A decomposition theorem and its implications to the design and realization of two-dimensional filters

Abstract

It is shown that an arbitrary rational 2-D transfer function can be expanded in first order terms, each one of which is a function of only one of the two variables. This method leads naturally to reconfigurable filters with great modularity and parallelism, which can realize any rational transfer function up to a given order.

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1562
A Decomposition Theorem and Its Implications to the
Design and Realization of Two-Dimensional Filters
IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING,
VOL. ASSP-33, NO. 6, DECEMBER 1985
Abstract-It is shown that an arbitrary rational 2-D transfer function
can be expanded in first order terms, each one of which is a function
of only one of the two variables. This method leads naturally to recon-
figurable fdters with great modularity and parallelism, which can re-
alize any rational transfer function up to a given order.
I. INTRODUCTION
T
(two-dimensional) is the fact that there is no fundamental
theorem of algebra for polynomials in two independent
variables. Factorization of 1-D polynomials into products
of first order terms plays a fundamental role in the devel-
opment of stability tests and stabilization, filter design al-
gorithms and implementation schemes, such as parallel and
cascade structures. Huang, Schreiber, and Tretiak
called the lack of the corresponding theorem
mental curse” of 2-D filtering. Bliss [2] mentioned that
“a two variable polynomial is not in general factorable
into first order polynomials; rather
nomial can be factored into irreducible factors, which are
themselves two variable polynomials, but which cannot be
further factored. ”
During the last few years considerable effort has been
directed by a number of authors to establish algorithms
and conditions for factorization [3]-[ 131. Chakrabarti et
al. in [3] treated the cases where two variable polynomials
can be expressed as a product or sum of special types of
two variable polynomials of lower order, and in [4] pre-
sented an algorithm to determine the
irreducibility of any arbitrary multivariable polynomial
with integer coefficients. In [5] the decomposability of 2-
D transfcr functions into I-D components was considered.
Morf et al. [6] proposed an algorithm for the case of prim-
itive factorization (i.e., in the case that one factor
function of one variable only and the other is a function
of both variables), and Bose 171, [8] gave a criterion to
determine whether any two multivariable polynomials are
relatively prime and an algorithm for the extraction of the
greatest common factor. In [9] and [lo], feedback was
used for the separation of some transfer functions into
HE most serious problem in the generalization of 1-D
(one-dimensional) digital filtering techniques to 2-D
[1]
“a funda-
a two-variable poly-
reducibility or the
is a
M&,luscript received July
work was supported by the Natural Sciences and Engineerin,
Council of Canada undcr Grant A7397.
A. N. Venetsanopoulos is with the Department of Electrical Engineer-
ing, University of Toronto, Toronto, Ont., Canada M5S 1A4.
B. G . Mertzios is with the Department
mocritus University of Thrace, Greece.
14, 1983; revised November
19. 1984. This
0 Research
of Electrical Engineering. De-
terms which are functions of one of the two variables.
Treitel and Shanks in [11] applied an approximation tech-
nique to the expansion of an arbitrary planar filter impulse
response into a converging sum of individually separable
2-D filters. Ekstrom and Woods [13] presented an alter-
nate technique, the spectral factorization approach, which
allows the factorization of 2-D transfer functions
terms with specified regions of analyticity. Their approach
is based on decompositions of the complex cepstrum.
While most earlier contributions were either special
cases or approximations to the general
problem, in this paper we propose a method for the exact
decomposition of a general 2-D real rational transfer func-
tion in first order terms, each one of which is a function
of only one of the two variables. The motivation behind
this approach is to give a general realization method,
which possesses a high degree of modularity.
Recent development of VLSI techniques have resulted
in enormous possibilities for the realization and
mentation of sophisticated algorithms of high complexity.
Since it is known that the most advantageous configura-
tions are parallel and cascade forms, consisting of second
order terms and real coefficients [14], we choose to com-
bine complex first order terms with appropriate conjugate
terms, in order to achieve second order terms
coefficients.
Present tendencies for the reduction of the cost of hard-
ware, coupled with an increase of the complexity of im-
plementation algorithms and applications, indicate
considerations leading to moderate
elements (registers, adders, multipliers, etc.) are
meaningful as they were in the past. Other considerations,
such as modularity, parallelism, regularity, flexibility, and
generality are of paramount importance and to that end
realizations exhibiting these properties become desirable.
The proposed method can be seen as a decomposition,
a special case of which, in the case of 1-D polynomials,
is simple factorization. While the fundamental theorem of
algebra expands a given polynomial of one variable into a
product of first order terms, the proposed method makes
use of array expansions to provide the additional degrees
of freedom required for the expansion of a polynomial of
two variables. Inasmuch as it is general, the method can
be used for the construction of general adaptive and re-
configurable filters. Such filters possess modular form and
are able to realize any rational transfer function up to a
given order, by varying the values of a set of parameters.
into
decomposition
imple-
with real
that
savings of dynamic
not as
0096-3518/85/1200-1562$01.00 O 1985 IEEE

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1563
11. A GENERAL
DECOMPOSITION
FUNCTION
OF A 2-D TRANSFER
In this section we shall establish a theorem to express
an arbitrary 2-D transfer function by first order terms of
one variable only.
Theorem 1: The 2-D rational transfer function of the
general form
p(z1, z2)
n2
c c pijz;zJ;
j=o j = o
mz
7
\'I
can be expressed by terms of the form
(Zl - Zli),
(z2 - z2;)
(4,
n2I.j = 1, 2, - * ' , max @ I ,
Proof: In the following, one of the possible proofs will
be given.
a) Numerator.
m2).
only, where zlir z2, are constants and i = 1, 2, - - , max
The polynomial q(zl, z2) can be written in the form
dz1, 22) = z T Q z 2
where
(2)
Q =
The polynomial (2) can be written equivalently as
dz1, z2) = ZTRSZ2
(3)
by writing the matrix Q as a product of two other matrices
R, S. The matrix R might be chosen arbitrarily as a non-
singular (nl + 1) X (nl + 1) matrix. In this case the
matrix S is determined by
s = R - ~ Q
and has dimensions (nl + 1) X (ml + 1).
We can readily see that
(4)
ZTR = [ro(zl), *
9 Ti(ZI),
* * * > rfll(zl)l
(5)
S z 2 = [so(z~), * * *
> si(z2), * * * , s,I(z2)lT
(6)
where
ri(zl) = roi + rlizl + - + riizj, + * * - rfl,izTl,
i = 0,1, -** 9 nl
t@-p--
- L J
Fig. 1. Transfer function of H&,, z2) = l/p(zl, z2).
P, (z,.z2)
4 q(z,,z*)]+~Tp7*
PI ( Z 1 J P )
Fig. 2. The feedback representation of the total transfer function.
Si(Z2) = sjo + SilZ2 + * * + s-zJ
v2 + * * * + Sirnlz!?',
i =0, 1, * . *
7 111
(7)
Making use of (5) and (6), the polynomial (2) may be
written as
~ ( z I ,
22) = [~o(zI), . *
> rikl), * * *
3 r n l ( z l ) l
[s0(~2),
* * +
3 si(z2)9
* * , ~ n I ( z 2 > [ ~
fll
=
ri(zl) si(z2).
(8)
Obviously, the polynomials
pressed as products of first order terms. We therefore con-
clude that q(zl, z2) can be expressed as a sum of products
of first order terms, each one of which is a function of
only one of the two variables.
b) Denominator.
We write the denominator of the given transfer function
as follows
ri(zl), and si(z2) can be ex-
i = O
P(Z1, z2) = c + Pl(Z1, z2)
(9)
where the polynomialp,(zl, z2) does not contain a constant
term. If we choose the constant llc in the forward branch
and the polynomial pl(zl, z2) in the feedback branch, we
obtain the configuration
1 lc
1
Pl(Zl9 z2)
HD(zl , z2) =
(10)
1 + ;
shown in Fig. 1.
Finally using the same technique used for the numera-
tor, we may write pl(zl, z2) in the form of an array (8).
The representation of the total transfer function is given
in Fig. 2. Note that the proof of Theorem 1 is just one
possible proof and does not necessarily lead to the most
economical and advantageous realization of the filter.
Lemma: Any physical realization of an arbitrary 2-D
rational transfer function H(zl, z2), given by (l), requires
a feedback configuration whose path transmittance in the
feedback branch is a polynomial in both zl, z2.
Prm$ Assume that H(zl, z2) can be realized by a gen-
eral lattice of terms of order one, in each one of the two
variables. Such a lattice does not contain feedback in both

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IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-33, NO. 6, DECEMBER 1985
variables. Therefore, H(zl, z2) is of the form
-
H(z1, z2) = c Hj1,
i= I
I
used in order to reduce the number of terms required for
the realization. To this end R is used if ml > n1 and S if
ml < nl.
Furthermore, a more economical decomposition can
* * , Hij,
* * , HiM
(11)
obtained if the auxiliary matrix R in (3) is chosen such
where Hg are factors, each one of which is a numerator or that the number of terms in the sum (8) is minimized. To
denominator term in one of the two variables only. In this
case, the denominator of the resulting transfer function
(11) is separable, which contradicts our hypothesis.
Q.E.D.
-
Note that ordinary factorization does not apply in gen-
eral to a function of the form (l), because it lacks the
necessary degrees of freedom required to represent this
function. The proposed decomposition provides the nec-
essary degrees of freedom to guarantee decomposition.
The realization proposed exhibits a high degree of par-
allelism with parallel branches which are separable. It is
shown in Section I11 that each of its parallel branches can
operate simultaneously on the input, resulting to a sub-
stantial increase in data throughout.
Theorem 1 can be easily extended to apply to half-plane
filters of the form [13]
this end, suppose now without loss of generality, that n,
I
ml. Then rank
= I
nl + and a more economical
realization can be obtained if a decomposition of the form
be
nl
ml
In this case the transfer function
pressed by terms of the form
H(zl, z2) can be ex-
(zl - z,J,
(z2 - z ~ ~ ) ,
where
I = 1, -1.
1
The proof is similar to that of the quarter-plane case.
We note that in special cases, a 2-D rational transfer
function can be realized with forward and feedback
branches with transmittance only in zl or z2. An illustra-
tive example for this latter situation is the case where the
given transfer function can be written in the form
The above transfer function can be
representing t(zl, zz), in cascade with a feedback loop
having s(zl, z2) in the forward branch and ~ ( z , )
back branch. However, since in the general case a 2-D
polynomial cannot be factored in lower order l-D and/or
2-D polynomials, the above is not the general case.
Clearly, the decomposition procedure used in the proof
of Theorem I is not unique. A generalization of this latter
decomposition is given by the following expression
realized by an array
in the feed-
q(z1, ~ 2 ) 1 ZTRR-IQS-ISZ2
(14)
where only one of the two matrices R, S is taken into ac-
count every time. The auxiliary matrices R and S can be
&I,
z2) = [ZTR-'l[RQZ,l (15)
is considered, where R is a nonsingular (nl + I) X (nl +
1) matrix, chosen to satisfy the following relation
RQ = [g]
where Q is an r X (nl + 1) matrix and 0 a null (nl + 1
- r) X (nl + 1) matrix. Taking into account (16), (15)
can be written as a sum of r terms of the form (8).
For 1-D polynomials, the order, i.e., the highest degree
of z determines the available degrees of freedom. For 2-D
polynomials we have two orders with respect to zI, z2,
which however do not determine the available degrees of
freedom. To be more specific, let the numerator polynom-
ial q(zl, z2) given in (2). Suppose that rank Q = r I
(nl + 1, ml + 1) where Q is the corresponding coefficient
matrix. Then the realization of this polynomial, based on
the decomposition theorem, needs only r parallel branches.
Thus, the number of the coefficients in the decomposed
form equals to t = r[(nl + 1) + (ml + l)], which is
greater than or equal to the number of coefficients (nl +
1) (ml + 1) in the given form.
At this point we want to clarify the fact that the condi-
tion rank Q = r < min (nl + 1, ml + 1) is not a sufficient
one for the primitive factorization of the polynomial q(zl,
z2) as follows:
min
dz1, z2) = ql(z1) q2(z2) q3(z1, 22).
(17)
Clearly, in the case of primitive factorization we have
rank Q = rank Q3 = r I
min (Al + 1, AI + 1) (18)
where
I
nl,
rizl I
ml, and (Al, AI) < (nl, ml).
Then the polynomial q(zl, z2) needs for its implementation
r parallel branches with
f = r[(R1 + 1) + ( A I + l)]
+ (nl - A,) + (ml - A,)
coefficients, since nl - A,, ml - AI are the orders of the
polynomials ql(zl), q2(z2), respectively. It can be readily
seen that t 2 2, where the equality holds for r = 1, which
is the case of exact factorization of the 2-D polynomial in
l-D terms. From the above results, the rank Q = r deter-
mines the smallest necessary number of parallel branches
for the realization of a 2-D polynomial. To this end, the

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VENETSANOPOULOS AND MERTZIOS: DECOMPOSITION THEOREM AND ITS IMPLICATIONS
1565
term “rank of decomposition” is proposed here to char-
acterize the number r.
In the following, some specific realizations based on the
decomposition theorem will be concisely reviewed. The
main difference among them is the specific choice of the
auxiliary matrices R, S.
A. The Jordan Form Decomposition [I51
A special form of Theorem 1 using the Jordan form is
possible. This approach is based on matrix diagonaliza-
tion. In the general case where nI # ml (i.e., Q is not a
square matrix), we consider the augmented square matrix
Q, defined by
Q = [QIOI,
n1 > m~
(20)
where 0 is an nl X (nl - ml) null matrix, or
where 0 is now an (ml - nl) X ml null matrix. We may
now write
Q = HJH-’
(22)
where H is the transforming matrix consisti!g of
genvectors of Q, and J the Jordan matrix of Q, consisting
of the eigenvalues of Q. Thus, the Jordan form decompo-
sition may be viewed as a special case of Theorem 1. In
this case, the augmented coefficient square matrix Q is
written as a product of two matrices R = HJ and S =
H - ’ . Then
the ei-
I
q(zl, z2) = C ai(zl> bi(z2)7
i=O
1 = max (nl, m ~ .
(23)
Note that the matrices R, S are determined in terms
coefficient matrix Q and cannot be arbitrarily
Furthermore the number of parallel branches is the max
(n1 + 1, m l + 1).
of the
defined.
B. The Singular Value Decomposition (SVD) [ll], 1161
In this case, the coefficient matrix Q of the 2-D poly-
nomial q(zl, z2) is expressed as a product of the form
Q = U[Al V
where A is a diagonal matrix whose entries are the sin-
gular values of Q, and U, and V are the row and the col-
umn eigenvalue system of Q, respectively [ 161. Due to the
orthonormal nature of U and V, Q can be written as
1/2
t
(24)
r
Q =
X!’2~i~f,
r = rank Q
(25)
where ui, and ui, and hi are the column vectors of U, V
and diagonal terms of A, respectively. The SVD tech-
niques have been used in the past for the approximation of
a filter in terms of a number of separable filters [ll] and
for image processing applications [ 161. The matrix Q need
i = I
not be square and the matrices R, S can be identified by
A ]‘I2, V‘ or U, A I’’~ v‘, respectively.
C. The Lower- Upper Triangular, (LU) Decomposition
71-[191
This is also a special case of the general decomposition
scheme where the matrix Q is written as
r
Q = LDU =
diliui
(26)
where L, 1; and U, ui are lower and upper triangular mat-
rices, respectively, and di scalars. As in the previous case,
R and S can be identified by LD, U or L, DU, respectively.
D. Other Decompositions
Furthermore, the matrix Q in (2) could be substituted
by QQ-Q, where “ - ” denotes one of the different gen-
eralized inverses. In this case, R and S can be identified
by Q, Q-Q or QQ- and Q, respectively.
However, it should be mentioned here
characteristics, such as roundoff noise reduction and re-
duction of the computational cost, can be obtained by using
orthogonal auxiliary matrices (therefore having known in-
verses) with integer elements such as Walsh matrices [20].
Generally, it is preferable to use matrices of a known sim-
ple structure with integer and zero elements
avoid internal multiplications.
that desirable
in order to
i = I
An important characteristic of the realizations based on
the decomposition theorem is the possibility to obtain a mul-
titude of them, depending on the choice of the matrices R,
S. Clearly, for example, the realizations suggested by (14)
are not unique. Since each of these realizations is of the
same general structure, we can choose in advance R and
S to achieve a desired objective, such as reduced round-
off noise, ease in the VLSI implementation, minimality,
etc.
111. IMPLICATIONS
OF THE DECOMPOSITION THEOREM
AND REALIZABILITY
In this section we consider the implications of the pre-
vious theorem to the design and realization of 2-D filters.
Throughout this paper, the z-transform of a 2-D sequence
x(n, m), is defined by
m m
X(z1, z2) = c c x(n, m) z;zy
n = - m * = - m
where zl, zz are delay elements [21].
The concept of feedback in 2-D filters is applied in a
way similar to that of 1-D filters [22], [23].
In this section, a corollary of Theorem 1 relevant to the
realization of 2-D filters is presented.
Corollary I : The 2-D filter described by a rational
transfer function of the form (1) can always be realized as
a cascade of three blocks. A cascade block (C), an array
block (A), and a feedback block (F) are defined in Fig. 3,
in a manner shown in Fig. 4.

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IEEE TRANSACTIONS ON ACOUSTICS, SPEECH,
AND SIGNAL PROCESSING, VOL. ASSP-33,
NO. 6, DECEMBER 1985
FEEDBACK
Fig. 3. Realization of general 2-D rational transfer function by the decom-
position theorem.
+
Fig. 4. (a). Realization of the cascade block
Hk,(z,, z2). (c). Realization of l/f&z1, z2).
of Fig. 3. (b). Realization of