Generalized Coiflets: A New Family of Orthonormal Wavelets

Transcription

1 Generalized Coiflets A New Family of Orthonormal Wavelets Dong Wei, Alan C Bovik, and Brian L Evans Laboratory for Image and Video Engineering Deartment of Electrical and Comuter Engineering The University of Texas at Austin, Austin, T USA wei@visioneceutexasedu; bevans@eceutexasedu; bovik@eceutexasedu Abstract We generalize an existing family of wavelets, coiflets, by relacing the zero-centered vanishing moment condition on scaling functions by a nonzero-centered one in order to obtain a novel class of comactly suorted orthonormal wavelets (we call them generalized coiflets) This generalization offers an additional free arameter, ie, the center of mass of scaling function, which can be tuned to obtain imroved characteristics of the resulting wavelet system such as near-symmetry of the scaling functions and wavelets, near-linear hase of the filterbanks, and samling aroximation roerties Therefore, these new wavelets are romising in a broad range of alications in signal rocessing and numerical analysis Introduction In the last decade, the discrete wavelet transform (DWT), which is imlemented by a multirate filterbank (FB), has been demonstrated to be a owerful tool for a diversity of digital signal rocessing alications Among the numerous wavelets that have been roosed, comactly suorted and real-valued wavelets having the orthonormal and erfect reconstruction (R) roerties have been the most widely used The associated FBs have finite imulse resonses (FIR) and real-valued coefficients In many alications (eg, image rocessing), one desirable roerty for FBs is linear hase, which corresonds to the symmetry or antisymmetry of the associated wavelets and the symmetry of the associated scaling functions However, it is well-known that there does not exist any non-trivial, symmetric scaling function in this family []; ie, in order to obtain symmetric scaling functions, at least one of the above roerties has to be given u One ossible solution to this dilemma is to construct nearly symmetric scaling functions while maintaining those useful roerties, thereby roducing FBs with nearly linear hases The articular class of wavelets known as coiflets are near-symmetric They have the same number of vanishing moments for both the scaling functions (centered at zero) and the wavelets In addition, coiflets have been shown to be excellent for the samling aroximation of smooth functions In this aer, we construct a new family of wavelets (we call them generalized coiflets) by relacing the zerocentered vanishing moment condition on the scaling functions of coiflets by a nonzero-centered one The merit of such a generalization is that it offers one more free arameter, ie, the center of mass of the scaling function (denoted by t ), which uniquely characterizes the first several zerocentered moments of scaling functions, and is hence related to the hase resonse of their FBs at the low frequencies and can be tuned to reduce the hase distortion In addition, by choosing a roer t, wavelets that are nearly odd-symmetric are obtained For a fixed t, we aly Newton s method to construct the lowass filters associated with the generalized coiflets iteratively We show that generalized coiflets are asymtotically symmetric and their filters are asymtotically linear hase as the order tends to infinity We formulate a general framework for minimizing the hase distortion of the lowass filters under various criteria We study the accuracy of generalized coiflets-based samling aroximation of smooth functions by develoing the convergence rates for L -norm of the aroximation error Due to sace limitation, we omit all the roofs of our results, which will be given in [] and [] Background We highlight fundamental results from wavelet theory [] on which this aer is based Let h(n) be the lowass filter in a two-channel orthonormal wavelet system The scaling function (t) is recursively defined by the dilation equation (t) = h(n)(t, n) () n

2 and the wavelet (t) is defined as (t) = n g(n)(t, n) where the lowass filter h together with a highass filter g constitute a air of conjugate quadrature filters (CQFs); ie, g(n) =(,) n h(n + n, n) () where [n ;n ]is the suort of the FIR filters h and g The orthonormal condition is given by h(n)h(n, k) = k () n for k,where k denotes the Kronecker delta symbol Definition Definition A wavelet is called a generalized coiflet of order l (denoted by l;t)ifforsome t R, thewavelet l;t and its scaling function (denoted by l; t) satisfy R (t, t ) l; t(t) dt = R t l;t(t) dt = for =;;;l, Note that t is the center of mass of the scaling function l; t When t =, the generalized coiflets reduce to the original coiflets constructed by Daubechies [] The vanishing moment conditions in the above definition are equivalent to n for =;;;l, Construction n h l; t(n) = t () (,) n n h l; t(n) = () n It was shown that a generalized coiflet system can be constructed by solving a set of multivariate nonlinear equations for the filter coefficients [] These equations can be derived from (), (), and () We aly Newton s method to solve them iteratively Define an N l vector h l; t = h l; t(,l);h l; t(,l +);;h l; t(n l, l, ) T where N l =bl=c and the suerscrit T denotes matrix transose Let f l R N l! R N l be a vector-valued function defined as f l (h l; t) = n h l;t (n), 6 n h l;t(n)h l; t(n +) nh l;t(n)h l; t(n + N l, ) n nh l;t(n), t n n h l; t(n), t n nbn l=c, h l; t(n), t bn l =c, n(,)n h l; t(n) n (,)n nh l; t(n) n (,)n n l, h l; t(n) where all the summations are from n =,l to n = N l,l, Therefore, the equation f l (h l; t) = Nl gives a set of N l indeendent equations in (), (), and (), where Nl denotes the zero vector of length N l The aroximate solution to this equation in the kth iteration is denoted by h k l;t With an initialization of h l;t, Newton s iteration becomes h k+ l;t = h k l;t, f l (hk l;t ), f l (h k l;t ) where f l denotes the Gateaux-derivative of f l and the oerator (), denotes matrix inversion The initial choice of h l;t is not arbitrary because some choices may cause the iteration to diverge In our design, we choose the original coiflet filter one order lower than the generalized coiflet filter we aim to construct as the starting solution; ie, (h l; ) T h l+;t = ( (h l; ) T T T if l is even if l is odd The iteration stos when the difference between h k+ l;t and h k l;t is small enough (eg, its norm is smaller than a given threshold) In our exeriments, with such an initialization scheme, the Newton iteration always converges Near-Symmetry and Near-Linear hase We use the hase of the Fourier transform of a scaling function to measure its symmetry If l; t is nearly symmetric, then the hase of b l; t is close to a linear function of 7

3 frequency The following roosition indicates how good this aroximation is at low frequencies Due to the lowass nature of l; t, in the frequency domain its energy is mostly distributed at low frequencies Therefore, l; t is nearly symmetric roosition If j!j is sufficiently small, then \b l; t(!) =, t!+cl; t! l + O(! l+ ) where the constant C l; t only deends on l; t; the scaling function of a generalized coiflet is asymtotically symmetric, ie, for each!, lim l! \ b l; t(!) =, t! Now we study the hase distortion of the lowass filters associated with the generalized coiflets Since the coefficients of these filters are real-valued, we only consider! [;] For a lowass filter, there are two tyes of symmetry If for some n, a filter h satisfies h(n) =h(n, n), then we say that h is whole-oint symmetric (WS) about n if n, andhalf-oint symmetric (HS) about n if (n+ ) In both cases, the hase resonse \H(ej! )=,n! If a filter is asymmetric, then its hase distortion can be measured as the deviation of the hase resonse from a linear function of frequency with some desired sloe There is a well-known fact [] regarding the relationshi between the symmetric tye of a wavelet and its associated lowass filter h ifh is WS, then is even-symmetric, and vice versa; if h is HS, then is odd-symmetric, and vice versa It has been observed that the lowass filters associated with the original coiflets are nearly WS [] In fact, for an lth-order original coiflet, its filter H l; (e j! ) has (l, ) zeros at! =, and hence a flat, near-zero hase in the neighborhood of DC In the following roosition, we show that in general, H l;n (e j! ) is close to linear hase at low frequencies roosition If j!j is sufficiently small, then \H l;n (e j! )=,n!+c hl;n!l + O(! l+ ) where the constant C only deends on h hl;n l;n; the lowass filter associated with a generalized coiflet ossesses asymtotically linear hase, ie, for each! [;), lim l! \H l;n(e j! )=,n! Though the above roosition is true for all real n, only integers and half-integers are of interest Thus, we define = fn n g An advantage of introducing halfinteger n is that filters close to HS can be constructed, which are more useful than WS filters in many alications 6 Minimization of hase Distortion From roosition we know that the filter h l;n, n, has nearly zero hase distortion if j!j is small enough However, the hase distortion at the other frequencies can be much larger The resulting hase resonse may not be satisfactory in many alications that require uniformly insignificant hase distortion over a broad frequency band The hase resonse at low frequencies is uniquely characterized by the first several moments of the scaling function, and hence by the arameter t in the case of the generalized coiflets Thus, we attemt to use this arameter to obtain smaller hase distortion Though, for any t 6 the roerty of near-zero hase distortion around DC will be lost, the gain lies in the fact that hase distortion can be largely reduced over a broad frequency band For a given t R, we exect that \H l; t(e j! ) is close to, [ t ]! While adjusting the arameter t mayalsoimrovethe near-symmetry of the scaling function l; t and the wavelet l;t, in this aer we restrict our attention to the hase distortion of the lowass filter h l; t Since in a tyical DWT-based alication the inut signal is convolved with a wavelet filterbank, the hase resonse of the outut signal is a sum of those of the inut signal and the filterbank Therefore, the hase distortion on the outut signal, which is additive and caused by the non-linearity of the hase resonse of the filterbank, can be viewed as the difference between the desired linear hase resonse for the filterbank and its actual hase resonse We define D [h] to be the measure of hase distortion of a filter h over [;], D [h] = W(!), \H(e j! )+n! (6) for some,, n, and some weighting function W [; ]! [; ] For the generalized coiflets, n = [ t ] In fact, the quantity D [h] is the weighted L - distance between the desired linear hase resonse,n! and the actual hase resonse \H(e j! ) From () we deduce that for a CQF air h and g of a finite suort [n ;n ], \G(e j! )=\H(e j(,!) ),(n +n )!+ Using this relationshi, we rewrite D [h] as D [h] = f W(!), \G(e j! )+n!+(n,) where n = n + n, n, f W is the mirror function of W about, ie, f W (!) = W (,!) for! [;],and,n!, (n, ) is the desired generalized linear hase resonse Therefore, the quantity D [h] also measures the hase distortion of g with resect to the weighting function f W Thus, such a metric is meaningful in not only the

4 DWT-based alications but also those based on wavelet acket transforms, where both lowass and highass subbands are decomosed iteratively With this quantitative measure, which is clearly a function of the arameter t for the generalized coiflets, we can formulate a class of otimization roblems by searching the otimal arameter t that minimizes D [h L; t] for given and W Such a general formulation allows the flexibility of choosing a roer arameter and a roer weighting function W in order to rovide an aroriate filterbank for a articular DWTbased alication In the following examles, we choose =, W (!) = if! [; ),andw(!) = elsewhere This imlies that we attemt to minimize the maxi- mum hase distortion over the lowass halfband In Figure, we lot the original coiflet and the otimal generalized coiflet of order as well as their scaling functions, where the subscrit w stands for WS The otimal generalized coiflet aears more symmetric than the original coiflet The otimal near-hs filters of the generalized coiflets with odd orders are quite similar to the filters of some biorthogonal sline wavelets, which are, in fact, the dual wavelets with resect to the Haar wavelet, and referred to as e h ;en in Table 6 in [] In Figure, we lot the order- biorthogonal sline wavelet dual to the Haar wavelet and the order- generalized coiflet having the minimal hase distortion, as well as their scaling functions, where the subscrit h stands for HS The two scaling functions are surrisingly similar to each other; so are the two wavelets 8 6 (c) 8 6 Figure Comarison between the rd-order biorthogonal sline wavelet dual to the Haar wavelet and the rd-order generalized coiflet having minimax hase distortion ; e (t); ; t h (t); (c) ; e (t); (d) ;t h (t) 7 Aroximation of Smooth Functions (d) An imortant issue in wavelet-based multiresolution aroximation theory is to measure the decay of aroximation error as resolution increases, given some smoothness conditions on the function being aroximated Let f be a smooth L function in the sense that f (l) is square integrable, and be an lth-order orthonormal scaling function Define i f to be the aroximation of f at resolution,i ; ie, the orthogonal rojection on the subsace sanned by f i;k g k, ( i f )(t) = = k + q=i+ hf; i;k i i;k (t) k hf; q;k i q;k (t) (c) Figure Comarison between the rd-order original coiflet and the rd-order generalized coiflet having the minimax hase distortion ; (t); ; t w (t); (c) ;(t); (d) ;t w (t) (d) where i;k (t) = i= ( i t, k), fori; k, and similar notation alies to It can be shown that the L -norm of the rojection error has the asymtotic form kf, i fk =C roj,il kf (l) k +O(,i(l+) )

5 where the constant C roj is given by C roj = m;m6= b (l) (m) = A In the above discussion, the exansion coefficients fhf; i;k ig k areassumedtobeavailablesothatthewavelet coefficients fhf; q;k i q; k ; q>igcan be efficiently comuted via Mallat s algorithm, on which the multiresolution analysis is based However, in ractice, only the uniform samles of a function rather than its exansion coefficients are often known, because the exlicit forms of the function and the scaling function are unknown (this is true for most wavelet bases), and the comutation of exansion coefficients usually requires the evaluation of numerical integrals, which are comutationally exensive If a generalized coiflet basis is used, then the function samles aroximate the exansion coefficients accurately We define a sequence of functions ff i;l R! R;i g, f i;l (t) =, i f(,i (k + i;k t )) l;t (t) k which can be viewed as successive aroximations of f with the scaled and translated scaling functions of an order-l generalized coiflet being used as the interolants Theorem If f R! R is (l +)times differentiable, f (l+) is bounded, and f (l) L (R), then the L -norm of the reconstruction error has the asymtotic form kf, f i;l k = C ar,il kf (l) k +O(,i(l+) ) where the constant C ar is given by C ar = l! (t, k, t ) l l; t(t, k) k! dt = In [], we show that for a generalized coiflet, the asymtotic constant C ar can be exressed as C ar = C roj + C sam where the constant C sam is given by C sam = l! R tl l; t(t) dt, t l The squared L -error for generalized coiflets-based samling aroximation can be reresented as kf, f i;l k = kf, ifk +k if, f i;l k which imlies that kf, f i;l k, kf, i fk,andk i f, f i;l k have the same convergence rate,il Thus, it is interesting to comare the associated asymtotic constants of ASYMTOTIC CONSTANT 6 x 6 SAMLING OFFSET ASYMTOTIC CONSTANT 9 x SAMLING OFFSET Figure Asymtotic constants for the samling aroximation errors vs the samling offset t of the th-order generalized coiflet C sam ;C ar the latter two The asymtotic constant for kf, i fk, which is the error due to rojection, is aarently C roj Therefore, C sam is the asymtotic constant for k i f, f i;l k, which is the error due to the aroximation of the rojection coefficients by the function samles In [], we roosed a numerical method to comute the asymtotic constants Figure illustrates the asymtotic constants for the samling aroximation error versus the samling offset t for the generalized coiflet of orders We find that C sam is much smaller than C ar This imlies that the L -error due to the aroximation of the rojection coefficients by the function samles is negligible comared to that due to the rojection, because the two tyes of errors have the same convergence rate 8 Conclusion We have resented a study of the generalized coiflets Since they ossess several remarkable roerties, they are excellent candidates of comactly suorted orthonormal wavelet bases in signal rocessing and numerical analysis References [] A Cohen, I Daubechies, and J-C Feauveau Biorthogonal bases of comactly suorted wavelets Commun ure Al Math, 8 6, 99 [] I Daubechies Ten Lectures on Wavelets SIAM, hiladelhia, A, 99 [] I Daubechies Orthonormal bases of comactly suorted wavelets II variations on a theme SIAM J Math Anal, ()99 9, Mar 99 [] D Wei and A C Bovik Generalized coiflets with nonzerocentered vanishing moments IEEE Trans Circuits Syst II, Secial Issue on Multirate Systems, Filter Banks, Wavelets, and Alications, to aear 998 [] D Wei and A C Bovik Samling aroximation of smooth functions via generalized coiflets IEEE Trans Signal rocessing, Secial Issue on Theory and Alications of Filter Banks and Wavelets, to aear 998