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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



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



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



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-



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



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



I


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"



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 .



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



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



,

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



.

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



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



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.



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



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



,


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




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



.

-00

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



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)



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.



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.




Figure 2. An Wncorrefated Picture

Figure 3 . A Correlated Picture Obtained by Processing the Uncorrelated

Picture of Figure 2



.

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



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



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




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



.

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



.

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




= 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



.


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




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 '



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



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,




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



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 &ample

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



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 .



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+




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




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




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



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



.

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



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



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 .




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




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



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




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




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




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



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 .



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




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 -



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



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



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




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



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.




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



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




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



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




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




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,




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.



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




YES

Figure 3 0 . Flow Diagram for the Crosscorrelation of 2 Pictures



.


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.

. .. . - .. . .




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



SPACE-GENERAL CORPORATION S G C 203R-4

Page 70

C A L L MATRIXw

Figure 31. Flow Diagram of Siationary Matrix Multiplication

~ _ _ _ _ _



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




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





.

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 "_

. . ,.



.

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




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’



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.



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



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 )



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



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



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



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.




Page 84

References

1. N . Abramson, D. Braverman, "Learning t o Recognize Pa tt er ns i n a Random

Environment," IRE Tra nsa cti ons on Inform ati on Theory, Vol. IT-8, No. 5,Sept., 1962, S 58-63, International Symposium on Information Theory.

2. R . C . Amara, "The Linear Least Squ ares Syn th es is of Continuous and Sampled

Data Multivariable Systems, 'I Stanford Electronics Labs e , TR40, Ju l y 28, 1958

3. A r m a r Bose, "A Theory of Nonlinear Systems," Research Laboratory of Elec-

t r o n i c s , MIT, TR 309, May 15, 1956.

4. D. J . Braverman, "Maehine Learning and Automatic Pa t t e rn Recogni tion ,"

Stanford Electronics Labs., TR 2003-1, February 17, 1961.

5 . Murray Eden, "Handwriting and Pa t te rn Rec ognit ion, " I R E Transactions onInformation Theory, Yol. IT-8, No. 2, February 1962, pp 160-166.

D. Gabor, e t a l . , "A Universal Nonlinear Fi l t e r , Predictor and Simulator

which Optimizes i t s e l f by a Learning Process," Proc. IEE, V o l . 108, p t B,6.

pp 422-438, Ju ly 1961.

7 . C. W. Harrison, "Ekperiments with Linear Pre dic tio n i n Tele visio n," BSTJ.,V o l . 31, pp 764-783, July, 1952.

8. H. Highleyman, "Linear Decision Functions, with Appli cation t o Patt er n

Recognition," Proc. IRE, Vol. 50, No. 6, P a r t 1, June 1962, pp 1501-1514.

9 . Ming-Kuei Hu, "Visual Pa tt er n Rec ogni tion by Moment Inv ar ia nt s, " I R ETran sacti ons on Inform ation Theory, Vol. IT-8, No. 2 , Fe bruary 1962,

PP 119-187.

D. A . Huffman, "A &thcd for t h e Co ns truc tio n of Minimum-Redundancy Codes"

Proc. IRE, Sep t. 1952, pp 1098-1101.10.

11. T . Kailath, "Correlat ion Detect ion of Signals Perturbed by a Random Channel,

Tra ns . IRE-PGIT-6, pp 361-366, June 1960.

12. R. E . Kalman, "New Wthods and Results i n Linear Predi ct ion and Fi l t er in g

Theory," RIAS, TR 61-1, 1961.

13. E . R. f ietzmer , "S ta t i s t ic s of Televis ion Signals ," BSTJ. , V o l . 31,PP 751-7639 July 1952.

E . R. Qetzmer, "Reduced Alphabet Repr esen tation of Te lev isi on Signa ls, "

1956 I R E Na tio nal Convention Record, p t 4, pp 140-147.

1 4 .



SPACEGENERAL CORPORATION

References (Continued)

SGC 203R-4

Page 85

15.

1 6 .

17.

18.

19.

20 .

21.

22 .

23

24 .

25 0

26.

P. M . Lewis 11, "The Ch ara cte ris tic Sel ect ion Problem i n Recognition

Systems," IFE Transact ions on Inform ation Theory, V o l . IT-8, No. 2,

February 1962, pp 171-1718.

C . S. Lorens, " O p t i m u m F i l t e r i n g , " JPL, EP 633, May 15, 1939.

C . S. Lorens, "The Doppler Method of S a t e l l i t e Tracking ," 3rd N a t l . Con.

M i l . Elec. , 1999 Proceedings, pp 255-263.

C . S. Lorens, "Aspects of Theoretical and Experimental Map-Matching,"

Space-General Corporation, RM 17, December 15, 1961

C. S o Lorens, A . M. Boehmer, J . Gallagher, "Theo retica l and Experimental

Processing of Lunar Telev ision Pic tu re s," Bi-Monthly Progress Report No.1, 2, 3, SGC 203R-1, 2, 3, &y, J u ly , S ep t. 2 4 196 2.

J . K. Lubbock, "A Self-Optimizing Nonlinear Filter ," Proc. IEE, Vol. 108,

p t B, pp 439-440, J u ly 1961.

Daniel D . McCracken, A Ghide t o FORTRAN Programming, John Wiley and Sons,

In c ., Ne w York, 1961.

W. W. Peterson, Error Correcting Codes, MIT Press and Wiley, 1961.

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