[IEEE 2012 International Conference on Wavelet Analysis and Pattern Recognition (ICWAPR) - Xian, China (2012.07.15-2012.07.17)] 2012 International Conference on Wavelet Analysis and

Proceedings of the 2012 International Conference on Wavelet Analysis and Pattern Recognition, Xian, 15-17 July, 2012

CONSTRUCTION OF TRIANGULAR BIORTHOGONAL WAVELETS USINGEXTENDED LIFTING

KENSUKE FunNOKI1,2 , SHUNSUKE ISIDMITSU2

1Department of Information Science and Technology, Oshima National College of Maritime Technology,Yamaguchi 742-2193, Japan.

2Graduate School of Information Sciences, Hiroshima City University, Hiroshima 731-3194, Japan.E-MAIL: [email protected]@hiroshima-cu.ac.jp

Abstract:We present new triangular biorthogonal wavelets by extending

two-dimensional lifting, which we call twist. The resulting trulytwo-dimensional biorthogonal filters defined on a triangular lattice inherit several nice features of the early triangular biorthogonal wavelet filters such as the hexagonal symmetry of low-pass filters, symmetrical arrangement of three high-pass filters on the lattice, both of which contribute to preserve isotropy of images in themultiscale wavelet decomposition. Besides, these filters have muchlarger support, providing much larger portions of the total energyto three detail components of decomposed images. This plays animportant role when extracting the edge structure of an image.

Keywords:Nonseparable wavelets; lifting; triangular lattice; isotropy.

1 Introduction

Among the recent developments in wavelet constructionmethods, one of the notable techniques is Swelden's liftingscheme [11, 12]. This gives a easy way to construct anybiorthogonal wavelets and allows a wavelet transform to bebuilt on more general domains such as geometric surfaces [1]and spheres [10].

The usual wavelet transform decomposes a one-dimensionalsignal into its multiscale components [6]. For a twodimensional signal such as an image, the transform is appliedto the horizontal and vertical directions of an image independently. As a result, in this tensor product form of the wavelettransform, the correlations of data in the horizontal and vertical directions are treated in the same way, but for other directions, such as that at 45° to the horizontal axis, the correlation may not be detected properly. That is, the conventional

978·1-4673·1535·7/121$31.00 ©2012 IEEE

wavelet transform does not maintain the rotational symmetry,or isotropy, of images.

To improve that situation, two-dimensional hexagonalwavelets would be suitable to keep isotropy of images, sincethe hexagon is the most symmetric polygon which fills upa two-dimensional plane. Sakakibara [9] proposed triangular biorthogonal wavelets that are defined on a triangular lattice and have a hexagonal support, with an almost straightforward generalization of the lifting to triangular lattice sites. Wehave developed the construction of the triangular biorthogonalwavelets applying an interpolation technique to the generalizedlifting, and it turns out that any order of the triangular biorthogonal wavelets can be constructed [3]. The resulting filters are abiorthogonal set of two-dimensional low-pass (LP) filters andthree isotropic high-pass (lIP) filters. In a preliminary study inwhich we applied our filters to several image processing tasks,we showed that the energy of decomposed images is distributeduniformly, which suggests that the triangular wavelets well respect isotropy of an image.

In this paper, we construct new triangular wavelets by applying a new operation to the lifting steps in our constructions,which we call twist. The twist mixes phase components of asignal before the lifting, providing filters that have much largersupports. Since this is an elemental extension of the lifting,we show that the resulting twist wavelet filters basically havethe same properties as those of the early triangular biorthogonal wavelets such as the hexagonal symmetry of LP filters, theisotropy of three independent lIP filters, and the uniform energy distribution of decomposed images.

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2 Triangular Biorthogonal Wavelets

(1)

Here we briefly review the construction of triangularbiorthogonal wavelets, including the Bravais lattice formalismused in solid state physics [4], and the generalized lifting, forour triangular lattice. To index a two-dimensional plane, wedefine the primitive translation vectors

t1=(1 O)T, t2=(-~ 1)T,that generate the regular triangular Bravais lattice

A = {t = n1t1 + n2t21 (n1,n2) E Z2},

Figure 1. Triangular Bravais lattice A generated bytwo primitive translation vectors t1 and t2 (left); thereciprocal lattice Agenerated by two reciprocal latticevectors A1 and A2 (right).

where P(W ) t is the Hermitian conjugate of the polyphase matrix P(w), formed by one LP filter i; and three HP filters9k,m, k = 1, 2,3 in the polyphase form. The inverse t,:ansform

can then be written using the dual polyphase matrix P(w) as

sembled similarly with dual filters hm and 9k,m. The conditionguaranteeing the perfect reconstruction of the original signal

(2) from its decomposition can then be written as

which is depicted in Figure 1. The hexagonal domain shown inFigure 1 is called the Wigner-Seitz cell, which corresponds toa pixel of an image.

The reciprocal lattice vectors are similarly defined by

>t1 = (0 ~)T, >t2 = (1 ~)T ,

that generate the reciprocal lattice

- 2A = {27r(A = n1A1 +n2A2)1 (n1,n2) E Z }.

To derive a set of filters that satisfy (5), we employ the lifting that enables a systematic design for constructing both thepolyphase matrix and its dual. In fact, the lifting corresponds tofactorizing a polyphase matrix into each lifting step [2], whichcan be generalized to our setting as

From the right, the matrices correspond to the predict, updateand scaling of the lifting steps. The objective of predictorsPm (w) is to predict the value of a signal, which calculates thedifference between an original signal and a predicted signal,and the objective of updaters um(w) is to preserve the averageafter the decomposition

(5)

P(w)t = (I ° 0 011K 0 0

0 11K 0x

0 0 11K

1 U1(W) U2(W) U3(W) 1 0 0 00 1 0 0 -P1(W) 1 0 00 0 1 ° -fi2(w) ° 1 00 0 0 1 -P3(W) 0 0 1

(6)

Cj(W) = LCj[t]e-iwt, W E R2 ,

tEA

then it can be represented with its four phase components

Cm,j(W) = LCj[2t+tm]e-iwot, m=O,I,2,3, (3)tEA

which is the polyphase decomposition of the signal.In one dimension, the discrete wavelet transform decom

poses a signal {Cj [k]}kEZ into its coarse component and detailcomponent with a LP filter h[k] and a HP filter g[k] after downsampling by a factor of 2 [5]. This can be directly applied toour situation, and the triangular wavelet transform that decomposes the signal Cm,j into its coarse component Cj -1 and detailcomponents dk,j-1, k = 1,2,3 can be written as

This corresponds to the Fourier domain of the Bravais lattice,and the Wigner-Seitz cell on the reciprocal lattice is the Brillouin zone. Additionally, vectors to = 0, t 3 = -t1 - t 2 , andA3 = A1 - A2 are defined for notational convenience.

Let the Fourier transform of a signal {Cj [t]}tEA with j E Zdenoting the resolution level defined on the triangular Bravaislattice as

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(11)

(10)3

h(w)h*(w) + L 9m(W)g:'n(w) = 4.m=l

and updaters can be defined in terms of the predictors

uN(w) = fit!,*(w) for N < N.m 4 -

One can immediately recognize that these matrices are all inwhich we call twist. The twist step plays a role of mixing thevertible, and hence, it is easy to construct the dual polyphase phase of a signal before the lifting. Using the twist, a signalmatrix as P(w)t- 1

, which guarantees the perfect reconstruc- Cm,j,m = 0,1,2,3, of the right hand side of the triangulartion (5). wavelet decomposition (4) can then be replaced as

The simplest choice

In the case of N = N = 2, the choice of the predictors andupdaters turns out to be

A 1 + eiw .t mA 1 + e-iw .t m

Pm(w) = 2 ,Um(w) = 8 .

K = 2, Pm(w) = 1, um(w) = 1/4, m = 1,2,3, (7)

.-. -1and P(w )t gives a system of triangular biorthogonal Haarfilters, which are the simplest two-dimensional biorthogonalwavelet filters defined on the triangular Bravais lattice. Ahigher-order filter can also be constructed using the Lagrange Crucially, the twist matrix is still invertible, guaranteeinginterpolation scheme [7] that reproduces the mid-value of a sig- the perfect reconstruction (5). Moreover, since the modificanal according to polynomial interpolation. For all L 2:: 1 and tion (12) has the form of simply adding the twist to normal-L < k ~ L, the coefficients have the explicit form lifting steps (6), the construction of the filters is straightfor-

2L ward. Consequently, one can construct any order of triangu-(_1)L+k-1 II (L - n + 1/2) lar biorthogonal twist wavelet filters by choosing predictors Pm

N [ ] n-1 according to (8) and updaters U m according to (9). Note thatPm ktm = (L + k - 1)1 (L - k)l(k - 1/2)' N = 2L, the resulting filters tum out to be orthogonal where the case

(8) N = N = 1, which has been constructed in [8] from a different derivation. Since this extension is simple, the twist waveletfilters will inherit all the nice features of our previously con

(9) structed triangular biorthogonal wavelet filters, such as hexagonal support of LP filters and three isotropic HP filters, whichallow isotropic signal decomposition. However, one can obtaintriangular biorthogonal filters and wavelets that are differentfrom those constructed by the untwist lifting.

For example, if we set the same choice as (11), we obtain atwist version of the triangular (2,2) biorthogonal wavelet fil-ters whose filter coefficients and frequency response are shownin Figure 2. Note that only the HP filters for m = 2 cases areshown; the other HP filters for m = 1 and m = 3 are simply271"/3 rotations of m = 2 cases in the reciprocal lattice. As described in the Introduction, the decomposition with respect totriangular wavelets enables isotropic signal processing becausethere are three independent HP filters that are symmetricallyarranged on the lattice. Owing to the effect of the twist, theprimal HP filters 9m are truly two-dimensional filters, whiletheir untwist versions have one-dimensional support on the lattice and a one-dimensional frequency response. Furthermore,all the filters have much larger support, which still satisfy (10).

3 Trinagular Biorthogonal Wavelet with Twist Lifting

As a result, we obtain a set of triangular (N, N) biorthogonalwavelet filters where (N, N) denotes the number of orders forthe primal and dual HP filters respectively. They are a set ofbiorthogonal perfect reconstruction filters {h, h, 9m,9m}, m =1, 2,3, which satisfy the halfband condition

We extend the generalized lifting (6) to construct a new fam-ily of triangular biorthogonal wavelets. We found that the fac- 4 Image Decompositiontorization of a polyphase matrix (6) can be modified as

1 0 0

P(w)t -+ P(w)t ~ -;1 !1o 1 1

o11

-1

We apply our filters to images to see how the triangular twistwavelets keep the isotopy of decomposed images and how the

(12) twist affects actual image processing tasks. We first present thedecompositions of concentric circles that has 512 x 512 pixels,using the triangular Haar, twist Haar and tensorial Haar filters,

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Figure 3. Decomposed images of concentric circles. From the left, each column shows the case of the triangulartwist Haar, triangular Haar and tensorial Haar.

as shown in Figure 3. The coarse approximation is shown attop left, whereas detail components ds, d3 and d2 are arrangedclockwise and correspond to components in the horizontal, diagonal, and vertical directions, respectively. The 512 x 512matrix data are mapped to rectangular pixels on the rectangularsites (n1, n2), n1, n2 = 0, 1,2, ... ,511. We treat the data as thevalue cg[n1t1 + n2t2], where the second primitive translationvector t2 is modified as (0,1) so that the both primitive translation vectors are defined on the rectangular lattice. In principle,the original data should represent hexagonal pixels arranged ina honeycomb structure. Unfortunately, however, such data arenot available in the standard image database, and hence, weemploy the above convention.

In Figure 3, we see in the case of the triangular wavelets forconcentric circles that the images are evenly decomposed intothree detail components compared with the tensorial case. Thiscorresponds to an energy (£1 norm) distribution over three detail components, which is shown in Figure 4. In Figure 4, thed1 and d2 components of all the filters share almost the samerate of energy. This implies that the image originally containsthe same density of energy in horizontal and vertical directionsas is clear from its visual shape. However, the d3 componentobviously differs between the triangular and tensorial cases,where the energy in the tensorial case is appreciably less thanthe energy in both cases of triangular wavelets. Note that theirstrong energy concentrations of d3 component are due to thefact that the original image Cg has been mapped to the squarelattice sites. Thus, the energy of the images decomposed withour triangular wavelets is more evenly distributed over the threedetail components, suggesting that the isotropy of the image iswell respected for both triangular filters.

This isotropic image decomposition is an apparently novelfeature of triangular biorthogonal wavelets and it is clearly in-

herited by the twist version. It should be noted that for the twocases of the triangular wavelets, there is no large difference intheir distribution rates but the amount of energy significantlydiffers. Each detail component of the triangular twist Haar hasmuch higher energy for both images, which means that more ofthe total energy goes to the three detail components in the twistHaar case. This implies that the detail components extract theedge structure of the images well.

To see that, we reconstructed Lena from only detail components after the one-level decompositions using the three typesof Haar filters used in the above demonstration. Each (2,2) filter is also used to observe the effect of increasing the order offilters. The Results are shown in Figure 5. In the Haar case,both triangular wavelets well extract the whole edge structuresof the image whereas the tensor product case contains jaggyparts at several locations such as in the hair of Lena. However,the observation differs somewhat for the case of the (2,2) filters that have a much larger support on the lattice. The twist(2,2) case provides much better edge extraction compared withthe twist Haar, but the untwist triangular case and the tensorialcase both result in a slight drop in the extraction quality relativeto the Haar cases. Thus, owing to the nature of the twist lifting,twist filters still preserve the isotropy of images, and producemore detail components that much contain image features suchas edges. This becomes more apparent when the order of thefilters increases.

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Figure 4. Energy distribution over the three detailcomponents of concentric circles.

5 Conclusion

A method for constructing new triangular biorthogonalwavelets by adding a twist operation to the generalized lifting was presented. The twist filters defined on a triangularlattice appear to inherit several advantages of our previouslyconstructed triangular biorthogonal wavelet filters, such as thehexagonal symmetry of LP filters and isotropic arrangementof three lIP filters, which allows the isotropy of an image tobe well preserved. Due to the effect of the twist, all the filters are truly two-dimensional biorthogonal wavelet filters andhave much larger supports on the lattice. This provides betterquality of extracting the edge structure of an image, especiallywhen the order of the twist filters increases.

References

Figure 2. Filter coefficients and their frequency responses for the triangular (N, N) biorthogonal twistwavelet filters (h,9m,h,gm) for N = N = 2. Thearrows represent the reciprocal lattice vectors Am,m =1, 2, 3 rescaled by a factor 271".

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Figure 5. Lena images reconstructed from only detallcomponents with various filters.

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[IEEE 2012 International Conference on Wavelet Analysis and Pattern Recognition (ICWAPR) - Xian, China (2012.07.15-2012.07.17)] 2012 International Conference on Wavelet Analysis and