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THEORETICAL AND EXPERIMEHI'AL PRCCESSING OF LUNAR TELEVISION PICTURES
Fi na l Repor t
SGC 203R-4
November 29, 1962
C . S. Lorens
A.M. Boehmer
J. Gallagher
Prepared for t h e
J E T PROPULSION LABORATORY
of the
NATIONAL AERONAUTICS A N D SPACE AININISTRATION
4800 Oak Grove Drive
Pasadena, Cal i fornia
Under Contract No. NAS 7-100(950249)
Walter F. Storer, Cognizant Engineer
Prepared by
SPACE -GENERAL CORPOLTION
9 0 0 E. F la i r D r iv e
E l Monte, California
TKi iWorE wa s petfarmd for he Jet PiapuhionLahw,California Institute of Techtdogy, sponsored by the
National Aeronautics and Space Adminiamtion under
Contract NAS7-100.Robert h. Stewart
Associate Manager
Research and Advanced
Development Division
i
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1
SPACE -GENERAL C ORPORATI ON SGC 203R-4
Page 1
Abstract
I t i s es tim a ted t h a t l a r g e q u an t i t i e s of l u n a r p l an et a r y p i c tu r e s a re
t o be obtained from forthcoming experiments. The proce ssing of pic tu re s through
d i g i t a l computer t echniques o f fe r s one po ss ib i l i t y f o r hand l ing th e la r ge number
of p ict ure s which are t o r es u l t from these exper iments.
repor t on a th eo re t i ca l and exper imental s tudy of techniques fo r process ing
p i c t u r e s .
Th is r epor t i s t h e f i n a l
An o u t l i n e i s presentedof
p o ss ibl e l i n ea r an d n o n -l i nea r t h e o r e t i c a l
work which can be made applicable t o processing lunar and p lane ta ry p ic tu res .
T h e t h eo r e t i c a l material of t h i s r epor t i s l i m i t e d t o l i n e a r p ro ces sin g. The
mat er ia l dea l s wi th ex p lo i t ing the geometry of p ic t u res p r i nc ipa l ly th rough th e
a ss um pt io n of s t a t i s t i c a l l y s t a t i o n a r y p r o p e r t i e s u nd er t r an s l a t i o n and r o t a t i o n .
P i c tu r e s a r e d e fi ned as fun cti ons of one index where th e index i s a 2-dimensional
vector corresponding t o th e 2-dimensional coordinates of t h e p i c tu r e . T h is n o t a -
ti o n makes i t poss ib le t o handle p ic tu res i n the conven t iona l l -d imens iona l no ta -
t i o n of c l a s s i ca l p ro ces sing t h eo r y w h i l e preserving th e 2-dimensional p rop er t ie s
of t h e p i c t u r e .
processing, matr ts , c o r re l a t io n W c t i o n s , s t a t i o n a r y and symmetry pro-
perties optimum l h e a r processing, trivial processing, orthogonal preprocessing,
minimum error, minimization with a cons t ra i n t , p reprocess ing wi th p red ic t ion ,
qu an tiz at ion , and convergence of i t e r a t i v e computations.
The repol..t d ea l s w i t h Pepresentat ion of t h e p i c t u r e s , l i n e a r
The experimental work of t h i s r e p o r t t e s t s t h e t h e o r e t i c a l r e s u l t s f o r
complexity, usefu lness, and correc tness.
capable of producing
correspond ing t o the 6 faces of th e cube onto which the shades of br ig ht ne ss were
pas ted.
Craters Era tos the nes and Archimedes i n 15 x 19 f i e l d s of e lement s. A r t i f i c i a l
p ic tu r es were a l s o cons truc ted hav ing con t ro l led cor r e la t ion fu nc t ion s . A number
of experimen ts were performed w ith the se pic tu res t o e xt ra ct i n an optimum manner
ano ther des i red p ic tu re . A v a r i e t y of FORTRAN programs are inc luded f o r even tua l ly
performing th e processi ng computations on a d ig i ta l computer.
An exper imenta l capab i l i ty w a s developed
p ic t u res wi th up t o 750 elements i n 6 shades of br ightness
This c a p a b i l i t y w a s then used i n the exper imental representat ion of the
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1 I
SPACE GENERAL CORPORATION SGC 203R-4
Page 2
In t oduct ion
The successf ul extra ct io n of s ign i f ican t da ta f rom lunar and planetary
televis ion pictures requires the survey and comprehens ion of present ly known t ech-
niques a s wel l as th e development of new techniques p a rt ic u la rl y di rec ted a t t e l e -
vis ion process ing.
th e f i e l d of processing pic tur es . Much of th e material of the report comes from
This repo r t deals with th eo re t i ca l and exper imental work in
3 previous repor ts 19.
The theore t ica l work i s d i r ec t ed a t producing techniques fo r ext rac t in gPre-processing ofata e i t h e r before t ransmiss ion or subsequent t o transmission*
t e l e v i s i o n p i c t u r e s ha s t h e p o s s i b i l i t y of minimizing th e transmissio n channel re -
quirements or maximizing the use of t h e transmission channel when the channel i s
f ixed and in f le xib le . Subsequent processing of te le vi s i on pic tur es a f t e r t r a ns -mission permits an organizat ion of t he dat a in to a form which i s more meaningful
as a f i na l produc t and a l s o permi t s hypo thesi s t e s t in g fo r the ex t r ac t i on of hy-
p o th e t i c a l d a t a f r o m th e p i c tu r e s .
The qua ntit y of dat a t o be gathered from lunar picture exper iments re-
i c r ed u ct io n and the automatic design of prototype reduction
t h e p i c t u r e s w i l l be used s o le ly f o r mapping areas where only
ent ly av a i l ab l e . Sc i en t i f i c e f f o r t s , however, w i l l be d i r e c t e d
a t measuring the dens i ty d i s t r i bu t ion of cra te r r ad i i and typ ing the debr i s a round
t h e c r a t e r s as w el l as w i th in t h e c r a t e r .
th e po ss ib il i t y of d is tin gui shin g between comet and as te ro id impacts as w el l as
t h e p o s s i b i l i t y o f t h e r eco g ni t io n of v o lcan ic c r a t e r s . S im i la r c r a t e r s t u d i e s
w i l l be performed on th e sc i e n t i f i c experiments dir ec te d a t the Planet Mercury.
Knowledge of the crater s t ructure has
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SPACE GENERAL CORPORATION
Geological ' s tu die s w i l l be in ter es t ed in t he uniformity and roughness of th e
sur face s t ruc tu re as wel l as the c l as s i f i ca t ion o f the su r face geo logy .
Theore t ica l S tud ies
Telev is ion p ic tu res are normally thought of as a sequence of pictures
capable of d isp la yin g moving obje ct s.
and planetary experiments is much broader tha n t h i s conce pt.
tures w i l l be i n d iv id u a l l y d i s t i l i c t i v e
Some of the p ic tu re s will b e r e l a t ed t o o th e rs t h r o b s er v at i on of substan-
t i a l l y t h e same exper iment from dif fer en t d i re ct io nd di f fe ren t t i m e s .
s e t s of pic tu res req uire s imoltaneous proce ssing which i s cons iderab ly d i f fe ren t
than t h a t r equ i red fo r the p rocessing of a sequence of pictures of mwing objects .
The use of te l ev is i on p ictu res i n lunar
Many o f t h e p i c -
an8 vi11 require appropriate processing.
Many optimum processing techniques are p r e s en t l y known f o r t h e p r o c -
es s ing of te lemetry .
e s s ing of p i c to r i a l d a t a .
t ech niq ues d i r e c t l y ap p l ic ab l e t o pr ocess in g t e l ev i s i o n p i c tu r e s w i th p a r t i cu l a r
emphasis on t h e use of d i g i t a l com putation f a c i l i t i e s f o r p r o cess in g pictures
Very few of these methods have ever been used i n the p roc-
The fol lowing theoret ical areas develop a number of
t ransmiss ion. The areas are l i s t e d in order of increasing
Linear Specia l F i l t er in & based on previous ly measured c orr el a t i on
funct ions can be used t o p reprocess da ta t o r educe i t s q u an t i t y as w e l l as t o p os t-
process data t o remove random ir re gu la r i t i es such as noise and t o enhance var ious
e f f e c t s f o r po s si b le l a t e r d e te c ti o n.
t h i s r e p o rt d e al s wi th t h i s area.
phone Laboratories and i s repor ted i n r e fe rences 7, 13, 1 4 and 24. Much of t h e
th e o r e t i c a l work i n 1 dimension a s presented i n refe rences 11and 16 i s changed
i n t o 2 dimensions in th i s rep or t . The concepts of ro tat io n and t r an s l at io n have
an expanded meaning i n 2 dimensions.
Most of th e th eo re t i ca l work presented i n
Some of th e o ri g i n a l work w a s done by B e l l Tele-
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SPACE -GENERAL CORPORATION SGC 203R-4Page 4
Linea r Learning Processes introd uce th e concept of measuring para-
meters f r o m se ts of hypothet ical ly cha rac ter is t ic data . The development of an
in t e l l i g e n t l e a r n in g p ro ces s h as t h e p o s s ib i l i t y of r ep l ac in g t h e h eu r i s t i c
guessing which has normally accompanied pi ct ur e proc essing i n the p as t.
ideas of References 1, and 4 have th e po ss ib i l i t y o f be ing app l icab le .
approaches are contained in References 8, 13, and 25.
The
Other
Nonlinear Sp at i a l F i l te r i ng and Learning i s d i r ec t ed a t process ing the
The shades-of-gray scale i s def inednheren t non l inear p roper t ies of a picture .
as t h e l o g a r i t h m of t h e b r i g h tn e ss scale t o cor re sp on d t o t h e human preception
of re la t i ve changes in the grey leve ls A th e r t h an t h e ab s o lu t e chang es i n t h e
lev e ls . The b r igh tness s ca le a l so has an upper and lower bound. I n ord er t o
hand le these non l inear e f fe c t s and poss ib ly o ther s , i t i s n ecess ar y t o s e r i o u s ly
stu dy techn iques of non lin ear processing. Much of th e ba si c the ory i s contained
i n Reference 27. Examples of th e implementation of t h i s the ory w i l l be found i n
References, 3, 6 , 9 , 1 7 , and 20.
Sequent ia l F i l te r i ng of P icture s which a re co rrel a te d with one anotheri n sequence is an expansion of t he amount of d at a used i n th e processing of a
s ing le point of a p ic tu r e .
it may be desirable t o pr oc es s th e movement and change of ob je ct s i n t h e p i c t u r e s .
The material of References 12 and 23 has poss ib le app l ica t ion .
The theory i s app l icab le t o scanning systems where
Extract ion of the presence or absence of dat a i n a picture from only a
hypothetical knowledge of what the data i s t o lo ok l i k e i s an area p a r t i c u l a r l y
d i rec ted toward the d i sc re te r ecogn i t ion or r e je c t io n o f the p resence of data i n
th e pictu re under observat ion.
t o co nt ro l of experiments and the acceptance of va li d commands.
re fe rences 5 , 8, and 18 can be a pp li ed t o t h i s s u bj e ct .
The techniques of t h i s are a are a l s o ap p l i c ab le
The material of
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I .
SPACE-GE"ERCIL C ORPORATION
Line ar Analog Techniques provide an a l t e r n a t e means of proc essi ng p ic -
tu res . Most the ore t ic a l work re su l t s in d isc re te processing on a digi ta l com-
puter .
t h e po s s i b il i ty of producing simple lin ea r equipment which may work a t consider-
ab ly highe r speeds tha n can be obtained with di g i t a l equipment. The or ig in al
work i s contained i n Reference 26.
A number of multi-dimension al analo g tech niqu es a l s o e x i s t which have
Further work i s in References 2, and 28.
The follow ing th e o re ti c a l and expe rimen tal work develops and t e s t s
a number of li n e a r technique s t o handle th e two-dimensional aspects of pictures
so tha t the goemetry of t h e p i c t u r e can be re ta in ed i n two dimensions ra the r i n
one dimension where a considerable amount of theory i s presen t ly w e l l known.
Represen tat ion of a Pic tu re
The follo wing th e o re ti c a l development has 2 object ives: 1. t o rep -
r e s e n t - a p i c t u r e and i t s processing i n c l as s i c a l vec to r and mat rix no ta tion ,
and 2. t o preserve the 2-dimensional p roper t ie s of t he p ic ture .
A sampled picture i s represented by the row vector x having elements
x
and i = (a,b) are 2-dimensional vectors.
t o w r i t e t h e elements of a row vector seq uen tial ly i n one row. However, i n t h e
no ta t ion of th i s repor t a row vector i s w r i t t e n i n a geometrical form, each of
th e e lements of t he index vector in dicat in g th e geometr ica l posi t ion of t he cor -
responding row ve ct or element. The elements of t h e re pr es en ta tio n of x then
correspond geometr ica l ly t o the p ic ture th at they repres ent .
resenta t ion of a row vector i s then
cor responding t o the samples of the p i c tu re where the ind ic ie s 1 = (1,l)L'i
I n c l a s s i c a l n o t a t i o n , i t i s customary
A possib le rep-
x = pl,i] = Pi] = X11 12 x13 x14
X21 22 x23 x24
x31 x32 x33 x34
X
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I
SPACE -GENERAL CORPORATION SGC 203R-4Page 6
Linear Processing
A second pic tur e y can be obtained fr om th e p ic tu re x b y l i n e a r l y
weighting and adding th e elementsof x t o form t he elements of t h e p i c t u r e y .
Notationally, t h e new picture i s computed by t h e ma tr ix formula
where
The summation i s over a l l t h e values of th e index i. I n r e a l i t y , t h i s is a
double summk lon . I€ s repregented here as a single sum because of i t s simpl i -
c i t y and i t s s i m i l a r i t y t o t h e c l a s s i c a l m a tr ix n o t at i on of t h e matrix product.
Linear processing can be used t o produce an i so la te d point i n a p i c -
t u r e or t o approximate some desired re s u lt based on t h e data conta ined i n th e
processed p ic ture .
Matr ix Transposit ion
The row vector x i s a degenerate form of a matrix h having elements
where the i nd ices i nd j ar e 2-dimensional ve ct ors . The transpose ht ofhi,
[ IThe transpose x
Note t h a t th e ind ic es have been transposed, not th e elements of t h e vecto r r ep -
resent ing t h e i nd ices .
as f o r the row vector x .
of the row vector x = xlJi i s def ined t o be a column vector.
The representation of the column vector xt i s t h e same
Xr11 x322 x3323 x34"
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SPACE -GENERAL C ORPaRATION
The i t h row of the m a t r i x h i s t h e row vec to r [hi, j! containing
f o r which i i s t h e f i r s t index. I n a l i k e manner t hel l the elements hi
Ar 9
j t h column of t h e matri x h i s the column vector h c on ta in in g a l l t h e e l e -1 i , j J
ments h fo r which j i s t h e second index.i , j
The vector indices i = (a,b) and j = (c ,d) are equal when t h e i r cor-
responding elements a re equ al. The ad dit io n of indic es i s also element by elemen
eq u a l i t y o f i n d i ce s : i = j +7 a = c and b = d
add i t io n of ind ice s : k = i + j <-7 k = (a*, b i d )
The sum, f , of 2 matrices h and k i n t h i s n o t a ti o n i s the matrix of
e lements equa l t o the sum of elements having id en t i ca l ind ices .
f = h + k
where
[ f . . ] = Ch + ki 1= , J i,j , j
The sum of th e two pic ture s x and y i s t h e sample-by-sample sum of
t h e p i c t u r e s .
Product of Li nea r Transformations
process ing of p ic tures by t h e r e l a t i o n s
and then by z = yg
l e ad s t o t h e product of the matrices h and g s inc e sub s t i t u t io n of y i n t o t h e
equat ion for z produces t h e r e l a t i o n
z = xhg = xc
where c = hg .
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SPACE GENERAL CO R P O R A T I O N
In terms of elements, t he matrix equations are
s o t h a t by su b s t i t u t i o n
SGC 203R-4Page 8
where
The product of t h e mat r i ces i s t hus a sum over the column index j of the f i r s t
matrix h and .t x j of the second matrix g.
A square m a . i x h i s one which has an equal number of row and column
ind ice s. The diagonal of a square matrix i s t h e se t of elements for which the
row and column indices are i den t i c a l . The ide n t i ty matrix I i s then defined t o
be the one which i s 1 on the diagonal and 0 elsewhere.
[Tl,j[Ei,
j]where '1, j
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SPACE -GENERAL C O F 2 O R A T I O N
Correlation Functions
The cros s-co rrel atio n function 'pxz between t he pi ctu res x and z
i s defined as t h e matrix product.
where th e bar denotes t h e ensemble average of each element i n t h e matrix product
tx z . Due t o the t ran spos i t io n, there i s only one index, 1 = (l,l), over which
the summation i s performed.
It should be noted that t h e two pictures x and z can have a different number of
samples and a re not r e s t r i c t ed t o be ing t h e same geometrical size.
co r r e l a t i o n f u nc t io n of a p ic tu r e x w i t h i t s e l f i s h o r n as t h e au to -cor re la t ion
funct on.
The cross-
- 7i
- -.. , j ] = rx i , l x l , j 1 = b i x j i
I n this 2-dimensional notation, t h e correlation functions have many of
Thet h e same properties as th e classical one-dimensional cor rel a t i on func t ion s .
t r an s p o s i t i o n of the cros s-ca rrel a t io n funct ion between x and z produces the
cross-correlation between z and x.
The auto-correlation i s also i t s own transpose.
= tot
%x xx
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,
SPACE -GENERAL C ORPORATION SGC 203R-4Page 10
Stat ist ical independence between th e two pictures produces the re-
su l t s t h a t t h e c ross - co r r e l a t ion func t ion i s t h e matrix product of the mean
values of t h e p i c t u re s
-t-x and z s t a t i s t i c a l l y in de pe nd en t= x z = x z
'pxz
where the mean value of t h e p i c t u r e x i s t h e row ve c to r made up of t h e mean
value of each sample of t h e p i ct u re . I n t h e p a r t i c u l a r c8se where the pic-
t u r e s are s ta t i s t ica l ly independent and one of them has a zero mean, t h e cro ss -
co r r e l a t ion func t ion i s zero.
The auto-correlat ion of t he sum of two pictures x and y i s t h e sum
of th e indiv i dual auto-cor re la t ions and the two cros s -cor re la t ions .
Q Vn + Qm + 9xy+
Wheye th e pie ta re g are s t a t i s t ; i c a l l y independent and one has a zero mean, the
auto-correlat i . f €%e sum is then just the slun of the indivi8Uat auto-
corre la t ion funct ions .
+ QYY= <pxx
The correlation f 'unction
ments a l l having
has many uses.
x x1 3
the same standard
of a picture made up of independent ele-
deviat ion u and mean m i s an example which
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.
SPACE-GENERAL CORPCRATION SGC 203R-4Page 11
The co r re l a t io n funct ion i s then
= u I + m U2
'pxx
where I i s t he iden t i ty mat r ix and U i s t he un i t ma t r ix o f a l l 1's.
The cross -corre lat ion function between two pi ct ure s y and y l i ne ar ly1 2
obtained from pi ctu re s x, and x i s a matrix operat ion on the cross-cor re la t ion2
2'unctions between xl. and x
I n the case where th e p ic t ures
y2 x2h2 + %
x and x have zer o mean th e cro ss -co rre la tio n1 2
funct ion cp has the p a r t i c u l a r l y simple form
y1y2
. .
A fur th er s impl i f ica t io n occurs in the computation of th e auto-cor re la t ion func-
t i o n 'p of t h e pi ct ure y obtained f r o m t h e pic tur e x when the pi ct ure x i s made
up of indep ende nt, zero-mean eleme nts.
cor re la t ion funct ion of t h e p i c t u r e y i s
YY2
I n t h i s c as e, 'pxx = u I so t h a t t h e a u to -
2 t t= a h h + k k .
cpYY
An example of t h e use of l i ne a r p rocess ing theory i s i n t h e c o ns t ru c -
t i o n of an a r t i f i c i a l p i ct u re from a random number table.
are of te n used i n experiments where it i s necessa ry t o know and control the
proper t i e s of t h e p i c t u re s .
A r t i f i c i a l p i c t ur e s
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SPACE-GENERAL CORF'ORATI ON SGC 203R-4Page 12
If it i s d es i red t o co n st r uc t an a r t i f i ' c i a l p i c tu r e w i th a prescr ibed
au to -cor re la t ion func t ion cp
symmetr ical ly fac tor i ng t h e matr ix 50
t ra n sf or m a s t a t i s t i c a l l y ind epe nd ent p i c t u re i n t o a n a r t i f i c i a l p i c t u re h a vi ng
t h i s co r r e l a t i o n f u nc t io n .
are not p resen t ly known.
a one-dimensional technique i s av a i l ab l e f o rYY
t o o bt ai n a l inear process h which w i l lYY
The two-dimensional implic ations of t h i s technique
A l a t e r s e c t io n d es c r ib e s e xp erim en ts i n a r t i f i c i a l l y c r e a t i n g c o rr e-
l a t e d p i c t u r e s from random n&er ta bl es and l i ne ar processing.
Pos i t ive Def in i te Condi t ion
A n ensemble of pictures x i s d e fin ed t o be l i n ea rl y dependent whenever
t h e r e e x i s t s a n o n t r i v i a l l i n ea r pr oces s h of t h e p i c tu r e x about i t s mean L
which w i l l produce a one-element p ic tu re y = x ' h having a ze r o co r r e l a t i o n
func t ion . -2
Q Y Y = y = O s-x ' = x - x.
A set of pictues in which the s m w o samples are always identical
is an exanple of an ensemble of l inear dependent pictures. When the ensemble of
Broducing
p i c t u r e s i s not l i ne ar ly dependent , it i s defined t o be linearly independent.
I n th e case of an ensemble of l i ne ar independent p ic tures ,every l i ne ar process h
a one-element pi ct ur e has a pos i t ive valued cor re la t ion func t ion
- t .- h q x l x , h > 0 f o r eve ry h .
VYY
An aut o-c orre lati on matrix thus i s d ef in ed t o b e p o s i t i v e d e f i n i t e
whenever it i s obtained from an ensemble of l i n e a r i t y independent pic tu re s.
Where t h e ma trix i s p o s i t i v e d e f in i t e , t h e d e te rm inan t Icp I w i l l be non-zero.xx
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SPACE-GENERAL CORPORATION
The imp lic ati on of the pic tu re x being l in ea r l y dependent i s t h a t at
least one sample xi i s es se nt ia l l y computable from th e r e s t of t h e p i c t u r e .
d e f i ni t i on , i f x i s l in ea r ly dependent , t he re ex i s t s a non - t r iv ia l h such th a t
By
where
This implies t h a t y i s e s s e n t i a l l y z e ro .
Since a t least one of the elements of h has t o be non-zero, assume th at
h f 0. This produces the resul t that1,1
Thus, i n the case where the aut o-corre la t ion matr ix Qn has a zero determinant
lqml = 0, the au to -cor re la t ion matr ix i s n o n- po si ti ve d e f in i t e s o t h a t t h e
c tu re a re l ine ar ly dependent . A t least one of the samples
can be produced from other s . In ' th e case of an au to -cor re la t ion f'unction
d IqxI = '0, t is always poss ib le t o drop +he de-
pendent samples with a l inear p rocess h so as t o produce a p i c t u r e w ' = x 'h wi th
a sma lle r number of li n e a r independent samples and an au to -c orr el ati on fun ctio n
which i s pos i t iv e de f i n i te and having a non-zero determinant.QW'W'
The new picture w ' w i l l contain a l l of the information contained i n the or i -
gi na l pic tu re x s i nc e th e l i ne ar ly dependent samples a re computa'ble from t h e
l i n e a rl y independent samples.
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SPACE-GENERAL CORPORATION SGC 203R-4'Page 1 4
Stat io nary C orre la t ion Funct ions
The i n i t i a l d e fi n i t io n of t he co r r e l a t ion func t ion
Qxz = [Fj]
permit ted each element i n the matrix (0 t o have a d i f f e r en t va lue . In many
p r a c t i c a l s i t u a t i o n s t h e c o r r e l a t i on f un c ti o n w i l l be independent of b o th t r a n s -
lettion and rotation, and dependent only on the d i s t ance of sep ara t ion between
xz
the elements x and z
.i j
The co rre la t io n function qxz i s defined t o be stat ionary wheh the ele-
ments of i t s matrix are functions only of th e diff ere nce between the ind ices. A
s t a t ionary co r r e l a t ion func t ion i s thus independent of t r an s l a t ion bu t no t of r o -
tation. Whenever qXz= 1-1 i s s t a t ionary , t he re e x i s t s qxz(k ) such th a t
L J
Qxz(k) = xizi+k i s independent of the index i. The addit ion and subtract ion of
indices i s a vecto r addi t icn and subt ract ion . A s takionary cross -cor re la t ion func-
t i o n s a t i s f i e s t h e r e l a t i o n
The s ta t ion ary auto-cor re la t ion funct ion 4pxx s a t i s f i e s t h e p e h tt i on
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S P A C E - GE NE FiA L C O R P O R A T I O N sc)c 203R-4Page 15
Symmetric Correlation Functions
The four elements xll, x12, x31, x32 of th e pic tu re
I2
22
-32
X42
X
x33
x43
-inormally have, a cor re la t ion func t ion such t h a t
h o r i zo n t a l
- v e r t i c a lx31 - x12 x32
diagonal
The f i r s t two r e l a t i o n s a r e s t a t i o n a r y r e l a t i o n s .
t h e d i f f e r en ce i n i n d i ce s ( 1 , 2 ) - (1,l) on the l e f t and (3,2) - (3 , l ) on ther i g h t , are t h e same.
t h e d i f f e r en ce of t on the l e f t i s (2, 1) and on t r i g h t (2,- 1)
I n order to inch& t h e diagonal symmetry of the correlation, a sta-
I n t h e h o r i zo n t a l r e l a t i o n
r e l a t i o n i s not a s t a t i o n a r y r e l a t i o n s i n ce
t i o n a r y co r r e l a t i o n f u n c t i on i s defined t o be symmetric whenever th e co rr el at io n
func t ion i s a func t ion of' the magnitude of t he e lements of th e dif f eren ces be-
tween the indices .
metr ic then p(kl) = cp(k2) whenever k1 = (al, bl) and k2 = (a2, b2) such th a t
lal\ =
1a21 and (bll = Ib2
I .funct ion, it i s s a id t o be symmetric.
That i s , i f t h e s t a t i o n a r y c o r r e l a t i o n f u n c t i o n p ( k ) i s sym-
When t h i s p ro pe rt y ho ld s f o r t h e e n t i r e c o r r e l a t i o n
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SPACE -GENERAL CORPORATION SGC 203R-4Page 16
Distance
Fur ther r es t r ic t i on s can be placed upon a s t a t i o n a r y co r r e l a t i o n f u n c -
t i on by spec i fy ing tha t the cor re la t ion i s only a funct ion of the dis tanc e be-
tween th e elements being correl ated.
having indices i = (il,i2)nd j = ( jl , j ,) i s d e fin ed t o be
The dis tan ce d between th e pic tu re elements
il - j,) 2 + (i2 3,) 2
TW Ocur r e la t ion e lements are d e f b e d t o be equal whenever they are t h e c o r r e l a t i o n
funct ion of picture e lements separated by t h e same dis tance .
A fur ther modif icat ion can be made by specifying t h e distance measured
i n terms of an e l l i p s e .
t i o n of scanned te levis ion pictures where the scanning process in troduces a d i s -
to r t io n in to the co r re la t ion f’unc tion.
This modification should have some use i n th e consid era-
St at io na ry and Symmetric Matri x Products
It would seem intuitive that t h e matr ix product of s ta t ion ary and sym-
metric matrices would also be stationary and symmetric. It i s fo r tuna te t h a t
t h i s i s the case s in ce It makes i t possible t o c o n st r uc t i n v a r i an t s-tric
process ing itevices .ub€ch are a function of the d i s t an ce and d i r e c t i o n from the
par t icu la r p ic tu re e lement which i s t o be cons tructed.
Whenever t h e ma tri ce s h and g are s t a t i o n a r y , t h e i r m a t r ix pr od uc t
c = hg i s a l s o s ta t ionary . Tha t i s i f
where h i s a funct ion of the d i f fe rence j - i and g i s a func t ion o f the d i f fe ren ce
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SPACE-GENERAL CORPORATION
k - j , then c i s a funct ion of the dif ference k- i .
In terms of th e dif fe rences between the indices k- i = r and k - j = s t h e matrix
product becomes the familiar convolution formulas.
=
[h ( r - s )
4 1= [ z h ( r + s ) g ( - s )1
Sc ( d
[E h ( s ' ) g ( r - s ' )
A s l ig h t ly more usefu l formular i n computation i s
c (d
Thus t o compute t h e element e(.), the f i e l d of elements of g i s r e f lec ted th rough
th e o r i g i n ana correspondingly mult ip l ied and sunned w i t h t h e f i e l d of elements h
For eiwrIp3.e where the matr ices aret a r t i n g at the appropr ia te e 2 e n t h(r).
fo
t h e f i e l d of g i s r e f l e c t e d t o giv e
=0 0
g1,o go, 0
g1,1 go, 1
0 .
h
0
1,o
0
0
0
g =
go, I
go, 0
0
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SPACE-GENERAZ, CORPORATION
h Oh - l , O g1,o ho,O g0,o 1,o
SGC 203R-4
Page 18
The term c
f i e l d s and summing.
of th e product i s obta ined by d i re ct ly mul t ip ly ing the0,o
hO,1°O'O 1
L L-
h o , O g0,o + h-l,O g1,o + hO,-l g0,1
The c a l c u l a t i o n of the element c of th e product i s obtained by moving
t h e f i e l d g ( -s) t o t he e lement h , , mul tip ly ing the f i e lds , and summing.
1 7 1
-11
1?1 -hO,l g1,o O g0,o
0.0
h O- 1 9 0
h O 0.00, -1
+ hl,O go,1o,O g l , l + hO,l g1,o
1 * O
-
Fur the r ca l cu la t ions would produce the matrix
fI
L C C0,-1 1,-1
where o n l y the non-zero terms have been indicated.
These ar e convenient formulas f o r both hand computation and
machine computation.
Born the matr ix expression it can be shown that the product of two
matri ces h and g which ar e th e i r own t ranspose (h t = h and gt = g ) , t h a t t h e
product c = hg i s not necessa r i ly a l s o i t s own transpose.
ct = gtht = gh f hg = c
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SPACE-GENERAL CORPORATION SGC 203R-4Page 19
However, when the matricee IT@ dependent o n l y on t he clifferenee between t h e i r
ind ices and t h u s s t a t i o n a r y , t h e p s ~ d u c t i l l be i t s own transpose when the
f a c t o r s a r e t h e i r own t ransposes .
pose h when h ( s ) = h ( - s ) .
A s t a t i o n a r y matrix h = h(s) i s i t s own t r ans -t
Thus, f ro m th e convolution formulas
1 J
where t h e first step is the change of var iab le s ' = -s and t h e second s t e p
i s the use of the s e l f transpose property on the factors h and g.
A simi lar r e s u l t i s th at the product of s ta t ion ary symmetric mt r i c e s
i s also symmetric.
under ref le ct io n of any of th e e lements of the vec to r r epresen t ing the index
d i f fe rence .
t io n ar y m t r i x g = [ g ( s ) ] i s symmetric i n t he f i r s t dimension when [ g ( s ) ] =[,(sf)].
For example
A s ta t i onary mat r ix i s symmetric when it i s i n v a r i an t
That i s , where the indices s = (a,b) and s ' = ( -a ,b) , the s ta-
1
i s symmetric i n the ho riz ont al dimension and not i n the ve rt ic al dimension.
Where both the s ta tio na ry fac to rs of a product have a p a r t i c u l a r
symmetry, th e product al so has t ha t symmetry.
a r e
In par t i c u la r where the ind ices
s = (a ,b) r = ( c , d )
s ' = (-a,b)
the matrix product i s
r' = (-c,d)
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SPACE-GENERAL CORPORATION SGC 203R-J+
Page 20
where the f i r s t s t e p i s a change i n t he o rde r of summation and the second
s t e p i s the use of the symmetry property of the f ac tor s h and g.
I t should be noted th a t fu l l y symmetric stat io na ry matr ices have
th e se l f t ranspose property . Thus, i n the matr ix product , th e f i e l d s can be
mul t ip l i ed d i r e c t l y toge the r wi thou t a ref le c t i on where both of th e fac t ors
are symmetric i n both dimensions.
Construct ion of a Correlated Picture
An interest ing example o f t he product of symmetric st at io na ry
matr ices i s the processing of a zero-mean picture x whose elements are un -
c o r r e la t e d t o o b t ai n t h e p i c t u r e y = xh. The corr ela t io n functi on Cp f o r
t h e p i c t u r e x is
xx
2% = a I
The c or re la t i on funct ion cp of the second pi ct ur e y i s t henYY
L
A specific example of a stationary symmetric matrix h i s
The. matrix nultiplication thexl produces the c o r r e l a t i o n f'unction
2= r J
'pyy
2' W I
This c o r r e l a t i o n i s then both sta ti on ar y and symmetric. The
normal ize cor re la t ion i s p l o t t e d i n Figure 1 f o r severa l va lues o f w.
The m a x i m u m $# occurs for w = 1/2.
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SPACE -GENERAL CORPORATION SGC 203R-4
Page 2 1
Figure 1, Theoret ica l C orre la t ion as a Function of
Distance fo r various Weighting Coef f icien ts of
th e Linear Process y = xh
Experimental ly th e uncorrelated p ict ure of Figure 2 w a s processed
wi th the s t a t iona ry p rocessor
corresponding t o th e weighting co ef f i c ie nt w = 112. This processing produced
the co r r e l a t ed p ic tu re in F igure 3. The smoothness of Figure 3 i s an indica-
t i o n o f t h e c o r r e l a t i o n i n c o n tr as t t o t h e sha rp ne ss of the independent pict ur e
of Figure 2.
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SPACE-GENERAL CORPORATION SGC 203R-4Page 22
Figure 2. An Wncorrefated Picture
Figure 3 . A Correlated Picture Obtained by Processing the Uncorrelated
Picture of Figure 2
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SPA CE -GENERAL CCRPORATION SGC 203B-4Page 23
Desired Pic tu res
Linear processing of the pic ture x produces another pic tur e y which
may have one sample o r many.
d i r ec ted a t obtaining a t h i r d p i c t u r e z , which i s known as t he des i r ed p ic -
t u r e , as i n F igure 4.
Normally, th e processing of th e p ic ture x i s
Processed
P ic tu re y
Desired
P ic tu re z
FXgure 4. Picture Processing Directed a t
Obtaining the Desired Picture z .
Exact const ruct ion of the desi red p ic ture z f rom the p ic ture x i s
normally prohibited by the inherent random differe nces ex is t i ng between x and z.
The e r r o r p i c tu re i n the p rocess ing i s defined as t he d i f f e r ence between t h e
pickures y and z.
e = y-z.
The mean squared er ro r of each sample of t h e e r r o r pic tu re i s t h e
diagonal of t h e au to -cor r e l a tion of the error p i c t u r e
'Pee
Optimum Processing
One approach t o the const ruct ion of a p i c t u r e y as c lo s e t o t h e p i c -
t u r e z i s t o make th e squared e rr o r between each sample of the two pic tures y
and z as small as possib le . I n th e case where t he proce ss i s a l i nea r p rocess ,
th e process can be represented by the matr ix re lat io n
y = x ' h ' f k
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SPACE-GENERAL CORPORATION SGC 203R-4
Page 24
where h and k are optimum matri ces t o be sp ec if ied and x ' i s t h e d i f f er e n c e
between the picture x and t h e mean picture F, x ' =
mean square error
-x - x. Variat ion of the
produces the.:relation
wee = 2 (aht x t t + akt) (x 'h + k - 2 )t
= 2ah ( ' ~ ~ ~ ~ : ~ h -xl,)
-+ 2akt (k - z )
The two relations
are su ff ic ie nt t o minimize each diagonal element i n t h e e r r o r c o r r e l a t i o n matrix
qee.IS . ' . 36: close t o each element of th e des i r ed p ic tu re z as can-be possible us&
l inear processing.
This implies t h a t in the mean square, each element of th e produced pict ure ;y
The processing which produces a minimum mean squa red e r r o r for each
element i s thus
-1Y = x' 5 0 x , x l ( P X l Z + z
This so lu t ion assumes th at the au to-cor re la t ion ( p x l x l has an inverse.
case where the auto-correlat ion (ox
i s zero ind ica t ing tha t it i s not p o si t iv e d e f i n i t e . I n t h i s c ase, t h e r e e x i s t s
a l inear process h which el iminates a se t of samples li n e a rl y dependent upon th e
I n t h e
has no inverse, th e determinant 19x,xl
0
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SPACE -GENERAL CORPORATION
r e s t of the samples. The new s e t of l i ne ar ly independent samples w con ta ins
a l l of the information of t h e o r i g i n a l se t of samp les. The optimum pr oc es s
i s then
-3 . 2
Qwlw' Qw'z= w '
-x' h [ht~ ~ , ~ , h ] - 't(px,z + z
where use has been made of t h e r e l a t i o n
W ' = x 'h
Simple Example
A simple example of l i ne a r p r o ces s in g r e s u l t s from t h e co n s id e r a ti o n
of p i c t u r e s z which have had stat is t ica lly -in de pe nd en t zero-mean no ise added
t o them. The pic tur e t o be processed i s t h en
x = z + n
an d t h e d e s i r ed p i c tu r e is z. The au to -cor re la t ion of t h e p i c t u r e i s t h e sum
o r r e l a t f o n ' funCtiuns of t h e desfred- picture z and the noise n
and the c ros s -cor re la t ion 'p between th e p i c tu re x and the des i red p i c tu r exz
z i s - - -t t t
= x z = z z + n z -'pxz - Qzz
The mean value of t h e p i c t u r e x i s equal t o t h e mean value of t h e d e s i r ed
p i c t u r e z
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SPACE -GENERAL CORPORATION SGC 203R-4Page 26
The optimum process is thus,
-11= x' (1 + vz tz t qn,, + z
where- -
Z' = z-z and x' = x-x,
Fpr small values of noise the process is approximately
-I'pz ' z 'pnn
' = x - X'
This is a useful result in picture processing since it essentially represents
a slight "touching up" of the o r i g i n a l picture x by the picture -x' q z l z lqnn.
Under normal operation of the transmission facilities, noise is usually small and
-1
statistically-independent, zero-mean.
At the other extreme, where the picture is predominantly lost in the
noise, the optimwn processor is
In the case of a one sample picture, the processing should be
-y = x' + z
p11+ 5 1
where
( b z l z l = cP,,l and 'Pnn = ru,,3
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SPACE -GENERAL CORPORATION SGC 20313-4
Page 27
I n t h e c ase of a two sample p ic tu re , t he p rocess i s the fo l lowing
operat ion i n terms of c la ss ic a l matr ix opera t ions .
where
and
Qnn
=E-In t he case where t he experimental pict ure x and desired pi ctu re z
are sta t ion ary , th e processing device using th e optimum h and k i s a l s o a
s ta t ion ary device. Under th e s ta t ionary condi t ion, h i s found from t h e solu-
t i o n of t h e se t of equations
o r
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SPACE-GENEFtAL CORPORATION SGC 203R-4Page 28
Since there is only one element in the stationary matrix k, the ma-
Also, since the matrix h is the product of the matrices
and pxtz re
In like manner the matrix h will be a function of distance when
trix k is symmetric.
and cp1
it will be symmetric when the matrices cpVX'X' x' ' x'x'symmetric.
the matrices Cp and qj are functions of distance. Further simplification
results in the equations for h where the matrices h, Cp
tions of only distance.
thus require only me equation.
x'x' x'zand ' p x l z are funa-
x'x'
A l l the h's at a particular distance are equal and
That is the s e t of equations
needs to be solved for only distinct distances in the index s.
In particular, for the auto correlation matrix
and crosscorrelation matrix
- [Y "4 Y]
9x1 z
0
the equations for the linear processor
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SPACE-GENERAL CORPORATION
= L u
'poho + +lhl 0a re
Solut ion of these equations produce the coeff icients
SGC 203R-4
Page 29
General iza t ion of t h i s so l u t io n i n d i ca t e s t h a t a l l c o e f f i c i e n t s a t
th e same di st an ce a re equa l. The number of simultaneous equa tions i n t h e solu-
t i o n f o r t h e c o e f f i c i e n t s i s eq ua l t o t h e nuTIlber of d i s t i n c t d i s ta n c es .
i s a p a r t i c u l a r l y p r a c t i c a l r e s u l t i n t h a t it i nd ica tes tha t the f i r s t s t e p
i n t he l i n e a r p ro c es s in g t h e p i c t u r e x i s t o f i r s t add a l l the samples a t a
pa r t i cu lar d is tan ce . This produces a computational reduction of about four
for rectan gular gr id s and up t o twelve f o r hexagonal gr id s.
"his
The hexagonal
mtrix of Figure s a computation red uctio n of 7 / 2 .
h =
F'IGURE 5 . A Linear Processor Based on a Hexagonal Matrix
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SPACE-GENERAL CORPORATION SGC 203R-4Page 30
The coeff ic ients f o r the hexagonal l i ne ar process a re
$('Po + *q + +pj + 4 - 6ol(Pl
Po((oo+
2=
0
+ 2 V ) + [PJ - 601
1 w p 0 - ao'pl2
i =
' p , ( (oo + *l + 2 ( o n + (o& -
for- very sample in the l i n e a r
process ing
y = x ' h + Y
where
-1h = (PXfX' ( P X f Z
The use of any other process
y = x ' (h + hs) + (E + ks)
produces the au to - co r r e l a t i o n of the error picture
t t(pee = e e = [ x ' ( h + hs) + 6 kg)-z] [ x ' ( h + hs) + ( y + kg)-z]
where the diagonal e nt r i es ar e the squared sample er ro rs .
w i th t h e ex p r e s s io n q x , x , h =qxrZ h e e r r o r i s
After some reduction
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SPACE-GENERAL CORPORATION SGC 203R-4Page 31
t ) ('z + ks) - ('it + kxt) - - -t ('z + ks)- + ('zt + k6Qee - 9 z z
Due t o the pos i t ive d e f i n i t e cond it ion , t he d iagona l en t r i e s of the term
h and k k are always positive except where h = 0 and k6 = 0.PX'X' 6 6 6 6
Thus, for a l i ne ar independent pic ture x', the mean error i s uniquely ob-
tained by the optimum process
y = x 'h + 'z
where
-1 -h = p X l x 1X lz nd x 1 = x-x.
The minimum mean error is the diagonal of the expression
t'pee - P z I z l - h'P,',lh
-1= c p 2'2' - Q& Q X l X l (bxlz '
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SPACE-GENERAL CORp(KIATI0N
I n t h e c as e o f independent ad di tiv e no ise , th e minimum e r r o r is
ee
SGC 203R-4Page 32
h- Qnn
The minimum e r r o r i n the one sample picture of a previous s e c t i o n is
I n th e two sample pi ct ur es , th e minimum e rr o r i s
2-- p11( al lu22- 5 2 + all (P11P22 - P;2)2-
1min + u )2(p11 + 011) ( P 2 2 + 5 2 ) - ( P 1 2
12
I n t h e case of s t a t i o n a r y co r r e l a t i o n f u n c ti o n s i n r o t a t i o n , t h e minimum error
for the two coef f ic ien t r ec tangu la r grid i s
- - w h - h l h lmin Pee - s o z l z l 0 0
and for the two coeff ic ient hexagonal g r i d ,
min ‘r , - - W h - h l h lr e e - ( P z ’ z ’ 0 0
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SPACE-GENERAL COFPORATIm
Tr iv ia l P rocess ing
The uniqueness of t he sol uti on t o th e optimum proce ssing can be used
t o id ent i fy the type of p ic tures which requi re no l ine ar processing o ther than
possib ly a change i n sca l ing.
i n s c a l i ng the o r i g i n a l p i c t u r e i s
The l inear process which i s r e s t r i c t e d t o a change
y = x 'd + zwhere d i s a diagonal matrix of constants which amplifies each sample independ-
en t ly . I n th e case where t he matrixd
represents the optimum linear process,-1
d =Px'x ' sox'z'
Since d i s th e unique solut ion, the only pictu res fo r which a change i n sca l in g
i s th e optimum process ar e where the re ex is ts a diagon al matrix d such t h a t
Tais m3ans t h a t a necessary and suff icient condit ion i s t h a t each column of
i s some multiple of the corresponding column of (0 .Vx'x' x'
[ i t h column of cp ] d . = [ith column of cp ] every ix 'x ' 1 x ' z
The mean error i n t h i s case is
-- a :ktz
min = ~ z ~ z '
A s an example, th e processing of a p ic ture z which has added t o it corre la ted
noise having a corre la t ion funct ion 9 equal t o a mul tip le k of t he co r r e l a -nn
t i o n f u nc t io n qzlzl f t he p ic tu re z would be processed by a simple change i n
s c al in g . I n t h i s c as e,
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SPACE-GENERAL CORPORATION
so th a t t h e optimum process i s
t h e minimum er r o r i s t h en
. *Orthogonal Pre-processing
SGC 203R-4Page 34
Another approach t o l i ne ar p rocess ing res u l t s from th e cons t ruc t ion
of an orthogonal maxtrix Q column-wise composed of th e normalized eige nve ctor s '
o f the c or re la t io n mat r ix 'p I n p a rt i cu l ar,x'x ' -
~ ~ f ~ i= QX
where
qi.
i s the d iagona l mat r ix of eigenvalues Xi corresponding t o th e e igenvec to r s
The normal or thogona l p roper ty of t he e igenvec to r s ind ica t es th a t
---Linear pree-processing ~f t h e p i c t u r e x w i th t h e matrix Q p r d u c e s t h e
p i c t u r e x = XQ which has the auto-co rrela t ion fun ct io n9
I
= x u~ = Q'x"~ Q = Q pXxQ ='px x 9 9
9 9
and a c r o s s co r r e l a t i o n f u n c t i o n
t t t t' p x z = x z = Q x z = Q c p .
9 XZ9
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SPACE-GENERAL C O F P O ~ I O N
Subsequent optimum li n e a r pro ces sing of the new picture x r eq u i r e s t h a tq
the mean e r r o r i s then
The use of or thogonal proeess ing appears t o be of use i n pre-
p o c e s s i n g p i c t u r es t o o b t ai n t t u r e x which i s compose
having zero c ros scor re la t ion and au to -cor re la t ion . Present
mental use of t h i s type of pre-processing i s l im i ted b y t h e n eces s i t y t o ob-
t a i n t h e matrix of eigenvectors &.
9'
i
A Stationary &le
A pa r t ic ul ar ly usef ul experimental example of s t a t i o n a r y l i n e a r p ro -
ces s ing i s the recovery of a co r r el a t ed p i c tu r e from a composite picture of
t h e c o r re l a t ed p i c t u r e added t o an u n co rr el at ed p i c tu r e .
could be Mne mu1 f of iri-dee transmission of a cam e la t ed pic tu re being cor-
The composite p i c tu re
ise,haVing a frequency bandwidth corresponding te h e
sampling ra t e of th e plcC e . Another example fs the recovery of t h e uncor-
re la ted picture f rom the composi te p ic ture of the sum of t h e co r r e l a t ed p i c tu r e
and uncorrela ted picture .
p ic t u re be ing corrup ted by a f luc tua t ing exposure lev e l r es u l t ing i n a corre-
la te d b ia s be ing added t o the p i c tu re .
This second si tu a ti o n corresponds t o a high-grade
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SPACE-GENE= CORPORATION SG C 203R-4
.Page 36
The correl a ted pic ture t o be used i n t h i s example i s t h e a r t i f i c i a l l y
constructed one of a pre vio us example. The pi ct ur e had zero-mean and t h e s ta-
t i o n a r y co r r e l a t i o n f u n c ti o n
where w =
114
1 112
2 1 1/4
/2 was used i n order t o produce th e h,ghest va,Je of normalized cqr-
r e l a t i o n a t a dis tance o f 1 sample.
t h e co r r e l a t ed p i c tu r e also had zero-mean and a s t a t i o n a r y co r r e l a t i o n f u n c t i o n
composed of o nly one non-zero elem ent.
The uncor rela ted p ic t u re t o be added t o
I
Since these two pictures a re independent of each other, the composite picture
made from t h e i r sum has a co r r e l a t i o n f u n c t i o n eq ua l t o th e sum of the cor re -
l a t i o n f un ct io n s (ol and 'p2.-
1/4
112
LThe composite picture was t h en used t o r ecov er e i t h e r t h e co r r e l a t ed p i c tu r e
o r th e u n cor r el a ted p i c tu r e .
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SPACE-GENEEAL CORPORATION
Since th e pict ur es ar e zero-mean and independent, th e cr oss corr ela t ion
between t h e composite pi ct ur e x and the c or re la ted p ic ture z i s cp , h e1 1
1x1
cor r e l a t ion func t ion of t he co r r e l a t ed p ic tu re . A s imilar r e l a t ionsh ip ho lds
f o r the independent picture z2'
-'pxz - 'p,
1
The matrix equat ions t o be solved are then
Since the correl&.im fuwt%ons+x,xT a& pi &re stat;iowry and s-tric, th?
so lu t ion h t o the equa tions i s also stationary and symmetric.
h =
h2
hl
h2 hl hO
hl
-
hJ2
hJ2
- h2
hJ2
hl
hJ2
By th e direct app l t ca t ion of-the method f o r matrix multiplication the follaTing
s e t of equations i s obtained:
2( 2 + a ) h o + 4 hl + 2 h0 + h2 = 2
2a
h = 1 0or
' h +P
1/2 ho +1/4 h +
+ (13/4+ a ) hl +0
h2 = 112 0
hl + (5/2 + a2 ) hJ2 +
0+ (2 + a ) h2 = 1/4hl+
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SPACE-GENERAL CORPORATION SGC 203R-4
Page 38
Only one equation i s needed f o r each of the elements of the matrix h .
The othe r equa tions a re i de n ti c a l t o these and thus produce no new equatio ns.
S ince the so l u t io n of th i s s e t of equat ions i s t he unique so lu t ion t o the op t i -
mizat ion of the l i ne ar processor , e lements of h ar e zero a t gre ater d is tan ce
than the extent of the cor r e la t ion funct ions cp
and des i r ed p ic tu re s . I n th i s case
and c p x , z of th e experimentalx'x '
h = Od
d > 2
Figure 8 i s the composi te p ic ture represent ing the add i t io n of th e
cor re la ted p ic t ure , F igure 6, and th e uncorrelated p ic tur e, Figure 7.
t i o n was performed with a
the uncorrela ted p ic ture had a cp
The correl ate d pi ct ur e i s thus th e dominant pi ctu re i n th e composite pict ure .
The addi-
2 2= 1 so tha t t h e co r r e l a t ed p ic tu re had a qoo= 20 ,
2 2= U , and the composite had a qoo = 30 .
00
The recovered corre la ted p ic ture i s F'igure 9 and the recovered uncor-
r e l a t e d p i c t u r e i s Figure 10.i s recovered wi th bet te r re la t i ve ' precis i on than the uncorre la ted p ic tu re .
i s i n t e r e s t i n g t u note U I E O P appearance of the recovered uncorrefated
picture from the highlr corre'iated composite picture.
As t o b e expected th e dominant co rrel ate d pic tu re
It- .
The minimum recovery e r r o r for l i n ea r p rocessing of a dd i t ive p ic tu res
i s the d iagonal of the co r re l a t io n funct ion of the e r r or p ic tur e .
where h ( 2 ) i s th e optimum proce ssor used t o recove r th e second pi ct ur e and h(1
i s th e optimum proc esso r used t o recover th e f i r s t p i c t u r e . The recovery error
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SPACE-GENERAL CORPORATION
Figure 6 . Des i red Cor re la tedP ic tu re z Figure 7. Desired Uncorrelated Picture1
o s i t e P i c t u re x of the Sum of the Cor re la ted P ic tu re z1 md
Uncorrelatelt 2
Figure 9 . Recovered Corr ela ted Pictu re y from the Composite Picture1
Figure 10 . Recovered Uncorrelated Picture y from the Composite Pic2
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SPACE-GENERAL CORPORATION SGC 203R-4
Page 40
i s t h e same fo r ei th er pi ct ure recovered. The correl ated and uncorrelated
pictures were both recovered with a t h e o r e t i c a l e r r o r of .52a . However, t h e
desi red cor re la ted p ic ture had a variance of 20 while th e desir ed uncorre-
2
2
2la t ed p ic tu re had a variance of CY ,of t he co r r e l a t ed p ic tu re .
Behavior of the Linear Processor
A genera l so lu t ion of the
processor can be worked out through
which indicates a b e t t e r r e l a t i v e r e c o v e r y
s e t of l i nea r equa t tons of t h e optimum
some fo r tuna te cance l l a t ions and f ac to r i -
za t ions . For the recovery of t h e co r r e la t ed p ic tu re , t he h ' s are
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SPACE-GENERAL CORPORATION SGC 203P-4
Page 4 1
The h ' s f o r the recovery of the uncor rela ted p ic tu re are c l o s e l y r e l a t e d .
a re
They
a2 - 3/4
a* + 1
2
hl+
l - t aL2) 31 2
A s t o be expec ted th e sum of t h e t w o recovered pictures y
t h e sum x of t h e co r r e l a t ed p i c tu r e z
and y2 i s eq u al t o1
and t h e u n co r re l a ted p i c t u r e z2.1
= xIh(') + x ' h (2 1y1 + y2
= z1 + z*
where 9 X ' X ' = 'pl + 'p2
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.
SPACE-GENERAL C ~ P O R A T I O N SGC 203B-4
Page 42
The behavior of t h e h ' s used t o recov er t h e co r r e l a t ed p i c tu r e i s
p lo t t ed i n Figu re 11.
cessor r e l i e s pr inc ipa l ly on the p ictu re e lement being processed t o produce
th e processed element. (ho 1, hl M hR h2 0) . A t higher l ev e l s o f
cor rupt ion the processor r e l i e s more on th e elements near th e element being
processed ra the r than on the e lement i t s e l f .
Where the additive corruption i s neg l ig ib le , the pro-
The behavior of the h ' s i n the r ecovery of th e uncor re la ted p ic tu re
i s plo t ted i n Figure 12 . Tt appears tha-l; a t a l l l e v e l s of corrupt ion, the
processor makes use of th e elements clust ere d around the element bei ng processed.-2
For recovery of e i t h e r p i c tu r e t h e t h e o r e t i c a l r e cov e ry e r r o r e i s
computed t o be
Figure 13. s a p l o t of t h i s e r r o r , a2h (I), normalized by the var iance of the
recovered correlated picture. If h i s optimum, t h e - v a r i a n c e of th e recovered
picture can be c alculat ed by the formula
0
2 2The var iance of the uncorrela ted pictu re , a u , normalized by the variance of
the composi te p ic ture (2 + a ) cr
of the amount of e r r o r presen t i n the composite picture x from which the pro-
cessed p ic tu re y i s obtained.
2 2i s a l s o p l o t te d i n F ig ur e 13 as an i n d i c a t i o n
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SPACE-GENERAT, CORPORATION SGC 203R-4Page 43
Figure 11. Coef f ic ien t s o f t h e O p t i m u m Linear Processor for the
Recovery of the Cor re la ted P ic tu re
. . .. . .. . -- ,
. .
. . .. *>.._
Figure 12. Coef f ic ien t s of t h e O p t i m u m Linear Processor f o r the
Recovery of the Uncorrela ted Picture
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SPACE-GEXEEUL CORPORATION SGC 203R-4
Page 44
FIGURE 13. Normalized Recovery Error
Processing t h e Crater Archimedes- . 5 .-
Eztheer .experimental
c ra t e r Archimedes, Figure 1 4 .
and addit ive noise of half the
mrk was performed on a sampled p i c t u r e of the
Figure 13 i s a composite p i c t u r e of t h e c r a t e r
variance of the o r i gi na l pi ct ure . Optimum li n-
ear processing then produced Figure 16, which appears t o be a decided improve-
ment over th e corrupted pi ctu re of th e c ra te r .
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SPACE-GENERAL CORPORATION SGC 203R-4Page 45
Figure 1 4 . The C ra te r Archimedes i n 287 Samples of Six Shades of Brightness_ 1
.. .I . . I. .
Figure 15. Corruption of the P ic tu re of Archimedes with Additive Independent
Noise Having a Variance of Half th e Variance of the Or ig ina l
P ic tu re
Figure 1 6 . Recovery of the Picture of the Crater Archimedes with an Optimum
Linear Processor
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SPACE-GENERAL CORPORATION SGC 203R-4Page 46
The pictu re of th e cr at er had th e normalized c orr ela t io n funct ion
.08
.32 .34 .32
.08 .54 1 .54 .
.32 .54 .32
.08-
which i s plo t ted i n Figure 17 as a func t ion of d i s t an ce .- _
Figure 17. Correl at ion Funct ion of t h e Crater Archimedes
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SPACEI-GENERELL CORPORATION SG C 203R-4Page 47
Addition of an independent noise picture with a var iance of half that ,
of th e or ig in al p ic t .ure produces th e fol lowing se t of equations for t h e d e t e r -
mination of the optimum processor.
1.50 h + 2.16 hl + 1.28 h
0.54 h + 2.22 hl + 1.08 h
0.32 ho + 1-08 l + 1.66 h
0.08 h + 0.54 hl + 0.64 h
+ 0-32 h2 = 1.00
+ 0.54 h2 = 0.54
+ 0064h = O"32
+ le30 2 = 0.08
0 J2
0 J2
J2 2
0 J2
which has- i o n
ho = .47
hl = .12
h = .O4J2
% = -.03
The la rg e value of ho indicates a ra t her poor suppress ion of t h e e r r o r .
the exper imenta l r esu l t s ind ica te a s a t i s f ac to r y r e t en t i o n of p ic tu r e f ea tu r e s .
However,
Perhaps, then, mean-square error i s not a good indicator of damage done t o a_ _
p i c t w noise .
Suppression of E r r o r s
The exper iment with a r t i f i c i a l p ic t ures produced very l i t t l e suppres-
s i o n of independent additive errors .
t h a t t h e n a e r o f pr oces sed samples was q u i t e s m a l l and the pic tur e was only
cor re la ted over a small d i s t an ce .
This i s t o be expected when it i s considered
An upper bound can be o btain ed f o r t h e amount of suppression of addi-
t i ve independent noise by assuming a high ly cor re la ted des i red p ic tu re .
co r r e l a t i o n f u n ct i o n of the des i red p ic tu re z i s assumed t o be o2 times t h e u ni t
m a tr i x of a l l 1's. The co rre la t ion funct ion of th e independent noise i s assumed
The
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SPACE-GENERAL CORPORATION SGC 203R-4Page 48
2 2to be u a times the identity matrix of a 1 on the diagonal. A l l the equations
for the coefficients of the processing matrix h are identical so that all the
coefficients are equal. The equation for the single coefficient is then
( n + a ) h o = l2or
1
n + a=
0
where there is a total of n coefficients in the matrix h. The variance of the
error in the optimum recovery is then
which should be normalized by the variance of the recovered picture
n 20
2n + a
Thus
c
2a
n
2- - -
'PYY
2 2In contrast, the variance of the error a u in the original picture
2 2normalized by the variance (l+a ) u of the picture is
2'pnn a
'pxx l + a
- = -2
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SPACE-GENERAL CORPORATION SGC 203R-4
Page 49
Tnese two rel at io ns a re plo t te d i n Figure 18 for a processor operat ing on five
samples.
i s obtained for s m a l l r e l a t ive l eve l s o f no i se. As the noise becomes l a rge r
th e l i n e a r proc es si ng bre ak s down and produces no improvement i n t h e r e l a t i v e
qua l i ty o f the p ic tu re .
A suppression of the independent noise by a f a c t o r of 5 in variance
10 T e d Limits to the S m e s s f o n of AdditiveVrrtse b r a L i m s p Pracessor Operating on 5 Picture Samples
The upper bound on the suppression of independent noise indicates
t h a t th e suppression i n magnitude of e rr or i s i nver se ly p ropwt iona l t o t h e
square root of the number of samples. Thus, i n o r de r t o o b ta in a suppression
i n magnitude by a f a c t o r of 1 0 the processor requires a t l e a s t , 1 0 0 samples t o
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SPACE-GENERAL CORPORATION SGC 203R-4
Page 50
recons t ruct one element i n the pi ctu re .
se n t ly unknown, as t h e bound given above i s computed under the assumption of
a p e r f ec t l y co r r e l a t ed p i c tu r e .
compute the error for a more real is t ic type of c o r r e l a t i o n i s unknown.
computation will r equ i re a t r i c k i n t he invers ion of the cor r e la t ion mat r ix.
The ac t u a l number of samples i s pre-
Whether or not it i s p os si bl e t o t h e o r e t i c a ll y
The
Minimization with a Constraint
I n t he optimum processing o f p i c t u r e s it i s o r t e n d e s ir a b l e t o s c a l e
t h e r e s u l ta n t p i c t u r e s t o a p a r t i c u l a r c o n t r a s t by having th e standard d ev i a t i o n
_ e q u a l t o t he s t a d a r d d e vi a ti o n of t h e d es i r ed p i e t u r e .
square error between the r esu l tan t p ic t u re and the 3 e w
s i r ed p i c tu r e s u b j ec t t o t h e co n s t ra in t t h a t t h e r e s u l t an t p i c tu r e h ave t h e same
s tandard deviat ion as the des i red p ic tu re r esu1 t. s i n a l i ne ar s ca l ing of th e
pi ct ur e produced under an unconstrained optimization.
The s tandard deviat ion i s given by the d iagonal terms of t h e cor re la -
t i o n f u nc t io n of t he zero-mean var ia bl es
d i a g . <p = cons tanty ~ y t = d i a g * Qz ' z '
. _
The minimizat2cm is then of the diagonal element-s of t h e matrix
u = < pee + h ( P y ' y '
where A i s an undetermined con sta nt and (b i s t h e e r r o r co r r e l a t i o n m a t ri xee
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SPACE- GE3lERAL CORPORATION SG C 203R-4
Page 51
The assumption that the output y i s obtained by a l i ne ar process on th e input
x r e q u i r e s t h a t
y = x ' h + k y ' = x ' h
-where x' = x-x and y ' = y-y.
Variat ion of the matrix U produces th e re su l t
+ 2 akt[k - l]
A s u f f i c i e n t condi t ion for zero va r ia t io n of t he d iagonal e lements of th e matr ix
U i s t h a t
and k = z
This i s th e same sol u t io n th a t w a s obtained fo r the unconstrained optimization
except for a sca l ing of th e output p ic t ure s y , producing the desi red s tandard
dev ia t ion .
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SPACE- GENEFL4L CORPORATION SG C 203R-[I
Page 52
Pre-Processing wi th Pred icti on
One method of transmitting a p ic tu r e w i th a sequence of independent
samples, i s t o s equen t ia l ly s can the p ic tu re , p red ic t the next e lement t o be
scanned, measure th e pred ict ion e rro r, and tran smit th e e rr or from which t h e
p ic tu r e can be reconstructed with a s imi la r p r ed i c t i o n .
Theoret ical ly th is method w i l l precisely produce a r ep l i c a o f t h e
or i g in a l p ic tu re . I n those cases where i t i s p o s s ib l e t o d o h igh q u a l i t y p re -
diction, consider able reduc%ion can be obtained i n the t o t a l amount of data
which must be t r ansmi t ted .
Quan t iza t ion of the t ransmit ted sequence of er ro r s in t roduces a guan-
t i z a t i o n d r i f t i n to t h e r econ s t ru c t io n p r ocess . T hi s d r i f t p r ev ent s t h e s ys tem
from running f o r any reasonable l ength of t i m e without having t o be r e s e t .
This sec t ion descr ibes a method of p red ict ion and rec ons tru cti on f r o m
quan tized e r ro r s i n such a manner as t o avoid t he quant iza t ion d r i f t i n t he re-
cons truct i on of th e d a t a .
The picture i s t o be recons tructed as i n F ig ur e 19 where the next
e lement i n the scan i s predicted from th e previous ly recons tructed pic tur e and
modif ied by ' a d i n g the e r r o r t o t h i s p r e d i c ti o n .
Quanti ed
E r ror.A Recovered
Addition 1 P ic tu r e
Figure 19. The Reconstruction of a Picture from Quantized
Pred ic tion E r r o r
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SPACE-GENERAL CORPORATION
The p red ic t ion e r r o r i s obtained as i n F ig ur e 20 by quan t iz ing the
er ro r between the ac tu a l p ic tu r e and t he p red icted p ic tu r e based upon a recon-
s t r uc t ion from th e quan t ized e r r o r .
manner a s it i s recons tructed i n Figure 19. The predict ion of t h e p i c t u r e i s
then based upon th e recons tructed pic ture ra th er than t h e pict ure which i s t o
be t r an s m i t t ed .
from t h e a c t u a l p i c t u r e .
The picture i s f i r s t r ecovered i n the same
The error i s obtained by subtract ing t he predicted p ic tu re
Transmission
Pre dic ted Element Recovered P i c tu re
Egure 20, Processing a P i c t u r e t o Obtain a Quantized
Pred ic t ion Error
The quant ized e r r o r s a re theo re t i ca l l y a sequence of l in ea r l y inde-
pendent samples.
i n g a sequence of independent samples i n t o a bina ry Huffman Code'' having a
maximum entropy pe r d i g i t .
na ry b i t s which must be t r ansmi t ted i n t r an smi t t ing a pic tu r e . Fur ther use of
var iou s block coding techniqu es as may be found i n Reference 22 w i l l produce
th e n ecess ar y r e l i a b i l i t y i n t h e t ran sm i ss io n of t h e b in a ry d a t a t o i n su r e
r e l i ab l e r eco n s t ru c t i o n of t h e p i c tu r e .
A l a t e r s e c t i o n p r e se n t s a F O R M s ub ro ut in e f o r t rans fo rm -
This code e ss e n ti al l y minimizes th e amount of bi -
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SPACE-GmERAL CORPORATION SGC 2 0 3 ~ - b
Page 54
Pred ic t ion
The theory of optimum linear processing provides a means of cons truc-
A preceding Section showed th a t th e optimum l i n e a r processori n g a p r e d i c t o r .
for p r ocess ing t h e p i c tu r e x t o ob ta in the p red ic t ion of the element z w a s
y = x ' h + k
where-
x ' = x - x
k = Z-
and
A preceding example used the s t a t io na ry cor rel a t i on matr ix
2( p x l x l = CY
The cross cor re la t i on mat r ix between th e p rev ious ly scanned p o r t i o n of t h e p i c -
tu re and the next sample i s then
2= C Y'px'z
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SPACE-GENERAL C OFiPORATI ON
where X marks the center of th e matrix. The matri x product (6 h i s obtained .x x'
from the formula
= L: ( p x l x , ( ' ) h ( r + dr
where th e pr ed ict ing processor has th e matrix
5h
-
-
h6
3 h2 h4
hl
h
X
where again X marks t h e c e n t e r of the matr ix.
t o t h e mosscorre la t ion produces t h e following se t of equations for t h e d e t e r-
mination of' the c o e f f i c i e n t s i n the matr ix of t he processor h.
Equating th e m atr i x product
+ l /2h2
+ 2 h2
h2
h2
+
h2
5h
3
3
3
+ h
+ h +
+ 2 h + 1/4h4 + 1/2h5
+ 1/4hj + 2 h4
h4
31/2h
+ l / 2 h j + 1/2h4
52 h
= 1
= 1
= 112
= 112
= 1 1 4
= 114
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SPACE- GEXI3RAT; CORPORATION SG C 2O3R-4Page 56
Solution of t h i s s e t of equations would then produce the c oe ff i cie nt s f o r th e
pred ic to r .
An approximate solution can ea si ly be obtained by assuming th at
h = 0 s o t h a t h = h and h = h6. The th i r d order s e t of equat ions resul t s4 1 2 5
i n t h e s o lu t io n
h =
-112
0-:;: :" ]The power i n the predic t ion e r r or i s obtained by the formula
, Quantization of P ictu res
Experimentally a p ic tu re i s sampled on a regular gird. It i s custo-
ma r y -Za i& q z d 3 . Q ~ pictures t o use anywhere from 32 t o 128 br igh tness l eve l s
i n the ~ W z a $ 2 a n : o f-%he samples. Due t o t h e l i m i t a t i o n i n experimental equip-
t u r e s presented i n t h i s report are quantized i n 6 brightness l e v e l s
(black, 2&, @, 6&, 8&, and w h i t e ) .
from the formula
The shade of grey g ca n be computed
= - logJ2
where b i s th e br ightn ess le ve l . The grey sca le and brightn ess lev el s have th e
r e l a t i o n
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SPACE-GENERAL CORPORATION SGC 203R-4Page 57
A geometr ical pat te rn i s formed by th e re l a t ive b r igh tness of the
elements of th e p ic tu re ra t he r than by th e precise br ightness l ev el s . The tone
of the p ic tu re i s t h e mean va lue o f the br igh tnes s leve ls . The standard devia-
t i o n i s a measure of the co nt ra st . I n numerous geo metr ical cases , it i s d e s i r -
ab le t o r ep resen t a s e t of pic tur es with a uniform mean and stan dard dev iat ion
of br ightness levels .
The processing of pic tures with l i ne ar and non-l inear processors pro-
duces a s e t of numbers which normally extend beyond t h e 0-1 ange of the br igh t-
ness 3eve l s . Se ve r a l methods are avai lable f o r quant iz ing a set of numbers s o
t h a t t h e y can be represented i n a f i n i t e number of br ightness le ve ls .
method senting a pic tur e wi th quantized br ightness IS
i s t o d i s t r i b u t e t h e q u an t iz a ti on l e v e l s s o t h a t a l l l e v e l s a r e r ep r ese nt e d w i th
equal frequency of occurrence.
Another method of represent ing a p ic ture i s t o l i n e a r l y d is t r ib u t e t he
This i s the method normallyuant iza t ion s teps over the range of sample values.
used i n qua ntizing the samples of an experimen tal pi ct ur e. However, pict ur es
which are the re su l t of numerical processing, ten d t o have a few excess ively
smal l and la rg e values which produce an excessive range. Line arly dis tr ib ut in g
this excessive range produces a p i c t u r e w i th m o s t
an- tep near *.The quan t i za tion of the p ic tu re s i n th i s r epor t i s based on a l inear
m s e pictures have very little con
system which d is t r ib ut es the four cent er shades of br ightness (2&, 40$, 60$,
8&) between plus and- minus one sta nda rd dev ia ti on i n br ig ht ne ss about th e mean
value of the p ic ture . A l l values below and above one st and ard dev ia ti on a re
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SPACE-GENERAL CORPORATION SGC 2O3R-4Page 58
made r es pe ct ive ly blac k and white.
as a pplied t o uniform and Gaussian dis tr ib ut io ns .
t h e mean frequen cy of occ urrence o f br igh tness l eve l s i s
Figure 2 1 depic t s t h i s type of quant izat ion
For these two dis t r ibut ions
Brightness Level Uniform Di st rib uti on Gauss an Dis t r ibu t ion
Black .212 * 159
20% .144 .150
.144
.I44
.I44
.191
.191
,191
mite .212 159
The frequency of occurrence of each of the s i x le ve ls i s very c lose t o
116 = .167.
The corr ela ted p ic tur e of Figure 3 had t h e d i s t r i b u t i o n of br ightness
Linear quantiza t ion between the standard devia t ionsevels shown i n Figure 22.
produces the resul t s of Figure 3 which clear ly show the correlat ion of the
p i c t u r e .
The principal disadvantage of t h i s method of repres enta t io n i s i n t he
emparison of different pictures having d i f f e r e n t standard deviations. Pic t t t res
are p&sented as though they had the stme standard deviations. The &GP€f;dbn-of
t he p ic tu res i n Figure 6 nd 7 t o g ive tha t of Figure 8 i s an example.
6, 7, and 8 have r e l a t iv e s tandard devia t ions of fi 1 a n d 6 Their standard
deviat ions are normalized s o t h a t a relat ive comparison of br ightness levels can-
not be d ir ec t l y made.
t i n g geomet ri ca l pa t t e rns wi th a high degree of c o n t r a s t i n a f a i r l y l i n e a r
manner .
Figure
The method, however, does prov ide a simp le way of re pr esen
Figures 23 and 24 are examples of t h i s typ e of quant izat io n i n th e rep
rese nta t ion of the cra te r Eratosthenes. The white area i n t he cen te r i s an il-
luminated inner w a l l which c as ts t he dark shadow extending t o th e edge of th e
Figure . The measured c or re la ti on funct ion i s Figure 25.
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SPACE-GENERAL CORPORATION SGC 203R-4Page 59
Figure 22. Dis t r ib in ion 0; R c d Ijmber’s o€ Llie Corr ela te d Pi ct ur e of Figure 3Along ,,i2Lll LP-C . ~ . i , l z a t i o nUsed i i i i t s Representat ion
F igure 21. Q i’ a Picture lnto S i x Brig ness Levels in R e la t ion
si311 acd UniL’orm D i s t r i b u t i o n
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SPACE-GENERAL CORPORATI N SGC 203R-4Page 60
Figure 2 3 . The Crate r Erato stene s i n 6 Brightness Levels
Figure 2 4 . The Crate r Eratostenes i n 6 Brightness Levels Distr ibuted
About t he Mean Plus and Minus One Stand ard De vi at ion
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SPACE-GENERAL CORPORATION SGC 203R-4Page 61
Figure 25 , Stat ion ary Correlat ion Function of th e Crater Eratos tenes
Machine Programming
The experimental work of t h i s r ep or t w a s performed b y hand computa-
The amDunt of computation for a very s imple picture i s a t t h e l i m i t ofi o n .
computation.
of t h i s t h e o r e t i c a l work has been t o cast it
i n a no ta ti on which can be e a s i l y programmed and run on a large automatic com-
puting machine.
pu t ing a l i ne ar processor fo r suppress ing undes ired e r r or s .
f low char t of a proposed experiment de al in g wit h th e removal of redundancy fromI
a p ic tu r e p r i o r t o t ran sm i ss ion .
Figure 26 represe nts th e f low cha rt s of t he experiment i n com-
Figure 27 i s th e
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SPACE-GENERAL CORPOFUTION
Compute ;1 Read Out i nomputeTerms o f
DistanceF Z l + R Z 2 4!#Y +4b,
*
SGC 2OjR-4Page 62
Generate Quantize and
Pic?.ures x 2 q a n a z2
R s d Out
e
Experiment a l l y
Qt* + k aCornpute
Linear Quantize andProcesswith h
to gat 9
*Read Out i n
M stanceTerm8 O f
Figure 2 6 . Flow Chart of an Experiment of the Suppression of
Undesirable Noise
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SPACE-GENERAL CORPORATION
Generat e
RandomPicture I
.
SGC 203R-4Page 63
‘r .Linear Compute
PI’OC088 4
with h E 2t o get x L
Figure 27. Flow Chart of an Experiment i n Removing th e Redundancyfrom a P ic tu r e
-.Quantize and
i Read Out
X
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SPACE-GENERAL CORPORATION SGC 203R-4
Page 64
FORT" Subroutines
The fo l lowing subrou tines have been wr i t t en t o f ac i l i t a te the exper i-
mental work i n p i c tu r e p r ocess in g .
ence 2 1 f o r most IBM 70 9 and 7090 compilers i s used.
The FORTR N convention contained i n Refer-
Crosscorrela t ion of Two P ic tu r e s
The cros sco rre lat io n of th e p i c t u r e X and Y i s p a r t i c u l a r l y u s e f u l
i n the des ign of optimum processors.
are assumed ta be of t h e same dimension as i n F ig ur e 28.
I n t h e fol lowing program these pictures
X ( 1 , J ) and Y ( 1 , J )
I = l , M J = l,Nt
Figure 28 , The Number of Elements i n a P ic tu r e
The cor r e la t ion t o be ca lcu la ted exper imenta lly has t h e dimensions of Figure 29.
The dimensions should be odd numbers due t o t h e si ng le v alu e of th e co r r e l a t i o n
a t t h e o r i g i n , (MO, NO) = (M C + 1 N C + L,
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SPACE-GENERAL CORPORATION SGC 203R-4
Page 65
C ( K , L )K = 1 , M C L = 1 , N C
Figure 29, The Number o f Elements i n th e C or re la tio n Function
The crosscor re la t ion i s computed by the formula
M N
MNL, / >( K , L ) = - '-
X(I,J)-XAVG) :: Y (I + K - MO,J + L - NO)-YAVG)
1=1 J=1
I n those cases where t he indice s o f Y ar e beyond t he range of the pic tu re, th e
average value of Y i s used r e su l t ing i n a zero t e r m being added in to th e
summation.
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SPACE-GENERAL CORPORATION SGC 2 0 3 R -4
Page 66
SUB ROU TIN E CRCOR (C, MC, NC, X, XAVG, Y, YAVG, M, N )
D I M E E X O N X 46,461 , Y(416,4#), C (46,416)
DO 1 I = l , M
DO 1 J = l , N
X(I , J ) = X(I , J ) - XAVG
Y ( 1 , J ) = Y ( 1 , J ) - YAVG
T = M s N
MO = (MC+1)/2
m-= m+1.)/2
DO 4 K = 1,MC
DO 4 L = 1 , NC
C (K, L ) = 6 .6DO 3 I = l,M
DO 3 J = l , N
I Y = I+K-MO
JY = J+L-NOI F (W 3,3,2
IF (Jy) 3 , 3 , 2
2 ,2 , 3
I F (JY-N) 2,2,3
2 C ( K , L ) = C(K,L) + X(I , J ) ::. Y(IY, J Y)
3 CONTINUE
4 C ( K , L ) = C(K,L) /T
RETURN
EM>
FORTRAN P r o g r a m for C a l c u l a t i n g t h e C r o s s c o r r e l a t i o n of Two P ic tu r e s X and Y
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SPACE-GENERAL CORPORATION SGC 203R-4Page 67
YES
Figure 3 0 . Flow Diagram for the Crosscorrelation of 2 Pictures
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.
SPACE-GENERAL CORPORATION
Stationarv Matrix MultiDlication
SGC 203R-4Page 68
The matrix product
c = hg
where c(r) = h(s) g(r-s)
is computed on the assumption that the dimension of each matrix is M x N with
appropriate subscripts.
S
The center is considered to be the integer part of
(MO, NO) =(y,The computation of the
M"H
C(I,J) =c H(K,L)
K=l L= l
_ I
product is then
'k G ( I - MCO + E O K + MHO, J - NCO + NGO - L + " 0 )
where the sumnation is only over the mutual range of H and G.
. .. . - .. . .
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SPACE-GENERAL CORPORATION
F O R T R A N P r o g r a m for C a l c u l a t i n g t h e Matrix Product
SUBROUTINE MATRIX (H, MH, NH, G, MG, NG, C, MC, NC)
D=SIm H 46,416), G 46,4151, c (46, 46)
MHO = (m+1)/2
"0 = (NH+1)/2
MGO = (m+1)/2
MCO = (MC+1)/2
NGO = (NG+1)/2
NCO = (NC+1)/2
SGC 20313-4
Page 69
p 1 0 = ~ + ! & 3 0 - ~
NO = NGO + NHO - NCO
DO 2 I = 1,MC
DO 2 J = 1,NC
C ( 1 , J ) = @.@DO 2 K = 1,MK
DO 2 L = 1,"
K G = I - K + M OL G = J - L + N O
IF (E) ,2,1
IF &+) 2,2,1
IF (KG - MG) 1,1,2
IF (IG - NG ) 1,1,2
1 C(I,J) = C(I,J) + H(K,L) :K G(KG,LG)
2 CONTINUE
RETURN
m
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SPACE-GENERAL CORPORATION S G C 203R-4
Page 70
C A L L MATRIXw
Figure 31. Flow Diagram of Siationary Matrix Multiplication
~ _ _ _ _ _
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S P AC E- GE" A C OR P OR ATI ON
C ONS TR UC ~ ON OF A RUFFMAN CODE"
SGC 203R-4Page 71
The conversion of a set of words in t o another se t of words
having m a x i m u m entropy per digi t i s a coding problem f o r which an
exact so lu t ion i s known i n terms of an algo rit hm .
The frequency of occurrence of t he o ri g i n a l words ar e
i n i t i a l l y l i s t e d from the smallest t o t h e l a r g e s t .
program assumes t h a t th e order ing process ha s a lre ady been performed.
The f i r s t two words are distinguished by a 0 f o r t h e f i r s t and a 1
for the second. They a r e then combined and tre at ed as a s in gl e word.
The following
The algorithm proceeds by r eo rdering th e new set of' words
d i s t ingu i sh ing the f i r s t tw o words with a 0-1 and combining.
word i s b u i l t up as a sequence of 0's and 1's depending upon the
algorithm .
Each
A n index matrix provides t h e correspondence between the
ordered se t of f requ encies and th e or i gi na l se t of words. The matrix
i n d i c a t e s t h a t t h e f i r s t , second, and four th words ar e asso cia ted
wi th t h e f i r s t frequency.
second frequency.
t o know which of t h e or ig in al words are t o be dis t inguished by0's and 1's.
The th ir d and si xt h are associa ted wi th the
With th i s type o f index matrix it i s then possib le
The program s t o re s th e code words rev er se d from r i g ht t o
l e f t w ith a 1 i gn i fy ing t h e end of the word.
obtained from th e decimal representation by expressing t h e representa t ion
i n binary, rever sing th e ordering and dropping t h e t e rmina l l .
The code words can be
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SPACE-GENERAL CORPORATION
A simp ler program may be po ss ib le by using t h e conce pt of "low
order pairing."
the table without any reorder ing or moving of th e r e s t of t h e t a b l e .
A se t of low order words are paired and moved up i n t o
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.
SPACE- ENERAL CORPORATION
FORTRAN PROGRAM FOR CONSTRUCTING A HUFFMAN CODE (CONTINUED)
34 K = I<+1
35 m Q io = S
36 Do 37 J = l,N
37 INDE X ( K , J ) = X ( J )
38 K = K+1
39 L = L-1
40 DO 44 I = K,L
41 I K = I+1
42 FREQ (I)= FREQ ( I K )
43 Do 44 J = l , N
44 NDE X ( K , J ) = INDEX ( I K , J )
45 PRINT 49 (WORD ( I ) / I = 1,N)
46 STOP
47 FORMAT
48 FORMAT
49 FORMAT
SG C 233R-4Page 74
. -
. . . .. . . _ ... ... I "_
. . ,.
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.
K = K + 1
L = L-1
SPAm-GENERAL CclRPORATION
4MOVE FREQ(1)
AND INDM(1,J)I = K,L
INDEX ( I , J )
WORD ( I )[,je
SUM INDM 0
AND FRJQ
4FIND K AND
SEll K = K-3
Figure 72, Flow Magram for the Construction of a H u f f m a n Code
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SPACE-GENERAL CORPORATION SGC 203R-4
Page 76
Addit ional FORTRAN Subroutines
The fol lowing subrout ines r e m i n ye t t o be w r i t t en :
1. Quantizat ion f or Display
This subrou tine should compute t h e qua nti za tio n cons tant s and
then quant ize th e p ic tu re , perhaps a s it i s being read out of the machine.
Provis ion should be made f o r en ter in g th e subrout ine with t he cons tants a l ready
given as may b e t h e ca s e i n f ix ed s ca l i n g .
2. Picture Generatian
Several randuxri rluii&ei- rcit’lries E LX neec?ec? fer producing a r t i f i -
c i a l p ic tur es with known pro pe r t i es . Uniform, and Gaussian di s t r i bu t i on s with
zero mean should be th e most u se fu l.
3 . Theoretical Correlation Computation
This i s th e computation of t he co rr el a t io n matr ix based on th e
l i ne ar p rocess ing o f a pic tu r e having a s p ec i f i ed co r r e l a t i o n m a tr i x.
4. opt$rmun Solution
Based on t h e inpu t comela t ion func t ion p and t h e cross-x ’ x ’
co r r e l a t i o n f u n c t i o n cp
t h e s o lu t i o n of a se t of simultaneous l i n e a r equat ions .
the optimum processor should be obtained. This i sx ’ 2’
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SPACE-- CORPORATION
CONVERGENCE OF I-lJERA!CIW S Y S W
One method of designing optimal processing systems I sthrough i te ra t i v e approximations .
is u sed t o cm p u te a b e t t e r app ro xim atio n I t e r a t i v e ly u n t i l t h e
optimal system is obtained. Sakrison(2?) has cons idered an i t e ra t ive
design of a spec i f ic op t imal system. This se ct io n considers some
of t be b asi c elements of h?-s ana lysi s of t h e convergence of a system
t o an optimal system.
A n approximation t o th e system
NORMS
I n order t o cons ider th e convergence of m i terat ivesystem t o an optimal system i t i s necessary t o have some measure of
the error between the approximation X and the optimum system G-
measure I s known as t h e-orm Et where t denotes the sequence of
approximations.
the case of a s in g l e v a r i ab l e or the average inner product of t h e
e r r o r I n the case of a mul t ip le variable system.
The
A very useful norm i s th e average squared er ro r i n
Other norms are based on weighting functions of t h e e r r o r s uch tha t
Et > O X f 0 -
E t = O X = e .
The behavtor of the nom I s o f p a r t i c u l a r im porta nc e I n
des crib ing t he convergence of the approximation t o optim al systerhs.
In th os e c ase s where t h e norm can be descr ibed by t h e f l r s t order
dif ference equat ion
hEt c a t + b E- t t
hE t = Et+l Et
q u i t e a b i t can be sa id about the convergence based on the p r o p e r t i e s
of the coef f ic i en t s at and bt.
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SPACE-CENERAL CORPORATION SG C 203R-4
Page 78
!Che ex ist en ce of th e di ffe re nc e eq uation which can be
e x p l i c i t l y s ol ve d is th e most im portant Idea of th is approach.
Where El is the so lu t ion o f the d i f fe rence equa t ion
= at + bt E;
and E is the actual norm which sat isf ies t h e d i f fe r e n ce r e l a t i o nt
&E at + bt Ett -
t h e nom Et is bounded by the solution E;
Et I l.
and a l l t whenever 0 - 1 + bt
If t h i s were no t so, then th ere would e xi s t a T for which
%+l +l
5 5 %
and
The dif f eren ce equat ions produce th e re la t i on s
Subtrac t ion produces th e ineq ual i ty
<eT+1- 5 (l+bT) (E T -
which is a con t rad ic t ion under the assumption that 1+ bT > 0 s in ce
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SPACE-GENERAL C O R P O R A T I O N
Thus for all i t e r a t i o n e t , the norm Et i e lese than o r equal t o
t h e so l u t io n E$ of the d i f ference equat ion .
SOLUTION OF TBE ITRST ORDER DIF’FEREmCE EQUA!l“OM
The f i r s t order difference equation
AE; = a + b t Et
has t he genera l so lu t ion
t
In the pa r t i cu la r case where b i s t h e c on st an t b and a i s t he cons tan t aJ i
raised t o the i t h power, the so lu t ion has the c losed form
iwhere a = ai
This so l u t i o n
where
al+b-
It i s easy t o
t + la1 - (m)
= ( l+ b ) t a + EA (1+ b)t+l
b = bj
canverges- o ze ro approximately
- ... (1+ bJt
< 1 a n d O < ( l + b ) < l
a s
see t h a t i f t = 0, t he genera l so lu t ion gives t h e
cor r ect s l u ti n
Ei = a + EA (1+ b o )
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SPACE-GENERAL CORFOFWCION SGC 203R-4
80
By i nduc t ion , t he s o lu t ion fo r t i s assumed corr ect
t-1 t-1 t-1= l a ll ( l + b j ) + EA n ( 1 + b )
j = i + l j=o Ji =o
E
From the d i f ference equat ion t h e so l u t io n f o r t + 1 s found t o be
= a + E; (1+ b t)Et+l t
Subs t i tu t ion of E produces the rela t i on
t t t
= 1i = o
fl (1 b ) + E* n (1 + b j )j = i + l j oj=o
Which i s the r e l a t i o n f o r t h e g e n er al so l u t io n of E .
Another property of the f i r s t order d i f ference equat ion ,
which makes it usef 'ul i n the analysis of i terat ive convergence, i s t h a t
t h e so l u t i o n E t converges t o zero whenever th e c o e f fi c i e n ts s a t i s f yt
th e cons t r a in t s
a > Oi -
a,
C a i = A < . ,
i = o
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SPACE-GENERAL C O R R M T I O N
t
( l + b ) <
SGC 203R-4
Page 81
E
3 -
The logarithm of the second tern of the genera l so lu t ion p roduces
sequence of r e l a t i o n st
This implles t h a t
t
Urn B~ IIt- j=o
(1+ bj) = o
Considerably more m3thematical r igor I s needed t o show t h a t t h e f i r s t
t e r n of the genera l so lu t ion a l s o comerges t o ze ro .
proof i s one which shuws t h a t f o r e- ie ry E > 0 t h e re e x i s t a T such tha t
A s u f f i c i e n t
(1+ bj) < E
I1i j 3 + li = o
for every t greater thaa T.
The proof needa se-reral s teps . F i r s t , s in ce 0 < 1+ b .(: 1,3 -
t(1 + b ) e 1 and s in ce the ai 2 0 there exists some I such that
j J + l 3 -
for every t .
These in eq ua l i t i es imply that
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SPACE-GENERAL C O R P O P A T I O N SGC 203R-4
Page 82
The second part of the proof re su l t s from knowing t h a t t he p roduc t
converges t o zero.
every t > T
That i s , t h e r e e x i s t s 8 o m T such that f o r
This inequ al i ty produces the re la t io n t h a t
I
i = o
Thus the t o t a l sum from 0 t o I and I + 1 o t i s l e s s t h a n E
whenever t ' .The ra te a t which I t converges t o zero depends on th e pa rt i cu la r
coef f icf ents of €he- dffference equation.
This implies that the second term converges t o zero.
PROPERTTES OF !FEE ITERATION
An example of t h e use of t h e f i r s t order difference equation
is i n t h e ca lcu la t ion of a norm equal t o t he inner product of t h e
error between th e system parameters X and the optimum parameters 8
r 7
En- e ) (Xn - e)]
The i t e r a t ive procedure f o r the determination of th e process is
an I t e r a t ive co r r ec t ion
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SPACE-GENERAL C O R P O R A T I O N
The error i s then
- e = x n - e + E n'n+l
so t h a t t h e norm i s
r
En+l = En + Average t c n . cn + 2(xn - O) .En }
This is then a dif ference equat ion of th e norm.
hEn= Average {en cn + 2 ( X n -
In those cases where t he re ex i s t s appropr ia t e a and bn such t h a t
nr
+ 2(xn - 5 an + bn Eten Enverage
The behador of th e norm En can be analyzed qu i t e e f f ec t ive ly wi th
the so lu t io n of t he d i f ference epuation.
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SPACE-GENERAL CORPORATION SGC 203R-4
Page 84
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Recommended