLEM
2.1
Aerial Photography
2.2
Rectification
2.3
2.2.1 Present Methods
2.2.2 Need for New Method of
Rectification
Study Philosophy
2.4
Definition of Terms Used in This
3.1
Report
General
14
3.2
Tilt
14
3.3
Non-Planar Focal Surfaces
16
3.4
Air Refraction
17
3.5
Lens Distortion
17
3.6 ,
Film Distortion
17
3.7
Motion of the Film During Exposure
17
3.8
Terrain Variations
18
3.9
Earth Curvature
18
3.10
Application to Mapping
21
IV
METHODS OF ELECTRONIC RECTIFICATION
4.1
Image Transfer Techniques
22
4.2
Scanning
24
4.3
Resolution
26
4.3.1 Vertical Resolution
27
4.3.2 Horizontal Resolution
27
4.3.3 Resolution of Proposed
System
30
4.4 Aperture Distortion
31
4.5 Local Scale Variations
32
4.6 Component Study
35
ENGINEERING REPORT NO. 5435 PAGE i
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TABLE OF CONTENTS (Continued)
4.6.1 Cathode Ray Tubes
35
4.6.2 Light Modulators
38
4.6.3 Choice of Modulator
41
4.6.4 Closed-Loop CRT Spot Position-
ing Servo
41
4.6.5 Alternate Video Pickup
43
4.6.6 Tonal Range
44
4.6.7 Contrast Modification
48
4.6.8 Accuracy
50
4.6.9 Testing of the Completed Rec-
tifier
52
4.6.10 F i Lm for.Recording Rectified
Photograph
53
5.1 General System Block
55
5.2 Ideal-System
58
5.3 The Computer
61
5.3.1 The High Speed Incremental
Digital Computer
6],
5.3.2 Radial Correction Computer
69
5.4 System Description - System 1
74
5.5 System II
81
VI RECOMMENDATIONS
83
APPENDIX I MATHEMATICS OF DISTORTION AND RECTIFICATION
84
LIST OF SYMBOLS
85
1 Distortions Due to Tilt and Sving
87
`2 Distortions of the Panoramic Camera
91
3 Distortions Corrected About the Nadir
101
4 Distortions Corrected About the Prin-
cipal Point
110
5 Miscellaneous Distortions
112
ENGINEERING REPORT NO. 5435 PAGE ii
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TABLE OF CONTENTS (Continued)
6 Other Applications of Rectification
Equations 124
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FLgure,No. Page No.
GEOMETRY OF VERTICAL PHOTOGRAPH
GEOMETRY OF THE TILTED PHOTOGRAPH
IMAGE DISPLACEMENT DUE TO TILT
RELIEF DISPLACEMENT DUE TO ELEVATION ABOVE OR
BELOW DATUM PLANS
EARTH CURVATURE CORRECTION
ORDER OF SCANNING OF PICTURE ELEMENTS
7
A.
LOSS OF VERTICAL RESOLUTION BY SCANNING APERTURE
OVERLAPPING EDGES OP IMAGE
28
B.
LOSS OF HORIZONTAL RESOLUTION RESULTING FROM
INSUFFICIENT BANDWIDTH TO REPRODUCE SQUARE WAVE
28
8
A.
MCPA19SION OF SCALE BY OVERLAPPING OF READ-IN
SCAN LINES
33
B.
COMPRESSION OF SCALE BY SPACING OF READ-IN SCAN
LINES
33
C.
EFFECT OF APERTURE DISTORTION
33
BRIGHTNESS TRANSFER CHARACTERISTIC
45
THE CHARACTERISTIC CURVE
46
GENERALIZED BLOCK DIAGRAM SHOWING INFORMATION FLOW
56
IDEAL SYSTEM
59
ANALOG DIFFERENTIAL CORRECTION COMPUTER
59
14 BASIC ANALOG BLOCK V4 = V2 V3
V1
15 A. BASIC DIGITAL SERVO
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LIST OF FIGURES (Continued)
B. SIN-COS GENERATOR
16
INCREMENTAL DIGITAL COMPUTER - Block Diagram for
Mechanization of
x' fx or In x' = In fx -1 ln(f2+y2)
65
f2. 2 2
17
INCREMENTAL DIGITAL COMPUTER - Block Diagram for
Mechanization of
yI f tan-l (y) or y = tan y'
(f) f f
67
18
INCRIIMNTAL DIGITAL COMPUTER - Block Diagram for
Mechanization of Coordinate Transformation Equations
68
19
ANALOG RADIAL CORRECTION COMPUTER BLOCK DIAGRAM USING
pit = f (r) PHOTOFORMER MASK
72
20
DIFFERENTIAL CORRECTION COORDINATE TRANSFORMER BLOCK
DIAGRAM
73
21
BLOCK DIAGRAM - ELECTRONIC RECTIFIER
22
RECTIFIED OBLIQUE PHOTOGRAPH
23
COORDINATE SYSTEM OF TILTED AND RECTIFIED PHOTO-
GRAPHS
86
24
PITCH, ROLL COORDINATE TRANSFORMATION
87
25
RELATIONSHIP BETWEEN PITCH AND ROLL, AND TILT AND OWING
88
26
GEOMETRY OF THE PANORAMIC CAMERA
92
27
GEOMETRY OF ROTATING, TRANSLATING, SLIT-SCAN CAMERAS
93
28
GEOMETRY OF MOVING, TILTED, PANORAMIC CAMERA
97
ENGINEERING REPORT NO. 5435 PAGE v
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LIST OF FIGURES (Continued)
29 ONE MILE SQUARE GRID ON GROUND SHOWN AS TAKEN BY
3" F. L. PANORAMIC CAMERA 100
30 EARTH CURVATURE AND AIR REFRACTION
DISTORTION FOR REPRESENTATIVE NON-PLANAR PLATEN 111
DISTORTION DUE TO PLANE WINDOW 113
REFRACTION BY BOUNDARY LAYER 114
DISTORTION CURVES OF SOME REPRESENTATIVE AERIAL
LENSES
DISTORTION FROM PRISM IN WINDOW OR LENS
TYPES OF LENS DISTORTION
Table 1 RELATIVE MAGNITUDES OF THE DIFFERENT TERMS OF THE
EARTH CURVATURE-AIR REFRACTION EQUATION
ENGINEERING REPORT NO. 5435 PAGE vi
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
c
This study was carried out in an effort to develop improved methods of
rectifying aerial photographs. The main emphasis is on methods of using elec-
tronic techniques to achieve rectification of all significant distortions pres-
ent in a photograph.
The areas studied were (1) the mathematics of rectification with special
attention to developing equations best suited to electronic computation tech-
niques, (2) the present methods of rectification and their disadvantages,
(3) the components and electronic techniques now available or likely to become
available in the near future which would be useful in the electronic recti-
fier, (4) the most suitable method of mechanizing a rectifier and the operat-
ing parameters to be expected.
The goals of the study appear to have been accomplished successfully. Al-
though requiring a substantial amount of engineering development, the design
and construction of a high accuracy, high resolution, electronic rectifier ap-
pears feasible.
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THE PERKIN-ELMER CORPORATION
ENGINEERING & OPTICAL DIVISION \11~
2.1 Aerial Photography
Aerial photographs are widely used as sources of information for mili-
tart' intelligence, mapmaking, road planning, forestry and hydrographic stud-
ies, determination 9f storage pile size or quantities of earth to be moved,
and many other applications. Probably the two most important uses are intel-
ligence and mapmaking.
When a photogrammetristtakes aerial photographs, he attempts to obtain a
point perspective whose central ray is truly vertical at the point of inter-
section with the ground. Unfortunately, this is never possible, although under
ideal conditions it may be closely approached. Under conditions likely to be
encountered by military aerial photographers, it may be impossible to even come
close to the desired conditions.
Many factors work against the photogrammetrist in his efforts to achieve
a vertical point perspective of the ground. With standard aerial cameras one
of the most important factors is lack of verticality of the lens axis due to
motion of the aircraft and to lack of an accurate vertical reference. In some
cases such as tri-metrogon systems the lens axis is deliberately given a large
tilt to provide more ground coverage. Other errors are caused by lens distor-
tion, air refraction, and motion of the aircraft.during exposure. These will
be discussed more completely in Appendix I. For vertical pictures the entire
camera assembly is often mounted on a gyro-stabilized camera mount which at-
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tempts to hold the camera vertical as the aircraft rolls and pitches. This
method is not entirely satisfactory even for vertical photographs mainly be-
cause this equipment is usually large and heavy. This often precludes its
use in the confines of the high performance aircraft needed for modern mili-
tary reconnaissance systems. it also corrects only one of the many errors
introduced into the photograph.
- In an effort to obtain a more nearly correct vertical point perspective,
photogrammetrists rectify, or correct, the photograph. This report discusses
improved methods of rectification.
2.2 Rectification
An aerial photograph is a perspective view of the ground similar to what
would be seen by a human eye from a single point above the ground.
A map is an orthogonal view in which each detail is indicated as if
viewed from directly above it.
A tall object, such as a smokestack, will show as a circle anywhere on a
map but will show as a line of definite length on a photograph except at the
one point which is vertically below the aircraft.
A vertical photograph is an aerial photograph taken with the principal
axis (optical axis) vertical.
The scale of a photograph is the ratio of distances on the photograph to
distances on the ground and on a theoretical vertical photograph is constant
at any point on the picture.
If a photograph does not have constant scale, it is said to be distorted
and should be rectified before use. Rectification may be considered as the
process in which a distorted photograph is converted to a vertical, point
ENGINEERING REPORT NO. 5435
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THE PERKIN-ELMER CORPORATION
ENGINEERING & OPTICAL DIVISION \11~
perspective taken from the original camera station. The scale will be constant
at any part of the photograph and will be equal to the focal length of the cam-
era divided by the altitude above the ground.
It is important that the geometric properties of a perspective?visv are
accurately preserved by the rectification process. A rectified photograph is
not a map.
A rectified photograph as discussed in this report may be considered as
being the equivalent to a photograph taken from the same camera station, through
a medium of uniform index of refraction, by a stationary camera having a
distortion-free lens, a known focal length, and whose principal axis is truly
vertical.
In addition to meeting these conditions, it is desirable to modify the
picture to correct for the effects of the earth's curvature. In this report
the earth's curvature adjustment will be considered to be-included under the
term rectification.
2.2.1 Present Methods
The method now used to rectify aerial photographs is to project an
image of the photograph onto light-sensitive paper as is done in any stan-
dard enlarger. In order to rectify the photograph, the negative, the
lens, the easel, or a combination of all three, are tilted.
The projection-type rectifiers suffer from many drawbacks. They
only correct for the tilt of the photograph and not for the many other
sources of distortion. They are very large and unwieldy and require dif-
ferent projection lenses for each taking lens. As the tilt angle becomes
very large, the light rays hitting the paper at glancing incidence are
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reflected instead of being absorbed where they can expose the print.
They are not capable of the high accuracy that is required by advanced
photogrammetrists, nor can they yield the full resolution inherent in
the original film. In many cases a photograph with large tilt must be
rectified in two stages, each with its inherent loss of accuracy, reso-
lution, and time.
2.2.2 Need for New Method of Rectification
Because of the great advances in the state of the photogrammetric
art, accuracy and resolution are outstripping the capabilities of the
present rectification process.
In recent years electronic techniques have made tremendous ad-
vances, especially in the communications and computer fields. Some
photogrammetrists have become interested in the great potential gains
that lie in a combining of photograsmnetry and electronics. It was this
line of thinking that led to this study.
Some people have done work in this field, and the results have
been good. A few of these will be mentioned for background information.
U. V. Helava of the National Research Council, Canada, has ap-
plied an electronic computer to position the plates of a stereo plotter
to remove distortion as the plot is made. Others in Canada have made an
electronic stereo perception attachment for stereo plotters. This de-
vice randomly scans the stereo plates, subjects the output signals from
each plate to electronic correlation techniques, and positions the plotter
by a servomechanism to plot the contours.
Professor A. McNair of Cornell University uses digital electronic
computers to solve analytic aerotriangulation problems, and many people
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are processing stereo-plotter outputs automatically with digital com-
The Fairchild Graphics Corporation, Syossett, New York, under a
contract with Rome Air Development Center, is building an electronic
rectifier to remove tilt from aerial photographs. This unit replaces
the conventional projection rectifier with one compact unit with no
need for changing lenses, since at a turn of a knob, any focal length
lens from 3 to 100 inches may be accommodated. In addition to rectify-
ing, this unit will enlarge or reduce by a factor of .3x to U.
The use of electronics in'the handling of photogrammetric informa-
tion is a rapidly growing field. The conventional plotters, rectifiers,
and other measuring instruments are optical and mechanical analogs requir-
ing large size and weight and present great manufacturing difficulties.
The use of electronic techniques permits the use of a minimum of
mechanical parts, such as lead screws, which can be chosen during design
as those which may be easily made to high accuracy.
As more and more complex methods and vehicles are used to obtain
aerial photographs, the rectification problem becomes more and more dif-
ficult. Reconnaissance aircraft are flying at higher speeds and alti-
tudes. Photographs taken from satellites and missiles will cover larger
distances in one exposure, balloons may be used as camera vehicles, and
non-photographic imaging systems and special-type cameras may present
very distorted photographs. All these will require rectification which
can only be done by a new type of rectifier.
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Negative
Figure 1
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I . Aa ___~
2.3 Study Philosophy
The aim of this study is to investigate, as thoroughly as possible, the
methods that could be. used to design and build a versatile electronic recti-
fier of high accuracy and resolution. The tentative requirements of .017.
geometric accuracy, 100 line pairs per millimeter resolution, 15 grey scale
tones of dynamic range, and 10-20 minutes' time of operation were established
as a goal. These areas were set down because they appeared to be reasonable
all-over goals for a device that would considerably advance the state of the
art of rectification. If no tentative limits were set except to study for
the "best)" the study could range far beyond the time and money available.
As an example, if several days were taken to rectify a photograph, a unit of
a few micron accuracy capability could be designed. This would be an imprac-
tical solution since the variables are not known to enough accuracy to justify
this precision and the time is not available if they were.
One goal of the study is to have a rectifier which is as versatile as
possible and is adaptable to future problems that may arise.
This dictates the input transducer (reading end) and the output trans-
ducer (writing end) do not contribute to the geometry of the picture but only
shift the reading and writing heads around, supply positional information,
image parts of the photograph, or supply density information. All of the
geometry must be handled in the computer section. The computer section will
rotate coordinate systems, vary coordinates and parameters, and solve all of
the geometry equations. Each distortion should be handled in a separate com-
puter block which is easily removable. In this way the unit can be modified
and further developed without major design changes.
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I- \
Vertical or Plumb Line Figure 2
ENGINEERING REPORT NO. 543
GEOMETRY OF THE
TflTED PHOTOGRAPH
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~, Negative
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Use of this type of systems will permit modification, addition, or re-
placement of computer blocks to solve new problems. If same as yet unde-
signed camera, with a different geometry than any now in use should be
used, it would only be required to design a computer block to correct its
pictures, build the block, and plug it in. As an example, it might be de-
sired to map underwater obstructions or reefs near a beach. A computer
block to correct for the bending of the light rays as they pass from the
water to the air could be constructed and used in the rectifier to correct
the geiommetry of the underwater photograph.
This philosophy was followed throughout the study, and it is recommended
that any rectifier built as a result of this study incorporate these prin-
ciples.
NOTE! Before reading section 3 it is suggested that the reader who
is not familiar with photogranunetric tarts study the definitions of
Paragraph 2.4 and the geometry of Figures 1 and 2.
2.4 Definition of Terms Used in This Report
DISTCRTICHJ Any deviation from a point perspective whose central
ray is vertical. Distortion results in local varia-
tions in the scale of the photograph.
ALTITUDE The vertical distance from the datum plane being
photographed to the interior perspective center of
the lens.
CRAB The angle between the projection Of the longitudinal
axis of the aircraft on the ground and its track.
FIDUCIAL MARKS Four index marks w.ch image on the film. The
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FOCAL LATH (f)
)
intersection of lines connecting opposite i'iducial
marks defines the principal point and, in this re-
port, the coordinate origin of the nnrectified photo-
graph.
The perpendicular distance from the film plane to the
interior perspective center of the lens. This is
often referred to as the calibrated focal length and
is so chosen as to distribute the effect of lens
distortion over the useful field of the lens. The
principal distance is similar to the focal length
and is used in place of f when measuring a photo-
graph which has been enlarged or reduced. The
principal distance is equal, to the product of the
focal length of the lens with which the photograph
was taken and the enlargement factor. If a photo-
graph taken with a 6-inch focal length lens is en-
larged by a factor of two, its principal distance is,
12 inches.
ISoCENW (I) A point defined by the intersection of the planes de-
fined by a tilted photograph, a vertical photograph
taken from the same camera station with the was lens,
and the plane defined by the Principal axis and ver-
tical (see Figure 2 ).
ISOLM A line through the ieocenter and perpendicular, to the
principal lies. If only tilt is considered, the scale
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is constant along this line and any line parallel to
it.
NADIR H The point at which the vertical line through the per-
spective center pierces'the ground or the photograph.
OBLIQUE PHOTOGRAPH A photograph taken with the principal axis intentionally
not vertical. Oblique photographs usually include
the horizon.
PRINCIPAl4 POET (P) The point at which a line perpendicular to the photo-
graph and through the interior perspective center
pierces the photograph. On a vertical photograph the
nadir and the principal point coincide.
PRINCIPAL AXIS The line connecting the.principal point and the in-
terior perspective center.
PRINCIPAL L NE The Line, in the plane of the tilted photograph,
connecting the nadir with the principal point.
RCTIPTCATICi The process of making the scale constant at every
point on the photograph. See page 3
SCALE The ratio of distance on the ground to a corresponding
distance on the photograph. The scale is numerically
equal to f/H. In this report local scale is used to
describe the scale at scene point on a distorted photo-
graph as opposed to the over-all scale which may be
changed by a simple enlarging process.
SWING The angle at the principal point of a photograph
measured clockwise from the +y axis to the principal
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TILT
line at the nadir. This definition is used throughout
this report because it appears to be the most commonly
used. The +x axis can also be used as a starting point.
In this case the tilt equations, while maintaining the
Same form, will ham some sign changes. Either system
can be incorporated into the rectifier.
The angle at the perspective center between the prin-
cipal axis (photograph perpendicular) and the plumb
line. The direction of tilt is specified by the sing
angle. :tee Fide 2 . In working with oblique
photographs, reference is often made tothe true
depression angle, which is the angle between the true
horizon and the principal Mi.s. This system is not
used here, but the simple conversion, tilt angle a 900
T.D.A., may be used to find the tilt.
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SECTION III
SOMCES OF DISTORTION
3.1 GENERAL
Any departure of the photograph from a vertical point perspective is
termed distortion. There are matte scums of distortion, and most of these
are present to some degree *in every aerial photograph. Some distortions
are large enough to require correction in every picture, and others are no small that they can always be neglected. A list of distortion sources is
given here, and each will be given a brief discussion. A detailed mathe-
matical analysis of the various sources of distortion is given in Appendix
I, and a cc mrison of magnitudes for typical conditions is given.
Tilt.
Non-planar focal surfaces.
Air refraction.
Lens distortion.
Film distortion.
Motion of film dozing exposare.
Earth curvature.
Terrain variations.
The distortions discussed belov are the most significant.
3.2 Tilt
The distortion due to tilt or lack of Verticality of the principal
axis may be seen in Figure 3 as the displacements and : As the
tilt angle becomes very large as in oblique photographs, the distortion
becomes so large as to approach infinity at the horizon. The distrtion
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Figure 3
Rectified
(Vertical)
Photo
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Principal
Principal Line
Axis Tilted Photo
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ENGINEERING & OPTICAL DIVISION l V1
Therefore, V4 = V2 V3
V7
Referring again to Figure 19, we see that this scheme is used to obtain
&xi and 6yi. One complication is introduced because k16t may go negative. To
ENGINEERING REPORT NO. 5435
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
avoid difficulty here, we introduce the constant k2 selected so that the in-
put k2 - kl R to the differential amplifier will always be positive.
The solution becomes meaningless at R = 0; and since R will often go to
zero, we must gate the output to prevent this solution from being utilized.
This presents no problems since the correction is always zero when R = 0.
The solution xi(k2 - ki R) is added to -xlk2 , and the result multiplied
R
by 1 to obtain the required result Axi = xi1R
k -~-
This solution for Axi will be corrected for earth curvature and air re-
fraction since both of these distortions will be included in the mask. Axi
must then be modified by the tilt angle coordinate conversion.
The basic coordinate transform equations of page 90 are:
x0 = ax + bf and y0 = -cx+dy+ef
T" -gx-hySkf T- -gx-hy+kf
The approximate differential corrections corresponding to the equations
"jx0 = aAxi and Apo = -cpxi+d&Yi
f -gxi-hyi-kf f -gxi-hyi+kf
Where Axi and 4yi are the differential correction inputs and 'xo. qyo
are the coordinate transformed differential corrections.
These equations are solved by the computer shown in Figure 20. The out-
puts of this computer form the inputs to another computer similar to Figure 19
which corrects for distortions about the principal axis.
The final corrections are added to the output of the digital to analog
converter and used to deflect the CRT spot.
ENGINEERING REPORT NO. 5435 PAGE 71
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ow
ENGINEERING & OPTICAL DIVISION
I",
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THE PERKIN-ELMER CORPORATION
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ENGINEERING REPORT NO. 3435
2
PAGE. 72
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0
0
w
14,
ENGINEERING REPORT NO. 5435 PAGE 73
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ENGINEERING & OPTICAL DIVISION ~~ THE PERKIN-ELMER CORPORATION
5.4 .stem Description System 1
The over-all block diagram of the rectifier which best appears to meet
the requirements of this study is shown in Figure 21. The two computers re-
quired are described in paragraphs 5.3.1 and 5.3.2. The system will be
described here starting with the rectified end and following in the general
direction of information flow.
A sheet of unexposed film is mounted on the drum and held in place by
spring-loaded pins or by a vacuum system. The picture to be rectified is
mounted on the input platen with the fiducial marks aligned to index marks
on the platen.
The variables of the problem, such as altitude and focal length, and
the precalculated constants of the coordinate transform equations are fed
into the computer; and the proper computer blocks (such as panoramic camera
correction) are switched on. The transducer is set at the proper position
and the system started. The drum will'`.rotate at 3600 RPM or 60 RPS, thus
scanning 60 lines each second. The lead screw will traverse the printing
light source along the drum at a constant rate to yield a line scan. The
system will apply an over-all enlargment factor of two or three to the photo-
graph being rectified to reduce the further magnification required to exploit
the photograph.
Two seta of transducers furnish position information to the computer.
One not of transducers produces pulses (Lx and ty)at fixed intervals as in-
puts to the incremental digital computer. In the x direction the pulses are
produced at maximum rate of 105 per second. In the y direction the pulses
are produced by a switch on the lead screw at a much lower rate. The x pulses
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ENGINEERING & OPTICAL DIVISION 'V r THE PERKIN-ELMER CORPORATION
A0
are produced by a photocell which is triggered by light passing through an
engraved scale fixed to the drum. A separate scale and photocell will yield
a pulse once each revolution to reset the computer, blank the CRT, and return
it to zero.
The other set of transducers provides analog voltages in x and y as in-
puts to the differential correction computer. The computer outputs form the
unrectified coordinates of the point to be printed.
The digital computer outputs are 17 bit binary numbers and, the outputs
of the analog computer are small analog corrections to the digital solution.
The digital computer integration rate is 105 per second. The spot scans the
film at an average rate of 180 in. per second to yield an accurate digital
solution every .0018 in. Since the picture elements are only about .0002 in.
apart, we must use a smoothing system to interpolate between the digital solu-
tions.
The digital solution for y' is fed into a binary subtraction unit.
The unrectified film platen position is monitored by a linear position
transducer whose 17 bit digital output is also fad into the binary subtraction
unit.
A Ferranti linear position transducer would probably be used on this axis.
This device consists of a grating the length of the platen with a fixed short
grating at a slight angle to it. A light shining through the grating is
picked up by a photocell. As the gratings move with respect to each other, a
Moire'fringe pattern with an approximately sinusoidal distribution is produced
as a result of the integrated interference pattern caused by the angular inter-
section of the individual lines on each grating.
ENGINEERING REPORT NO. 5435 PAGE 75
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
The Moire fringes are detected by the photosensitive element and the
waveforms used to form a digital measuring system. This system is not af-
fected by error or wear in the screws used to move the table and will be ac-
curate to .0002 inches over a maximum usable travel of 26 inches. The max-
imum speed of one inch per second is completely adequate for this application.
The picture will be covered in strips three inches wide and the full
length of the film. A 9 x 9 format must be covered in three passes and a
75 mm format in one pass. The entire three inch width is covered by the CRT,
and no servo positioning of the table is required in the x direction. The x
table motion is entered by hand after each pass by turning a crank to one of
three index points.
As mentioned before, the desired table position in y' is compared with
the actual position in the binary subtraction unit.. The difference is used
to drive a servomotor to position the table. Any difference between actual
table position and desired table position is fed to the digital to analog
converter. The converter output is changing in steps at the rate of 105
steps per second. The output feeds into a smoothing unit containing predic-
tion circuitry which converts the step input to a smooth curve. The output
of the smoothing circuit is added to the analog output of the differential
correction computer and forms the input to the y' deflection circuit of the
CRT. With this system the high speed scan of the CRT is in the x' direc-
tion. The platen moves slowly along in y' under the CRT to form a line scan.
Any lag in the platen position is taken up by the y' deflection of the spot.
The x' circuitry is similar except that, as mentioned previously, no
servo drive is required in x' because the entire 3 inch motion in this direc-
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ENGINEERING & OPTICAL DIVISION \TffF/ THE PERKIN-ELMER CORPORATION
tion is supplied by the CRT scan.
The CRT will be about 7-9 inches in diameter, have 15,000 lines resolu-
tion across the face, and, with proper yokes, have a linearity of 0.1%. A
demagnified image of the tube face will be projected onto the negative to
be scanned. This image will be 3 inches in diameter, thus covering a 3 inch
wide strip of film at 5,000 lines/inch.
Assuming an 8 inch diameter tube, a linearity of .1X results in a spot
uncertainty of .004 inches which, when imaged on the film at a reduction of
8 to 3, will result in a spot error of .004.x 3/8 = .0015 inches. This fig-
ure is somewhat optimistic since a .1% linearity is difficult to achieve and
a slightly larger tube may have to be used to allow for not being able to use
the tube out to the edge.
A phototube can be used to monitor the brightness of the spot. As the
spot travels across the tube brightness, changes may occur due to phosphor
variations. The velocity of the spot will vary considerably as local scale
changes occur.
The brightness of the spot will vary with the velocity and would result
in a velocity modulation of the spot if left uncorrected. The output of the
phototube is amplified and used in a closed loop to maintain constant bright-
ness of the spot. This system requires that the phosphor decay time be very
short since the tube measures all the light emitted by the screen. This
causes no problem since the decay time must be shorter than the picture ele-
ment period of about 10-6 second/element. The P-24 phosphor which decays to
10% of original brightness in 1 microsecond would be adequate.
ENGINEERING REPORT NO. 5435 PAGE 77
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ENGINEERING & OPTICAL DIVISION
_W_
THE PERKIN-ELMER CORPORATION
As the scanning spot passes over the negative, an amount of light pro-
portional to the transparency of the film passes through it and, after col-
lection by the condensing system, falls on the photomultiplier tube.
The output of the photomultiplier is amplified and modified as described
in paragraphs 4.6.6. and 4.6.7. for proper exposure of the film. The amplified
signal is fed to the exposing light source which is monitored by a phototube
in a closed loop to eliminate the effects of non-linearity and to extend the
usable frequency range of the light source.
The time required to rectify a photograph by this method depends not
only on the size of the photograph but on the scale change required.
The output drum speed will be about 3600 RPM or 60 RPS. This will write
60 lines per second regardless of the width of the rectified photograph. A
panoramic photograph which has been rectified out to 600 on each side of the
horizon is about 8 inches long with no over-all enlargement factor. At 5000
lines per inch, this will require 8 x 5000 - 670 seconds or 11 minutes.. Print-
60
ing out at an over-all enlargement factor has no effect on the time because
the scan lines will be twice as wide.
A 75 mm x 75 mm near vertical photograph will require about 4 minutes.
A 9 x 9 photograph must be covered in three 3 inch wide passes since the CRT
can only scan 3 inches in the x' direction at one time. This means 27 linear
inches must be covered in the y' direction. For a near vertical 9 x 9 format
5000 x 27 m 2,250 seconds or 37 minutes. This will yield three separate rec-
60
tified photographs, each of which will represent a 3 x 9 inch strip of the
original 9 x 9 photograph.
ENGINEERING REPORT NO. 5435 PAGE 78
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4 L?71T1J,0. FOC, 9L Lf/ 7
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
First Second I Third
Pass }Pass I Pass
Onrectified
First Pass Second Pass
Third Pass
1 r
Rectified
The upper drawing shows how a 9 x 9" oblique photograph is covered,in
three passes by the 3" scanning line of the C.R.T. The lower drawing
shows the three separate reproductions of the oblique photograph produced
by the rectifier. (Not to scale.)
Figure 22 ? RECTIFIED OBLIQUE PHOTOGRAPH
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ENGINEERING & OPTICAL DIVISION IV THE PERKIN-ELMER CORPORATION
5.5 Svst_II
An alternative to System I is to cover the unrectified photograph by
scanning it with a very reduced image of a C.R.T. This reduced image would
be about 1/8 to 1/4 inch in diameter. The small image would be scanned in
lines across the photograph and the C.R.T. spot would scan at high speed
across the main scan lines. At the output and the line scan is supplied
by a rotating prism. Most of the computer system is quite similar since
the basic method of operation is similar.
This system has the advantage of better accuracy of spot positioning
on the unrectified film. If a 5-inch diameter tube were optically reduced
20 times to a 1/4 inch diameter image on the film, a linearity of about lx
would be required to achieve a spot position accuracy of .001 inch on the
film. The linearity could be held better than this, and, since the table
position can be very accurately established, an accuracy of .001 could
probably be achieved.
An advantage of System II is that an ordinary C.R.T. could be used in
place'of the special, high linearity, high resolution, tube required for
System 1. A drawback of System II is that the output line length would
have to be held to picture element size accuracy to prevent overlapping of
elements with consequent banding and loss of resolution.
System I lends itself to future improvement by incorporation of a spot
position servo system at some time in the future while System IL, although
having better accuracy initially, would be more difficult to modify without
major design changes.
It should be.noted here that both systems will require heavy machine
tool type design in order to maintain accuracy.
ENGINEERING REPORT NO. 5435 PAGE 81
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ENGINEERING R nPTIrAi nivlcin.,
I "L- rLr r'rl-tLMtK CORPORATION
There do not appear to be any fundamental limitations to increasing
the resolution of either system to approximately 200 line pairs per m.m.
This change would result in a longer time of operation. Although four
times the number of bits must be processed, the time per rectification
would only increase by a factor of about two, since the bandwidth of the
video electronics could be increased somewhat. Increasing the resolution
of System I would result in a smaller coverage per pass unless a larger
C.R.T. can be obtained.
ENGINEERING REPORT NO. 5435
PAGE
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ENGINEERING & OPTICAL DIVISION
THE PERKIN-ELMER CORPORATION
I G FT
Recommendations:
It is our recommendation that the rectifier described as System I be
constructed to provide a universal rectifier which substantially advances
the state of the rectification art, is adaptable to further development,
and can be completed with a reasonable amount of engineering development.
This unit will have a resolution capability of about 100 line pairs
per m.m., a computational accuracy of .01%, and an information pickoff
accuracy of about .003 inches or better. This should result in a rectified
accuracy for near vertical photographs of about .02% of the 9 x 9 format.
As was discussed in Paragraph 4.6.8,.the accuracy of rectification of
tilted photographs varies with the amount of local scale change.
The computer blocks to be included and the range of focal lengths
and format sizes depend on the needs of the individual customer. An
overall enlargement factor of two is recommended because this will result
in reasonably sized rectified photographs and will permit use of reasonably
fast films such as Plus?X Aerecon and consequent use of a glow modulator
tube.as an exposing light source.
ENGINEERING REPORT NO. 5435 PAGE 83
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THE PERKIN-ELMER CORPORATION
ENGINEERING & OPTICAL DIVISION \111~
. ff
MATHEMATICS OF DISTORTION AND RECTIFICATION
The equations in this section were derived because a search of the liter-
ature failed to disclose the equations required. It appears that very little
work has been published concerning the mathematics of distortion, especially
in the less significant cases, such as air refraction and image motion.
In order to obtain a linear, uniform, scan at the output, the inputs to
the computer must be the rectified coordinates of the read-out light source.
All of the equations present the unrectified coordinates of a point as a func-
tion of its rectified coordinates. Thus, the rectified coordinates are inputs
to the computer, and the outputs are the unrectified coordinates, or the co-
ordinates that the scanning spot should take to pick up the proper information
for the output end to print.
The equations are derived as if no other distortions were present. Thus,
distortion due to tilt is derived as if no air refraction and earth curvature
were present. This does not result in any error, since the inputs to the tilt
section are the outputs of the earth curvature computer. As explained in Sec-
tion 5.1, all of the distortions except tilt have either the principal axis or
the vertical (nadir) axis as a coordinate origin and tilt may be corrected by
a coordinate rotation from one to the other.
The following distortions will be treated in the order listed:
1. Distortion due to tilt and swing.
2. Distortions found in the panoramic camera due to basic
method of operation and to aircraft velocity, crab angle,
rotation, and image motion compensation (S distortion).
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APPENDIX I
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ENGINEERING & OPTICAL DIVISION \11~ THE PERKIN-ELMER CORPORATION
I . je
4. Distortions due to non-planar focal surfaces.
5. Miscellaneous Distortions.
a. Distortion due to plane window.
b. Correction for rapidly moving boundary layer.
c. Film distortion.
d. Lens distortion.
e. Distortion due to prism of lens or windows.
6. Other applications of rectification equations.
a. Scanning accelerations.
b. Small areas.
c. Maximum reduction in scale.
LIST OF SYMBOLS
The symbols most frequently used in this report are:
x', y' are coordinates determined by the fiducial marks on photograph
to be rectified. The origin of coordinates is at the principal point. The
*x' axis is in the general direction of flight, and the +y' axis is in the
general direction of the left wingtip.
x, y are coordinates on the rectified photograph. The origin of
coordinates is at the nadir.
X, Y are coordinates on the ground.
H altitude
h height above or below datum plane
t tilt angle
s e swing angle
p - pitch angle of aircraft
r roll angle of aircraft
f . focal length of camera
N nadir point
I isocenter
P principal point
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ENGINEERING & OPTICAL DIVISION \TJ THE PERKIN-ELMER CORPORATION
General. Direction of Plight
Rectified (8orisontal) Photo
Axis of Tilt
COORD]MATS SYSTRK OP TILTED AND RECTIPIRD PHOTOGRAPHS
Figure 23
ENGINEERING REPORT NO. 5435
Unrectified (Tilted) Photo
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
P= tch dny/e
lr _ roll etn9 le
.H-Z =dlt,tvde
canw*r4 fKeYl
/erj th
P. d;rc t~?oh o
f / i j / 7;ath oh.
1rc~~:ra'
vr c 24
D,'s tar t,'ons c~c c tc.
X , Y, Z'= gr~irr/ coor~/>'ndtcs cr' ob~~rv~
1o rr . t . ~ ? ~p r,'~ ~'n a t /7 47
_. tr"ani j ~t rnr:/ r;~ordi'/tom ~: P; D=
Un rec, t /' * '?d .c h t,e) ra:~?, A
coord/'notes o Q
X ~/ - = r(' (t f /'c'r/ />/, o r J ro
' (icrdfn rr.r; ~,~
Y
P1 r h; ./' l/( ~: ! I:", 7 Trans f,KtI;~ti l/ ,;
_
_
These des fort,'on5 -~ /l/ /t-57` pe 4ct/Yiel
it) e'er/ns of-
/';i tc/1 c nd t'D// dnd t heh be. ~X`prP. s. / i n t -crP s r "
tilt 2ltlq'
ENGINEERING REPORT NO. 5435 PAGE 87
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
Figure 25. RILATIONSHIP BETRSEN PITCH AND BOLL, AND TILT AND SWING
ENGINEERING REPORT NO. 5 PAGE 88
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ENGINEERING & OPTICAL DIVISION Wi,1'W THE PERKIN-ELMER CORPORATION
The coora//n4 to tr4n5 fortis for P/tch gild ro// -~Z P(
(Sec f,'~ ..t )
P~ t:c h
X coSP -k s;n
Rol/
Xz Y,
y, = Y Xz= y,cosr-Z,s;nr
Z, = SC s i ny' 1 ~ (or L~ r ~- Z~ c c.: r
So,
X2= XcoSP - Z?;np
yz= Ycosr -,Ys;rPsinr - co ,p ;,,r
Zx = Ysnr *.X s;nP coSr c^sr
a/so X'=7 X
So,
XcoS,o
comer t Y s,'n r +- f cos, ces r
X snps/',nr t Ycosr.-:j ~nc
XS; rip cos r t- ys'"nr r f cos/ cos{
To. cohvcr't. these Cod' iCfe/,
ff . ..
w-e.rCfer.to 2 5 from
GZ= co 5(S-/T)=.-Cosy a _t
; d = t in P ., . 7 t .
c=..Sinr-
dos = 1 , - I
f ; /t and sw~'i~9
terms, of.,
which we see
= P, ~ttan~P
Sl n 2r - -7-t--TT- = Got?
4n, r
(l ) Ca S ~o
, f . CGS'iS t 4p t
/ t cos2S dh~C
01-S'A1tSihZS
(.v. P= t I~ Cn;~P - f ~, h tS; h S = -S,"n 7 si S (from c1~ _
(3) si/i s,'nr S~'nt-s;n~ co. s ~n~Z t S"') "'r 0
(q) Gosr=
t co:?s all
ENGINEERING REPORT NO. 5435
PAGE _"
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
(S) CosPsinr = tos
s
-r ri l -s,'n
~
-
t6) Sif,Pcosr= c
(7) Sin r = - co; sn,t~
/- S,h SSi'/I~i-T
( ) cos co r = l`S,rJZS ' .~ t COS
=-CoSS Sih t
cos t
Svbst, to ~;n9 (lJ-(S) i/J the cxpre~s/ s for dnd4/,
/~ /' /? ` 'yam / ~-- ~ _--- /' /~ _.._yJ......
A
t
S//1 '2r"Sin_SCoS t rn.~'~`_" ' __ v+ CDS S S%/1 L` f
. S ~'h t s,'n s cos
', ht`CoSS _._ y Cost
a T
or
_ ~/- sin =Z' ~rnzs) X t Sin=S(l-SiAzZ`Sil?
-.-
Sih tsi/)SCDSt X Ccs
1 - Sih~Z` Slits
COSS x tCo$t os5'S%h t (/ Sin
S,inZ'S,nS,cDst X -- S,/1t coSS tGOS '(l-Sid t`:S.ihRS~
,de
G S,'' 2 (' ,'n 5 toss.
d- Cost`
e s . COS S!s ih t O-Si?n 2TS,?hts)''a-
/,rs< efv8t;ohS C-Skt thf forts
Sihts%/J$CO.Sr
a.X+ bf
-cxtdt ef-
-jXby tK
Vw, V,
a_, b)C)d, e, h) K are
co1l s rant for~hy Cr,e pho togra,ph.
ENGINEERING REPORT NO. 5435
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ENGINEERING & OPTICAL DIVISION 'J THE PERKIN-ELMER CORPORATION
2. DISTORTIONS OF THE PAI4ORAt4IC CAMERA
(a) First, we consider the distortion of the panoramic camera image
when the camera is mounted in a level vehicle mpving at constant alti-
tude B. ground speed V. and direction X, but rotating about a vertical
axis with angular velocity fL - . The scan rate of the camera is
_9T and the film image motion compensation velocity is 111i
zZ
Since scan begins at 9= - , we have e . -- 1- A,d t. The origin of
the rectified coordinate system is this nadir when 60 . The time,
tn, at which 0= 0 is found from the relation/ k dt . L7~ . The azi-
o
muthal angle is given by = tj5 d t . Thus d t
a o
Figures 26 and 27 show the relationships among the different earth, air-
craft, camera, and unrectified film coordinates,. From these figures we
see that the following relationships hold, where Xc , Yc are camera
coordinates and y' the coordinates on the film itself.
`x,, Xx, Y,, y.:) Ml XL, Y,YY)
Cos 6
H t Y4 f f t yx
tan
Y ..=~ tan-'
x.XCoSO =..x`:+l.vt dt
o.
X,= XCo$ -ySih~/n4
y, L x S;~ -7Co$
X)- C f_Y(t-tn)1Cos'- yr r1
H
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ENGINEERING & OPTICAL DIVISION
THE PERKIN-ELMER CORPORATION
ENGINEERING REPORT NO. 5435 PAGE 92
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.ENGINEERING & OPTICAL DIVISION 1~7 THE PERKIN-ELMER CORPORATION
GENTRY OF ROTATING, T.ANSLATING., SLIT-SCAN CABS
ENGINEERING REPORT NO. 5435 Figure 27
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
In the special case of constant rotation and constant scan rate, we
12 = G> _ C n nS ~7 Ir
3 ?t = K= c,n5t-dn.r
t-fry = g to = f = J~ '' t _
L e t- 1.) = Vo r o_ 7/',, t h
11 e
z
X. Xcos9, - y ,.h ~o
Y~ = X S1, r,
X;, (x-L)cosCpo -4%
HK
: + rvf ~b th) 1-
rl
fyCOS (fPct ci';
These last two are the equations to be solved for x' and y' to rectify
a panoramic picture taken from.a constantly rotating craft.
Specializing still further to the case of no rotation and. constant
seen-rate) we have with
o.s B .
?rf = Lk' (
H
So thet
Cos = dt-)secE K
(x- nrK js,h ?Xo trCOS y 7:. t-sn v
Y
S, t1 n = 7? , CoS /n =/
For very small values of ~/o
these equations can be solved immediately for x' and y' in terms of x
and y and used for rectification. If Vlo is not so small, a more com-
plicated iterative procedure may he used.
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r)Sec 9
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ENGINEERING & OPTICAL DIVISION 7 THE PERKIN-ELMER CORPORATION
If we are interested in the error due to crab angle J, , we must
compare x' and y' with xl, yl instead of x2, y2 to avoid introducing
the translation error which is included is the so-called S-distortion.
The magnitude of both these distortions will be found. .
Putting /to=o and solving for x' in the last equations, gives us
the normal panoramic rectification equations
X' XC^_. (lt~in9-- eiS j
t N t
so the S-distortion error is
1f' /tK X'
Putting the crab-angle equations in terms of xl, yl, we get
Xi = f~ c0 ` a f (x'
M K
y,= tvi ,c")/" f taro .
from which we see that the errors in ground coordinates are
x
= f V,q (C ? o -~~ _ f V 9 ~cst? ('loS -~.
H~'iyX'f.4Clt~~~ln,- X'~
H
Ay-
.ham. VB S in
? Mk tal,
A case for which the error is relatively large for both of these is
given by the conditions:
H= 6rl%le$ = 3/68 ft.
Y= ihilrs 'Per
K='~trri+~4./.fed l~ v., ,..?~ r _,(
L4 = 11_.. r /. -. ,.;
Xr = `~Smrri
f= 3". -1y. f.
Z -,
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
Under these conditionst-
X /S-J; s t b f.7
y)crdb - 6~~ n -
(b) We consider next the distortions produced in the panoramic camera
by tilt and swing in-the photograph. We first do this in terms of pitch
and roll and then express the results in terms of tilt and awing.
Referring to Figure 28, ve see that the following relations hold:
C? = -! f K t Vj = ro'tid ~ rdc tf itc/cr,' f.
v9 IT -+ V9 `-
x; = X XN
11; H-WSinp1V~ts;hp=
de f,'n e
K
.X,_?j xz
Z~
Z,4
z1
The relationship between the (2)-system an the (i)-system,is the same
as that between the (x2, etc.)-system and the (X, etc.)-system, in the
previous treatment of tilt and swing. And we have:
%; coSF - S,hP
Y cos r - X; s - n~ ;, n r - ~; ? cost S- b r
Z. Y r,'h/' t .~; s,/,r c r r /. 1, c.Sr,
ENGINEERING REPORT NO. 5435
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
.Az
Aj p
Zi
Xn
Instantam
Nadir/
1
x ?Vgx
(Scan begins)
r/I)
1/.
j / Y2
I X m Vgx X
us 2K }
(Scan ends)
GEOMETRY OF MOVING, TILTED, PANORAMIC CAMERA
Figure 28
ENGINEERING REPORT NO. 5435 PAGE 97
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ENGINEERING & OPTICAL DIVISION ? THE PERKIN-ELMER CORPORATION
The transformation equations for a tilted panoramic photograph can now
be written down as follows:
y' =
f B
X~: Zz Cos9
S c d r;c p M. ron,.'C
QlrCr4'[r , o !r
2117
The difficulty with getting these equations into a rectifiable form is
that 9 not only occurs directly in the equations but also x2, Z2 are
functions of B while 0 is a complicated function of itself.' The situa-
tion is not so bad, however.
9stVi~yZ _ tan_l
Defining 6 'as f Vj_ s;n p(#-cusp) we have. 6 (?4 F where
(r3sr
BS = tan-"I
ysin r r x Si'h~ cost t f Cc;P~oSr - r S;n p(j-co:,) ros r
defines BS for the stationary panoramic,' tilted photograph.
Assuming E to be small so that higher.poigers can be, neglected and
expanding B in a Taylor's series about A we have:
B
B - tin-'L~7f nr)~;- cosr~-/1 _ tdo-I'(~ ;nr: l1
1 L`
ENGINEERING REPORT NO. 5435
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Y, c& -X; Sit sihr Cn ~~ i, r
?-
C,s;~. +X'51hfcosr Z; r'.
Ycosr~(X "~;'S/n~sinl' -(Htesl'n~ CcS?s~'nr
Y srn r t (-1- COSr i- (ii k si n/') C,-SID cos r
ten-' ycosr - Xsr- feo,10 r t~ f S;h p0-cosP) sin
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
tdn-~% r E /}CO l-~` . n J = t.~n-' t fV~As;nP~i-rn::~OXAio.Sf' "Ts~rhl,
f}'f B I H K A2tg'-
solving for B ,
In a typical case:
y
r
[?- 6(..2) ry(.(9R) 3(98)2=
G 6 ,r Zg?b1-2.3.d
So we see that for all practical purposes A = ?3i, A and the trans-
formation equations can be used in the given form for rectification.
.An example of a rectified panoramic transformation is given in Figure 29.
ENGINEERING REPORT NO.
PAGE 99
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ENGINEERING & OPTICAL DIVISION
Figure 29
ENGINEERING REPORT NO. 5435
THE PERKIN-ELMER CORPORATION
H s 15,840 ft.
Tilt = 11 1/20
Swing 2700
(11 1/2e pitch, 00 roll)
No earth curvature
Flight Direction
Unreetified
ONE MILE SQUARE GRID ON GROUND
SHOWN AS TAKEN BY 3" F. L.
PANORAMIC CAMERA
1/2 Scale
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ENGINEERING:& OPTICAL DIVISION
uler's egvkt
E ioh for this prob/em %S
3. DISTOETIONS CORRECTED ABOUT THE NADIR (Earth Curvature and Air Refraction)
According to Permat's principle, the fundamental principle of geometrical
optics, the path taken by a light ray between two points is the path of "quick-
eat arrival." Expressed mathematically
(1) E/ nds= 0
where do is an increment of the ray path and n is the refractive index in the
region of do. Euler's differential equations solve (1) and from them can be
derived the laws of refraction and reflection and the solution to our problem.
If we use, the system of spherical coordinates shown in the figure, we
have, assuming n (r; P) B) = n(r)
ds= drz~- r ~dp~
n = n (r)
hdS = d n r.- t r' cf.
(3)
_ z
Sin ~l,
h
r
P
p
or }
Ir
At. o ; r= rp = Ye.f/
nz
THE PERKIN-ELMER CORPORATION
r cot' or c rr
'See, for instance, Margenau, H. and Murphy, e., "The mathematics of Physics
and Chemistry," D. Van Nostrand Co., New York, 1943, Pages 195, 196, 199.
PAGE 101
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ENGINEERING & OPTICAL DIVISION
THE PERKIN-ELMER CORPORATION
- center of Earth
a radius of Barth
- altitude
- angle of arrival of refracted
light ray GP
apparent point of origin of GP
apparent origin of GP referred
? to tangent plane
s, 8' = surface distances to G$ G'
r(rp) a point on trajectory of ray GP
p = angle between nadir and r
(polar angle)
polar angle of G
polar angle of G'
n(r) = 1 + .(r) a refractive index of
air
np = l + p a - refractive index of
air t P
r - R
r a dr
d~
E
re
H
Y/
G'
u'
BARTH CURVATURE AND AIR RETRACTION
Figure 30
ENGINEERING REPORT NO.
5435
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
Letting D= Le /n i we heRvt
1- 9 ode:-,'h t' c r; . a 11
JIII z 1 B - ~ `
Jre 1~t~ It 0
Ex
rP
Lj~
_ D ~C + L--L9) D pz(:~a-4~; d r
r
the
)'T'(2 -2)~ pz(za-p?) --~ r
a~; ~) (i _1)r
r
Since for n ?
that equation
rc
constant - npp the ray path becomes
A6) becomes
a
a straight line, we see
If we change the variables to v-= r 1 e , d->- _ dr,, r
Also, we is* from the figure by the law of sines
001) +
Solving (10) for we have
/ _ Sh,-/ s nY'COS -11i I t f "y /-~~(.2i tti n
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
17
Putting the results of (9) and (11) in (8), we have, since to h N a~z .9 the
exact expression relating points on a photograph, u', to actual earth surface
distance, 8, as influenced by earth curvature and altitude-dependent variations
in atmospheric refractive index.
(14
J lfS ./--~~) L? CI-0)
(-)'(_1"_ .() I
> ; 'Z~
z
E , # p are very small compared to
to great accuracy by a few terms of B = / -(# -,I& - E tE =P
great accuracyt
can be expressed
. Taking
terms up to the third powers of c , Ep in the integrand of (12), we get to
-
+~;F-~
'
S = re Sr'0_ Sihrr.:,;
If H is small compared to re - say H is less than 400 miles, the first term
in (13)--pure curvature term--and the integrand of (13) can be expanded in
powers of If. this .is done, the final form of the expression for 8 will
.be
(/t).
H
where the C's are functions of the, altitude, H, only. If H is not small con-
pared with r0,.it would be best to. express ,5 in powers of v. A. few terms.
should express E , to the desired degree of accuracy. The integrand of (13)
could then be reduced to integrals of the form
fz(Iz2)dz
These can be found directly in tables of indefinite integrals. The first
term on the right of (13) must now be used "as is" and the integral in (13)
ENGINEERING REPORT NO. 5435
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ENGINEERING & OPTICAL DIVISION NNFINW THE PERKIN-ELMER CORPORATION
integrates to a complicated function of y_. The resulting expression, however,
a
would hold at any distance from the earth.
As an indication of orders of magnitude, we will consider terms up to
the third power in and up to the second power in E, Ep . The first term
of (13) becomes
C
rcS -) !?.7~I1^~ r'{ f -J t' z,/:.~?' ;tYr J~? m `'~` )'-~lvgra
P
I ?>
4- 1
=t tj Sit) lc:'?')ti-.v.~",.,3 Y!/~'I~)trthz )
/F ;rl rc~ anYL l T -;t;
Using the power series expansion of sin lx,
Htxh~'Llt t. -v nz'~t -an~1f(lf3tar..;~'% }' ~hn"(ter
In order to sialuate the integrand of (13), we note that
D_ Sr'J t(t ~Sin1/i
r )-5Cc
l( p'')~ =tangy"ri{~P1C1-Ep(~tEp:tdn?.~t~(~t~,).Sec??~'~-~
i~?T7 7, PC~7` 'l t .w i( r'r ~/ ~[/-~fi(.Zf~P} tdh~~i~rf S(i ~G 1 a C'i z)~j~'R
go, to the desired order, the integral in (13) is
ftnrscc?(Ep1Ee_(' tt /3t~r)c, t S'/ ( ri'
r
* f?it s/"(~t;"t n^ )~~ t 2 t/s-~ f?C?-
H tahr ec' ` (E=LFp)~.:?.-N ter,YSc~~ N!-#(#-~~) ~/t~}Irk
~o f ( 1, 1
+ N tari~!Sec'' `i(!t3 tg. n?r) /(-f) r"rjy' it
t /
t 'ec (e2fs tah
p)
ENGINEERING REPORT NO. 5435
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~n"fIJI =t 5/fl :~t.Yr,f
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I ,..? . _ ~ l j i , , _ ? r,l "T,;, , -.1 - H . / 0 ` , . -1. , -I- -
ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
' H
we can express (18) as
L`?
Letting H'= tan ?p we may now give the complete expression for S for the
second-third order appxoiaation:
z r
-H"iro 6''~ {lt3/ ?~?1' +-
tl.
}- i t: '
1. j ,
-'
`.
. .1~::~
a
~
~i * }i,~i { .il:LrY ,Lrj
? t
~i
r'I
??
Y
r~-
.
jr~ 11~?~TE I' :11y.S YL~
1/
~i
~' ?.n2
~..
... _..~. M_
A __.1__.
.1. ?}
~'
rhr
'
PAGE 111
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ENGINEERING REPORT WO. .5435 Figure 31..
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
If Z , 4 ? 1 , then
r(1
r
In this particular example
r'-
Corresponding to .257 error
For L ? 150, we approximate
Z > l ,' : c t
r ? 6.43", Z = .105"
Corresponding to .427. error
MUCSLLANIOUS DISTORTIONS
ENGINEERING REPORT NO. 5435 11 n`. PAGE 112
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 Fos Sne115 Law-
i
fah = rtdr -
Jr =
~Tl1
V i4MP [I
hi5ec~P-?a
r'= [rt to~ (PD-
H I+(b4t 0 N
Nt tA
NrCt N LIr I L
h It. nl-I
wovsl( case is C.A e..
.T teii
r-
_~'N_ f~,s
tto
C-orrec-+1o61 Is hcy~r~ib~e
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ENGINEERING & OPTICAL DIVISION jffFj W THE PERKIN-ELMER CORPORATION
H
e.9. Zf t _ 311
G/'{# fides cfreafer
~N4N ZS00 f:
D15TORTION DUE TO PLANE WINDOW
F,ure 32
ENGINEERING REPORT NO. PAGE 113
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ENGINEERING & OPTICAL DIVISION \% THE PERKIN-ELMER CURPURATIUN
In Liepmann's Paper on the deflection of a light ray by the boundary layer
of air moving past a rapidly moving aircraft, it was considered that the velocity
distribution was a function of the normal distance from the skin only; and this
distribution was in,plane parallel layers. It was considered that if w,, was the
angle at which the ray"enters.the boundary layer and ;'s the angle at which the
ray leaves the layer, then Snell's law holds so that
I Y.
C 1~
1~.: r
Figure 33. REFRACTION BY BOUNDARY LAYER
,From aerodynamic considerations and the perfect gas law., Liepmann says
where fi` is the ratio of specific heats.,,,- is a constant related to the Prandtl
numbers and M. is the free stream Mach number. From this he calculates the angle
CORRECTION FOR RAPIDLY MOVING BOUNDARY LAYER
and plots -
W., ..r3
as a function of t"! and altitude.
lO Liepmann, H. W.,, "Deflection and Diffusion of a Light Ray Passing Through a
Boundary Layers"Douglas Aircraft Company, Inc., Report SM-14397, 16 May 1952.
ENGINEERING REPORT NO. 5435 PAGE 114.
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ENGINEERING & OPTICAL DIVISION \11~ THE PERKIN-ELMER CORPORATION
I . ja ___
However, if in addition the ray goes through a plane glass plate in order to enter
the craft, the final direction of the ray will depend only on the free stream re-
fractive index and that inside the aircraft. If the inside temperature and pres-
sure are equal to that of the free stream, there will be no angular deflection.
ENGINEERING REPORT NO. 54-1" PAGE ".15)
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ENGINEERING & OPTICAL DIVISION - `_ THE PERKIN-ELMER CORPORATION
C. FILM DISTORTION
From the instant of exposure the film begins changing its size and shape.
This distortion is due to many causes, the most important of which are humid-
ity changes, temperature changes, changes during processing, and changes due
to long-term storage.
If the changes in size are uniform along and across the roll or sheet,
the distortion is only a change in scale and is equivalent to a different
focal length or altitude. Unfortunately, most films have different coeffi-
cients of expansion across and tong the film, thus giving rise to a true dis-
tortion.
All of these distortions can be corrected by the electronic rectifier if
the magnitudes are known. The correction is quite simple since the x and y
coordinates of the rectifier correspond to length and width of the film. All
that is required is that the over-all multiplying factors for the picture be
entered into the computer. All x and y coordinates will be multiplied by
these factors during the rectification. There are interesting possibilities
for correcting for distortion of the rectified film before it is printed. It
will usually not be necessary to correct for temperature and humidity changes
of the rectified picture since it will probably be used at the same ambient
conditions in which it was rectified, but the processing shrinkage can be pro-
grammed into the computer so that the rectified picture will be correct after
processing.
All of the temperature and humidity conditions of the film during expo-
sure may not be known. It is, therefore, desirable to have accurate reference
ENGINEERING REPORT NO. 5435 PAGE 116
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TICA!... DI\ISI)*: THE PERKIN.EL.~+.;:R CORPORATION
ENGINEERING 8 O
P
marks, such as the distance between fiducial marks, on each expo4ure to aid
in finding the correction factors for the computer.
As. an example, let us assume that a photo is taken on Kodak Aerographic
film and is to be rectified onto the same film. If film were exposed in the
aircraft at 10?F and 107. R.H. and rectified at the Kodak recommended condi-
tions of 70OF and 507. R.H., the following changes in film size will occur:
The rectified photo will be used at 70?P and 507. R.H. so it is desired
to correct for the temperature and humidity changes in the original and for
the processing shrinkage in both the original and rectified photos.
Humidity
50% - 10% - 40% R.H. change. 11
'
Length (y) correction m 40
X 8.5 x 10-5
3.4%
Width (x) correction a 40 x 9.0 x 10-5
3.67.
Temperature
?
700 - 100 = 60?F temperature change
Length (y) correction = 60 x 4.2 x l0_5.257.
Width (x) correction a 60 x 4.4 x 10 .267.
Processing Shrinkage
Length (y) correction
.05x
Width (x) correction
.06%
The rectified photo must be reduced in size by the humidity and temperature
corrections and increased in size by the processing shrinkage correction. The
processing correction must be applied twice; once for the shrinkage of the
original and once for the shrinkage of the reproduction.
11 Values of film distortion factors are taken from "Kodak Materials for Aerial
Photography," 4th Ed., Page 9, Eastman Kodak Co.
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ENGINEERING & OPTICAL DIVISION \1
_j THE PERKIN-ELMER CORPORATION
Over-All Corrections
Length (y) corrections = .9966 x .9975 x 1.0005 x 1.0005 a .9951
Width (x) corrections = .9964 x .9974 x 1.0006 x 1.0006 = .9950
This process can be extended to cover all of the listed distortions of
both original and rectified films. If desired, long-term distortion could be
included so that the, rectified photograph would be distortion free after a
year's storage.
It is recommended that Dupont Chronar base films be used for the recti-
fied photograph since this base has excellent temperature and humidity coef-
ficients, good optical clarity, high strength, and since it contains no sol-
vents or plasticizers, has good long-term aging characteristics.
John Centa12gives the following coefficients for Chronar base:
Humidity coefficnent: 1.0 - 2.0x 10-5 in/in/`, Q.S.
Thermal coefficient: 2.0 x 10-5 in/in/?F.
He also states that accelerated and normal aging tests show no indica-.
tion of base change or deterioration.
In an instrument of the precision described in this report, every effort
should be made to prevent degradation of the results from external sources.
Chronar base films will aid in attaining this goal, and their use is highly
recommmended.
In the worst possible cases the distortion due to film changes will prob-
ably never exceed 17., and .1% is probably a typical figure. All of these dis-
tortions in either the original or the rectified photo can be corrected with
an electronic rectifier.
12 Cents, J. M., "Performance Characteristics of 'Chronar' Polyester Photo-
graphic Tilm Bass," PHOTOGRAleMTRIC ENG., Vol. 2, No. 4, Sept. 1955,
Page 539.
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
Lens distortion may, under some conditions, cause serious errors in the
geometry of the photograph.
Aerial lenses very from extremely low distortion lenses, such as the
Wild Avigon, to lenses with very high distortion, such as the Zeiss Pleon.
The Pleon is a very wide angle (136?) lens which is designed with a large
amount of negative distortion in order to obtain better edge illumination.
The two important types of lens distortion are: radial or linear dis-
tortion, which is a linear displacement of the image point radially toward
or away from the principal point. The positive direction is taken as being
away from the center (See Figure 36).
Tangential distortion is a displacement of the image perpendicular to
radial lines from the center of the field. Tangential distortion causes a
straight line through the center of the field to image as a curved line.
Improper centering of the element causes bent axis distortion. This
is equivalent to a small wedge in front of the lens and is discussed on
page ?
Although the best mapping lenses have distortions so low as to be neg-
ligible (the Wild Aviotar is claimed to have a maximum radial distortion of
5? or .005 mm), it may not always be possible to use such a lens. It appears
that lens designers could design lenses with better resolution if they could
let the distortion increase. In view of this, it is desirable to have the
ability to correct for lens distortion in the rectifier.
Figure 34 shows the radial distortion curves of two mapping lenses, the
Planigon and the Metrogon, and of the wide angle Pleon.
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ENGINEERING & OPTICAL- DIVISION \V44 THE PERKIN-ELMER CORPORATION
1 71 - - ----
Other examples of magnitudes of distortion that may be expected may be
found in military specifications. Mil-L-4325A(ASG) is for a 36-inch, f/8,
9 x 18 format, aerial reconnaissance and spotting lens. This spec calls for
a distortion not exceeding'10 mm. Correction during rectification would
greatly improve the accuracy of photos made with this lens though it ob-
viously would not be used intentionally as a mapping lens since it has a
distortion of about 4%.
Mil-L-7367B(ASG) is for a 6-inch, f/6.3, 9 x 9 format cartographic lens.
The maximum tangential distortion is .02 mm, and the maximum radial distor-
tion is -.17 mm at 450. This results in a radial positional error of .117.
and a tangential error of .013%. Obviously, the radial distortion, as is
usually the case, is the more troublesome.
Distortion characteristics of lenses are usually given as curves of dis-
tortion vs. radial distance as shown in Figure 34. Correction of distortion
in the electronic rectifier is quite simple for radial distortion but is con-
siderably more complex for tangential distortion. Although feasible, it is
probably not advisable to correct for tangential distortion since it is so
low in good mapping lenses.
As a typical example of a good mapping lens, the Planigon is duscussed
here. Mil-L-6637B(ASG) covers a Planigon aerial cartographic lens 6 inches,
f/6.3 for a 9 x 9 format. This spec calls for a maximum tangential distor-
tion of .008 ass and a maximum radial distortion of .012 mm. The maximum tan-
gential distortion usually occurs at the maximum. radius DT = .008 = .0052',x,.
152
The maximum radial distortion occurs at 130 mm radius (Figure 34) DR m .012
130
ENGINEERING REPORT NO. 5435 PAGE 120
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ENGINEERING & OPTICAL DIVISION ' \' THE PERKIN-ELMER CORPORATION
6 in. .f/6.3 Planigon
+2
3
100 120- 140
\
Diatance from Optical Axis. in ease.
20 40
20
40
Distance from Optical Axis in =a.
6 in. f/6.3 Metrogon
80
f/8 Pleou 1360 Field
60 80
Distance from Optical Axis is mm.
DISTORTION CURVES OF SOME REPRESENTATIVE AERIAL LENSES
ENGINEERING REPORT NO. 5435 Figure 34
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ENGINEERING & OPTICAL DIVISION Lv7 THE PERKIN-ELMER CORPORATION
A rectifier operating at an accuracy of .017. could not improve these
figures.
ENGINEERING REPORT NO. 5435 PAGE 122
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ENGINEERING & OPTICAL DIVISION
THE PERKIN-ELMER CORPORATION
N
Figure 35. DISTORTION FROM PRISM IN WINDOW OR LENS
Radial Distortion
Tangential Distortion
Figure 36. TYPES OF LENS DISTORTION
ENGINEERING REPORT NO. 5435
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ENGINEERING & OPTICAL DIVISION N~r
THE PERKIN-ELMER CORPORATION
Distortion due to prism of lens or windows. (bee Fig. 35)
If the prism angle is x and a is small, thdn quite closely
6.
!. ,,...., 13
The distortion is worst at large angles ,-, and since the largest
may be around 60?-800, we have for a bad case, ~~.;.
This results in an error
go if
OTHGR APPLICATIONS OF RICTIFICATION BQTIONB
(a) One schemae.of rectification, consists of a read-out drum rotating
at uniform speed with the read-out printing spot traveling at uniform
speed parallel to the axis of the dram so that it 'travels one line
width in one rotation. The lcoputer.would position the read-
in drum to
the proper spot for pick-up. In order to gat an idea of'the velocities
and accelerations required of the read-in drum for this scheme, we mast
calculate the partial derivatives involved in the equations:
13 Washer, F. E., "The Sffect of Prism on the Location of the Principal
Point," P) TOGSA IC NMGIN88BINC, Vol. 23, June 57, Page 520,
ENGINEERING REPORT NO. 5435 PAGE
124
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ENGINEERING & OPTICAL DIVISION THE PERKIN-ELMER CORPORATION
ate .dsirtations as follows using the transformation for a stationary
camera as we are mainly concerned with the large distortions.
We have
Similarly,
Vy = a ~X 1- y'''
X Y
It is likely that the tilted panoramic photograph will be one of the
most difficult to follow by this system, so we calculate the appropri-
Le.t
JSin t5ihi
a j
b = .i y'
Co5
C ht Cc,
~ y Z
?! `C r
ENGINEERING REPORT NO. 5435
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ENGINEERING & OPTICAL DIVISION F TW THE PERKIN-ELMER CORPORATION
Then
d2thz.
? -k2 = ~ls%n?too=- jrcos-f n
h o
-q ia
de- hk = c
1j-(7 7 i -T7
~ aX vx 't