Discrete wavelet transform implementation in Fourier domain for multidimensional signal

Journal of Electronic Imaging 11(3), 338–346 (July 2002).

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Discrete wavelet transform implementationin Fourier domain for multidimensional signal

Frederic NicolierIUT de Troyes

LAM, URCA–Universitede Champagne-Ardenne9 Rue de QuebecBoite Postale 396

F-10026 Troyes, Cedex, FranceE-mail: [email protected]

Olivier LaligantFrederic Truchetet

IUT Le CreusotLE2I, Universitede Bourgogne

12 Rue de la FonderieF-71200 Le Creusot, France

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Abstract. Wavelet transforms are often calculated by using theMallat algorithm. In this algorithm, a signal is decomposed by acascade of filtering and downsampling operations. Computing timecan be important but the filtering operations can be speeded up byusing fast Fourier transform (FFT)-based convolutions. Since it isnecessary to work in the Fourier domain when large filters are used,we present some results of Fourier-based optimization of the sam-pling operations. Acceleration can be obtained by expressing thesamplings in the Fourier domain. The general equations of thedown- and upsampling of digital multidimensional signals are given.It is shown that for special cases such as the separable scheme andFeauveau’s quincunx scheme, the samplings can be implementedin the Fourier domain. The performance of the implementations isdetermined by the number of multiplications involved in both FFT-convolution-based and Fourier-based algorithms. This comparisonshows that the computational costs are reduced when the proposedimplementation is used. The complexity of the algorithm isO(N log N). By using this Fourier-based method, the use of largefilters or infinite impulse response filters in multiresolution analysisbecomes manageable in terms of computation costs. Mesh simplifi-cation based on multiresolution ‘‘detail relevance’’ images illustratesan application of the implemenentation. © 2002 SPIE and IS&T.[DOI: 10.1117/1.1479701]

1 Introduction

Space-scale analysis is a very popular tool for signal pcessing especially for two-dimensional~2D! signal process-ing. Many applications deal with image compression, mtiresolution edge detection and more generallyalgorithms that benefit from the multiresolution approasuch as Markovian modelization or texture segmentatThe multiresolution analysis algorithm developedMallat1 has largely contributed to this popularity. In th

Paper 99066 received Nov. 17, 1999; revised manuscript received Sep. 4,accepted for publication Nov. 29, 2001.1017-9909/2002/$15.00 © 2002 SPIE and IS&T.

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algorithm, a signal is decomposed into approximations adetail signals by a cascade of filtering and downsamploperations.

In all these applications, wavelets are associated wlinear phase filters, this property being largely admittedbe a crucial point for good results in image processiOne-dimensional linear-phase filters associated withthogonal bases have an infinite impulse response~IIR!. Inmultidimensional processing, finite impulse response~FIR!and linear-phase filters associated with orthogonal analare theoretically possible but involve very constrainiconditions. The problem is often solved by using a bithogonal basis that leads to compact linear phase fil~FIR!.2 However, due to the maximal independence of tsignal components obtained, the interest in an orthonorbasis seems to be obvious. Kovacˇevic and Vetterli havepresented such filters associated with orthogonal analbut the corresponding wavelets are not accurately localiin the frequency domain.3

The difficulty is that implementation of IIR filters withclassical methods is hardly ever in accordance with retime applications. When large filters are used in signal pcessing, it is necessary to work in the Fourier domainorder to have reduced computational time. A well-knowresult is that digital filters with more than 32 taps, in ondimensional signal filtering, the Fourier domain providreduced computational time.4 A sensible implementationwould use Fourier-based convolution consisting of afast Fourier transform~FFT!, complex multiplications forthe filtering and a 2D inverse FFT~IFFT!. In this way, thefiltering step of the algorithm can be accelerated. We hlooked for some efficient method by which to acceleratesampling step. If the samplings can be expressed inFourier domain, then some FFT and IFFT can be avoidTherefore, we have looked for efficient implementation

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Discrete wavelet transform implementation . . .

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the whole Mallat algorithm in the Fourier domain. Somresults on sampling in the continuous Fourier domainbe found in Refs. 5–9. We have extended these resultdigital multidimensional signals. The algorithm obtainednot tailored for a specific wavelet. It allows fast compution even with FIR filters such as Daubechies ones.

Presentation as well as an evaluation of this approacproposed in this paper. For digital multidimensional snals, the general transcription in the Fourier domain of stial sampling is given. Furthermore, several special saplings such as the one-dimensional, two-dimensioseparable and quincunx cases are studied.2 For the quin-cunx scheme, two representations are developed andshow that in the case of Feauveau’s representation10 theFourier implementation is quite worthwhile in termscomputational cost.

This paper is organized as follows. The transcriptionmultidimensional sampling in the discrete Fourier domis described in Sec. 2. Special cases of multidimensiosamplings are studied in Sec. 3. The performance ofimplementation in the Fourier domain of the whole wavetransform algorithm is presented in Sec. 4 by comparcomputational costs of the FFT-based-convolution metwith those of the Fourier-based method presented. Anplication of the implementation is given in Sec. 5.

2 Multidimensional Sampling

Multidimensional samplings are an extension of ondimensional ones. The sampling is expressed by multiping the signal sample coordinates by a matrixJ denotingthe scaling~or dilation! operation. For example, in 2D thcoordinates of a given sample are transformed by

S x8y8 D5JS x

yD . ~1!

For instance, in the separable case, the sampling matrgiven by

J5F2 0

0 2Gand in the quincunx sampling matrix by

J5F1 1

1 21G .2.1 Sampling in the Continuous Fourier Domain

As previously shown by Viscito, Dubois, Vetterli anco-workers,5,6,9 downsampling in the continuous time domain can be expressed like in Eq.~2!; the transcription inthe continuous Fourier domain of these operations is shin Eq. ~3! whereXF is the Fourier transform ofx:

y@k#5x@Jk#, ~2!

YF~v!51

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D21

XF~J2Tv22pJ2Tvl !, ~3!


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where D5udetJu, v5(v1 ,...,vS)T is an S-dimensionalreal vector andvl denotes the vectors associated with tcosets derivated fromJ. The operations involved in upsampling can also be expressed in the time and Fourier domas shown in Eqs.~4! and ~5!.

y@k#5H x@J21k#, if k5Jn, nPZS,

0 otherwise,~4!

YF~v!5XF~JTv!. ~5!

2.2 Sampling in the Discrete Fourier Domain

One can notice in Eqs.~3! and~5! that a Fourier transformin the continuous domain is used. Since digital signal pcessing needs discrete signals and operators, here in2.2 the sampling operations are transcribed into equatin the discrete Fourier domain. The discrete Fourier traform ~DFT! of a multidimensional signalx@k# is denotedby x@m#, wherek and m are vectors. In the discrete domain, sample indices of the signals have to be integersnot real numbers since they have to be with the use ofv.Thus Eqs.~3! and~5! need to be rewritten according to thiExplicit developments are given only for upsampling bthey can be done easily for downsampling. Examplestaken for a 2D signal~image!. A dilation and a rotationcentered on the origin are involved in the transformatrealized by the matrix multiplications. As shown in Fig.an N3N imageXF is transformed into anM3M imagedenotedYF and a rectangular lattice is used for storagEquation~5! is rewritten as

YF~v!5XF~v8! withH v52p

Mm,

v852p

Nm8,

~6!

wherem is related tom8 according tom85 (N/M ) JTm.So Eq.~5! can be rewritten to give upsampling for discremultidimensional signals:

y@m#5 xF S N

MJTmDmodNG , ~7!

where modN is added because the DFT assumes sigperiodicity. It can be verified that a similar demonstratican be conducted to express downsampling:

Fig. 1 Two-dimensional upsampling in the Fourier domain: Dilationand rotation are involved in the transformation.

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3 Special Two-Dimensional Samplings

It was shown in Sec. 2 that any multidimensional samplcan be transcribed into the discrete Fourier domain. HerSec. 3, two examples dealing with 2D sampling are givin detail: the separable scheme and the nonseparablecunx scheme. Feauveau’s quincunx representation isstudied and is shown to lead to very simple operationsthe Fourier domain.

3.1 Separable Case

From Eq. ~5!, the upsampling in the discrete Fourier dmain in the separable case is expressed byYF(v1 ,v2)5XF(v18 ,v28) wherev1852v1 and v2852v2 . In this op-eration signals are dilated by 2 and there is no rotatinvolved. An M3M signal is transformed into a 2M32M one. If v1 andv2 are defined in discrete terms acording tov15(2p/2M ) m1 and v25(2p/2M ) m2 , thenv185(2p/M ) m18 and v285(2p/M ) m28 . Since v1852v1

andm15m18 ~v2852v2 andm25m28! the transformation inthe discrete Fourier domain of the separable upsamplingiven by

y@m1 ,m2#5 y@m1 modM ,m2 modM #. ~9!

3.2 Quincunx Rotating Scheme

From Eq. ~5!, upsampling in the discrete Fourier domafor the quincunx case is expressed as

YF~v1 ,v2!5XF~v11v2 ,v12v2!. ~10!

A & dilation and ap/4 rotation are involved in thisoperation. As shown in Fig. 2, anN3N signal is trans-formed into anM3M one. In continuous terms, becauXF is an N3N square andYF is an N&3N& rotatedsquare,M is related toN by M52N. If equation Eq.~10!is rewritten:

YF~v1 ,v2!5XF~v18 ,v28! with 5 v152p

Mm1 , v25

2p

Mm2 ,

v185v11v252p

Nm18 , v285v12v25

2p

Nm28 .

~11!

So m1 andm2 are related tom18 andm28 according to

Fig. 2 Quincunx upsampling in the Fourier domain: Dilation androtation of an N3N signal XF in an N&3N& signal YF . The ro-tated YF signal is contained in a M3M image.



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H m185N

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m285N

M~m12m2!.

~12!

Equation~10! can be rewritten to give quincunx upsampling for digital multidimensional signals as

y@m1 ,m2#

5 xF N

M~m11m2! modN,

N

M~m12m2! modNG , ~13!

As shown in Fig. 2 the rotation involved in the transformtion is centered on the origin of the image. Thereforetransformed imageYF obtained is distributed in severablocks. In order to produce a one-block signal, anM /2translation into the discrete Fourier domain~i.e., ap trans-lation into the continuous Fourier domain! must be in-cluded to change the image’s origin. The result of thtranslation is shown in Fig. 3 wherey is contained in anM3M signal and, as the space is periodized, quarters oyare duplicated~seea–d in Fig. 3!. So the final transforma-tion in the discrete Fourier domain of the quincunx ovsampling is given by

y@m1 ,m2#

5 x@n1 ,n2# with H n15 12~m11m2! modN,

n25 12~m12m2! modN.

~14!

Since this transformation is applied onx samples, onlysamples ofy that correspond to even (m11m2) and (m1

2m2) terms are calculated~i.e., whenn1 andn2 are inte-gers!.

3.3 Feauveau’s Quincunx Scheme

One way to represent quincunx sampling is the previoone, where the images of two successive scales are ro

Fig. 3 Complete quincunx upsampling in the Fourier domain of anN3N signal x in an N&3N& signal y contained in a M3M image.Due to Fourier domain periodicity, parts a, b, c and d are dupli-cated. The origin of the signal is translated to produce a one-blocksignal y.


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by p/4. Another representation was proposedFeauveau,10 in which the signal axes are kept steady butfilters are rotated and two types of downsampling andsampling are used. The analysis algorithm is shown in F4. For even scale computation, convolution with normfilters and keep samples where (k11k2) is even~downsam-pling a:↓a!. For odd scales, convolution with rotated filteand keep one column and one row out of two~downsam-pling b:↓b!. For the synthesis, the upsampling is codedputting zeros between each column and each row.

These samplings can easily be conducted in the discFourier domain as shown in Eqs.~15!–~17! ~see the Appen-dix!.

y↓a@k1 ,k2#k1 ,k2P[0,N21]

51

2 S x@k1 ,k2#1 xFk11N

2,k21

N

2 G D , ~15!

y↓b@k1 ,k2#k1 ,k2P[0,N/2 21]

51

4 S x@k1 ,k2#1 xFk11N

2,k2G1 xFk11

N

2,k2G

1 xFk11N

2,k21

N

2 G D . ~16!

Fig. 4 Feauveau-quincunx scheme: Decomposition of a digital ap-proximation at scale s51 into approximations at coarser scales (s52 and 3). The samples kept are indicated by the big dots. Zerosare put at small dots.


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y↑@k1 ,k2#k1 ,k2P[0,N21]5 x@k1,k2# with H k15k1modN

2,

k25k2 modN

2.

~17!

Equation~15! translates downsampling↓a into the dis-crete Fourier domain, Eq.~16! is for downsampling↓b andEq. ~17! deals with upsampling. A geometrical interprettion of these equations is shown in Fig. 5. Downsampl↓a is interpreted as an addition betweenx andxI , wherexI isobtained by permutation ofx quadrants~1–4 and 2–3!. y↓b

is obtained by the addition of all the quadrants ofx. Finallyy↑ is obtained by a duplication ofx.

4 Performance

In the earlier discussion the simplicity of processing saplings in the Fourier domain has been pointed out. Tfollowing discussion will characterize the advantages ofing the Fourier domain in terms of computational co~i.e., the number of real additions and multiplications! forall the operations involved in a multiresolution analysThe results given are general and do not take into accoany optimization and therefore do not benefit from asymmetry of the signals or from special samplings.

2D signals of finite sizeM3M are considered. Computational costs are evaluated up to scaleL. According to thesamplings, at scales ~s varying from 0 toL! the signal sizeis approximated byM /Ds/23M /Ds/2 whereD5udetJu. Sothe computations at scales of both detail and approximation are only done forM2D2s samples.

The comparison is done between the DFT-based conlution algorithm and our Fourier algorithm. These algrithms are synthesized in Figs. 6 and 7. For convenienonly the approximation of each scale is calculated.

Fig. 5 Feauveau-quincunx scheme: Geometric interpretation of thesamplings in the Fourier domain.

Fig. 6 DFT-based convolution algorithm: Decomposition of a digital approximation (As) into approxi-mations at coarser scales (As11 and As12). Filtering is realized in the Fourier domain. Sampling isrealized in the time domain.




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Fig. 7 Fourier domain algorithm: Decomposition of a digital approximation (As) into approximations atcoarser scales (As11 and As12). Filtering and sampling are realized in the Fourier domain.

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4.1 Computation Cost in the Fourier Domain

Implementation of the whole wavelet transform in the Forier domain consists of computing the Fourier transformthe input signal, filtering~complex multiplications!, pro-cessing the Fourier domain equivalent of the spatial ssamplings and finally computing inverse Fourier transforfor the output signals~approximations and details!. Thecomputational cost of the FFT of anM3M signal is

N FFTx 54M2(log2 M22). The output must be given in th

time domain, so the cost of the IFFT of the details of scs is given by

N IFFTx ~s!54D2sM2@ log2~MD2 ~s/2!!22#. ~18!

The filtering operations are just complex multiplicatio~i.e., two real multiplications and four real additions! withthe filter in the Fourier domain, so the cost of filteringsignal in Vs is N f

x(s)52D2sM2. The downsampling isprocessed by a matrix multiplication@see Eq.~8!# for eachsample of the destination signal. One 232 matrix multipli-cation involves four real multiplications and two real addtions. So to obtain a signal at scales, the multiplication

cost for the downsampling isN↓x(s)54D2sM2. Finally the

computational costs are summed over scales to obtainfinal multiplication cost of anL scales two-dimensionawavelet transform in the Fourier domain:

N Fx ~L !5NFFT

x 1 (s50

L21

Nfx~s!1(

s51

L

@N↓x~s!1NIFFT

x ~s!#.

~19!

4.2 Computational Cost Using DFT-basedConvolutions

An efficient and classical implementation of the wavealgorithm consists of processing the convolution in tFourier domain. The downsampling costs are the same athe Fourier domain algorithm@N ↓

x(s)5N ↓x(s)#. The costs

of the convolutions are the costs of a FFT, plus the costcomplex multiplications for filtering, plus the costs of aIFFT.

For one-scale processing, the computational cost oFFT ~or an IFFT! is N FFT

x (s)54M2D2s@ log2(MD2 (s/2))22#. The filtering operations are just complex multiplictions ~i.e., two real multiplications and four real additions!,thus the cost of filtering a signal inVs is Nf

x(s)52D2sM2. These computational costs are summed oscales to obtain the final multiplication cost of anL scaletwo-dimensional transform using DFT-based convolutio

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N Cx ~L !5 (

s50

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~2NFFTx ~s!1Nf

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L

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4.3 Comparison Between the Algorithms

In Secs. 4.1 and 4.2 we have specified howN Cx andN F

x areevaluated. The performance of the algorithm is evaluaby analyzing cost ratios. The ratio of the FFT-based conlution algorithm cost to the Fourier algorithm costR(L)5 @NC(L)/NF(L)#. This ratio depends both on thimage sizeM3M and on the determinantD of the dilationmatrix.

As shown in Figs. 8 and 9, two ratios are calculated:R2

for the quincunx sampling andR4 for the separable sampling. Four curves are drawn for each scheme, and tcorrespond to an analysis made from scale 1 to scalL(L51,2,3, or 4!. For the two samplings, the ratio is abo1.8 for the deeper scale analysis (L54).

These comparisons are made with nonoptimized alrithms. In the case of the quincunx algorithm, the saplings can be simplified by using Feauveau’s scheme~seeSec. 3.3!. Figure 10 shows the ratio of the quincunx algrithm to Feauveau’s algorithm cost using the Fourier impmentation. The curves show a little decrease in the comtational cost~about 6%! but this ratio does not take intoaccount the advantage of implicit periodization of the Feveau sampling.

The implementation costs are summarized in Fig.The memory requirements given include all the filte

Fig. 8 Comparisons between algorithms: Ratios of FFT-based con-volution algorithm costs to Fourier algorithm costs as a function ofthe image size M and the analysis depth L. Quincunx sampling.


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needed. These filters can be preloaded into memory inder to avoid hard drive access. To be efficient, the fquency responses of the filters are used in order to apointless FFTs. The number of operations given corsponds to 32 bit real multiplications.

We have made a comparison of our implementation wpolyphase factorization. The filter size used in our impmentation is equal to the image size. The complexity ofpolyphase implementation has thus been evaluatedthis size filter. The complexity of standard polyphase fatorization isO(N2).9 The complexity of our implementation is O(N logN). A comparison of these two complexitieshows that our implementation is more efficient thpolyphase factorization.

5 Application

We have used our Fourier implementation in a mesh splification algorithm previously developed.11 This meshsimplification process inherently deals with multitextur

Fig. 9 Comparisons between algorithms: Ratios of FFT-based con-volution algorithm costs to Fourier algorithm costs as a function ofthe image size M and the analysis depth L. Separable sampling.

Fig. 10 Comparison between Feauveau schemes: Ratio of theclassical-quincunx scheme cost to the Feauveau-quincunx schemecost as a function of the image size M and the analysis depth L.


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data sets using multidimensional feature space. Figureshows a flow chart of the simplification process. Each teture in the data set is a dimension in feature space. A tture of particular interest is what we call the ‘‘detail reevance’’ image. This image is the result of applying the 2quincunx-wavelet transform to a three-dimensional~3D!mesh. Detail extraction is achieved using multiresolutianalysis~MRA! based on the wavelet cascade approaThe MRA process extracts detail information at varioresolutions and produces a texture image that showsrelevance of a vertex to the mesh structure. The userinput in this process to select which resolutions are mimportant. This approach is useful for filtering noise, prserving discontinuities, mining surface details, reducidata, and many other applications. The mesh simplificatprocess is the process of reducing the number of vertithat compose a mesh. Our process is based on ancollapsing process that collapses two vertices into one.decision to collapse one edge over others is made in patspace~or feature space!. Each pattern is a dimension in thispace. Pattern space is composed of multiple types offormation and measurements of the mesh. For this stuthey are restricted to geometrical coordinates, curvatunormal, and more importantly the results of multiresolutianalysis interpreted as binary information~0: absence ofdetail, 1: presence of detail!.

The binary detail relevance image is obtained from dtail images of multiresolution analysis of the range imagThe range image is decomposed over scales using quincanalysis. Each detail image is then reconstructed at thetial scale. This reconstruction is realized via an invertransform where the detail signals corresponding to otscales are set to zero. Each detail image is reconstrucedthis process. Finally, one information image is obtain

Fig. 11 Number of operation (real multiplications) and memory re-quirements of our Fourier domain wavelet transform implementationfor three level decomposition (L53). The memory requirements arecomputed using a float 32 number representation.

Fig. 12 Flow chart of the mesh simplification algorithm. The usercontrols the simplification by selecting relevant detail images. Thedetail images are obtained from a quincunx multiresolution analysis.




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Fig. 13 Simpification of a digital elevation map (DEM) model: (a) initial wire frame (12 000 vertices,texture image 5123256 and 15000 faces), (b) model with thermal texture (texture image 5123256),size: 0.7 Mb, (c) model display with no texture, (d) simplification of the model based on textureinformation (thermal area in red on the color texture is kept unchanged), (e) its reduced wire frame(reduction factor is 5), size: 124 kb, (f) simplification of the model based on geometric information, (g)its reduced wire frame (reduction factor is 5) constructed with segmented detail images from scale 6,size 124 kb.

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The computational time of the information image is veimportant. This time was reduced using our optimized Frier domain implementation. The filters were computedline and are loaded in memory at the beginning of theplication. The image size is 2563256, the filter size is 25325, the depth analysis isL56 and since a quincunxmethod is used,D52. The computational time ratio between the space domain implementation and our implemtation is about 41. This means that the computation timethe information image is a few seconds in simple MATLAimplementation.

6 Conclusion

Some new considerations dealing with implementationthe Fourier domain of multiresolution analysis onorthornormal basis were presented. Since it is more ecient to work in the Fourier domain when large filters aused, in this paper we have emphasized a Fourier equlent of up- and downsampling operations for multidimesional digital signals. It has been shown that these proc

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ings can be easily be performed by simple operatio~spectrum repetitions and additions for the Feauvescheme!. The resulting scheme for multiresolution analyswas described and was compared with the classical oThe improvement obtained in terms of computational cis noticeable. In this way, the use of large or IIR filtersmultiresolution analysis is quite manageable in termscomputational costs. The performance of the implemention was given in terms of the number of operations amemory requirement estimations. The complexity of talgorithm is O(N logN) which is lower than that ofpolyphase factorization. The proposed implementation wapplied to mesh simplification based on quincunx muresolution analysis. The resulting computational time isfew seconds.

Since the performances given were presented withsimplification we think that in special applications, opertions could be directly implemented in the Fourier domain order to avoid some FFTs. For example, in an imacompression–decompression algorithm, quantization~sca-lar or vectorial! could be achieved with benefit directly iFourier space, especially when a complete set of coecients is set to zero~diagonal details of the finest scales, f


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example!, in this case, a Fourier implementation scheseems to be particularly worthwhile. Other benefits cantaken from the Fourier domain. For instance, as the pericity of a signal is implicitly included in the definition o


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the Fourier domain, when signals are processed there arrequirements to make any periodization of the signal, escially when using the separable or Feauveau quincschemes.

Appendix

Downsampling a

y~k1 ,k2!k1 ,k2P[0,N21]512 x~k1 ,k2!~11ej (k11k2)p!,

y~n1 ,n2!n1 ,n2P[0,N21]51

2 (k150

N21

(k250

N21

x~k1 ,k2!~11ej (k11k2)p!e2 ~ j 2pk1n1 /N!e2 ~ j 2pk2n2 /N!,

51

2 (k150

N21

(k250

N21

x~k1 ,k2!e2 @ j 2p(k1n11k2n2)/N#

11

2 (k150

N21

(k250

N21

x~k1 ,k2!e2 ~ j 2pk1 /N![n12~N/2!]e2 ~ j 2pk2 /N![n22 ~N/2!] ,

51

2x~n1 ,n2!1

1

2xS n12

N

2,n22

N

2 D .

Downsampling b




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Upsampling

y~k1 ,k2!k1 ,k2P[0,N21]51

2xS k1

2,k2

2 D ~11ej (k11k2)p!,

y~n1 ,n2!n1 ,n2P[0,N21]51

2 (k150

N21

(k250

N21

xS k1

2,k2

2 De2 j 2p @(k1n11k2n2)/N#

11

2 (k150

N21

(k250

N21

xS k1

2,k2

2 De2 j 2p @(k1n11k2n2)/N#ejk1pejk2pH p15k1

2

p25k2

2

↔ H k152p1 ,k252p2 ,

y~n1 ,n2!n1 ,n2P[0,N21]51

2 (p150

N/2 21

(p250

N/2 21

x~p1 ,p2!e2 j @2pp1n1 /~N/2!#e2 j @2pp2n2 /~N/2!#

11

2 (p150

N/2 21

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References1. S. G. Mallat, ‘‘A theory for multiresolution signal decomposition: Th

wavelet representation,’’IEEE Trans. Pattern Anal. Mach. Intell.11,674–693~1989!.

2. A. Cohen and I. Daubechier, ‘‘Nonseparable bidimensional wavbases,’’Rev. Matematica Iberoamericana9~1!, 51–137~1993!.

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4. O. Rioul and D. Duhamel, ‘‘Fast algorithms for discrete and contious wavelet transforms,’’IEEE Trans. Inf. Theory38, 569–586~1992!.

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7. Y. Kok, S. Lee, and H. Tan, ‘‘Periodic orthogonal splines and walets,’’ Appl. Comput. Harmon. Anal.2, 201–218~1995!.

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11. M. Toubin, C. Dumont, F. Truchetet, and M. Abidi, ‘‘Multiresolutiosimplification of 3D multimodal objects using a 2D quincunx waveanalysis,’’Proc. SPIE3837, 280–288~1999!.

Frederic Nicolier received his master’sdegree in applied physics from Universitede Bourgogne in 1995 and PhD degree inimage processing at the same university in2000. He is now an assistant professor atLAM, Universite de Reims Champagne–Ardenne (URCA). His research interest in-cludes wavelets transform, 3D problemsand shape recognition.



Olivier Laligant received his PhD degreein 1995 from the Universite de Bourgogne,France. He is an assistant professor in theComputing, Electronic, Imaging Departe-ment (Le2i) at the Universite de Bour-gogne. His research interests are focusedon multiscale edge detection, merging ofdata, wavelet transform and recently mo-tion estimation.

Fred Truchetet received his master’s de-gree in physics from Dijon University,France, in 1973 and PhD in electronics atthe same university in 1977. He was withThomson-CSF for two years as a researchengineer and he is currently full professorand the head of the image processinggroup at Le2i, Laboratory of Electronic,Computer and Imaging Sciences, Univer-site de Bourgogne and CNRS (FRE 2309),France. His research interests are focused

on image processing for artificial vision inspection particularly onwavelet transform, multiresolution edge detection and image com-pression. He is member of SPIE and IEEE.


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