IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
749
Lifting Structures for Quincunx FIR Filter-Banks
With Quadrantal or Diagonal Symmetry
Bhushan D. Patil, Pushkar G. Patwardhan, and Vikram M. Gadre
Abstract—Lifting structures, also referred to as ladder structures, have been used for the design of linear phase (centro-symmetric) 2-D Quincunx filter-banks. In this letter, our aim is to use
lifting structures for the construction of Quincunx filter-banks
with the filters having the higher order symmetries viz. quadrantal
or diagonal symmetries. We derive conditions on the predict and
update steps of the lifting structure so that the filters in the
filter-bank have quadrantal or diagonal symmetry. Using these
conditions, we construct example structures for the predict and
update steps which yield Quincunx filter-banks with quadrantally
or diagonally symmetric filters.
Index Terms—lifting, Quincunx filter-banks, two-dimensional
filter-banks.
Fig. 1. Lifting structure for Quincunx filter-bank.
I. INTRODUCTION
ARIOUS approaches for the design of 2-D linear phase
Quincunx filter-banks have been considered in the literature. Design using cascade structures has been considered in [1],
whereas [2]–[8] use lifting structures. Design approaches using
the method of generalized McClellan transformation have been
presented in [9] and [10]. All these approaches impose the condition of linear phase in the filters of the Quincunx filter-bank
which implies that the filter impulse response has centro-symmetry. As observed in [12] and [13], higher order symmetries
are possible in the 2-D impulse response (2-D signals in general). Quadrantal and diagonal symmetries are some of the more
common symmetries. These symmetries are also particularly
useful in the context of Quincunx filter-banks, since the desired ideal passbands of the filters in a Quincunx filter-bank have
quadrantal and/or diagonal symmetry. For the case of 2-D Quincunx filter-banks, filters having these symmetries are required to
“preserve” signal symmetries in 2-D signal extension schemes
[14], [15]. One of the requirements for preserving signal symmetries, as discussed in the above references, is to design Quincunx filter-banks with the filters having quadrantal or diagonal
symmetries.
Cascade structures which yield Quincunx filter-banks with
the filters having quadrantal or diagonal symmetry have been
proposed in [15]. In particular, [15] derives a set of conditions
on the polyphase matrix of the Quincunx filter-bank such that
the filters in the filter-bank have quadrantal or diagonal symmetry. In this letter, our aim is to consider lifting structures for
the design of Quincunx filter-bank with quadrantal or diagonal
V
Manuscript received March 11, 2008; revised May 25, 2008. The associate
editor coordinating the review of this manuscript and approving it for publication was Dr. Nikolaos Boulgouris.
The authors are with the Department of Electrical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India
(e-mail:
[email protected];
[email protected];
[email protected]).
Digital Object Identifier 10.1109/LSP.2008.2001807
symmetry. Using the results of [15], we derive a set of conditions on the “lifting steps” (the predict and update steps) which
guarantee that the final filters have the desired symmetry.
The rest of this letter is organized as follows: In Section I-A,
we review the lifting structure for the Quincunx filter-bank, and
in Section I-B, we briefly review the characterization of quadrantal and diagonal symmetries. In Section II-A, we consider
quadrantally symmetric Quincunx filter-banks, and we derive
conditions on the lifting steps for the filter-bank to have quadrantal symmetry. In Section II-B, we present conditions on the
lifting steps for diagonally symmetric Quincunx filter-banks.
We present design examples in Section III.
Notation: Boldfaced lowercase letters are used to represent
vectors, and boldfaced uppercase letters are used for matrices.
denotes the transpose of
denote the inverse
of , and
denotes the determinant of . A vector
raised to a matrix power
is
defined as a vector
is a vector whose th entry is
,
where
. We use
to refer to the specific Quincunx
sampling matrix
.
A. Review of Quincunx Filter-Bank and the Lifting Structure
Fig. 1 shows the lifting structure for the Quincunx filter-bank.
Here,
and
are the analysis and synthesis polyphase
matrices of a “base” perfect reconstruction (PR) Quincunx
filter-bank. The predict and update lifting steps can be used
to increase the order of the polyphase matrix (and thus of the
filters) while maintaining PR. The predict and update steps
involve adding the following factors to the analysis polyphase
matrix:
The corresponding synthesis matrix is given by
1070-9908/$25.00 © 2008 IEEE
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on January 14, 2009 at 01:12 from IEEE Xplore. Restrictions apply.
750
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
.
b)
Now, our aim is as follows: Suppose we have a polyphase
matrix
We note that multiple predict and/or update steps could be
and
functions, while
added, possibly with different
maintaining PR of the filter-bank.
, also
referred to as the “lazy wavelet,” is a typical choice as the
“base filter-bank” before applying the lifting steps. However, in
general, any PR filter-bank could be used as the starting point.
In this letter, the problem that we address is as follows: Given
a PR Quincunx filter-bank with quadrantally or diagonally symmetric filters, what are the conditions on the predict and update
and
, respectively, so that the symmetry of
factors,
the filters in the filter-bank is maintained?
satisfying the conditions
of Proposition-1. We now wish to increase the order by using a
lifting step (either a predict or an update step). The question we
want to address is: What are the conditions on the lifting steps
such that the new polyphase matrix obtained after the lifting step
also satisfies Proposition-1?
First consider the predict step. The new polyphase matrix obtained after the predict step is
(1)
Note that the predict-function
From (1), it is evident that
is a 2-D transfer function.
B. Review of Quadrantal and Diagonal Symmetry
Characterization [14], [15]
(2)
A quadrantally symmetric 2-D sequence, with the
, can be characcenter of symmetry
terized as [14], [15]
where
Further, we have from (1)
and
A diagonally symmetric 2-D signal, with center of symmetry
can be characterized as follows [14], [15]:
(3)
and
to be both
Now, first of all, we would like
diagonally symmetric. This can be achieved as follows.
Let
be diagonally symmetric. We will denote the center
as
. From (3), we have
of symmetry (COS) of
. We would
. We can achieve
like to have
this if we can ensure that
where
and
Thus, we require that
II. QUADRANTALLY SYMMETRIC QUINCUNX FILTER-BANKS
In this section, we derive conditions on the lifting steps so
that the filters in the Quincunx filter-bank have quadrantal
or diagonal symmetry. We consider quadrantal symmetry in
Section II-A and diagonal symmetry in Section II-B.
but, since the above equation satisfies Proposition-1, we have
and
.
Thus, we choose
(4)
A. Conditions for Quadrantally Symmetric Quincunx
Filter-Banks
From [15, Proposition-1], the conditions on the polyphase
analysis for a quadrantally symmetric Quincunx filter-bank are
as follows.
Proposition-1: [15]: Let the analysis polyphase
matrix
be
.
Then,
should
have
diag(i.e.,
onal symmetry with symmetry parameters
with symmetry, and not anti-symmetry), and with centers of
symmetry as
, and
respectively, and satisfying the following constraints:
a)
, where is an arbitrary integer vector,
and
With this, we have
(5)
It can be verified that with the choice of
have
as in (4), we also
(6)
From (3)–(5), it follows that
also satisfies Proposition-1.
Lemma-1: Given a polyphase matrix
satisfying Propoobtained by the presition-1, the new polyphase matrix
dict lifting step as
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on January 14, 2009 at 01:12 from IEEE Xplore. Restrictions apply.
will also satisfy
PATIL et al.: LIFTING STRUCTURES FOR QUINCUNX FIR FILTER-BANKS WITH QUADRANTAL OR DIAGONAL SYMMETRY
Proposition-1 if the predict function
is chosen to be diagonally symmetric with center of symmetry as
.
Now consider the update step: The new polyphase matrix obtained after the predict step is
(7)
Again, the update function,
From (7), it is evident that
, is a 2-D transfer function.
(8)
Further, we have from (7)
751
III. DESIGN EXAMPLES
We now present examples of lifting steps which satisfy the
conditions derived in Section II and which can be used to construct Quincunx filter-banks with quadrantal or diagonal symmetry.
Example-1: In this example, we construct lifting steps for
a quadrantally symmetric Quincunx filter-bank. As the “base
filter-bank,” we start with the PR quadrantally symmetric Quincunx filter-bank from [15]. We then give examples of lifting predict and update steps which can be used to increase the order
of the polyphase matrix, while maintaining PR and symmetry.
From [15, Section III-A], we have the following polyphase matrix:
(9)
We would like
and
to be both diagonally
symmetric. Following similar arguments as done for the predict
step, we can achieve this by having
(10)
It can be verified that, with (10), we have
Here
, and are parameters subject to some constraints [15]. For this polyphase matrix we have
and
We note that the above
includes the identity matrix as a
special case with
and
.
Thus, from Lemma-1, we should choose a diagonally symmetric
such that
and
(11)
also satisfies PropoThus, from (8) and (11), it follows that
sition-1. We can summarize this as follows.
Lemma-2: Given a polyphase matrix
satisfying propoobtained by the upsition-1, the new polyphase matrix
will also satisfy
date lifting step as
is chosen to be diagProposition-1 if the update function
onally symmetric with center of symmetry as
.
A possible choice of
is
where and are parameters.
With this, the new polyphase matrix is given by
where
B. Conditions for Diagonally Symmetric Quincunx
Filter-Banks
In this section, we present conditions on the lifting steps for
Quincunx filter-banks with diagonal symmetry. For diagonally
symmetric Quincunx filter-banks, the analysis polyphase matrix
should satisfy [15, Proposition-2].
Lemma-3: Given a polyphase matrix
satisfying [15,
Proposition-2], the new polyphase matrix
obtained by the
predict lifting or update steps as
, or
, will also satisfy [15, Proposition-2]
if the lifting steps satisfy the following conditions.
is quadrantally symmetric with
a) The predict function
.
center of symmetry as
b) The update function
is quadrantally symmetric with
center of symmetry as
.
and
It can
satisfy Proposition-I, and this gives
be verified that
quadrantally symmetric analysis filters.
With the above predict step, we increased the order of resulting analysis highpass
, while maintaining quadrantal
symmetry and PR. To increase the order of resulting lowpass
filter
, we can use the following update stage:
So this choice of the update step satisfies Lemma-2.
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on January 14, 2009 at 01:12 from IEEE Xplore. Restrictions apply.
752
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
used only a single predict and update step. Further, the symmetry aspect of the filters is not explicitly addressed in [3].
IV. CONCLUSION
Fig. 2. Plots for design Example-2. (a) Analysis lowpass filter. (b) Analysis
highpass filter.
It can be verified that the polyphase matrix obtained after the
update step also maintains the quadrantal symmetry of the Quincunx filter-bank.
Example-2: The purpose of this example is to show that
Quincunx filter-banks with good frequency characteristics can
be obtained using the lifting steps proposed in Example-1.
Using the base filter-bank and ten lifting steps of Example-1,
we use Matlab’s fminunc function to optimize the parameters
of the lifting steps. We use an objective function involving the
squared error in the passband and stopband of the analysis lowpass and highpass filters, using the diamond shaped response
as the desired ideal lowpass passband. Fig. 2(a) and (b). shows
the frequency response of the designed filters.
Example-3: For this example, we consider a Quincunx filterbank having both diagonal and quadrantal symmetry. This is
also referred to as octagonal symmetry.
It is interesting to note that starting with
, the predict
and update steps presented in [3] satisfy Lemmas 1–3. We note
that
trivially satisfies Propositions 1 and 2 with
and
Thus, the Quincunx filter-bank designed in [3] has octagonal
symmetry. For example, consider the following specific choices
for the predict and update steps from [3]:
and
For this case,
and
are both octagonally (quadrantal
and diagonal) symmetric with
and
.
Thus, from the results of Section II, the above choice of
and
will yield octagonally symmetric Quincunx
filter-banks (i.e., the polyphase matrix will satisfy both Propositions 1 and 2).
Further, based on our results of Sections II-A and III-A, we
could use multiple stages of the lifting step. We note that [3]
In this letter, we presented conditions on the lifting steps to
design Quincunx filter-banks with quadrantal or diagonal symmetry. In particular, we showed the following:
a) to design quadrantally symmetric Quincunx filter-banks,
the lifting steps should have diagonal symmetry with a
constraint on their center of symmetry; and
b) to design diagonally symmetric Quincunx filter-banks,
the lifting steps should have quadrantal symmetry with a
constraint on their center of symmetry.
We presented examples of quadrantally and diagonally symmetric Quincunx filter-banks, by using lifting steps satisfying
the conditions derived in this letter.
REFERENCES
[1] G. Karlsson and M. Vetterli, “Theory of two-dimensional multirate
filter banks,” IEEE Trans. Signal Process., vol. 38, no. 6, pp. 925–937,
Jun. 1990.
[2] J. Kovacevic and M. Vetterli, “Nonseparable multidimensional perfect
reconstruction filter banks and wavelet bases for
,” IEEE Trans. Inf.
Theory, vol. 38, no. 2, pp. 533–555, Mar. 1992.
[3] J. Kovacevic and W. Sweldens, “Wavelet families of increasing order
in arbitrary dimensions,” IEEE Trans. Image Process., vol. 9, no. 3, pp.
480–496, Mar. 2000.
[4] Z. Lei, A. Makur, and X. Zhiming, “On lifting factorization for 2-D
LPPRFB,” in Proc. IEEE Int. Conf. Image Processing, Oct. 2006, pp.
2145–2148.
[5] A. Gouze, M. Antonini, and M. Barlaud, “Quincunx lifting scheme for
lossy image compression,” in Proc. Int. Conf. Image Processing, Sep.
2000, vol. 1, pp. 665–668.
[6] A. Gouze, M. Antonini, M. Barlaud, and B. Macq, “Design of
signal-adapted multidimensional lifting scheme for lossy coding,”
IEEE Trans. Image Process., vol. 13, no. 12, pp. 1589–1603, Dec.
2004.
[7] S.-M. Phoong, C. W. Kim, P. P. Vaidyanathan, and R. Ansari, “A
new class of two-channel biorthogonal filter banks and wavelet bases,”
IEEE Trans. Signal Process., vol. 43, no. 3, pp. 649–665, Mar. 1995.
[8] I. Shah and T. Kalker, “Ladder structures for multidimensional
linear phase perfect reconstruction filter banks and wavelets,” in
Proc. SPIE Conf. Visual Communications and Image Processing,
Boston, MA, 1992, pp. 12–20. [Online]. Available: http://citeseer.ist.psu.edu/kalker92ladder.html.
[9] D. Tay and N. Kingsbury, “Flexible design of multidimensional perfect
reconstruction FIR 2-band filter-banks using transformation of variables,” IEEE Trans. Image Process., vol. 2, no. 10, pp. 466–480, Oct.
1993.
[10] I. Shah and A. Kalker, “Theory and design of multidimensional QMF
sub-band filters from 1-D filters and polynomials using transforms,”
Proc. Inst. Elect. Eng., vol. 140, pp. 67–71, Feb. 1993.
[11] P. P. Vaidyanathan, Multirate Systems and Filter Banks, ser. Signal
Processing. Englewood Cliffs, NJ: Prentice-Hall, 1992.
[12] H. C. Reddy, I. Khoo, and P. K. Rajan, “2-D symmetry: Theory and
filter design applications,” IEEE Circuits Syst. Mag., vol. 3, no. 3, pp.
4–31, 2003.
[13] M. N. S. Swamy and P. K. Rajan, , S. G. Tzafestas, Ed., “Symmetry
in Two-Dimensional Filters and its Application,” in Multidimensional
Systems: Techniques and Applications. New York: Mercel Dekker,
1986, pp. 401–468.
[14] P. G. Patwardhan and V. M. Gadre, “Preservation of 2-D signal symmetries in quincunx filter banks,” IEEE Signal Process. Lett., vol. 14,
no. 1, pp. 35–38, Jan. 2007.
[15] P. G. Patwardhan, B. Patil, and V. M. Gadre, “Polyphase conditions
and structures for 2-D quincunx FIR filter banks having quadrantal or
diagonal symmetries,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol.
54, no. 9, pp. 790–794, Sep. 2007.
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on January 14, 2009 at 01:12 from IEEE Xplore. Restrictions apply.
R