Bi-orthogonal filter banks with directional vanishing moments [image representation applications]

Abstract

In this paper we study 2D nonseparable filter banks that annihilate information along a certain discrete direction. This is done by having filters with directional vanishing moments (DVM). We study the approximation property of such filters and the design problem providing conditions for its solvability. In particular, we completely characterize the solution and propose a design procedure utilizing the mapping technique. Nonlinear approximation experiments with the contourlet transform indicate that compared with the traditional filters, the new filters designed with DVM provide gains in SNR and visual quality due to their short size.

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BI-ORTHOGONAL FILTER BANKS WITH DIRECTIONAL VANISHING MOMENTS
Arthur L. da Cunha and Minh N. Do
University of Illinois at Urbana-Champaign
Department of Electrical and Computer Engineering
Coordinated Science Laboratory
Urbana, IL 61801
Email:{cunhada,minhdo}@uiuc.edu
ABSTRACT
In this paper we study 2-D nonseparable filter banks that annihi-
late information along a certain discrete direction. This is done
by having filters with directional vanishing moments (DVM). We
study the approximation property of such filters and the design
problem providing conditions for its solvability. In particular we
completely characterize the solution and propose a design proce-
dure utilizing the mapping technique. Nonlinear approximation
experiments with the contourlet transform indicate that compared
with the traditional filters, the new filters designed with DVM pro-
vide gains in SNR and visual quality due to their short size.
1. INTRODUCTION
Recently there has been a wide interest in image representations
that efficiently handle geometric structure. This comes from the
recognition that wavelets essentially fail to take advantage of ge-
ometric regularity, a common feature in natural images. In 1-
D however, wavelets are known to be optimal in approximating
piecewise smooth signals, a feature that is attained by the presence
of vanishing moments in the transform. It is thus natural to ask the
question of which equivalent feature would be responsible for an
optimal approximation of piecewise smooth images.
In[1]thecontourlettransformisproposedasameanstofixthe
failure of wavelets in handling geometry. It is shown that nonlinear
approximation using contourlets can achieve the optimal approxi-
mation rate for piecewise C2images. This however relies on the
presence of directional vanishing moments in the contourlet frame
element. This condition can be stated in terms of a filter design
constraint.
In this paper we study two channel filter banks with DVM.
Two-channel filters are attractive since they are simpler to design
and can be used in a tree structure to generate more complicated
systems such as the directional filter bank [2]. Our goal is to im-
pose directional vanishing moments in the contourlet basis func-
tion without resorting to long filters. That is, we attempt to cancel
direction information using DVM’s instead of good frequency se-
lectivity, thus working with shorter filters and avoiding the Gibbs
phenomena. Potential applications of the filters designed in this
work are in the contourlet transform of [1], the CRISP-contourlet
system [3] and directionlets [4].
Thanks to CAPES and NSF for funding.
Notation: Throughout the paper we use boldface and capital
boldface characters to represent 2-D vectors and 2 × 2 matrices
respectively. Thus, a discrete 2-D signal is denoted by x[n] where
n = (n1,n2)T. The 2-D z-transform of a signal x[n] is denoted
by X(z), where it is understood that z is short hand for (z1,z2).
If u = (u1,u2)Tis a vector in Z2, then we denote zu= zu1
whereas, zS= zs1zs2with s1 and s2 being the columns of the
matrix S. We also denote the modulation vector Wk
for a 2 × 2 matrix S and an integer vector k.
1zu2
2
S:= ej2πSk
2. DIRECTIONAL EIGEN-SIGNALS
Before we begin, let us formally define directional vanishing mo-
ments.
Definition 1 Let C(z) be a discrete filter and u = (u1,u2)Ta
2-D vector of co-prime integers. We say C(z) has a DVM of order
d along direction θ = tan−1 u2
u1if it factors as
C(z1,z2) = (1 − zu1
For contourlets, the filter C(z) is a composite one, which in-
volves the Laplacian pyramid filters and the polyphase compo-
nents of the directional filters [1].
In view of the previous definition and the 1-D case, it is nat-
ural to ask what signal would be annihilated (i.e., completely fil-
tered out) by the filter in (1). Such signal is an eigen-signal of the
complementary branch of a two-channel FB where C(z1,z2) is an
analysis filter. In fact, those signals can be related to samples of
polynomial surfaces separated by one or more edge discontinuities
along the direction of the vanishing moment as the next theorem
shows.
1zu2
2)dR(z1,z2).
(1)
Theorem 1 Suppose that x[n] is the signal obtained by sampling
a continuous time signal xc(t), that is,
x[n] = xc(∆Tn)
where xc(t) is a piecewise polynomial of degree less than d sepa-
rated by lines of slope θ = tan−1 u2
C(z).
u1. Then x[n] is annihilated by
Proof: See [5]. Figure 2 illustrates a typical signal described in
Theorem 1 and its filtered version. The filter has a DVM along the
edge discontinuity direction.
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Fig. 1. Illustration of line zero moments as an edge annihilator.
The piecewise polynomial image in (a) was filtered with a 2-D
filter H(z1,z2) =
?1 − z1z2
2
?3. The output image (b) has no
edges.
3. TWO-CHANNEL FILTER BANKS WITH
DIRECTIONAL VANISHING MOMENTS
S
SS
S
x[n]
x[n]
Fig. 2. Two-channel Filter Bank.
A typical 2-channel filter bank is the Fan filter bank illustrated
in Figure 2. For the FIR case with an arbitrary down-sampling
matrix S, the synthesis/analysis filters are related by
(H1(z), G1(z)) =
?
zk1G0(Wk1
S−T◦ z),z−k1H0(Wk1
S−T◦ z)
?
where(·)Tstandsforvectortransposition, “◦”denotesdirectprod-
uct of vector entries and k1is the nonzero integer vector in the set
N(S) := {STx,x ∈ [0,1) × [0,1)} ([6] pp. 575). The analy-
sis/synthesis relation gives the perfect reconstruction condition:
H0(z)G0(z) + H0(Wk1
S−T◦ z)G0(Wk1
S−T◦ z) = 2.
(2)
Thus, our problem consists in designing filters H0(z) and G0(z)
such that the above PR condition is met and in addition the filters
have the form in Definition 1, that is
H0(z) = (1 − zu1
1zu2
2)NaRH0(z)
and
G0(z) = (1 − zu1
1zu2
2)NsRG0(z)
where Na and Ns denote the order of the DVM in the analysis
and synthesis parts respectively. Substituting these filters in (2) we
obtain the design equation
(1 − zu)LRP0(z)+
?
1 − (Wk1
S−T◦ z)u?L
RP0(Wk1
S−T◦z) = 2
(3)
where RP0(z) := RH0(z)RG0(z) and L = Na+ Ns.
It can be shown [5] that an FIR filter RP0(z) satisfying the
above equation exists if and only if uT2S−Tk1is an odd integer.
If that is the case, the following proposition greatly simplifies the
problem.
Proposition 1 Consider the filter equation (3) with u such that
uT2S−Tk1is odd. Then there exists a unimodular1integer matrix
1A square matrix is unimodular if its determinant is equal to one.
≡
H0(ω)
H0(RTω)
RR
Fig. 3. Change of variable is equivalent to a pre/post re-sampling
operation plus filtering with modified filter.
R such that if R(z) solves (3) then˜R(z) = R(zR) solves
(1 − z1)L˜R(z) + (1 + z1)L˜R(Wk1
Conversely, if˜R(z) is a solution to (4), then there exists a matrix
R such that R(z) =˜R(zR−1) is a solution to (3).
R−1S−Tz) = 2
(4)
The previous Proposition asserts that it suffices to consider the
design of filters with DVM’s along the horizontal direction. It can
be interpreted in terms of pre/post re-sampling operations as Fig-
ure 3 illustrates. In addition, we need to consider (4) with only
two possible forms for˜R(Wk1
˜R(−z1,−z2) corresponding to the rectangular and quincunx lat-
tices. The other possible lattice is the transpose of the rectangular.
The next proposition provides a complete characterization of the
solution to (4).
R−1S−Tz), namely˜R(−z1,z2) or
Proposition 2 Let a ∈ {1,−1}, then an FIR filter˜R(z1,z2) is
solution to the equation
(1 − z1)L˜R(z1,z2) + (1 + z1)L˜R(−z1,az2) = 2
if and only if it has the form
(5)
˜R(z1,z2) = RL(z1) + (1 + z1)LRo(z1,z2)
(6)
with RL(z1) being a univariate solution given explicitly by
RL(z1) =
L−1
?
i=0
?
L + i − 1
L − 1
?
2−(L+i−1)(1 + z1)i
(7)
and Ro(z) satisfying
Ro(z1,z2) + Ro(−z1,az2) = 0
(8)
Remarks:
1. The above Proposition is akin to its 1-D counterpart which
is used to construct compactly supported wavelets (see e.g., [7]).
The distinction occurs in the higher order term which now can
be any two-dimensional function satisfying (8). This higher or-
der term will make the filter a truly two dimensional one, meaning
a filter with a nonseparable support. Moreover, this higher order
term can be used to control the shape of the 2-D frequency re-
sponse.
2. This already provides a class of biorthogonal filters with
DVM in which each one of the filters in the analysis and synthesis
is a degenerate 1-D solution. Thus, the filters obtained as in Propo-
sition 2 can be seen as a 2-D generalization of the bi-orthogonal
spline wavelet filters of [7].
Proposition2providesacompletecharacterizationoftheprod-
uct filter H0(z)G0(z) for a filter bank having DVM’s. To obtain
the filters one needs to factorize this product which is difficult in
multiple dimensions. Next we show a way to overcome this need
by using transformation of variable.
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4. DESIGN
One successful method for multidimensional filter design is the
transformation of variable or mapping technique discussed in [8].
This method has the desirable feature of easy control of the fre-
quency and phase responses. It consists in replacing the complex
variable z in a 1-D biorthogonal pair of filters, or some 1-D func-
tion of it, by a function defined on the C2plane. Specifically, we
may write the 1-D filters as H(1D)
0
use the mapping f(z) ?→ M(z), the same applying for the re-
maining filters. It is not hard to see that perfect reconstruction
condition will be kept provided [8]
(f(z)),G(1D)
0
(f(z)) and then
f(z) = −f(−z), and, M(z) = −M(Wk1
For FIR solutions, one typically has f(z) = z + z−1for zero-
phasefiltersorf(z) = z forfiltersconsistingofonlypositivepow-
ers of z. Notice that in both cases, one cannot have G(1D)
H(1D)
0
(z−1).
In the context of DVM, the goal is to devise a mapping func-
tionM(z)suchthateachofthe2-DfiltersH0(z) := H(1D)
G0(z) := G(1D)
0
(M(z)) has a given number of (1 − z1) factors.
In addition we require M(z) so that perfect reconstruction is kept
after mapping. The next proposition shows an explicit form of the
required mapping function.
S−T◦ z).
(9)
0
(z) =
0
(M(z)),
Proposition 3 Let H(1D)
1-D filter pair and a ∈ {1,−1}. Suppose M(z) is an FIR map-
ping function such that
0
(f(z)),G(1D)
0
(f(z)) be a bi-orthogonal
M(z1,z2) = (1 − z1)L˜R(z1,z2) + c0
where c0is such that H(1D)
0
G(1D)
0
(f(z)) has a factor (f(z)−c0)Ns/L, and˜R(z1,z2) satisfies
the valid mapping equation:
(10)
(f(z)) has a factor (f(z) − c0)Na/L,
(1 − z1)L˜R(z1,z2) + (1 + z1)L˜R(−z1,az2) = 2c0.
Then
(11)
1. H0(z),G0(z) form a bi-orthogonal pair, i.e., they satisfy
H0(z1,z2)G0(z1,z2)+H0(−z1,az2)G0(−z1,az2) = 2.
(12)
2. The mapped filters factor as
H0(z) = (1−z1)NaRH0(z), G0(z) = (1−z1)NsRG0(z).
(13)
The previous proposition together with the characterization of
the mapping function of Proposition 2 allows for an easy design
methodology that can be summarized in the following steps:
Design a 1-D filter pair H(1D)
Na and Ns zeros at some point c0 ∈ C. If solution needs
to be FIR, then constrain the filters to have positive powers
of f(z) only.
Step 2 If necessary, perform a change of variables to convert the
problem to the form in (4).
Step 3 Let M(z) = (1−z1)L˜R(z)+c0with˜R(z) =˜RL(z1)+
(1 + z1)L˜Ro(z1,z2) and˜Ro(z1,z2) = −˜Ro(−z1,az2)
Step 1
0
(f(z)),G(1D)
0
(f(z)) with
Step 4 Substitute f(z) ?→ M(z) in the 1-D filters and obtain the
2-D ones.
The next example illustrates how a typical design is carried
out. Refer to [5] for more details.
Example 1 In order to get a solution with four directional vanish-
ing moments (for the Quincunx lattice), we use the minimal degree
complementary filter to (1 − f(z))4, which from (7), gives the
product filter
P(1D)(f(z)) =
1
16(16−29f(z)+20f(z)2−5f(z)3)(1−f(z))4,
where we let f(z) = (z + z−1)/2 to obtain zero-phase solutions.
We factor the above polynomial assigning two zeros for one filter
and one for the other. Moreover, the fourth order zero at f(z)
is equally split between H(1D)
0
mapping function we choose it to have second order zero moment
term. The mapping function we use is
(f(z)), and G(1D)
0
(f(z)). For the
M(z1,z2) =?2 + z1+ z−1
where to guarantee that the map satisfies the valid mapping con-
dition (10) we pick
1
?R(z1,z2) − 1
R(z1,z2) = 1 + (2 − z1− z−1
For simplicity we choose Ro(z1,z2) = α(z2 + z−1
α = 0wesetthe9-7filters. Figure4illustratetheanalysis/synthesis
pair frequency response when we set α =√2.
1 )Ro(z1,z2)
2). With
Fig. 4. Frequency response of example analysis and synthesis
filters designed with 4-th order directional vanishing moment.
5. EXPERIMENTS
In order to illustrate the applicability of the directional vanishing
moment filters proposed, we perform an experiment in which we
replace the DFB filters of the contourlet construction with the fil-
ters designed with zero moments, following the discussion in [5].
In order to assess the filters performance, we reconstruct the de-
tail subspace Wj with the N largest coefficients and compute the
corresponding SNR. The directional expansion tree we use in each
scale is one that leads to a maximum number of distinct DVMs and
is the same for all test images, hence the expansion is fixed. The
analysis filter is zero-phase with 7×13 coefficients and the syn-
thesis with 9×17, also zero-phase. As a comparison, we use the
quincunx/fan filters of [9] (PKVA), where the analysis filter has
23×23 taps and the synthesis 46×46.
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(a)
00.020.040.06 0.080.10.120.140.16
0
2
4
6
8
10
12
Percentage of coefficients retained (%)
SNR (dB)
DVM
PKVA
WAVELET
(b)
Fig. 5. NLA Approximation on the detail subspace. (a) Synthetic
piecewise polynomial image. (b) SNR curves for the Synthetic
image.
Figure 5 displays the curve of a piecewise polynomial image.
For this synthetic image, a significant improvement is observed.
Such improvement is due to the fact that the synthetic image has
directional information in a very small set of directions, which,
due to DVM’s, are well represented in the expansion. For natu-
ral images the DVM filters render a slightly better performance
than the PKVA filters. However, because the DVM filters are con-
siderably shorter, we observe less ringing artifacts when compared
against the PKVA filters, even when both give similar SNR. Figure
6 shows the “Peppers” image reconstructed with 2048 coefficients
using both of the filters. As can be seen, the image reconstructed
with the DVM filters exhibit less ringing artifacts.
6. CONCLUSION
We have studied 2-channel biorthogonal filter banks in which one
filter bank channel kills information along a prescribed direction
by means of directional vanishing moments. Such filters have
proven to be useful as building blocks for more complicated de-
composition structures such as the contourlet expansion. We stud-
ied the DVM filter bank design problem and provided a complete
characterization of the product filter. Using the mapping design
methodology, we proposed a design procedure in which the map-
ping filter can be calculated explicitly. Experimental results indi-
cate that filters with DVM’s can be as good as filters designed with
frequency response as a primary criteria and, in some cases, yield
better results. In addition, because the filters are short, the Gibbs
phenomena is considerably reduced.
(a) Old PKVA filters(b) New DVM filters
Fig. 6. “Peppers” Image reconstructed with 2048 coefficients. (a)
Old PKVA filters, (b) New DVM filters. The image on the right
shows less ringing artifacts.
7. REFERENCES
[1] M. N. Do and M. Vetterli, “The contourlet transform: An efficient
directional multiresolution image representation,” IEEE Trans. Img.
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[2] R. H. Bamberger and M. J. T. Smith, “A filter bank for the directional
decomposition of images: Theory and design,” IEEE Trans. Signal
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[3] Y. Lu and M. Do, “Crisp-contourlet: a critically sampled directional
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regular multi-directional wavelet bases with one-dimensional filter-
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[5] A. L. da Cunha and M. N. Do, “Biorthogonal two-channel filter banks
with directional vanishing moments: Characterization, design and ap-
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[6] P. Vaidyanathan, Multirate Systems and Filterbanks.
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[7] I. Daubechies, Ten Lectures on Wavelets.
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