TEN GOOD REASONS FOR USING SPLINE WAVELETS

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1 Proc. SPIE Vol. 369, Wavelets Applicatios i Sigal ad Image Processig V, 997, pp TEN GOOD REASONS FOR USING SPLINE WAVELETS Michael User Swiss Federal Istitute of Techology (EPFL) CH-05 Lausae, Switzerlad. ABSTRACT The purpose of this ote is to highlight some of the uique properties of splie wavelets. These wavelets ca be classified i four categories: othogoal (Battle-Lemarié), semi-orthogoal (e.g., B-splie), shift-orthogoal, ad biorthogoal (Cohe-Daubechies- Feauveau). Ulike most other wavelet bases, splies have explicit formulae i both the time ad frequecy domai, which greatly facilitates their maipulatio. They allow for a progressive trasitio betwee the two extreme cases of a multiresolutio: Haar's piecewise costat represetatio (splie of degree zero) versus Shao's badlimited model (which correspods to a splie of ifiite order). Splie wavelets are extremely regular ad usually symmetric or ati-symmetric. They ca be desiged to have compact support ad to achieve optimal time-frequecy localizatio (B-splie wavelets). The uderlyig scalig fuctios are the B-splies, which are the shortest ad most regular scalig fuctios of order L. Fially, splies have the best approximatio properties amog all kow wavelets of a give order L. I other words, they are the best for approximatig smooth fuctios. Keywords: splies, wavelet basis, biorthogoal wavelets, regularity, smoothess, time-frequecy localizatio, approximatio properties.. INTRODUCTION Researchers are ow faced with a ever icreasig variety of wavelet bases to choose from. While the choice of the "best" wavelet is obviously applicatio-deped, it ca be useful to isolate a umber of properties ad features that are of geeral iterest to the user. The purpose of this paper is to preset a list of argumets i favor of splies, which are uique i a umber of ways. Splies have had a sigificat impact o the theory of the wavelet trasform. The earliest example is the Haar wavelet which is a splie of degree 0. This costructio was exteded to higher order splies by Strömberg 4, evethough his work remaied largely uoticed util wavelets became what they are today. Perhaps the best kow examples of splie wavelets are the orthogoal Battle-Lemarié fuctios 5,, which ca be see as percursors of Mallat's multiresolutio theory of the wavelet trasform 3. Splies have also bee used to illustrate may of the later costructios of o-orthogoal wavelet bases (semi-orthogoal, biorthogoal, ad more recetly, shift-orthogoal). Droppig the orthogoality requiremet was foud to be advatageous for recoverig may desirable wavelet properties that could ot be achieved otherwise. Noteworthy examples are the B-splie wavelets 6, 9 which are compactly supported ad achieve a ear optimal time-frequecy localizatio. Fially, the most popular represetatives of the Cohe-Daubechies-Feauveau class of biorthogoal wavelets 7 are splies as well. This is because the iteratio of the biomial refiemet filter which is a crucial compoet i ay wavelet costructio coverges to the B-splie which is the geeratig fuctio for polyomial splies.. SPLINES AND WAVELETS Splie wavelets stad apart i the geeral theory of the wavelet trasform. Their costructio starts with the specificatio of the uderlyig multiresolutio fuctio spaces (polyomial splies). Thus, splie wavelets ca be characterized explicitly; this is i cotrast with most other costructios where the scalig fuctio is specified idirectly via a two-scale relatio. The mai advatage of a explicit costructio is that oe does ot have to worry about the delicate issues of the covergece of the iterated filterbak. It also makes the study of regularity much more trasparet.

2 - -. Polyomial splies A polyomial splie of degree is made up of polyomial segmets of degree that are coected i a way that garatees the cotiuity of the fuctio ad of its derivative up to order -. The joiig poits betwee the polyomial segmets are called kots. I the cotext of the wavelet trasform, the kots are equally-spaced ad typically positioed at the itegers. Oe ca thus defie a hierarchy of splie subspaces of degree, { V i } i Z, where V i is the subspace of L -fuctios that are (-) times i i cotiuously differetiable ad are polyomials of degree i each iterval [ k, ( k + ) ), k Z. The spacig betwee the kot poits i is cotrolled by the scale idex i. Clearly, a fuctio f ( x) V i0 that is piecewise polyomial o each segmet i 0 i0 k, ( k + ) 0. Thus, we have the followig iclusio property [ ) is also icluded i ay of the fier subspaces V i with i i L L V V L V L { }. () 0 i 0 Furthermore, it is well kow that oe ca approximate ay L -fuctio by a splie as closely as oe wishes by lettig the kot spacig (or scale) go to zero ( i ). This meas that the above sequece of ested subspaces is dese i L ad therefore meets all the requiremets for a multiresolutio aalysis of L i the sese defied by Mallat 3, 4. This implies that it is ideed possible to costruct wavelet bases that are polyomial splies. The best way to proceed is to use Schoeberg's represetatio of splies i terms of the B-splie basis fuctios 7, 8. I order to satisfy the multiresolutio iclusio property for ay degree, we will use the so-called causal B-splies which ca be costructed from the (+)-fold covolutio of the idicator fuctio i the uit iterval (causal B-splie of degree 0) where 0 0 ϕ ( x) = ϕ L ϕ ( x), () 4 43 ( + ) times, 0 x < ϕ 0 ( x) = 0 otherwise. The B-splie of degree satisfies the two-scale relatio 3 ϕ ( x / ) = h ( k) ϕ ( x k), (4) where h ( k) is the biomial filter of order + whose trasfer fuctio is z + z h ( k) H ( z) = + (3). (5) I 946, Schoeberg proved that ay polyomial splie of degree with kots at the itegers could be represeted as a liear combiatio of shifted B-splies 7. Thus, our basic splie space V 0 ca also be specified as V0 = s0( x) = c( k) ϕ ( x k) c l, (6) where the weights c(k) are the so-called B-splie coefficiets of the splie fuctio s ( x). I additio, it ca be show that the B- { } costitute a Riesz basis of V splies ϕ ( x k) k Z 0 0 i the sese that there exist two costats A > 0 ad B < + such that c l, A c c( k) ϕ ( x k) B c. (7) l L The lower iequality implies that the B-splies are liearly idepedet (i.e., s0( x) = 0 c( k) = 0 ). The upper iequality guaratees that V0 L. Hece, ay polyomial splie has a uique represetatio i terms of its B-splie coefficiets c(k). Schoeberg also proved that the B-splies are the shortest possible splie fuctios 9. This, together with the fact that these fuctios have a simple aalytical form, makes the B-splie represetatio oe of the preferred tools for the study ad characterizatio of splies 0. l

3 Biorthogoal wavelets I the most geeral case, the costructio of biorthogoal wavelet bases ivolves two multiresolutio aalyses of L : oe for the aalysis, ad oe for the sythesis 7. These are usually deoted by { V i ( ϕ )} i ad { V Z i ( ϕ) } i, where ϕ ( x ) ad ϕ( x ) are the Z aalysis ad sythesis scalig fuctios, respectively. Note that ϕ ad ϕ ca be arbitrary solutios of a two-scale relatio ad ot ecessarily the causal B-splies ϕ defied previously. The correspodig aalysis ad sythesis wavelets ψ ( x ) ad ψ( x ) are the cotructed by takig liear combiatios of these scalig fuctios ψ ( x / ) = g ( k) ϕ( x k) (8) k ψ( x / ) = g( k) ϕ( x k). (9) They form a biorthogoal set i the sese that with the short form covetio ψ k ψ i, k, ψ j, l = δi j, k l, (0) i, k i / = ψ( i x k). This allows us to obtai the wavelet expasio of ay L -fuctio as f L f = f, ψ i k ψ i k. (),,, i Z Note that the uderlyig basis fuctios are usually specified idirectly i terms of the four sequeces h( k), h ( k ), g( k) ad g ( k ), which are the filters for the fast wavelet trasform algorithm..3 Splie wavelets We have a splie wavelet trasform wheever the sythesis fuctios ( ψ( x ) ad ϕ( x )) are polyomial splies of degree. This meas that the sythesis wavelet ca also be represeted by its B-splie expasio ψ( x / ) = w( k) ϕ ( x k). () It is importat to observe that the uderlyig scalig fuctio ϕ( x) V 0 is ot ecessarily the B-splie of degree uless h(k) is precisely the biomial filter (5). This fuctio is usually specified idirectly as the solutio of the two-scale relatio ϕ( x / ) = h( k) ϕ( x k), (3) where h(k) is the correspodig (lowpass) recostructio filter. However, i the splie case, there will always exist a sequece p(k) such that ϕ( x) = p( k) ϕ ( x k). (4) Such specific B-splie characterizatios for various kids of splie scalig fuctios (orthogoal, dual, or iterpolatig) ca be foud elsewhere 3. Note that the sequece p(k) defies a ivertible covolutio operator from l ito l which performs the chage from oe coordiate system to the other (i.e., ϕ to ϕ ). The basic requiremet for { ϕ( x k) } k Z to form a Riesz basis of V is that there exist two costats A > 0 ad B j 0 p p < + such that Ap P( e ω ) B almost everywhere, where P e j ω p ( ) deotes the Fourier trasform of p. If we combie (9) with (4), we obtai the B-splie coefficiets of the wavelet ψ( x ): w( k) = ( p g)( k) z W( z) = P( z) G( z). (5) These splie wavelets are quite attractive because they are extremely regular. I fact, ay splie wavelets of degree is (+) times differetiable almost everywhere. It has a Sobolev regularity idex s max =+/ meaig that all its fractioal derivatives up to s max are well-defied i the L -sese 3. A very importat wavelet parameter is the order of the represetatio determied from the zero-properties of the refiemet filter h. By defiitio, the order L is the largest iteger such that the trasfer fuctio of h ca be factorized as, where Q(z) is the

4 - 4 - z-trasform of a stable filter. Thus, splies have a order of approximatio L=+ which is oe more tha the degree. This order property has some remarkable cosequeces such as the vaishig momets of the aalysis wavelet, the ability of the scalig fuctio to reproduce polyomials of degree =L-, ad the special eige-structure of the two-scale trasitio operator (cf. Strag- Nguye 3, Chapter 7). May kids of splie wavelets have bee described i the literature. The four primary types ca be differetiated o the basis of their orthogoality properties; they are summarized i Table. Four correspodig examples of cubic splie wavelets ad their duals are salso show i Fig.. TABLE I: CLASSIFICATION OF SPLINE WAVELETS WITH THEIR MAIN PROPERTIES. Wavelet type Orthogoality Compact support Key properties Implemetatio Orthogoal splies (Battle-Lemarié, Mallat) Semi-orthogoal splies (B-splies) (Chui-Wag, User-Aldroubi) Shift-orthogoal splies (User-Théveaz-Aldroubi) Biorthogoal splies (Cohe-Daubechies-Feauveau) Yes No Symmetry & regularity + Orthogoality Iter-scale Aalysis or Symmetry & regularity Sythesis + Optimal time-frequecy localizatio Itra-scale No Symmetry & regularity + Quasi-orthogoality + Fast decayig wavelet No Yes Symmetry & regularity + Compact support IIR/IIR Recursive IIR/FIR IIR/IIR FIR/FIR Orthogoal splie wavelets: The wavelets i this first category were costructed idepedetly by Battle 5 ad Lemarié. They were also ivestigated by Mallat 3, 4 to illustrate his geeral multiresolutio theory of the wavelet trasform. Like most splie wavelets, the Battle-Lemarié fuctios are very regular, smooth ad symmetric. Ufortuately, they are ot compactly supported, evethough they decay expoetially fast. Semi-orthogoal splie wavelets: The wavelets i this category retai the iter-scale orthogoality, but there is o requiremet for the basis fuctios to be orthogoal to their traslates withi the same resolutio level. Chui-Wag 6 ad User-Aldroubi-Ede 9 idepedetly costructed the first such example: the compactly supported B-splie wavelets (cf. Fig. b), which are the wavelet couterparts of the classical B-splies. It was the realized that oe could still geerate may other semi-orthogoal splie wavelets by takig suitable liear combiatios 3. These wavelets are versatile because is possible to choose the sequece p(k) i (4) ad w(k) i () such that the uderlyig fuctios have some specific property 3 (e.g. iterpolatio, orthogoality or optimal time-frequecy localizatio). I fact, it is possible to costruct semi-orthogoal splie wavelets with virtually ay prescribed shape. Some of the wavelets ad scalig fuctios i the semi-orthogoal case ca be compactly supported; however, the dual aalysis fuctios, which are splies as well, are geerally ot. Thus, the implemetatio usually requires some ifiite impulse respose filters. We ote, however, that a recursive implemetatio is usually possible 3, except i the orthogoal case where the filters eed to be trucated to fiite legth.

5 - 5 - SYNTHESIS WAVELET ψ( x) ANALYSIS WAVELET ψ ( x) (a) Orthogoal splie (b) Semi-orthogoal B-splie (c) Shift-orthogoal splie (hybrid degree = 3 ad ñ = ) t t (d) Biorthogoal splie ( L = 4, L = 6) Fig. : Examples of four differet types of cubic splies wavelets ad their correspodig duals.

6 - 6 - Biorthogoal splie wavelets: Biorthogoal wavelet basis were itroduced by Cohe-Daubechies-Feauveau 7 i order to obtai wavelet pairs that are symmetric, regular ad compactly supported. Ufortuately, this is icompatible with the orthogoality requiremet that has to be dropped altogether. Biorthogoal wavelets build with splies are especially attractive because of their short support ad regularity. These wavelets tur out to be quite popular for codig applicatios 36. I particular, the symmetry ad short support properties are very valuable for reducig trucatio artifacts i the recostructed images. It is obviously also possible to costruct other o-compactly supported splie wavelets sice this type of costructio is the least costraied of all. Shift-orthogoal splie wavelets: This category explores a last possibility which is to retai the itra-scale orthogoality requiremet aloe 35. This idea was first ivestigated with the costructio of splie wavelets of hybrid degree allowig for a very direct visualizatio of the two uderlyig multiresolutio; for example, piecewise liear for the aalysis ad cubic for the sythesis 34, 35 (cf. Fig. c). Their mai advatage is that it is possible to reduce the decay of the wavelet while essetially retaiig the orthogoality ad approximatio properties of the Battle-Lemarié splie wavelets. These features ca potetially be of iterest for subbad codig. The orthogoality property is required for the quatizatio error i the wavelet domai to be a exact idicator of the recostructed image's fial distortio. Orthogoality with respect to shifts, i particular, is cosistet with the idea of idepedet chael processig (scalar quatizatio). Likewise, havig shorter wavelet sythesis filters is advatageous for reducig recostructio artifacts (e.g., spreadig of codig errors, rigig aroud sharp trasitios). These possibilities remai to be tested experimetally. 3. SPLINE AND WAVELET PROPERTIES We have see that all splie wavelets are liear combiatio of B-splies. Thus, they will iherit most of the properties of these basis fuctios. The list below start with the most basic properties that are commo to all splies ad the proceeds with others that are more specific. Because of the iitial title of this paper, we have limited ourselves to te such properties. We gladly ivite the reader to fid more. 3. Closed form solutio The B-splies, which have bee defied as the (+)-fold covolutio of a uit box fuctio, have simple explicit forms i both the time ad frequecy domai. To derive such formulae, we start by expressig the covolutio property () by a product i the Fourier domai, which yields 0 + ϕˆ ( ω) ϕˆ ( ω) = ( ) = e jω where ϕˆ ( ω) deotes the Fourier trasform of ϕ ( x). This may also be rewritte as ( + ) jω jω +, (6) ˆ + ϕ ( ω) = e sic ( ω / π), (7) which ivolves the (+)th power of the sic fuctio. Next, we expad (6) usig the biomial expasio + jωk ϕˆ + k e ( ω) = ( ) ( ω ) + k j. (8) k = 0 The crucial step is the to idetify ( ) ( jω +) as the Fourier trasform of the (+)-fold itegral of the Dirac delta; i.e., the fuctio ( x) + /! where ( x) + = max( 0, x). By iterpretig the complex expoetials as shift factors, we ca get back to the time domai + ϕ k + ( x) = ( ) ( x k). (9) +! k k = 0 This formula shows that ϕ ( x) is piecewise polyomial of degree. It is also clear from (8) that ϕ ( x) ca be differetiated times before oe starts ucoverig discotiuities at the iteger kots. This is because the (+)th derivative of ϕ ( x)

7 - 7 - (multiplicatio by ( jω ) + i the frequecy domai) is a sequece of Dirac impulses with the alteratig biomial weights + k ( ). k Aother iterestig observatio is that (9) ca also be iterpreted as the (+)th forward differece of the oe-sided power fuctio ( x) +. I other words, we have ϕ + x + ( x) = ( ), (0) where + deotes the (+) iteratio of the forward differece operator f ( x) = f ( x) f ( x ). The importat poit here is that these formulae together with () provide a explicit characterizatio of all splie wavelets. By cotrast, most other scalig fuctios are oly defied through a ifiite product i the frequecy domai 9, 3, 36 + ˆ ω / ϕ( ω) = H( e j i ). () i= Applyig this latter result to the B-splie case yields the idetity + jω + e + e = jω i= i jω / +. () 3. Simple maipulatio Splies are piecewise polyomial, which greatly simplifies their maipulatio 0, 6. I particular, it is straightforward to obtai splie derivatives ad itegrals. For istace, the first derivative of a causal B-splie of degree is give by dϕ dx ( x) = ϕ ( x) ϕ ( x ). (3) This result may be derived by multiplyig (8) by jω (differetiatio i the Fourier domai). Thus, differetiatio correspods to a reductio of the splie degree by oe. Similarly, itegratio will result i a correspodig icrease i the degree. This type of relatio may be useful if oe uses splie wavelets for solvig differetial equatios. 3.3 Symmetry The B-splies are symmetric. It is therefore easy to costruct splie wavelets that are either symmetric or ati-symmetric by selectig the sequece with the appropriate symmetry i (). The advatage is that the correspodig wavelet trasform ca be implemeted usig mirror boudary coditios which reduces boudary artifacts 7. This is usually ot possible for o-liear phase wavelets such as the celebrated Daubechies wavelets Shortest scalig fuctio of order L We recall that the refiemet filter for a Lth order wavelet trasform ca be factorized as 9 H( z) = ( + z ) L Q( z), (4) where Q( z) is the trasfer fuctio of a stable filter. The B-splie of degree =L- correspods to the shortest Lth order refiemet filter with Q(z)=. Oe ca therefore coclude that the B-splie of degree =L- is the shortest possible scalig fuctio of order L. 3.5 Maximum regularity for a give order L I geeral, the regularity (or Sobolev smoothess) of a scalig fuctio caot be greater tha the order L. We also kow that the Sobolev smoothess 3 of a B-splie of degree =L- is s = max L. However, the B-splies are are ot the smoothest scalig fuctios of order L they are oly optimal if we take ito accout the filter legth. For istace, we ca cosider the refiemet filter (4) with Q( z) = ( + + z ε ), which, for ε>0 sufficietly small (but o-zero), ca achieve the maximum smoothess

8 - 8 - smax = L. But this also shows that the B-splies are the smoothest scalig for a refiemet filter of a give legth: the example above with ε=0 is a B-splie of order L+ which has a smoothess smax = L + this is / better tha aythig else of the same legth! 3.6 m-scale relatio The B-splies satisfy a two-scale relatio for ay iteger m. Ulike most other scalig fuctios, m is ot restricted to a power of two. We will derive this property by first cosiderig a B-splie of degree 0 expaded by a factor of m, which ca obviously be represeted as m 0 0 ϕ ( x / m) = ϕ ( x k). (5) k = 0 We rewritte this equatio i the Fourier domai as ϕˆ ( mω) = H ( z) ϕˆ ( ω) m 0 k where H ( z) = z / m. If we ow apply the covolutio properties of B-splies, we get m m k = 0 or, equivaletly, where 0 + m ϕˆ ( mω) = ( H ( z) ) ϕˆ ( ω), ϕ m ( x / m) = m hm ( k) ϕ ( x k), (6) Hm( z) = m m k = 0 z k +. (7) Thus, (6) provides us with a two-scale relatio that is valid for ay iteger m. I additio, the refiemet filter h m( k) ca be iterpreted as a cascade of (+) movig sum filters. Each of these filters ca be implemeted very efficietly usig a recursive updatig strategy that requires o more tha two additios per sample poits. This property ca be advatageous for implemetatio purposes. For istace, it is the basis for a efficiet filterig procedure for zoomig up sigal or images by a iteger factor m 8. Aother applicatio is the fast implemetatio of the cotiuous wavelet trasform with iteger scales Variatioal properties Splies provide a "atural" sigal iterpolat that is optimal i the sese that it has the least oscillatig eergy. This property is a cosequece of the first itegral relatio, which states that for ay fuctio f(x) whose mth derivative is square itegrable, we have + ( ) = ( ) + ( ) + + f ( m ) dx s ( m ) m m dx f ( ) s ( ) dx where s( x) is the splie iterpolat of degree =m- such that s( k) = f ( k). I particular, if we apply this decompositio to the problem of the iterpolatio of a give data sequece f(k), we see that the splie iterpolat s(x) miimizes the orm of the mth derivative amog all possible iterpolats f(x), which is a rather remarkable result 9. I this sese, the splie is the iterpolatig fuctio that oscillates the least. For m=, the eergy fuctio i (8) is a good approximatio to the itegral of the curvature for a curve y=f(x). Thus, cubic splie iterpolats exhibit a miimum curvature property, which justifies the aalogy with the draftma's splie, or Frech curve. This latter device is a thi elastic beam that is costraied to pass through a give set of poits. (8)

9 Best approximatio properties i For a Lth order wavelet, the approximatio error decreases with the Lth power of the scale a = (cf. Strag, ). Specifically, we ca derive the followig asymptotic relatio 6 lim a 0 f P f = C a f a L ( L) ϕ, (9) where P f a deotes the projectio (orthogoal or oblique) of f oto the multiresolutio space at scale a. This error formula becomes valid as soo as the samplig step a is sufficietly small with respect to the smoothess scale of f(x). The costat C ϕ is the same for all splie wavelet trasforms of a give order L, ad is give by 6 C ϕ = B L ( L)!, (30) where B L is Berouilli's umber of order L. This turs out, by far, to be the smallest costat amog all kow wavelet trasforms of the same order L (cf. Table I i the above metioed referece). This is partly due to the fact that the B-splies basis fuctios are very regular. For compariso, the Daubechies costats are so much worse that oe would eed to sample the sigal at more tha twice the rate to reach the same asymptotic L -error. Thus, it appears that splies are the best for approximatig smooth fuctios. Aother iterestig cosequece is that the asymptotic approximatio error will be the same irrespective of the orthogoality properties of the trasform. 3.9 Optimal time-frequecy localizatio The direct wavelet couterparts of the B-splies are the B-splie wavelets which are compactly supported as well. We have show i our earlier work 9 that these wavelet coverge to a cosie-modulated Gaussia (or Gabor) fuctio as the degree of the splie goes to ifiity. Specifically, if ψ ( x) deotes the B-splie wavelet of degree, the we have the followig approximate formula (cosie-modulated Gaussia) ψ ( x) + 4b π ( + ) σ w cos ( πf0( x ) ) exp ( x ) σ ( + ), (3) with b= , f 0 = ad σ w = The quality of this Gabor approximatio improves rapidly with icreasig ; for =3, the approximatio error is less tha 3%. The implicatio is that there are splie wavelet bases that ca be optimally localized i time ad frequecy. I other words, we ca get as close as we wish to the time-frequecy localizatio limit specified by the ucertaity priciple. For the cubic splie example i Fig. b, the product of the time ad frequecy ucertaities is already withi % of the limit specified by Heiseberg's ucertaity priciple. 3.0 Covergece to the ideal lowpass filter Splies provide a coveiet framework that allows for a progressive trasitio betwee the two extreme cases of a multiresolutio: the piecewise costat model (with =0) ad the badlimited model correspodig to a splie of degree ifiite 4, 5, 0. I particular, it ca be show 30 that the Battle-Lemarié scalig fuctios coverge to sic(x) the impulse respose of the ideal lowpass filters as the order of the splie goes to ifiity. Likewise, their correspodig orthogoal wavelets coverge to the ideal badpass filter 3 (modulated sic). I practice, their approximatio of a ideal filter starts to be good for 3 (relative error < 5 %). This type of covergece properties may be relevat for codig applicatios. For istace, it has bee show that the ideal badpass decompositio is optimal i the sese that it maximizes the codig gai for all statioary processes with o-icreasig spectral power desity 5. Refereces. P. Abry ad A. Aldroubi, "Desigig multiresolutio aalysis-type wavelets ad their fast algorithms", J. Fourier Aalysis ad Applicatios, Vol., No., pp , J.H. Ahlberg, E.N. Nilso ad J.L. Walsh, The Theory of splies ad their applicatios, Academic Press, New York, 967. w

10 A. Aldroubi ad M. User, "Families of multiresolutio ad wavelet spaces with optimal properties", Numerical Fuctioal Aalysis ad Optimizatio, Vol. 4, No. 5-6, pp , A. Aldroubi, M. User ad M. Ede, "Cardial splie filters : stability ad covergece to the ideal sic iterpolator", Sigal Processig, Vol. 8, No., pp. 7-38, August G. Battle, "A block spi costructio of odelettes. Part I: Lemarié fuctios", Commu. Math. Phys., Vol. 0, pp , C.K. Chui ad J.Z. Wag, "O compactly supported splie wavelets ad a duality priciple", Tras. Amer. Math. Soc., Vol. 330, No., pp , A. Cohe, I. Daubechies ad J.C. Feauveau, "Bi-orthogoal bases of compactly supported wavelets", Comm. Pure Appl. Math., Vol. 45, pp , I. Daubechies, "Orthogoal bases of compactly supported wavelets", Comm. Pure Appl. Math., Vol. 4, pp , I. Daubechies, Te lectures o wavelets, Society for Idustrial ad Applied Mathematics, Philadelphia, PA, C. de Boor, A practical guide to splies, Spriger-Verlag, New York, A. Haar, "Zur Theorie der orthogoale Fuktioesysteme", Math. A., Vol. 69, pp , 90.. P.-G. Lemarié, "Odelettes à localisatio expoetielles", J. Math. pures et appl., Vol. 67, No. 3, pp. 7-36, S.G. Mallat, "Multiresolutio approximatios ad wavelet orthogoal bases of L (R)", Tras. Amer. Math. Soc., Vol. 35, No., pp , S.G. Mallat, "A theory of multiresolutio sigal decompositio: the wavelet represetatio", IEEE Tras. Patter Aal. Machie Itell., Vol. PAMI-, No. 7, pp , M.J. Marsde, F.B. Richards ad S.D. Riemescheider, "Cardial splie iterpolatio operators o l p data", Idiaa Uiv. Math J., Vol. 4, No. 7, pp , P.M. Preter, Splies ad variatioal methods, Wiley, New York, I.J. Schoeberg, "Cotributio to the problem of approximatio of equidistat data by aalytic fuctios", Quart. Appl. Math., Vol. 4, pp , -4, I.J. Schoeberg, "Cardial iterpolatio ad splie fuctios", Joural of Approximatio Theory, Vol., pp , I.J. Schoeberg, Cardial splie iterpolatio, Society of Idustrial ad Applied Mathematics, Philadelphia, PA, I.J. Schoeberg, "Notes o splie fuctios III: o the covergece of the iterpolatig cardial splies as their degree teds to ifiity", Israel J. Math., Vol. 6, pp. 87-9, G. Strag, "Wavelets ad dilatio equatios: a brief itroductio", SIAM Review, Vol. 3, pp , G. Strag ad G. Fix, "A Fourier aalysis of the fiite elemet variatioal method", i: Costructive Aspect of Fuctioal Aalysis, Edizioi Cremoese, Rome, 97, pp G. Strag ad T. Nguye, Wavelets ad filter baks, Wellesley-Cambridge, Wellesley, MA, J.O. Strömberg, "A modified Frakli system ad higher-order splie system of R as ucoditioal bases for Hardy spaces", Proc. Coferece i Harmoic Aalysis i hoor of Atoi Zygmud, pp , M. User, "O the optimality of ideal filters for pyramid ad wavelet sigal approximatio", IEEE Tras. Sigal Processig, Vol. 4, No., pp , December M. User, "Approximatio power of biorthogoal wavelet expasios", IEEE Tras. Sigal Processig, Vol. 44, No. 3, pp , March M. User, "A practical guide to the implemetatio of the wavelet trasform", i: A. Aldroubi ad M. User, ed., Wavelets i Medicie ad Biology, CRC Press, Boca Rato, FL, 996, pp M. User, A. Aldroubi ad M. Ede, "Fast B-splie trasforms for cotiuous image represetatio ad iterpolatio", IEEE Tras. Patter Aal. Machie Itell., Vol. 3, No. 3, pp , March M. User, A. Aldroubi ad M. Ede, "O the asymptotic covergece of B-splie wavelets to Gabor fuctios", IEEE Tras. Iformatio Theory, Vol. 38, No., pp , March M. User, A. Aldroubi ad M. Ede, "Polyomial splie sigal approximatios : filter desig ad asymptotic equivalece with Shao's samplig theorem", IEEE Tras. Iformatio Theory, Vol. 38, No., pp , Jauary M. User, A. Aldroubi ad M. Ede, "A family of polyomial splie wavelet trasforms", Sigal Processig, Vol. 30, No., pp. 4-6, Jauary M. User, A. Aldroubi ad M. Ede, "The L polyomial splie pyramid", IEEE Tras. Patter Aal. Mach. Itell., Vol. 5, No. 4, pp , April M. User, A. Aldroubi ad S.J. Schiff, "Fast implemetatio of the cotiuous wavelet trasform with iteger scales", IEEE Tras. Sigal Processig, Vol. 4, No., pp , December M. User, P. Théveaz ad A. Aldroubi, "Costructio of shift-orthogoal splie wavelets usig splies", Proc. SPIE Cof Wavelet Applicatios i Sigal ad Image Processig IV, Dever, CO, pp , August 6-9, M. User, P. Théveaz ad A. Aldroubi, "Shift-orthogoal wavelet bases usig splies", IEEE Sigal Processig Letters, Vol. 3, No. 3, pp , March M. Vetterli ad J. Kovacevic, Wavelets ad Subbad Codig, Pretice Hall, Eglewood Cliffs, NJ, 995.