1982 sequential convolution techniques for image filtering
TRANSCRIPT
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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
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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.
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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.
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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
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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$’s)
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TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNA LROCESSING, VOL. ASSP-30 NO. 1 , FEBRUARY 19
such that
f = / / A ( m , )
W m ,
(I11 1)
is
minimum, where
and
(1112)
p q m , n) =
p;;)W$"+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
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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
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8 IEEE TRANSACTIONS ON ACOUSTICS. SPEECH,NDIGNAL PROCESSING,
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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.
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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:
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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