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1

A Directional Extension for Multidimensional

Wavelet TransformsYue Lu∗ and Minh N. Do

IP EDICS: 2-WAVP (Wavelets and Multiresolution Processing).

Abstract

Directional information is an important and unique feature of multidimensional signals. As a result

of a separable extension from one-dimensional (1-D) bases, multidimensional wavelet transforms have

very limited directionality. Furthermore, different directions are mixed in certain wavelet subbands. In

this paper, we propose a simple directional extension for wavelets (DEW) that fixes this subband mixing

problem and improves the directionality. The building block of the DEW is a two-channel 2-D filter

bank with a checkerboard-shaped frequency partition. The DEW works with both the critically-sampled

wavelet transform as well as the undecimated wavelet transform. In the 2-D case, it further divides

the three wavelet subbands (i.e. horizontal, vertical, and diagonal) at each scale into six finer directional

subbands. The DEW itself is critically-sampled, and hence will not increase the redundancy of the overall

transform. Though nonseparable in essence, the proposed DEW has an efficient implementation that only

requires 1-D filtering. Meanwhile, the DEW can be easily generalized to higher dimensions. In a nutshell,

the proposed directional extension provides an optional tool to efficiently enhance the directionality of

multidimensional wavelet transforms. Numerical experiments show that certain wavelet-based image

processing applications will benefit from this improved directionality.

Index Terms

Wavelet transform, directional information, checkerboard filter bank, filter design, multidimensional

signal processing, image denoising, feature extraction.

Y. Lu is with the Department of Electrical and Computer Engineering and the Coordinated Science Laboratory, University of

Illinois at Urbana-Champaign, Urbana IL 61801 (e-mail: [email protected]; WWW: http://www.ifp.uiuc.edu/∼yuelu).

M. N. Do is with the Department of Electrical and Computer Engineering, the Coordinated Science Laboratory, and

the Beckman Institute, University of Illinois at Urbana-Champaign, Urbana IL 61801 (e-mail: [email protected]; WWW:

http://www.ifp.uiuc.edu/∼minhdo).

This work was supported by the National Science Foundation under Grant CCR-0237633 (CAREER).

April 1, 2005 DRAFT

2

I. INTRODUCTION

Directional information is a unique feature of multidimensional (MD) signals. Recently, the importance

of directional information has been recognized by many image processing applications, including feature

extraction, enhancement, denoising, classification, and compression.

The wavelet transform [1], [2], [3] has a long and successful history as an efficient image processing

tool. However, as a result of a separable extension from one-dimensional (1-D) bases, wavelets in higher

dimensions can only capture very limited directional information. For instance, 2-D wavelets only provide

three directional components, namely horizontal, vertical, and diagonal. Furthermore, the 45◦ and 135◦

directions are mixed in diagonal subbands.

There have been a number of approaches in providing finer directional decomposition. Some notable

examples include 2-D Gabor wavelets [4], the steerable pyramid [5], the directional filter bank [6], 2-D

directional wavelets [7], brushlets [8], complex wavelets [9], [10], [11], curvelets [12], and contourlets

[13], [14]. However, the wavelet transform is still very attractive for image processing applications for

a number of reasons. First, the wavelet transform has a critically-sampled implementation, and can be

easily extended to multidimensional cases. In contrast, most of the approaches mentioned above are

expansive systems with various redundancy ratios. For example, the complex wavelet transform is 4-

times redundant for images, and in general 2N -times redundant for N -dimensional signals. In terms of

implementation complexity, the multidimensional wavelet transforms can be implemented efficiently in

a separable fashion. In contrast, systems such as the directional filter bank involve nonseparable filtering

and sampling and have high computational complexity. Last but not least, the theory and applications

of wavelets have already been extensively studied, offering us a plethora of ready-to-use filters and

processing algorithms.

Therefore the natural question is: “Can we extend the wavelet transform with finer directionality, while

still retain its structure and desirable features?” We give an affirmative answer in this paper by proposing

a simple directional extension for wavelets (DEW). Based on a nonredundant checkerboard filter bank,

the proposed DEW works with both the critically-sampled wavelet transform as well as the undecimated

wavelet transform. In the 2-D case, the DEW leads to one lowpass subband and six directional highpass

subbands at each scale, and fixes the subband mixing problem of wavelets (see Figure 1). Being critically-

sampled itself, the DEW will not increase the redundancy of the overall transform. Though nonseparable

in essence, the DEW has an efficient implementation based on 1-D operations only. Finally, the DEW

can be easily generalized to higher dimensions.

DRAFT April 1, 2005

LU AND DO: A DIRECTIONAL EXTENSION FOR MULTIDIMENSIONAL WAVELET TRANSFORMS 3

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LHHH HH

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ω2 (π, π)

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(a)

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Fig. 1. Division of the 2-D frequency spectrum. (a) The frequency decomposition of the wavelet transform. (b) The frequency

decomposition of the wavelet transform with the proposed directional extension.

The outline of the paper is as follows. Section II presents the filter bank construction of the directional

extension in 2-D, with emphasis on intuition and illustration. The generalization of the results to higher

dimensional cases is given in Section III with a more rigorous treatment. Section IV discusses filter

design and efficient implementation. We will present some numerical results in Section V and conclude

the paper in Section VI. Some preliminary results of this paper were reported in an earlier conference

paper [15].

II. THE DIRECTIONAL EXTENSION IN THE 2-D CASE

The traditional way to construct 2-D wavelets is to use tensor products of their 1-D counterparts.

The advantage of this approach is its simple separable implementation. Unfortunately, this also imposes

serious limits on the directionality of the resulting frequency partitioning. As shown in Figure 1(a), the

2-D wavelet transform produces one lowpass subband (LL), and three highpass subbands (HL, LH, HH),

corresponding to the horizontal, vertical, and diagonal directions. Furthermore, diagonal subbands mixes

the directional information oriented at 45◦ and 135◦. The main idea here is to find some directional

extension to further divide each highpass subband of the wavelets into two branches. In particular,

we want to have a system with the frequency partitioning shown in Figure 1(b), which contains six

directional subbands roughly oriented at 15◦, 45◦, 75◦, 105◦, 135◦ and 165◦. This is the same frequency

decomposition provided by the 2-D dual-tree complex wavelet transform [9], [10], [11], which has been

shown to be successful in several image processing applications. However, the 2-D complex wavelet

transform is 4-times redundant and uses a different filter bank structure compared to wavelets. In the

following, we will discuss our directional extension for wavelets in two cases, i.e., when the wavelet

transform is critically-sampled or undecimated.

April 1, 2005 DRAFT

4

horizontal

vertical

LL

LH

HL

HH

D2

D2

D2

D2

D1

D1

F0(z2)

F0(z2)

F0(z1)

F1(z2)

F1(z2)

F1(z1)

Fig. 2. The filter bank construction of the 2-D DWT for one level of decomposition. The filtering and downsampling operations

are performed in a separable fashion. F0 and F1 denote the 1-D lowpass and highpass filters, with dark regions representing the

ideal passband. D1 = diag(2, 1) and D2 = diag(1, 2) are the 2-fold downsampling matrices along the horizontal and vertical

directions, respectively.

A. Working with the Critically-Sampled Discrete Wavelet Transform (DWT)

This case corresponds to the most widely-used wavelet implementation in which the lowpass and

highpass filters are always followed by a 2-fold downsampling. Figure 2 shows the filter bank imple-

mentation of the DWT for one level of decomposition. For notations used in the figure and hereafter,

we use lower-case letters x[n], where n = (n1, n2)T , to denote 2-D discrete signals or filters. We use

the corresponding uppercase letters X(z1, z2) and X(ejω1 , ejω2) for their z-transforms and discrete-time

Fourier transforms, respectively. When the signal or filter is 1-D, the above notations will be simplified as

x[ni], X(zi) and X(ejωi), with i = 1 or 2 specifying the particular dimension. We use the downsampling

matrices

D1 = diag(2, 1) =

⎛⎝ 2 0

0 1

⎞⎠ and D2 = diag(1, 2) =

⎛⎝ 1 0

0 2

⎞⎠

to represent the 2-fold downsampling operations along the horizontal and vertical directions, respectively.

In general, the multidimensional (MD) downsampling operation [16] is specified by an integer matrix

M as y[n] = x[Mn], where x[n] and y[n] are the MD input and output signals.

To see what the desired directional extension should be, we can examine the frequency contents in the

wavelet subbands. For simplicity, we first consider the case of using ideal filters, where the frequency

response of each filter is constant in the passband and exactly zero in the stopband. We know the

diagonal subband (HH) captures certain directional highpass frequency information (illustrated as regions

{a, b, c, d} in Figure 3(a)) in the input image, where {a, d} and {b, c} correspond to directional information

oriented at 45◦ and 135◦, respectively. With the downsampling operations in the wavelet transform, these

DRAFT April 1, 2005


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ω1

ω2 (π, π)

−(π, π)

a b

c d

(a)

ω1

ω2 (π, π)

−(π, π)

ab

cd

(b)

Fig. 3. (a) The diagonal highpass frequency regions of the input signal. (b) The frequency contents of the diagonal subband

(HH) after being downsampled by 2 in each dimension.

H0(z1, z1)

H1(z1, z2)

G0(z1, z2)

G1(z1, z2)

D1

D1 D1

D1

Fig. 4. The two-channel 2-D filter bank with a checkerboard-shaped frequency partition. The dark regions represent the ideal

passband.

frequency regions will be scrambled and mapped to the actual frequency contents in the HH subband, as

shown in Figure 3(b). Now to separate regions {a, d} from {c, b}, we can see that a natural choice is to

use a two-channel 2-D filter bank with a checkerboard-shaped frequency partition, illustrated in Figure 4.

The downsampling matrix used in the filter bank is the simple diagonal matrix D1. 1

Similarly, it can be checked that the same checkerboard filter bank can also be used to divide the

other two wavelets subbands (HL and LH). This suggests that we can use the checkerboard filter bank

as the building block for a directional extension for the DWT to improve its directionality, as shown in

Figure 5, where only the analysis part is given. The original wavelet transform is kept in the first two

levels. In the third level, the three highpass subbands are further split by the checkerboard filter bank.

Applying the multirate identities [16] in multirate signal processing, we can get the equivalent filter of

each subband and verify that the system indeed achieves the desired frequency partitioning (Figure 1(b))

with ideal filters.

Since each individual component of the proposed system, i.e., the DWT and the checkerboard filter

1We can also use D2 as the downsampling matrix.

April 1, 2005 DRAFT

6

2

1

3

4

0LL

HL

HH

LH

separable 2−D wavelet transform directional extension

5

6

D1

D1

D1

D1

D1

D1

WT

Fig. 5. The filter bank construction of the proposed system for one level of decomposition. The system can be iterated on the

lowpass subband 0 for multiple levels of decomposition. The synthesis part, i.e., the inverse transform, is given by a concatenation

of the synthesis part of the checkerboard filter banks and the inverse wavelet transform.

bank, is critically sampled, the overall system is also critically sampled. Furthermore, if we design the

checkerboard filter bank to be perfect reconstruction, then the whole system is also perfect reconstruction.

A nice property of the wavelet transform is that it has an efficient separable implementation. As shown in

Section IV, the proposed directional extension also has an efficient 1-D implementation in the polyphase

domain, and hence will only moderately increase the computational complexity of the wavelet transform.

We recognize that the proposed system has a fundamental limit in its frequency response, when we

use non-ideal filters. Consider the chain of operations in a certain branch, say subband 5, in Figure 5.

D1 D1D2

F1(z1) F1(z2) H0(z1, z2)

Here, F1(z1), F1(z2) are the wavelet highpass filters along the horizontal and vertical directions, and

H0(z1, z2) is one of the checkerboard filters. Using the multirate identities, we can rewrite this branch

in its equivalent form

F1(z1)F1(z2)H0(z21 , z2

2) M

where the equivalent downsampling matrix is M = D1 · D2 · D1 = diag(4, 2).

DRAFT April 1, 2005


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0

1

2

(a) |F1(ejω1)F1(e

jω2)|

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0

0.5

1

(b) |H0(ej2ω1 , ej2ω2)|

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0

1

2

(c) |Feq(ejω1 , ejω2)|

Fig. 6. The magnitude frequency responses of F1(ejω1)F1(e

jω2), H0(ej2ω1 , ej2ω2) and Feq(e

jω1 , ejω2) for the DWT case.

In the Fourier domain, the equivalent filter of this branch is

Feq(ejω1 , ejω2) = F1(ejω1)F1(ejω2)H0(ej2ω1 , ej2ω2). (1)

In Figure 6(a) and Figure 6(b), we show the typical magnitude frequency responses of some non-ideal

F1(ejω1)F1(ejω2) and H0(ej2ω1 , ej2ω2). The magnitude frequency response of the equivalent filter Feq is

shown in Figure 6(c) as the multiplication of the two previous responses. We can see that |Feq(ejω1 , ejω2)|is concentrated mostly along the 135◦ direction. However, there are also some unwanted bumps (aliasing

components) along the 45◦ direction. A similar phenomenon was mentioned in [11] for a nonredundant

construction of the complex wavelet transform. Similar to the observations made there, it is only possible

to reduce the area of those bumps by using longer filters with sharper cut-off behavior; however, the height

of the bumps cannot be reduced. It is expected that this aliasing problem will have a negative impact

on the performance of the proposed system. However, as shown in Section V, the overall system still

benefits from the improved directionality in some applications, since the equivalent frequency responses

are still mostly concentrated along single directions.

B. Working with the Undecimated Wavelet Transform (UWT)

Another useful wavelet transform is called the undecimated wavelet transform [17], [2], which discards

all the downsampling operations in the DWT and use appropriately upsampled filters via the “algorithme a

trous” [17]. With J denoting the number of decomposition levels, the UWT is a (3J +1)-times redundant

transform in 2-D. Note that the lowpass/highpass filters in coarser levels are upsampled accordingly so

that the overall equivalent filters are the same as those from the DWT. Similar to the discussion above, we

can improve the directionality of the UWT by attaching checkerboard filter banks (directional extension)

to its highpass branches. The construction is similar to the one shown in Figure 5, with the DWT replaced

April 1, 2005 DRAFT

8

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0

0.5

1

(a) |H0(ejω1 , ejω2)|

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0

1

2

(b) |Feq(ejω1 , ejω2)|

Fig. 7. The magnitude frequency responses of H0(ejω1 , ejω2) and Feq(e

jω1 , ejω2) for the UWT case.

by the UWT in the wavelet part. Since the checkerboard filter banks are still downsampled, the directional

extension will not increase the redundancy of the overall system.2

For the UWT case, the frequency aliasing problem is eliminated. For example, the chain of operations

in subband 5 of Figure 5 will now be

D1

F1(z2)F1(z1) H0(z1, z2)

The Fourier transform of the equivalent filter is

Feq(ejω1 , ejω2) = F1(ejω1)F1(ejω2)H0(ejω1 , ejω2).

Compared with (1), here we are using H0(ejω1 , ejω2) (Figure 7(a)) instead of its upsampled version

H0(ej2ω1 , ej2ω2) (Figure 6(b)) to divide the wavelet subband. This makes a big difference when we use

filters with non-ideal frequency responses. As shown in Figure 7(b), the frequency response Feq(ejω1 , ejω2)

for the UWT case is single directional and free of the unwanted aliasing components.

C. Connections with Other Related Systems

As mentioned before, the 2-D wavelet transform with the proposed DEW achieves the same frequency

partitioning as that from the 2-D dual-tree complex wavelet transform (CWT). However, there are several

differences that are important to mention. The CWT produces complex coefficients and hence provides

the phase information, which has be shown to be very useful for some signal processing applications.

2However, the overall system is no longer shift-invariant due to the downsampling.

DRAFT April 1, 2005


As a price to pay for that, the CWT is an expansive system, being 4-times redundant for images, and in

general 2N -times redundant for N -dimensional signals. In contrast, the proposed DEW is a real-valued

transform, and hence discards the phase information. However, it can be nonredundant (when working

with the DWT) for arbitrary dimensions. More importantly, the scaling filters in the two trees of the

CWT have to satisfy the half-sample delay condition, and hence cannot be arbitrarily chosen. This leads

to new challenging joint filter design problems. However, our proposed DEW can work with any existing

wavelet filters to suit different applications. No filter design on the wavelet part is needed. Actually this

difference reflects the philosophy of this work: we try to improve the directionality of wavelets in an

efficient and unintrusive way. Any existing software or hardware implementation of the wavelet transform

can continue to be used. Just attach the directional extension!

In [18], [19], Fernandes et al. proposed a new non-redundant version of the complex wavelet transform

through either a pre-mapping or a post-mapping step. In that interesting construction, the phase informa-

tion can be obtained without sacrificing in redundancy. One major difference between the two approaches

is that the inverse pre-projection and post-projection steps need to use infinite impulse response (IIR)

filters to achieve perfect reconstruction. In contrast, our proposed system has perfect reconstruction with

linear-phase finite impulse response (FIR) filters.

III. GENERALIZATION TO HIGHER DIMENSIONS

In this section, we generalize the proposed directional extension for wavelets to arbitrary dimensions.

Let us first take a look at the 3-D case. We know the 3-D discrete wavelet transform gives one lowpass

subband and seven highpass subbands. Just like its 2-D counterpart, it suffers from limited directionality

and the subband mixing problems. Figure 8 shows that the situation here is actually even more severe.

Analog to the 2-D case, our idea is to find a DEW to improve the directional selectivity of these highpass

subbands.

A. Partitioning the N -Dimensional Frequency Spectrum

We start with some notations. Let B denote the set of frequencies in [−π, π). The frequency spectrum

of N -dimensional (N -D) discrete signals is the Cartesian product BN = B × B × . . . × B︸︷︷︸N

. Let B+0 =

[0, π/2), B−0 = [−π/2, 0), B+

1 = [π/2, π), and B−1 = [−π,−π/2). One can easily verify that the

one-level separable N -D wavelet transform (both the DWT and the UWT) decomposes BN into 2N

April 1, 2005 DRAFT

10

(a) (b)

Fig. 8. Frequency-domain support of two 3D-DWT subbands. Dark regions represent the ideal passband. In the extreme case,

information corresponding to four completely different directions is mixed in subband (a).

subbands

BN =⋃

ki∈{0,1}B±

k1× B±

k2× . . . × B±

kN

def=⋃

ki∈{0,1}B±

k1B±

k2. . . B±

kN, (2)

where B±ki

def= B+ki∪ B−

kifor i = 1, . . . , N . In other words, if we calculate the equivalent filter 3 of each

wavelet subband, the frequency-domain support of that filter will be one of the 2N regions in (2). For

each highpass subband B±k1

B±k2

. . . B±kN

with at least one ki = 1, we can expand it using the distributive

law

B±k1

B±k2

. . . B±kN

def= (B+k1

∪ B−k1

)(B+k2

∪ B−k2

) . . . (B+kN

∪ B−kN

)

=⋃

si∈{+,−}Bs1

k1Bs2

k2. . . BsN

kN(3)

=⋃

si∈{+,−}

(Bs1

k1Bs2

k2. . . BsN

kN∪ B−s1

k1B−s2

k2. . . B−sN

kN

). (4)

We group the 2N terms in (3) into 2N−1 different polarized pairs in (4), so that each of them corresponds

to the frequency support of some real-valued filter. For the 3-D case, we show in Figure 9(a) and

Figure 9(b) two possible subbands from (4). We can see that each of them corresponds to a single

direction, which is in sharp contrast to the situation in Figure 8(a). In addition to the “full” splitting in

(4), it is also possible to have other “partial” splitting. For example,

B±k1

B±k2

. . . B±kN

=⋃

si∈{+,−}

(Bs1

k1. . . B

sn+1

kn+1∪ B−s1

k1. . . B

−sn+1

kn+1

)B±

kn+2. . . B±

kN, (5)

where n = 1, . . . , N − 1. The “full” splitting is a special case when n = N − 1. Figure 9(c) shows

one of the possible subbands from the “partial” splitting in 3-D. These subbands are useful, since they

3In this section we only consider ideal filters.

DRAFT April 1, 2005


(a) (b) (c)

Fig. 9. Frequency-domain support of several subbands, with dark regions representing the ideal passbands. Part (a) and Part

(b): Two possible subbands from the “full” splitting in (4). Each of them corresponds to a single direction, which is in sharp

contrast to the situation in Figure 8(a). Part (c): One possible subband from the “partial” splitting in (5).

correspond to filters that can capture signals with singularities lying on some low-dimensional manifold

structures (e.g. curves, surfaces, etc.) [20].

B. The Directional Extension for Wavelets in N -D

The main result in this section is that we can achieve the subband splitting in (4) and (5) by attaching

some directional extension to the highpass subbands of the wavelet transform. The basic building blocks

of the N -D DEW are checkerboard filter banks shown in Figure 10(a). For notations, we use CBi,j

to denote a checkerboard filter bank whose filtering operations are carried out along the ith and jth

dimensions. The downsampling matrix is defined to be Di = diag(1, . . . , 1︸︷︷︸i−1

, 2, 1, . . . , 1), which keeps

every other sample along the ith dimension. The two channels of the (analysis) filter bank are denoted

by CBi,j(0) and CBi,j(1), respectively.

Lemma 1: Suppose we attach the filter bank CBi,i+1 (1 ≤ i ≤ N − 1) to one of the highpass

subbands of the N -D wavelet transform (either the DWT or the UWT) with an ideal frequency sup-

port B±k1

B±k2

. . . B±kN

. If all filters used are ideal, the wavelet subband will be split into two channels

with passband support B±k1

. . . B±ki−1

(B+ki

B+ki+1

∪ B−ki

B−ki+1

)B±ki+2

. . . B±kN

and B±k1

. . . B±ki−1

(B+ki

B−ki+1

∪B−

kiB+

ki+1)B±

ki+2. . . B±

kN, respectively.

Remark 1: This is the generalization of the 2-D result in Section II and is proved by applying the

multirate identities. See Appendix for details.

As shown in Figure 10(b), an n-level DEW is a binary tree-structured expansion of filter banks

CB1,2, CB2,3, . . ., and CBn,n+1, for which we have the following theorem:

Theorem 1: Suppose we attach the n-level DEW (1 ≤ n ≤ N −1) to one of the highpass subbands of

the N -D wavelet transform (either the DWT or the UWT) with an ideal frequency support B±k1

B±k2

. . . B±kN

.

April 1, 2005 DRAFT

12

1

0

H0(zi, zj)

H1(zi, zj)

CBi,j

Di

Di

(a)

00

0

0

0

1

1

1

1

1

CB1,2

CB2,3

CB2,3

CBn,n+1

CBn,n+1

2n

subbands

n-level DEW

(b)

Fig. 10. (a) CBi,j is the analysis part of the checkerboard filter bank. The subscripts i, j (i �= j) specify that the filters H0(zi, zj)

and H1(zi, zj) operate along the ith and jth dimensions. (b) The n-level DEW is a binary tree-structured concatenation of

checkerboard filter banks CB1,2, CB2,3, . . ., and CBn,n+1. Notice the arrangement of subscripts for the simple expansion rule.

If all filters used are ideal, the wavelet subband will be split into 2n channels as in (5).

Remark 2: This is proved by successively applying the result in Lemma 1. See Appendix for details.

Note that the “partial” splitting in (5) is along dimensions 1, 2, . . . , n+1. However, through a permutation

of indices, we can construct a similar n-level DEW to generate “partial” splitting along n + 1 arbitrary

dimensions d1, d2, . . . , dn+1, where di ∈ {1, . . . , N}. Moreover, the “full” splitting in (4) is a special

case when n = N − 1.

In some applications, e.g., enhancing a particular direction in the signal, one might be only interested

in getting a single frequency region. Instead of using the full binary tree expansion of 2n−1 checkerboard

filter banks in Figure 10(b) to get all 2n subbands, we can use a connected sequence of n checkerboard

filter banks for that specific subband. Such a sequence of checkerboard filter banks is specified by the

following proposition, which can be verified in the proof of Theorem 1 (Appendix B).

Proposition 1: Suppose the subband of interest is(Bs1

k1. . . B

sn+1

kn+1∪ B−s1

k1. . . B

−sn+1

kn+1

)B±

kn+2. . . B±

kN,

where si ∈ {+,−}. The desired “adaptive” DEW for that subband will be

CB1,2(c1) → CB2,3(c2) → . . . → CBn,n+1(cn),

where CBi,i+1 is connected to channel # ci−1 (either 0 or 1) of CBi−1,i (i = 2, . . . , n) with the following

expansion rule:

for DWT:

ci =

⎧⎪⎨⎪⎩

1 − δ(ki − ki+1), if si �= si+1

δ(ki − ki+1), if si = si+1

;

for UWT:

ci =

⎧⎪⎨⎪⎩

0, if si �= si+1

1, if si = si+1

,

where i = 1, . . . , n, and δ(·) is the Kronecker delta function.

DRAFT April 1, 2005


H0(z)

H1(z)

G0(z)

G1(z)D1D1

D1 D1

x x

y0

y1

Fig. 11. A maximally decimated two-channel 2-D filter bank with analysis filters H0(z) (zdef= (z1, z2)), H1(z), synthesis

filters G0(z), G1(z), and the downsampling matrix D1. x and x are the input and reconstructed signals, respectively. y0 and

y1 are the two output signals of the analysis part.

H(z) G(z)z−11

z1

D1

D1

D1

D1x x

y0

y1

Fig. 12. The equivalent polyphase form of the filter bank in Figure 11. H(z) and G(z) are 2-by-2 polyphase matrices.

IV. FILTER DESIGN AND EFFICIENT IMPLEMENTATION

As stated before, there is no need to modify the wavelet transform part and we can choose any existing

wavelet filters to suit different applications. The remaining task is to design a 2-D checkerboard filter

bank with the desired frequency response and perfect reconstruction. Here we will present a design

based on a parameterization of the polyphase matrices. The advantage of this design is that the resulting

filter banks have efficient implementation that only requires 1-D filtering. A more general design of

checkerboard filter banks, in particular the ones that are tailored to work in combination with wavelets,

will be investigated in a future work.

A. Design by Polyphase Matrices Parameterization

Let us consider a maximally decimated two-channel filter bank in 2-D with the downsampling matrix

D1, shown in Figure 11. The well-known equivalent polyphase form of the system is given in Figure 12.

To simplify notation, we denote z = (z1, z2). The relationship between the analysis and synthesis filters

{Hk(z), Gk(z)} and the polyphase matrices H(z) and G(z) can be expressed as [16]

Hk(z) = Hk,0(z21 , z2) + z−1

1 Hk,1(z21 , z2) (6)

Gk(z) = G0,k(z21 , z2) + z1 G1,k(z2

1 , z2),

April 1, 2005 DRAFT

14

for k = 0, 1. We can see from Figure 12 that a sufficient condition for the filter bank to have perfect

reconstruction is

H(z) · G(z) = I, (7)

where I is the identity matrix. In our design, we choose H(z) and G(z) to be

H(z) =√

2

⎛⎝ 0.5 0

−0.5α(z) 1

⎞⎠⎛⎝ 1 α(z)

0 z1

⎞⎠ (8)

and

G(z) =1√2

⎛⎝ 1 −z−1

1 α(z)

0 z−11

⎞⎠⎛⎝ 2 0

α(z) 1

⎞⎠ ,

where α(z) is a free design parameter. It is easy to verify that the perfect reconstruction condition (7)

is structurally guaranteed for arbitrary choice of α(z). This form of polyphase parameterization was

first proposed by Phoong et al. [21] for the 1-D filter bank. However, the 1-D to 2-D mapping method

proposed in that work can only be used to design 2-D filters banks with a single parallelogram-shaped

support, such as the diamond shape. While the checkerboard shape we want here does not belong to this

class. In the following, we will propose a novel 1-D to 2-D mapping for the checkerboard filter bank in

the polyphase domain.

Substituting (8) into (6), we get

H0(z) =1 + z−1

1 α(z21 , z2)√

2(9)

Similarly, we can also write down H1(z), G0(z) and G1(z). Actually, they are all related to H0(z) as

follows.

H1(z) = z1

(√2 −

(√2H0(z) − 1

)H0(z)

)(10)

G0(z) = −z−11 H1(−z1, z2) (11)

G1(z) = z−11 H0(−z1, z2). (12)

If H0(z) is the ideal filter with the desired checkerboard-shaped frequency support shown in Figure 4,

i.e., if its Fourier transform takes the constant value√

2 in the passband and 0 in the stopband, we can

then verify from (10) - (12) that the other three filters H1(z), G0(z) and G1(z) will also achieve the

desired frequency response. For instance, G1(ejω1 , ejω2) = e−jω1H0(ej(ω1−π), ejω2), i.e., G1(ejω1 , ejω2)

is obtained by shifting H0(ejω1 , ejω2) horizontally by π and multiplying a linear phase.

DRAFT April 1, 2005


ω1ω1 ω1

ω2 ω2ω2(π, π)(π, π) (π, π)

−(π, π) −(π, π) −(π, π)

+j+j

+j

+j

−j−j

−j

−j

+1

+1

−1

−1

M(z1) M(z2)z−11 α(z2

1 , z2)

Fig. 13. The separable decomposition of z−11 α(z2

1 , z2). The values of the 2-D Fourier transform of the filters are shown in the

figure.

Therefore, we only need to design H0(z) to approximate the ideal filter on the checkerboard support.

In turn, this implies that the Fourier transform of the filter z−11 α(z2

1 , z2) should take constant values

(−1, 1,−1, 1) in the four quadrants of the 2-D frequency plane, as illustrated in Figure 13. Since this is

a separable filter, we can decompose it as the product of two 1-D filters M(z1) and M(z2), i.e.,

z−11 α(z2

1 , z2) = M(z1) · M(z2). (13)

If we further constrain M(z) to have real coefficients, i.e., M(ejω) = M∗(e−jω), then one of the only

two choices is 4

M(ejω) =

⎧⎪⎨⎪⎩−j, for ω ∈ (0, π]

+j, for ω ∈ (−π, 0]. (14)

Meanwhile, the decomposition form in (13) also implies that

M(z) = z−1β(−z2), (15)

for some 1-D filter β(z), and

α(z) = z−12 β(−z1)β(−z2

2). (16)

From (14) and (15), we want β(z) to be an allpass filter with half-sample delay specified as:

|β(ejω)| = 1, ∀ω (17)

∠β(ejω) = 0.5ω, ∀ω ∈ (−π, π); (18)

In summary, the proposed filter bank design process is given as follows.

(1) Design a filter β(z) to approximate the conditions in (17) and (18).

4The only other choice is to use −M(ejω).

April 1, 2005 DRAFT

16

TABLE I

THE COEFFICIENTS OF THE FILTERS β(z) WITH d = 3, 4, 5, 6.

b(n) d = 3 d = 4 d = 5 d = 6

b(0) 0.6141 0.6217 0.6266 0.6300

b(1) -0.1515 -0.1705 -0.1835 -0.1929

b(2) 0.0473 0.0670 0.0839 0.0972

b(3) -0.0240 -0.0383 -0.0526

b(4) 0.0161 0.0272

b(5) -0.0144

(2) Obtain the analysis and synthesis filters H0, H1, G0, G1 from (16) and (9) - (12).

Note that step (2) is only needed for filter bank analysis. As we will see later, knowing β(z) is sufficient

for actual implementation (see Figure 16).

B. Designing the Allpass Filter with Half-Sample Delay

The half-sample delay conditions (17) and (18) can only be realized by an IIR filter with an irrational

system function. There are various ways to design FIR filters to approximate the conditions. See [22]

for a review. Here we use a simple approach by choosing β(z) to be a type-II linear phase filter [23] of

length-2d:

β(z) = b0(1 + z) + b1(z−1 + z2) + . . . + bd−1(z−(d−1) + zd).

Note that the phase condition (18) is satisfied exactly, and we need to design the coefficients b0, . . . , bd−1

such that the magnitude of the frequency response

|β(ejω)| =d−1∑k=0

2bk · cos(

(k +12)ω)

is as close to 1 as possible.

In Table I, we list the coefficients of the filters β(z) of different lengths designed by the Parks-

McClellan algorithm. We show in Figure 14 the magnitude responses of those β(z). From (16), (9),

and (10), we can verify that the resulting 2-D filters H0 and H1 are of sizes (4d − 1) × (4d − 1) and

(8d− 3)× (8d− 3), respectively. In Figure 15, we give the magnitude response of the 2-D filter H0(z)

obtained from a β(z) with d = 6. We can see that it is a good approximation to the ideal checkerboard

filter.

DRAFT April 1, 2005


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−70

−60

−50

−40

−30

−20

−10

0

10

Normalized Frequency (×π rad/sample)

Mag

nitu

de (

dB)

Magnitude Response (dB)

d = 3

d = 4

d = 5

d = 6

Fig. 14. The magnitude frequency responses of β(z)

with d = 3, 4, 5, and 6. Note that the phase responses

(not shown in the figure) satisfy (18) exactly.

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

0

0.5

1

1.5

Fx

Fy

Mag

nitu

de

Fig. 15. The magnitude frequency response of the

filter H0(z) obtained from a β(z) with d = 6.

z−12 β(−z1)β(−z2

2) z−12 β(−z1)β(−z2

2)−z−12 β(−z1)β(−z2

2) −z−12 β(−z1)β(−z2

2)

√2z1

1√2z−11

z1z−11

1√2

√2

H(z) G(z)

D1

D1 D1

D1

x xy0

y1

Fig. 16. The efficient polyphase implementation of the 2-D checkerboard filter bank.

C. Efficient Implementation

A nice property of the proposed filter design is that the structure is similar to a ladder network [21].

In Figure 16, we show the polyphase implementation of the analysis and synthesis parts of the filter

bank. Although the designed 2-D filters are nonseparable, their polyphase components are separable.

Meanwhile, the downsampling matrix D1 = diag(2, 1) is also separable. Therefore, the entire system

can be implemented by efficient 1-D operations only.

To have a rough estimate of the computational complexity of the proposed DEW, we can count the

number of arithmetic operations needed for each new input sample to the DEW. In 2-D, the DEW is

just a single-level checkerboard filter bank implemented in the polyphase domain as in Figure 16. For

simplicity, we assume the convolution operations are implemented as straight polynomial products, while

similar analysis can be done for FFT-based convolutions. Suppose the filter β(z) used in Figure 16 is of

April 1, 2005 DRAFT

18

TABLE II

NUMBER OF ARITHMETIC OPERATIONS/INPUT SAMPLE FOR DIFFERENT SYSTEMS. BOTH THE DWT AND THE UWT

PERFORM A J -LEVEL DECOMPOSITION OF A N -DIMENSIONAL SIGNAL. THE TWO 1-D FILTERS USED IN THE WAVELET

TRANSFORM ARE OF LENGTH L1 AND L2 . THE CHECKERBOARD FILTERS USED IN THE DEW ARE DERIVED FROM A β(z) OF

LENGTH 2d.

# of multiplications/input sample # of additions/input sample

DWT (L1+L22

)N · 1−2−NJ

1−2−N (L1+L22

− 1)N · 1−2−NJ

1−2−N

DEW for DWT (2d + 1)(N − 1)(1 − 2−NJ) (4d − 1)(N − 1)(1 − 2−NJ)

UWT (L1 + L2)(2N − 1)J (L1 + L2 − 2)(2N − 1)J

DEW for UWT (2d + 1)(2N − 1)J (4d − 1)(2N − 1)J

length 2d. Since it is a symmetric filter, the filtering by β(−z1) requires d multiplications/input sample

and 2d − 1 additions/input sample. Moreover, the filtering by β(−z22) can be efficiently implemented in

the polyphase domain and still requires d multiplications/input sample and 2d−1 additions/input sample.

We can verify that the total number of operations needed for the analysis part of the DEW is 2d + 1

multiplications/input sample and 4d − 1 additions/input sample. The number of operations needed for

the synthesis part is the same, since the analysis and synthesis parts are duals of each other. For the

general N -dimensional case, the DEW (for “full” splitting) is a (N − 1)-level binary tree expansion

of checkerboard filter banks. The number of operations becomes (2d + 1)(N − 1) multiplications/input

sample and (4d − 1)(N − 1) additions/input sample.

For comparison purposes, we show in Table II the number of arithmetic operations/input sample for

the DWT, the UWT, and the corresponding DEW in either case. We assume both wavelet transforms are

implemented in the polyphase domain 5 and perform a J-level decomposition of a N -dimensional signal.

The lowpass and highpass filters used in the wavelet transforms are of length L1 and L2, respectively. We

do not assume special structures, such as linear phase or orthogonality, on them. The DEW is attached

to all the highpass subbands of the wavelet decomposition tree. Here we just list the numbers without

derivation. A detailed analysis on the computational complexity of multirate systems can be found in [2].

We can see from Table II that the relative increase in computational complexity (by attaching the DEW) is

mainly determined by the ratio d/(L1+L2). In 2-D and for some typical numbers L1 = 9, L2 = 7, d = 6,

5This is especially important for the computational efficiency of the UWT.

DRAFT April 1, 2005


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0

2

(a) DWT: frequency

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0

5

(b) DWT-DEW: frequency

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0

0.2

(c) UWT-DEW: frequency

(d) DWT: spatial (e) DWT-DEW: spatial (f) UWT-DEW: spatial

Fig. 17. A few basis images from different transforms. (a) - (c): Some basis images in the frequency domain. (d) - (f): Some

basis images in the spatial domain.

applying the DEW to the DWT corresponds to an increase of 61% multiplications and 123% additions.

Similarly, applying the DEW to the UWT brings an increase of 81% multiplications and 164% additions.

V. NUMERICAL RESULTS

All experiments in this section use the checkerboard filters designed in Section IV with a β(z) of

length 12 (i.e. d = 6). These checkerboard filters are close to having the ideal frequency response and

thus can provide better separation of the directional information. Meanwhile, the sharp cut-off of these

filters can help alleviate the aliasing problem mentioned in Section II. For the wavelet transform, we use

three levels of decomposition and employ Daubechies’s symmlet of length 12 [1]. As before, we use

DWT and UWT to represent the critically-sampled and the undecimated wavelet transforms, respectively.

Correspondingly, we use DWT-DEW and UWT-DEW to denote the two wavelet transforms equipped

with the proposed directional extension.

A. Basis Images

Figure 17 shows a few basis images from different transforms in both frequency and spatial domains.

The basis images of UWT are not shown here, since they are equivalent to those of DWT. We can

see that the basis images of DWT-DEW and UWT-DEW have better directionality than those of DWT.

April 1, 2005 DRAFT

20

(a) (b)

(c) (d)

Fig. 18. (a) A synthetic image consisting of diagonal lines. (b) The diagonal subband of the DWT. (c) Subband 5 of the

DWT-DEW. (d) Subband 6 of the DWT-DEW.

As expected, the critically-sampled DWT-DEW suffers from the frequency aliasing problem, resulting

in some unwanted bumps in the opposite direction (Figure 17(b)). Although the height of these bumps

cannot be reduced, it is possible to make the area of the bumps small by using filters with sharp cut-

off. However, we should see that most of the subband energy of DWT-DEW concentrates on a single

direction, and hence it might still be useful for some image processing applications. Figure 17(c) shows

that the aliasing problem is eliminated for UWT-DEW, but at the cost of a redundant system.

In the spatial domain, the basis image of DWT from the diagonal (HH) subband exhibits an ap-

parent “checkerboard” artifact (Figure 17(d)). In contrast, DWT-DEW (Figure 17(e)) and UWT-DEW

(Figure 17(f)) succeed in isolating different directions, and no “checkerboard” effect appears.

B. Directional Feature Extraction

We first apply the DWT and DWT-DEW on a synthetic image (Figure 18(a)), which consists of lines

oriented at both diagonal directions. The result of the UWT-DEW is not shown here, since it is very close

to that of the DWT-DEW. Figure 18(b) shows the diagonal (HH) subband of the DWT. As discussed

before, 45◦ and 135◦ directions are mixed in this subband and the separable wavelets cannot discriminate

DRAFT April 1, 2005


(a) (b)

(c) (d)

Fig. 19. (a) The cameraman image. (b) The diagonal subband of the DWT. (c) Subband 5 of the DWT-DEW. (d) Subband 6

of the DWT-DEW.

between them. In Figure 18(c) and Figure 18(d), we show the subbands 5 and 6 of the DWT-DEW.

Clearly the subband mixing problem is solved, and the two subbands correctly capture the corresponding

directional information.

We then do the same experiment on the cameraman image. As shown in Figure 19(c) and Figure 19(d),

the DWT-DEW correctly captures and separates the directional information at 45◦ and 135◦ in two

subbands.

C. Denoising

The improvement in directionality of the transform will lead to improvements in many applications.

As an example, we compare the performance of the DWT and the UWT, both with and without the

proposed DEW, in image denoising. In this experiment, we apply additive white Gaussian noise of

different variances σ2 to the Barbara image. A simple hard thresholding scheme [24] with threshold

T = 3σ is used for all 4 transforms. In Figure 20, we show the peak signal-to-noise ratio (PSNR) of

denoised Barbara images using different transforms under various noise levels. We can see that in all

cases the DEW help improve the performance of the original transforms. For a noisy image with PSNR

April 1, 2005 DRAFT

22

16 17 18 19 20 21 22 23 24 2522

23

24

25

26

27

28

29

30

31

PSNR (dB) of Noisy Images

PS

NR

(dB

) of

Den

oise

d Im

ages

UWT−DEWUWTDWT−DEWDWT

Fig. 20. PSNR (dB) of denoised Barbara images by using different transforms under various noise levels.

= 20.19 dB, the improvement (by using the proposed DEW) is about 0.30 dB for DWT, and a more

significant 1.01 dB for UWT.

Figure 21 displays a “zoom-in” comparison of the denoising results. We can see that edges and

directional textures are better restored when we apply the proposed DEW to the DWT and UWT. Despite

the relatively small improvement in PSNR, the DWT-DEW clearly outperforms the DWT (especially in

regions with directional textures) in terms of visual quality.

VI. CONCLUSION

In this work, we proposed a simple directional extension for wavelets that equips the multidimensional

wavelet transform (both the DWT and the UWT) with finer directionality. The filter bank construction of

the overall system is a concatenation of the separable wavelet transform with a sequence of checkerboard

filter banks. No modification on the wavelet part is necessary and hence any existing wavelet implemen-

tation can continue to be used. In 2-D, the proposed DEW further divides the three wavelet subbands

(i.e. horizontal, vertical, and diagonal) at each scale into six finer directional subbands. The DEW is

nonredundant, and can be easily generalized to higher dimensions. Moreover, the DEW can be optional

and adaptive to specific directions. We also proposed a design of the checkerboard filter banks based

on a parameterization of the polyphase matrices and a novel 1-D to 2-D mapping. The resulting filter

banks have efficient implementation that only requires 1-D filtering. Numerical experiments indicate the

DRAFT April 1, 2005


(a) Original image (b) Noisy image: PSNR = 20.19 dB (c) DWT: PSNR = 24.84 dB

(d) DWT-DEW: PSNR = 25.14 dB (e) UWT: PSNR = 26.72 dB (f) UWT-DEW: PSNR = 27.73 dB

Fig. 21. Comparison of denoising results when applying DWT, DWT-DEW, UWT, and UWT-DEW on the Barbara image.

potential of the proposed DEW in several wavelet-based image processing applications.

APPENDIX

A. Proof of Lemma 1

We assume the wavelet transform is the DWT. The proof for the case of UWT is similar. Suppose we

attach CBi,i+1 to one of the wavelet highpass subbands with an ideal frequency support B±k1

B±k2

. . . B±kN

.

The chain of operations from the input to channel #0 of CBi,i+1 is

→ Fk1(z1) → ↓ D1 → . . . → FkN(zN ) → ↓ DN︸︷︷︸

N -D DWT

−→ H0(zi, zi+1) → ↓ Di︸︷︷︸CBi,i+1

→,

April 1, 2005 DRAFT

24

ωi

ωi+1(π, π)

−(π, π)

Fig. 22. The frequency-support of H0(ej2ωi , ej2ωi+1). Dark regions represent the passband.

where Fkjwith kj ∈ {0, 1} are the ideal lowpass and highpass wavelet filters, and H0 is one of the

checkerboard filters in Figure 10(a). Applying the multirate identities, we can rewrite this channel as

→ Fk1(z1) . . . FkN(zN ) · H0(z2

i , z2i+1)︸︷︷︸

equivalent filter Feq(z1, . . . , zN )

−→ ↓ D1 . . .DN · Di︸︷︷︸equivalent downsampling matrix

→ .

The frequency-domain support of the equivalent filter is

supp (Feq) = supp(Fk1(e

jω1) . . . FkN(ejωN ) · H0(ej2ωi , ej2ωi+1)

)= supp

(Fk1(e

jω1) . . . FkN(ejωN )

) ∩ supp(H0(ej2ωi , ej2ωi+1)

)= B±

k1B±

k2. . . B±

kN∩ supp

(H0(ej2ωi , ej2ωi+1)

). (19)

The frequency support of H0(ej2ωi , ej2ωi+1) is shown in Figure 22. One can easily verify that

supp (Feq) =

⎧⎪⎨⎪⎩

B±k1

. . . B±ki−1

(B+ki

B+ki+1

∪ B−ki

B−ki+1

)B±ki+2

. . . B±kN

, if ki �= ki+1

B±k1

. . . B±ki−1

(B+ki

B−ki+1

∪ B−ki

B+ki+1

)B±ki+2

. . . B±kN

, if ki = ki+1.

(20)

In either case, the two channels of CBi,i+1 split the wavelet subband into two subbands as described in

Lemma 1.

B. Proof of Theorem 1

Again, we assume the wavelet transform is the DWT. The proof for the case of UWT is similar. The

n-level DEW has 2n subband channels. The chain of operations from the input to any one of the channels

is

→ N -D DWT −→ Hc1(z1, z2) → ↓ D1 → . . . → Hcn(zn, zn+1) → ↓ Dn︸︷︷︸

one channel of the n-level DEW

→,

DRAFT April 1, 2005


where ci ∈ {0, 1}. Applying the multirate identities, we can get the frequency-domain support of the

equivalent filter as

supp(Fn

eq

)=(B±

k1B±

k2. . . B±

kN

) ∩ supp(Hc1(e

j2ω1 , ej2ω2) . . . Hcn(ej2ωn , ej2ωn+1)

)=(B±

k1B±

k2. . . B±

kN

) ∩(

n⋂i=1

supp(Hci

(ej2ωi , ej2ωi+1)))

=n⋂

i=1

(B±

k1B±

k2. . . B±

kN∩ supp

(Hci

(ej2ωi , ej2ωi+1)))

(21)

= supp(Fn−1

eq

)⋂(B±

k1B±

k2. . . B±

kN∩ supp

(Hcn

(ej2ωn , ej2ωn+1)))

, (22)

where F ieq (i = 1, . . . , n) is the equivalent filter from an i-level DEW. Note that each term in (21) is

the frequency support of a subband produced by attaching a single-level DEW (CBi,i+1) to the wavelet

subband (e.g. see (19)). This implies that we can successively apply the one-step splitting result in

Lemma 1. More formally, we will use an induction proof.

(1) The claim in Theorem 1 is true for n = 1, as ensured by Lemma 1.

(2) Now by induction, we suppose that the (n − 1)-level DEW (2 ≤ n ≤ N − 1) can split the wavelet

highpass subband into 2n−1 regions as⋃si∈{+,−}

(Bs1

k1. . . Bsn

kn∪ B−s1

k1. . . B−sn

kn

)B±

kn+1. . . B±

kN, (23)

and we want to check if the splitting result works for the n-level DEW.

(3) From (22), (19), and (20), each subband in (23) will be split into two in the n-level DEW:((Bs1

k1. . . Bsn

kn∪ B−s1

k1. . . B−sn

kn

)B±

kn+1. . . B±

kN

)⋂(· · · (B+

knB+

kn+1∪ B−

knB−

kn+1) · · ·

),

and ((Bs1

k1. . . Bsn

kn∪ B−s1

k1. . . B−sn

kn

)B±

kn+1. . . B±

kN

)⋂(· · · (B+

knB−

kn+1∪ B−

knB+

kn+1) · · ·

),

which are equivalent to(Bs1

k1. . . Bsn

knBsn

kn+1∪ B−s1

k1. . . B−sn

knB−sn

kn+1

)B±

kn+2. . . B±

kN,

and (Bs1

k1. . . Bsn

knB−sn

kn+1∪ B−s1

k1. . . B−sn

knBsn

kn+1

)B±

kn+2. . . B±

kN,

respectively. Therefore, the 2n subbands of the n-level DEW can be written as⋃si∈{+,−}

(Bs1

k1. . . B

sn+1

kn+1∪ B−s1

k1. . . B

−sn+1

kn+1

)B±

kn+2. . . B±

kN,

April 1, 2005 DRAFT

26

which proves the claim in the theorem for all n ∈ {1, . . . , N − 1}.

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DRAFT April 1, 2005


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