Design of 6-Band Tight Frame Wavelets With Limited Redundancy

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Gertrude Montgomery
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1 Desig of 6-Bad Tight Frame Wavelets With Limited Redudacy A. Farras Abdelour Medical Physics Memorial Sloa-Ketterig Cacer Ceter New York, NY Telephoe: () Abstract I this paper we explore the desig of tight frame wavelets with a dilatio factor M = 4. The resultig limit fuctios are sigificatly smoother tha their orthogoal couterparts. A advatage of the proposed filters over the dyadic filterbaks is that the proposed filterbaks result i a reduced redudacy whe compared with dyadic tight frames, while maitaiig smoothess. The proposed filterbaks geerate five wavelets ad a scalig fuctio with the uderlyig filters related as follows: H 5 i(z) =H i( z), i=...5. I. INTRODUCTION Dyadic (M =) wavelets have bee extesively studied ad aalyzed [5], [], ad wavelets based o critically sampled FIR filterbaks have witessed various applicatios [4], []. The tight frame variatio allows for additioal desig degrees of freedom ad smooth limit fuctios. Dyadic tight frames have bee well documeted ow. Some of the earliest papers discussig compactly supported dyadic tight frames are [7], [7]. Tight frames with wavelets possessig more tha oe vaishig momet each were developed i [8]. The desig of 4-chael dyadic ad symmetric limit fuctios has bee addressed i [], [3], [6], [5], [6]. Symmetric tight frames with two geerators have bee addressed i [], [9] where symmetry ad compact support require the lowpass filter H (z) to satisfy a additioal costrait, amely the roots of H (z)h (/z) H ( z)h ( /z) must be of eve multiplicity [6]. Tight frames have bee show to exhibit a early shift-ivariat behavior, which is importat for applicatios such as deoisig. Orthogoal symmetric limit fuctios have bee obtaied uder M = 4 [], but the resultig limit fuctios lack smoothess. I this paper we seek 6-bad tight frame FIR filters {h,h.h,h 3,h 4,h 5 } with dilatio factor M =4. The filters are related as follows: H i (z) =H 5 i ( z), i =...5. The decimatio rate of 4 limits the throughput redudacy of the resultig filterbaks, while takig advatage of the desig flexibility of frames. The filterbak desig is accomplished usig Gröber basis [8], [9], [3]. The resultig filterbaks ejoy a limited redudacy whe compared with their dyadic tight frame couterparts,.66 for M = 4 versus a redudacy of 3 (for three dyadic wavelets) ad (for two dyadic wavelets), while maitaiig smooth limit fuctios. We preset examples of frames with osymmetric, symmetric, ad iterpolatig symmetric scalig fuctios. II. BACKGROUND THEORY I this sectio, we discuss some defiitios ad properties pertaiig to the case M =4tight frame filterbaks geeratig five wavelets. A set of wavelets {ψ l (4 m ), l =...5} costitutes a frame whe for A B < ad ay fuctio f L we have [] A f 5 f,ψ i (4 m ) B f, i= m, where A ad B are kow as frame bouds. The special case of A = B is kow as tight frame. Moreover, whe A = B =, ad ψ l,j,k = l, j, k, ψ l,j,k costitutes a orthogoal basis. I geeral, a wavelet system cosistig of oe scalig fuctio ad 5 wavelets with dilatio 4 defies the followig spaces: with V j = Spa{φ(4 t )}, W i,j = Spa{ψ i (4 j t )}, i 5 V j = V j W,j W,j... W 5,j ad the correspodig scalig fuctio ad wavelets satisfy the followig multiresolutio equatios: φ = h ()φ(4t ), ψ i = h i ()φ(4t ), i 5. The bouds A ad B ow take o the value [4] A = B = 4 5 h. = A fuctio f L ca the be expaded as follows: f = k c(k)φ k + 5 l= j= d l (j, k)ψ l,j,k, () k

2 where φ k φ(t k), ψ l,j,k ψ l (4 j t k), ad the coefficiets c(k) ad d l (j, k) give by c(k) = d l (j, k) = fφ k dt fψ l,j,k dt, l =...5. Give a set of N 4 filters {h,h,...,h } with a dilatio factor 4, the perfect recostructio coditio ca be writte as follows [5]: = H (z)h (/z) = 4, H ( z)h (/z) =, = H (jz)h (/z) =, = H ( jz)h (/z) =, = where j =. The filters take o the followig form: H (z) = ( +z + z + z 3) K Q (z) H i (z) = ( z ) K i Qi (z), i=...5 where K ad K i are the umbers of vaishig momets of the limit fuctios. A. Redudacy Tight frame wavelet trasforms are expasive, i.e., they augmet a data vector of legth N ito N wavelets coefficiets where N >N. The redudacy rate depeds o the dilatio factor M, the umber of filters N, as well as the umber of filterig stages L. The redudacy rate is expressed as []: R = N ( ) M M L + N M L. From the above equatio we have R M as L.For the case M =4ad N =6the redudacy R is bouded by 3 R 5 3. We ote that the proposed wavelets are less redudat tha the dyadic tight frames, where the upper boud of R is for the case of two wavelets, ad 3 for dyadic tight frames with 3 wavelets. B. Near Orthogoality The filters described i this paper are ear orthogoal relative to their shifts by 4m, m Z. This also reflects the approximate orthogoality of the limit fuctios {φ, ψ,...,ψ 5 } ad their iteger shifts. We shall use θ(h i,h j, 4m) to idicate the agle betwee two vectors h i ad h j shifted with respect to each other by 4m, defied as ( ) hi (),h j ( 4m) θ(h i,h j, 4m) = arccos. () h i h j C. Desig Approach We use Gröber basis for our filterbak desig. I order to reduce the computatioal burde of fidig the Gröber basis, we first fid the lowpass filter h separately. The resultig Gröber basis ca the be icorporated with the equatios describig the etire filterbak. This two-step approach reduces the computatio time sigificatly. For the purpose of the lowpass filter desig, two coditios eed to be satisfied, give the desired vaishig momets K ad K mi : ( +z + z + z 3) K H (z), (3) ( z z ) K mi 4 H (z)h (/z). (4) The first coditio guaratees regularity of the resultig scalig fuctio, while the secod coditio imposes a regularity of the associated wavelets. The resultig equatios are solved usig Gröber bases method. We require that the filters be related as follows: H 5 (z) = H ( z) (5) H 4 (z) = H ( z) (6) H 3 (z) = H ( z). (7) III. EXAMPLES I this sectio we preset examples of tight frame fiite support filterbaks with vaishig momets. We list oly the coefficiets of filters {h,h,h }, with the remaiig filters {h 3,h 4,h 5 } obtaied from equatios (5-7). A. Example We cosider the case K =4,K mi =ad osymmetric filters. We seek a filterbak where the scalig fuctio vaishig momet is give by K = 4, i additio to a miimum wavelet vaishig momet of K mi =.Theset of vaishig momets is the {K,K,K,K 3,K 4,K 5 } = {4,,,,, 4}. Notice that K = K 5 =4, this is due to the fact that H 5 (z) =H ( z). The lowpass filter is give by H (z) = ( +z 7 + z + z 3) 4 ( α + αz ), ( ) where we have α = ±. Both values of α result i filters ( related trivially: ) H (z) ad z H (/z). Choosig α = +, the resultig scalig fuctio has a Sobolev regularity parameter.9. Filters h, h, h 4, ad h 5 are of legth 4, while filters h ad h 3 are of legth oly. The filters coefficiets are listed i table I. Fig. shows the frequecy respose of the filters. The resultig limit fuctios are depicted i Fig.. Table II lists a subset of the agles formed betwee the filters ad their shifts by multiples of 4 usig equatio (). The filters geerally approximate orthogoality, with a exceptio for θ (h,h, 4) = 66.. B. Example I this example we seek limit fuctios with symmetric scalig fuctio ad with vaishig momets K =4ad K mi =. The lowpass filter h is give by H (z) = 8 ( +z + z + z 3) 4 ( +z ).

3 Frequecy Respose TABLE I EXAMPLE, CASE K =4AND K mi =, WITH legth h =legthh =4, AND legth h =. h () h () h () Fig.. Frequecy resposes of the filters described i Example. TABLE II EXAMPLE, A SUBSET OF ANGLES BETWEEN SPACES GENERATED BY FILTERS IN FIGURE AND THEIR SHIFTS shift h, h 3 h, h 4 h, h 5 h, h 3 h, h 4 h, h φ. 4 ψ 4 ψ 4 ψ 4 ψ 3 4 ψ 5 Notice how while both lowpass filters i this example ad the previous example are of the same legths, h i this example is symmetric for the same umber of momets K. The resultig scalig fuctio is highly smooth, with a Sobolev parameter of 4. However, the miimum umber of wavelet vaishig momets has ow reduced to K mi =. The TABLE III EXAMPLE, CASE K =4AND K mi =, WITH SYMMETRIC SCALING FUNCTION. WEHAVElegth h =legthh =4, AND legth h =. h () h () h () / / / / / / / / / / / / / / Fig.. Limit fuctios of Example, with {K,K,K,K 3,K 4,K 5 } = {4,,,,, 4}. highpass filter h 5 is atisymmetric. The remaiig wavelet filters {h,h,h 3,h 4 } are approximately (ati-)symmetric. Similar to Example, filters h ad h are of legth 4, while filter h is of legth. Filters coefficiets are listed i table III, while Fig. 3 shows the frequecy respose of the filters.

4 .4. Frequecy Respose Fig. 3. Frequecy resposes of the filters described i Example..6 φ 4 ψ 4 ψ ψ 4 ψ ψ 5. 4 Fig. 4. Limit fuctios of Example, with {K,K,K,K 3,K 4,K 5 } = {4,,,,, 4}. The resultig limit fuctios are depicted i Fig. 4. The filters exhibit a ear orthogoality behavior, with the majority of the agles foud usig equatio () are above 8 degrees, with oe exceptio i the case of θ (h,h, 4) = C. Example 3 This example presets the case of limit fuctios where the uderlyig filters are of odd legths, ad the scalig fuctio φ is symmetric ad possesses iterpolatig property. I order for a scalig fuctio to be iterpolatig, it is ecessary that φ be of the form φ() =δ( ), {, } Z. (8) The filters are thus shifted by samples so as to satisfy equatio (8). From Fig. 6 the scalig fuctio φ is zero at itegers except at = 3, where we have φ(3) =. The lowpass filter h is give as follows: H (z) = z ( +z + z + z 3) 4 ( 5+z 5z ). 56 Notice the z factor, accoutig for the filter s two samples shift. I additio to the filter s symmetry ad iterpolatio properties, the filter s coefficiets are dyadic. The scalig TABLE IV EXAMPLE 3, CASE K =4AND K mi =, WITH SYMMETRIC INTERPOLATING SCALING FUNCTION. h () h () h () 5/ / / / / / / / / / / / / fuctio s Sobolev smoothess i this case is.. Itisworth otig at this poit that the orthogoal scalig fuctio with dilatio factor M =4ad legth 5 has K =oly, ad lacks the smoothess of the scalig fuctio i this example, with Sobolev smoothess of oly.9 []. Filters h ad h are of legth 5 coefficiets each, while h is of legth 3. Thus for the same h filter legth, K icreases by two whe compared with the M =4symmetric orthogoal couterpart. Table IV lists filters coefficiets, while Fig. 5 shows the frequecy respose of the filters. The resultig limit fuctios are depicted i Fig. 6. Table V shows a subset of agles betwee the filters ad their shifts. The filters are geerally ear orthogoal, with a few exceptios, such as θ (h,h, ) = 68.8.

5 Frequecy Respose IV. CONCLUSION I this paper we have preseted a family of tight frame limit fuctios with dilatio factor 4 ad six FIR filters which are of miimal legth for a give umber of vaishig momets. The resultig scalig fuctios ad wavelets are smooth, ad have a throughput redudacy limited to.66. For future research, we will examie the desig of a similar tight frame structure, amely with five wavelets ad a dilatio factor of 4, but with (ati-)symmetric limit fuctios Fig. 5. Frequecy resposes of the filters described i Example 3. φ 4 6 ψ 4 6 ψ ψ 4 6 ψ ψ Fig. 6. Limit fuctios of Example 3, with symmetric scalig iterpolatig fuctio ad {K,K,K,K 3,K 4,K 5 } = {4,,,,, 4}. TABLE V EXAMPLE 3, A SUBSET OF ANGLES BETWEEN SPACES GENERATED BY FILTERS IN FIGURE 5 AND THEIR SHIFTS shift h, h 3 h, h 4 h, h 5 h, h 3 h, h 4 h, h REFERENCES [] A.F. Abdelour. Wavelet Desig Usig Gröber Basis Methods. PhD thesis, Polytechic Uiversity,. [] A.F. Abdelour. Symmetric tight frames with shifted wavelets. I Proceedigs of SPIE, volume 594, pages 33 4, Sa Diego, 3 July - 4 August 5. [3] A.F. Abdelour ad I.W. Selesick. Symmetric early shift ivariat tight frame wavelets. IEEE Tras. o Sigal Processig, 53(), Jauary 5. [4] H. Bölcskei, F. Hlawatsch, ad H.G. Feichtiger. Frame-theoretical aalysis of oversampled filter baks. IEEE Tras. o Sigal Processig, 46(): , December 998. [5] C.S. Burrus, R.A. Gopiath, ad H. Guo. Itroductio to Wavelets ad Wavelet Trasforms. Pretice Hall, 997. [6] C.K. Chui ad W. He. Compactly supported tight frames associated with refiable fuctios. Applied ad Computatioal Harmoic Aalysis, 8(3):93 39, May. [7] Z. Cvetković ad M. Vetterli. Oversampled filter baks. IEEE Tras. o Sigal Processig, 46(5):45 55, May 998. [8] G.-M. Greuel ad G. Pfister. A SINGULAR Itroductio to Commutative Algebra. Spriger Verlag,. [9] G.-M. Greuel, G. Pfister, ad H. Schöema. Sigular Referece Maual. I Reports O Computer Algebra, umber. Cetre for Computer Algebra, Uiversity of Kaiserslauter, May [] B. Ha. Symmetric orthoormal scalig fuctios ad wavelets with dilatio factor 4. Advaces i Computatioal Mathematics, (8): 47, 998. [] J.-C. Ha ad Z.-X. Cheg. The costructio of M-bad tight wavelet frames. I Proc. of It. Cof. o Machie Learig ad Cyberetics., volume 6, pages , 6-9 August 4. [] Q. Jiag. Parameterizatios of masks for tight affie frames with two symmetric/atisymmetric geerators. Adv. Comput. Math., 8:47 68, February 3. [3] J. Lebru ad I. Selesick. Gröber bases ad wavelet desig. Joural of Symbolic Computig, ():7 59, February 4. [4] T. Morris ad D. Britch. Biorthogoal wavelets for itra-frame video codig. I IWISPA, pages 9 4,. [5] A. Petukhov. Explicit costructio of framelets. Applied ad Computatioal Harmoic Aalysis, :33 37, September. [6] A. Petukhov. Symmetric framelets. Costructive Approximatio, 9():39 38, Jauary 3. [7] A. Ro ad Z. She. Costructio of compactly supported affie frames i L (R d ). I K.S. Lau, editor, Advaces i Wavelets. Spriger Verlag, 998. [8] I.W. Selesick. Smooth wavelets tight frames with zero momets. Applied ad Computatioal Harmoic Aalysis, ():63 8, March. [9] I.W. Selesick ad A.F. Abdelour. Symmetric wavelet tight frames with two geerators. Applied ad Computatioal Harmoic Aalysis, 7(): 5, September 4. [] G. Strag ad T. Nguye. Wavelets ad Filter Baks. Wellesley- Cambridge Press, 996. [] B.E. Usevitch. A tutorial o moder lossy wavelet image compressio: foudatio of JPEG. IEEE Sigal Processig Magazie, 8: 35, September.

Filter banks. Separately, the lowpass and highpass filters are not invertible. removes the highest frequency 1/ 2and

Filter banks. Separately, the lowpass and highpass filters are not invertible. removes the highest frequency 1/ 2and Filter bas Separately, the lowpass ad highpass filters are ot ivertible T removes the highest frequecy / ad removes the lowest frequecy Together these filters separate the sigal ito low-frequecy ad high-frequecy