1982 Sequential Convolution Techniques for Image Filtering

8/20/2019 1982 Sequential Convolution Techniques for Image Filtering

1/10

IEEERANSA CTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING,

VOL. ASSP-30,

NO.

1 ,

FEBRUARY

1982 1

SequentiaI ’Convolut ion Techniques for

Image Filtering

JEAN-FRANCOIS ABRAMATIC,

MEMBER, IEEE, AND

OLIVER D. FAUGERAS,

MEMBER, IEEE

Ahtrac t-Sequ entially convolving images with small size operators is

a promising idea

for

performing image filtering. Various classes

of

two-

dimensional inite mpulse esponse FIR) i l ters hat

can

be imple-

mented with such echniques are studied. Design procedu res are pre-

sented

for

each

of

the classes. They are based

upon

minimizing

L 2

and

Lm

riteria. Results are assessed in an image processing conte xt.

W

INTRODUCTION

HEN digital image processing becam e a viable area of

research in the early sixties, a lot of effort was directed

towa rds image restor ation, image coding, and feature extrac-

tion (mainly edge detection).

A

basic tool in this c ontext ha s

alwaysbeen image filtering. The mple men tation of two-di-

mensiona l 2-D) ilters using general-purp osecomputers was

first nvestigated.Fourier echniques using the fastFourier

transform algorithm were adapted to the 2- D case. Problems

related to image transposition and overlap-save techniques were

studied (see Ekstrom and Mitra [11). Recursive filters have

great potentiality

in

computational savings, thus a great amount

of work was done to solve problems related to the 2-D struc-

ture of 2-D recursive filters (see Huang [ 2] , Abramatic

e t

al.

[3]). Recent developmen ts in hardware technology suggest a

study of 2-D filters that can be adapted to real-time image con-

volution. More precisely, modern image display systems (see

Pra tt 4]) are able to perform csmall” size convolutions

at TV rates. Herewe study various classes of 2-D finite m-

pulse response (FIR) filters th at have “large” size impulse re-

sponses but can be implem ented by sequentially using “small”

size operators. Meck lenbraiiker and Mersereau [ 5 ] first studied

the implementation of 2-D finite impulse response (FLR) fil-

ters designed b y m eans of th e McClellan transformation. The y

showed howonecould mplement hesefilters by using se-

-quentialconvolution echniques.Starting rom mplementa-

tion ra ther th an design issues, we will define various classes of

2-D F IR filters that will be larger than the classes of FIR fil-

ters designed b y the generalized McClellan transfo rmatio n. We

will propos e design algorithms for these classes of filters based

upon the minimization of L 2 and ;“ riteria . We will assess

implementation schemes of these frlters in anmage processing

context.

Manuscript received October 29, 1980;evised June 22, 1981.

J.-F. Abramatic is with he nstitutNationalde Recherche,en

In-

formatique et en Autom atique, Dom aine deVoluceau-Rocquencourt,

B.P. 10 5, 781 50 Le Chesna y, France.

0

D. Faugeras

is

with the University of Paris South , and the Insti tut

National de Recherche en Informatique et en Automatique, Domaine

de Voluceau-Rocq uencourt, B.P. 105 , 78150 Le Ches nay, rance.

1.

h

,K

FILTERS:

EFINITIONS ND PROPERTIES

The

SGK Concept

Implementation schemes for I-D FIR f$ rers have been widely

investigated (Oppenheim-Schafer [ 6 ] ) , 3abiner-Gold7]).

When fixed-pointarithm etic is used, tl

,

m a d e s t ruc ture i s

recommended as

it

allows tu ning of the ,.‘:plementation , pre-

vents overflow errors, and minimizes roun doff errors. IfH( z)

denotes &k e -transformof he real talued mpulse esponse

h(k)of c .D filter

Q,

H ( z )

=

2

h(k)Z-k

1

(11)

k = - Q ,

then

H z)

can be decomposed as

Q

H(z)

= n

Pi(Z) (12)

i

1

where

P i @ )

=

a ,z-l a afz. (13)

This result comes from the fact that a polynomial of degree

n

with real coefficients can be decomposed int o a product of n

polynomials with real or complex conjugate coefficients. Gath-

ering conjugate complex coefficients and pairs of real coeffi-

cients leads t o (12). The cascade implementation of I-D FIR

filters can then be described as in Fig. 1

When fixed-point arithmetic is used, one can take advantage

of his mplementation scheme. Carefully ordering the Pi(z)

can reduce the overall roundoff error. Scaling the Pi(z) while

preserving the overall gain ma y avoid overflow errors. Finally,

we want to emphasize that the numb er of arithme tic opera-

tions required is 3Q, while the direct implementation requires

2 Q t operations. The use of such a schemewill thus be inter-

esting when fixed-point arithmetic implementation is enforced.

Another reason for using the cascade structure may be tha t the

scheduling of the implem entation allows one to perform , say,

three operations at a time rather than 2Q

t

1.

In the 2-D case, the problem is quite different. The decom-

position of (12) is no t available in general for 2-D z-transform

H(z,

, z).

This is related to th e fact that 2-D polynomials of

degree n have, in general, an infinite noncountable number of

zeros. We are thus led to consider the 2-D FIR filters whose

transferfunctionsaredecompo sable. We thus define various

We will suppose th at

Ql = Q = Q

in the following to simplify nota-

tions.

’ 0096-3518/82/0200-0001 00.75 982 IEEE


2/10

2

IEEE TRAN SACTIONS ON ACOUSTICS, SPEE CH, AND SIGNALROCESSING, VOL. ASSP-30, NO. 1, FEBRUARY 1

INPUT O ITPVT

Fig.

1.

Cascade implementat ion o f a 1-D

FIR

filter.

INPUT OUTPUT

+= -p

_ - - -

4

Fig.

2.

Cascade implementation of

a

2-D basic SGK filter,

INPVT

-:-=- j

OUTPUT

_ _

~

~ ~ ~

Fig. 3 . Implementat ionofan SGK filterderived f rom

a

McClellan

transformation.

classes o f 2-D FIR filters th at we call “small” generating ker-

nels (SGK) filters. Referring to (I2), the first class called basic

SGK filters is defined as follows.

A

2-D

FIR

filter belongs to th e class of basic SGK filters

if

its transfer function can be written s

where

The implementation can be described by the bloc k diagram of

Fig. 2.

(2PQ

t

arithmeticperationserutput sample are

needed to implement such filters using the direct implem en-

tation whereasequentialmpleme ntation will require

Q(2P

l)z

of them. The interesting feature here is that wha t

one oses ngenerality’may besomewhatcompensated or

by appealingmplementationproperties.Furthermore,one

can define various classes of SGK filters th at will have com -

pari.de implem entation schemes, bu t will be different as far

as approximation procedure is concerned.

Aside from the basic SGK filters we already introduc ed, we

study two other classes of SGK filters.

SGK filters derived

from

McClellan transformations:

The

firstone was introduced b y Mecklenbrauker and Mersereau

[ S I .

It appears to 6e the class of 2-D FIR filters generated by

McClellan transformations. Implementation of such filters is

described in Fig. 3.

Their transfe r f unction s are given by

where

N P U T

I t t

Fig.

4.

Implementat ion of general SGK filters.

Fig. 5. Basic SGK filter memory handling.

Fig. 6 . General SGK filter memory han dling direct form.

They require Q(2P f 1)’ t Q operations per output sample

implementation.

General SGK Filters

The second class

we

study includes the

T W O

previous on

Its filters use different kernels and produce the output as a

earcom bination of the nter me diate results. Fig. 4 show

block diagram of this implementation.

It is easy to see tha t this class includes the two other on

Filters

of

this type alsorequireQ(2P

f f

Q arithmetic o

erations for implementation.

Implementation Issues

Aside

from

thenumberof operation s involved, sequen

convolution technique s need to be assessed in terms of stora

requirements. Thinking of an image display terminal, mem

storage is the mo st costly par t of the device. Thu s, special c

has t o be devoted

to

this feature. We will assume that the

pu t image has to be preserved. Oftentimes the user will like

choose the 2-D filter interactively and will thus need the in

image t o perform various tries. Memo ry requirements for th

basic SGK imp leme ntation are d escribe d in Fig. 5.

IM

and

OM

denote the input and output mem ory storag

S1 symbolizes the initialization step where

IM

is fed into

O

S 2 symbolizes the Q iteration s. The general SGK filter has

more complex structure described by Fig. 6.

The scratch memory SM is needed to store intermediate

sults. S1 and S 2 drive th e initialization while S 3 and

S4

c

trol he schedulingofoperations. The useof this cratc

memory can be avoided by transform ing the implem entatio

in to th e so-called transposed form (Fig.

7).

One is thus led to the mem ory handling situation describ

in Fig. 8.


3/10

A B R A M A T I CD FAUGERAS: S E Q U E N T I A L

C O N V O L U T I O N TECHNIQUES 3

WrPUT

Fig. 7. Transposed form m plementation.

t

h

I

ig.

8.

General

SGK

filter memory handling transposed form.

I t

thusappears that he transposed mplementation which

was rejected yMecklenbrauker

.

and Mersereau [SI and

McClellan and Chan [8] for fixed-point implementation issues,

has a very mpor tant feature regarding memory requirements.

11. DESIGNOF SGK FILTERS DERIVE D FROM

MCCLELLAN

TRANSFORMATIONS

McClellan transformations are used to design 2-D zero-phase

FIR

filters from 1-D zero-phase,

FIR

filters. The basic idea is

to define a transformation of the 1-D frequency domain into

the 2-D frequency domain. If h(n) and H e J W ) enote the im-

pulse response and the transfer function of a zero-phase 1-D

filter, respectively, they are related by

Q

H(eiw)

= h(0)

2h(k) cos

ok (111

1

k =

Using th e Moivre form ula, this can be rew ritten as

The generalized McClellan transformation is defined as

P P

cos o t(m,) cos m o l cos no2 ’

m = o n = o

2-D

FIR

filters obtained through this transformation are thus

defined by

This can be rewritten as

2Mersereau

[ 9 ]

recently proposed another transformation that leads

to any 2-D zero-phase

FIR

filter.

Comparing (113) and (16) indicates that 2-D FIR filters de-

rived by McClellan transformations have the sequentialcon-

volution prop erty (for more details see [5]). The use of such

transformations eads oefficie nt design algorithms or2-D

FIR filtersespecially n he case where the “shape” of he

transfer unctions is not oo complicated. As an xample,

circular symmetric filters will use a transformation where

P = 1

- (0, 0)

=

t(0, 1)

=

t(1,O)

=

t (1 , I >

= 3.

The design of ‘such filters only requiresdesigning the

1

D fil-

ter def ining the ‘‘radial’’ characteristics of the 2-D filte r. When

the “shape”gets more complicated, optimization algorithms

are needed to determine

t ( m ,

n), and the design procedure be-

comes more tedious. More details are given in Mersereau et d

[ l o ] . O u rapproach is somewh atdiffer ent. We start from a

prototype nd use approximationechniques to findhe

member of the class of SGK filters which minimizes an error

criterion.

Approximation Techniques for SGK Filters Design

Our approach to designing SGK filters consists in solving the

following approximation problem,

Given P and 2, wo integers defining the “order” of the SGK

filter

H ( z , , z 2 ) ,

the ransfer unction

of

aprototype filter

whose impulse response is of size (2PQ t

1)

X (2PQ + l), find

such that

and

P(z1 ,

2

= piiz;iz;j.

+ P + P

(116)

-P P

This results n a well-defined approximation problem once

the norm of 114) is chosen.

We will present three algorithms. The

first

is based upon the

choice of L 2 norm.The twoothersare elated to the

L“

norm, one of them s suboptimum.

L 2 Norm-The choice of an

L 2

norm leads to the following

criterion.


4/10

where W(zl,

z 2 )

s a real positive w eighting func tion , and

r l

and

r z

are the unit circles of the

z 1

andz, planes, respectively.

Discretization

of

the Problem

After choosing

N 2

samples

(N

2

2PQ

1) on the uni t ircles

rl

and I?,we get a discretized version of the problem where

the criterion becomes

m = o

n = o

where H ( m ,

n )

and A ( m , n ) are the DFT's of the proto type

and the syn thesized im pulse esponses, respectively,

with

Since the criterion is quadr atic n he variables { h k } , they

can be obtained from the proto type and a set of given { p i i } by

solving a linear system. The problem can then be reduced to

finding the {pi j} . We now restate the criterion in matrix form.

Matrix Formulation

Stacking

H ( m , n ) , P ( m , n ) ,

k in column s we define

- 1

( N -

1 , N - 1)

P ( 0 ,O) .

*

P Q ( 0 , O )

P ( N - 1 , N - l ) * . * P Q ( N - , N -

?= (1110)

c is an N 2 dimensional column vector, ? is an N 2 X Q matrix.

We will also need

and

@=diag{W(O,O);.. ,W(N- 1 , N - 1

a t 1 dimensional vector and an N 2 X

N 2

diagonal matrix.

The criterion can then be written s

E = ( - H ) ~ @ ( ~ A - H )

5

(I11

1

and A*(H,Y), the optimu m value

of

A for given H and 9 is

3

F(m,n )

will be

a reduced notation for

F(e(2inmiN),

e(2inn'N)

s

W M

denotes

as

usual

e(2niiN).

The problem reduces to finding the

{ p i i } .

This will be d

by means of a gradient algorithm. We shall now proceed

calculate the gradient of the criterion with respect o pij.

Gradient of the Criterion

Differentiating

I11

1)

gives

4

IEEE TRANSACTIONSNCOUSTICS,PEECH, AND SIGNAL PROCESSING,OL. ASSP-30,

NO. 1,

FEBRUARY

j--.

x

denotes

the

conjugate

of the

complexumber

x

respectively.

As hk is chosen to be optimum, the third termvanishes.

Since

P ( m , n )

s

the DFT of pii, we easily get

Defining

e ( m ,n )

= 2

k h : P k - ( m , n>

[ x

{?

: P k ( m , n ) H ( m , n ) W ( m , n )

a€

ae

apz (m

n )

apR ( m n )

=

Re

{ e ( m , n ) }

= -1m

{ e ( m , n ) } .

Combining (II13) , (II14), and (1116) we obtain

€

DFT-' { e ( m , n ) } .

aPii

Gradient Algorithm

(I1

(11

(I1

(1

(11

At each iteration,we need t o calculate

(ae /ap i i ) ,

hat is

1)

calculate

P ( m, n ) .

We will not use an FF T algorithm

cause {pii} is a "small" kernel.

2)

Calculate

A*

using (1112).

3) Calculate e ( m , n ) .

4) Calculate (&lapii). Again we will not use an FFT a

rithm.

L m

Norm-In the 1-D case, the

L"

norm is often used in

ter design. The approximation problem is then refered to

Chebyshe v pproximation roblem.Experimen ts resented

6PR(m,

)

and

Pz(rn, n are

t h e

real

and imaginary parts

of P(m


5/10

ABRAMATIC AND FAUGERA S: SEQUENTIAL CONVOL UTION TECHNIQUES

5

later will also prove the practical nterest of his choice. We

present here two Chebyshev design procedures. The first one

is related only to the choice of the

{ h k } ,

the approximation is

called a inear Chebyshev approximation. Specific algorithms

are available for th is case. We shall use a Remes-ty pe algorithm

following th e appro ach described n Kamp and Thiran

[l

11

for the design of 2-D F IR circular symmetric filters. The sec-

ond approach is general and uses nondifferentiable optimiza-

tion algorithms.

Suboptimal Chebyshev Design

The discretized problem can be state d as follows.

Given

H(m, n )

the DFT of the prototype impulse response;

P(m,

n )

the DFT of the “small” generating kernel of size

Q the numb er of sequential convolutions.

2 P t 1;and

Find

{ h k } ;

k =

0,

such that

e, = max

W(m,

n )

E(m,

n (1117)

m , n

is minim um, where

a m ,

n )

=

p(m, n) H(m,

n)J

Q

A(m,

n )

= kpk(m,

.

(I118)

k = 0

W(m,

n )

s again a

real

positive weighting function.

This problem is solved by means o f a one-for-one exchange

algorithm that goes as follows.

1 )

Choose a so-called reference

R

th at is a subset of Q

2

points of the N X

N

frequency grid.

2)

Solve the Chebyshev approximation for

R ,

hus obtaining

a set of coefficients

{X:};

k

=

0, . . ,Q.

3)

Use the

{A:}

toevaluate hemaximumerror and he

point where it is reached on the whole

N X N

grid.

4) If this point belongs to the reference, then the optimu m

is reached or else exchange it with a point of the reference and

go back to

2).

Solving the Chebyshev approximation for R requires the so-

lution o f two linear systems. This is the most com putationally

expensive part of the algorithm. In our case one of those lin-

ear systems is a Vandermonde system. Thus it can be solved

by means of the Cramer formulas [see Appe ndix

11-21.

This

speeds up considerably th e exchange algorithm.

Optimal Chebyshev Design

{ h k } ’ ~nd

{ p ~ } ’ s

y using an

L“

criterion.

We can tr y to adjust the whole set of parameters that is the

Given

H(m, n )

he DFT

of

the prototype impulse response.

P , Q

two integers giving the “order” of SGK filter.

Find

{ A t ) ; k

= 0 Q .

{ p i j } ; - P < i < t P ;

- P < j < + P

such that (1117) and (1118) hold.

We can use gen eral techniques to solve nond iffere ntiable op-

timization problem s (see Lem arechal

[12]).

These techniques somewhat generalize gradient-typemethods

to problems for which gradients may not always be defined.

In our case, the algorithm runs as follows.

At iteration r , let us call

A,

d$ the values of the unknow ns.

For these values, one looks at the point

s)

(m*,

n * )

n the fre-

quency domain where the maximum of the

E(m, n )

is reached.

(Notice that there may be several such points.) We then cal-

culate (aE(m*,

n*)/ah,)

and

(aE(m*,n*

> l a p , ) for this (or

these) particular ( m * ,

n * )

and call them “subgradients.” The

descent algorithm consists here in choosing new values hi+ ,

pG+ based upon thesesubgradients.

Finally, ne sees tha t we need to be able to calculate

(aE(m*,

n*

)/ah,),

(aE(nz*,

n * ) / a p i j )

or

a

fixed chosen point

where

(1119)

(1120)

(1121)

Q

. kXiP, -’(m, n)

W

mi+ni ) . (1122)

k = O i

These formulas allow us to set up the optimization algorithm.

111.

DESIGN

O F

GENERA L GK FILTERS

Design Problem

Designing a general SGK filter consists of solving the fo llow-

ing discretized approximation problem.

Given

H(m,nj, the transfer function of the prototype filter,

P ,

wo integers defining the “orde r” of the SGK filter.

Find

{ p p ’ } ;

1 < k < Q

- P < i < t P ; - P < j < t P

(X’s

have been “included” in

p$’s)


6/10

6

IEEE

TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNA LROCESSING, VOL. ASSP-30 NO. 1 , FEBRUARY 19

such that

f = / / A ( m , )

W m ,

(I11 1)

is

minimum, where

and

(1112)

p q m , n) =

p;;)W$"+in).

+P + P

i=-p

j=-P

Again, we solve this approxim ation problem with

L 2

and L"

norms.

Block Relaxation Method

In b ot h cases, we use an iterative procedure to solve the ap-

proximationproblem.Each teration is decompo sed nto Q

steps. At each step we fix

Q

-

1

kernels and minimize crite-

rion (1111) with respect to the Qthemaining kernel.

More precisely the iterative algorithm is described as follows.

1)

Choose initial conditions

p ;);

-

P < ,

G

t P , k =

1 ,

* ,

2 ) k + 1.

3)

For

L 2

criterion, solve a inear system in (2 Q

1 2

un-

For L" criterion, solve a inearChebyshev approximation

4) k k t

1. If

k


7/10

ABRAMATIC AND FAUGERAS: SEQUENTIAL CONVOLUTION TECHNIQUES

PROTOTYPEILTER

Lab

SYNTHESIZED FILTER

x

3

K E R N E L

a = 12 d =

4 .5

Fig. 9.

Low-pass exam ple: transfer function.

PROTOTYPEILTER

S G K FILTER 3 x 3 ‘{ERNEL a = 5

S G K F I L TER (Chebyshev)

Fig.

11.

Bandpass example: transfer function .

( 4 (dl

Fig. 10. Low-pass example: images. (a) Original mage.

b)

Filtered

by

prototyp e. (c) Filtered by the SGK filter. (d) Difference.

We t he n filtered a test image wh ich is an aerial view. It was

chosen because it presents a variety of situations: mall details,

high contras ts, large textured field. Fig. 10 shows the nput

image (a), the images filtered by the pro toty pe fiiter (b), and

the

SGK

filter (c). Finally, as the eye is not too clever abo ut

seeing differences between blurred images, we present the ab-

solutedifferencebetween he wooutput images scaled by

a factor of 100 (d). As expected, differences occur close to

edges, but they are mall on the order of 1 percent).

The second example deals with a bandpass filter. We check

here the influence of the criterion. Poorer results are picked

so that heeyecould perceive differencesbetween images.

This means that a small number of kernels are used and thus

that the implem entation would be cheap. Fig.

11

shows trans-

fer functions of he prototype filter and SGK fi te r of the

McClellan class chos en using

L 2

and L“ criteria. Fig. 12 pre-

sents crosssections of the transfer functions showing the poor

results of th e

L

procedure in the low-frequency domain.

Fig. 13 shows the original image (a) and the three outputs

associated with the transfer functions

of

the Fig. 11. The eye

cannot see the difference between the prototype (b) and the

‘ 4 ~

outputs (d) but the

“L2

’ output (c) looks different in

the high contrasty areas.

This

illustrates the influence of the

choice of the criterion.

The third example deals with a deconvolution problem. The

original image has been blurred by a Gaussian blur.

A

decon-

volution filter

is

derived using the power spectrum equaliza-

tion echniq ue (see Cannon

[13]).

The ransfer function of

the fiiter is shown in Fig. 14. In this case we used general

SGK

filters (with different kernels). This transfer function ob tained

from the

L

procedure is also shown in Fig. 14. Fig. 15

shows

the blurred image, the three outputs of the deconvolution

fil-

ters. In this case the

“L2”

output looks much better than the

“L 0 ” one.

V.

CONCLUSION

New developmen ts n ntegratedcircuits echnology have

provided new potentialities n the area

of

image display sys-

tems. The new generation of such devices s able to do basic

operations on a 512X 5 12 image within

&

of a second. This

delay is the time allowed between tw o images on the screen.

Any operatio n done at hat rate provides itsresults“imme-

diately,” which means that th e observer will not see any tran-

sitioneffectsand will

just

feel that he operation has been

done in “real time.” Among such basic oper atio ns, than ks to


8/10

8 IEEE TRANSACTIONS ON ACOUSTICS. SPEECH,NDIGNAL PROCESSING,

VOL.

ASSP-30,

NO.

1 , FEBRUARY 1 9

GAUSSIAN BLUR

Atmospheric Turbulence)

PROTOTYPEILTER

PROTOTYPE FILTER

S

K

FILTER

m s e

approx imat ion)

SG K FILTER Chebyshev

approx imat ion)

Fig. 12. Bandpass example: cross sections.

5

SGK

F I L T E R

P , I a = 7

Fig. 14. Deconvolution example: transfer functions.

(C) (d )

Fig. 15 . Deconvolutionexamples: images. (a)Original image.

(b)

F

teredby heprototype. c)Fi l teredby the L z SGK filter. (d) F

tered by the L“

SGK

filter.

another memory. This provides th e way of sequentially usi

basic operations. The next question

is,

ho w to cascade variou

basic operation s t o perform “nonbasic” operations. Our pap

provides an answer to this question .

A

variety of procedur

Fig. 13. Bandpass example: mages. (a) Original mage. (b) Filtered t

the prototype. (c) Filtered by L 2 SGK filter. (d) Filtered by the L”

are

described that design sequences

Of

basic

x

Operatio

SGK filter.

whose results pproxim ate hose of a prototype nonbas

very fast multipliers, one can use

3 X 3

convolutions. Further-

Our approach needs to be compared to the one proposed

more. the results displayed on the screen can be “frozen”

in

Pratt

141.

He decomposes theprototype FIR filter into

operation.


9/10

ABRAM ATIC AND F AUG ERAS: SEQ UENTI AL CO NVO LUTI O N TECH NI Q UES

9

sum of eparable FIR filters using the singular value de-

composition (SVD), theorem. acheparableilteranhen api?

@,

l

be

“SGK”

expanded since it is built out of 1-D filters tha t al-

ways have the

“SGK”

property.he design procedure is thus Q

very simple andhasette r characteristics thanurptimiza-

H ( m , n ) - h;Pk’ m,

tion techniques. The drawback however is the implementation

scheme whic h requires a scratch memo ry to store intermediate similarly,

results. Againwe are’faced with

a

tradeoff (implementation

versus design) thatneedsmoredata bout he relative econo- aE(m,n, = - 2 Im

mies to be resolved.

apI m,

{ k =

to our filters.

filters?

set of parameters (cutoff frequency of a low-pass fiter, for

example)?

aE(m’ n = - Re

x

khkPk- ( m ,

{ k = 0

Q

__

k = O

khkPL-’ m, n

Q

To

motivate further work, let us pose two questions related

1)

How does one set up fixed-point implementation of

SGK

2) How doesone design

SGK

filters starting from a small whichcompletes hederivation.

(m, n

-

k

f

O hx.pkf m,n))} (A1o)

APPENDIX

CH EBYSH EV APPROXIMATIONN A REFERENCE

APPENDIX The most costly step in the exchange algorithm consists in

Theproblemconsists

in

differentiating with respect to is a

Q

t

2

sample subset

(mi,

ni;

=

7

*

‘

,

Q

+

2

o f t h e

fie-

PR m, and P; m, )

quency grid. Following the approach described in Kamp and

G R A D I E N T

O F

THE

L z

C R I T E R I O N solving thepproximationroblemneference R which

e2 = w(m, n E(m, n

Thiran

[ 1 11

, he algorithm goes s follows.

1 Solve the following Q t

i

X Qt

1)

linear system

m r n

= W(m,n)

E(m,n> can be rewritten as

(A4) B Q P =

B 1 * 032)

This is a Vanderm onde system as row j

of

the matrix is com-

posed of the elements of the first row raised to the power j

A5) and the right-hand vector also has this property . One can then

solve this system by using Cramer’s formula, Vanderm onde’s

determinant being easy to calculate.

2)

Solve the seco nd

(Q + 1) X

( Q

t 1)

linear system

We then use the following relationship:


10/10

10

IEEEXANSACTIONS ON ACOUSTICS,PEECH, AND SIGNALROCESSING, VOL. ASSP-30, NO. 1 , FEBRUARY

19

Thissecondsystemhas no particularpropertyand equires

standard routines to be solved.

REFERENCES

S . K. Mitraand M P. Ekstrom, “Two-dimensional digital signal

processing,”BenchmarkPapers in Electric alEngineering nd

Computer Science

,

vol. 20. Dow den, Hutchin son and Ross Inc.,

1978.

T.

S . Huang, “Picture processing and digital filtering,” Topics in

Applied Physics, vol. 6. New York : Springer-Verlag, 1975.

J . F. Abramatic, F. Germain, and E. Rosencher , “D es i~ n f two-

dimen sional separable denom inator ecursive filters,”

ZEEE

Trans.

Acous t., Speech , Signal Processing, vol. ASSP-27, pp. 445- 453,

Oct. 1979.

W. K. Pratt, “An intelligent image processing display terminal,”

inf’roc. SPIE Tech. Symp., vol. 27, SanDiego, CA, Aug. 197 9.

W. F. G. Mecklenbraiiker and R. M. Mersereau, “McClellan trans-

formation s or wo-dim ensional digital iltering:Part I-imple-

mentat ion,”

ZEEE

Trans. Circuits Syst., vol. CAS-23,pp. 414-

422, July 1976.

A.V. Oppe nheim and R . W. Schafer, Digital Signal Processing.

Englewood Cliffs, NJ: Prentice-Hall, 197 5.

L. R. Rabiner and B. Gold, Theory an d Application of Digital

Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975 .

J. H. McC lellan and D. S .

K.

Chan, “A 2-D F IR filter structure

derived from he Chebyshev ecursion,” ZEEE Trans.Circuits

R.

M.

Mersereau,“Thedesignof arbitra ry 2-D zero-phase FI R

filtersusing ransformations,” ZEEE Trans. CircuitsSyst.,vol.

CAS-27, pp. 142-144, Feb. 1980 .

R.

M.

Mersereau, W. F. G. Mecklenbrauk er, and

T.

F. Quatieri,

“McClellan transformations for two-dimensional digital filtering:

I-design,’‘ ZEEE Trans. Circuits Syst., vol. CAS-23, pp. 405- 414,

July 1976.

Y.

Kamp and J . P. Thiran, “Chebyshev app roximation for two-

dimensionalnonrecursivedigital ilters,” ZEEE Trans. Circuits

C. L emarechal, Non -Sm ooth Optim ization , R. Mifflin, Ed.New

York: Pergamon, 1979.

Syst . , V O ~ .CAS-29, July 1977.

Syst . , V O ~ .CAS-22, pp. 208-218, 1975.

Jean-Franqois Abramatic (“78) graduated fr

the Ecole des Mines de Nancy n 1971 . He

ceived theDocteur-Ingkieur degree rom he

University aris XI, Orsayn 1975 and

Docteur &s Sciences degree from the Univers

Paris VI in 1 980.

He is cu rrently nginieurdeRecherche t

INRIA (Inst i tut National de Recherche en In-

formatiquetutomatique) , Lehesnay,

France, where he has been working in the Im-

age Processing Group since 1974.Dur ing he

year 197 9 he was on sabbat ical leave in the Image Processing Inst i t

of th e University of Sout hern California. His curre nt intere sts are o

signal processing and image processing and analysis.

Im pu lse Response Arr ays of Disc rete-Space System

Over a Fini te Field

Abstract-The main thrust of this paper is directed towards an analy-

sis of the s tructure a nd propert ies f the impulse response array

f

two-

dimensional

(2-D)

discrete-space ystems,characterizable by atio nal

functi ons having coefficients n a field Zq It is shown that suchan

array exhibi ts a row (column) type of periodici ty . Expressions for the

Manuscript received Decembe r 5, 1 98 0; revised August 3 , 1981. This

workwassupported by the U.S. Air ForceOfficeofScientific Re-

search under Grant AFOSR-78-3542 and the National Science Founda-

t ion under Grant ENG.78-23141.

K. A. Prabhu is with he Visual Commu nicat ions Research Depart-

ment, Bell Laboratories, Holmdel, NJ 07733 .

N.K. Bose is with heDepartments ofElectricalEngineeringand

Mathematics, University of Pittsburgh, Pittsburgh,PA 15261 .

period are derived. Arrays with a maximum of three levels in the a

correlat ion funct ion are dent if ied and explic i t expressions for hese

levels are given.

W

I. INTRODUCTION

HILE a significant am ount of literature on systems t

are linear with respect to the field of real numbe rs

available, the area of systems that are linear over a finite fi

has,by no means,beenoverlooked. A reasonablycompr

hensive list of papershasbeen eferenced n

[ l ]

[ 2 ] .

substantiated in [ 2 ] , hile binary sequences with special au

0096-3518/82/0200-0010 00.75@ 1982 IEEE

Documents

1982 Sequential Convolution Techniques for Image Filtering