Optimal Representation of Piecewise Hölder Smooth Bivariate Functions by the Easy Path Wavelet Transform

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1 Optmal Representaton of Pecewse Hölder Smooth Bvarate Functons by the Easy Path Wavelet Transform Gerlnd Plonka 1, Stefane Tenorth 1, and Armn Iske 2 1 Insttute for Numercal and Appled Mathematcs, Unversty of Göttngen, Lotzestr , Göttngen, Germany {plonka,s.tenorth}@math.un-goettngen.de 2 Department of Mathematcs, Unversty of Hamburg, Hamburg, Germany ske@math.un-hamburg.de Abstract The Easy Path Wavelet Transform (EPWT) [25] has recently been proposed by one of the authors as a tool for sparse representatons of bvarate functons from dscrete data, n partcular from mage data. The EPWT s a locally adaptve wavelet transform. It works along pathways through the array of functon values and t explots the local correlatons of the gven data n a smple approprate manner. In ths paper, we am to provde a theoretcal understandng of the performance of the EPWT. In partcular, we derve condtons for the path vectors of the EPWT that need to be met n order to acheve optmal N-term approxmatons for pecewse Hölder smooth functons wth sngulartes along curves. Key words. sparse data representaton, wavelet transform along pathways, N-term approxmaton AMS Subject classfcatons. 41A25, 42C40, 68U10, 94A08 1 Introducton Durng the last few years, there has been an ncreasng nterest n effcent representatons of large hgh-dmensonal data, especally for sgnals. In the one-dmensonal case, wavelets are partcularly effcent to represent pecewse smooth sgnals wth pont sngulartes. In two dmensons, however, tensor product wavelet bases are no longer optmal for the representaton of pecewse smooth functons wth dscontnutes along curves. Wthn the last few years, more sophstcated methods were developed to desgn approxmaton schemes for effcent representatons of two-dmensonal data, n partcular for mages, where correlatons along curves are essentally taken nto account to capture the geometry of the gven data. Curvelets [2, 3], shearlets [12, 13] and drectonlets [34] are examples for non-adaptve hghly redundant functon frames wth strong ansotropc drectonal selectvty. For pecewse Hölder smooth functons of second order wth dscontnutes along C 2 -curves, Candès and Donoho [2] proved that a best approxmaton f N to a gven functon 1

2 f wth N curvelets satsfes the asymptotc bound f f N 2 2 C N 2 (log 2 N) 3, whereas a (tensor product) wavelet expanson leads to asymptotcally only O(N 1 ) [21]. Up to the (log 2 N) 3 factor, ths curvelet approxmaton result s optmal (see [7, Secton 7.4]). A smlar estmate has been acheved by Guo and Labate [12] for shearlet frames. These results, however, are not adaptve wth respect to the assumed regularty of the target functon, and so they cannot be appled to mages of less regularty,.e., mages whch are not at least pecewse C 2 wth dscontnutes along C 2 -curves. Just recently, the N-term approxmaton propertes of compactly supported shearlet frames have been studed also n the 3D case, [19]. Usng two smoothness parameters α and β n (1, 2] controllng the classcal as well as the ansotropc smoothness, nearly optmal approxmaton rates are proven n [19]. In [11], a general framework called parabolc molecules has been presented coverng most of the curvelet- and shearlet-lke constructons and showng that these systems all provde the same approxmaton results. In the bvarate case, for pecewse Hölder smooth functons of order α 2, one may rather adapt the approxmaton scheme to the mage geometry nstead of fxng a bass or a frame beforehand to approxmate f. Durng the last few years, several dfferent approaches were developed for dong so [1, 5, 6, 8, 9, 14, 18, 20, 22, 25, 26, 27, 29, 30, 33]. In [20], for nstance, bandelet orthogonal bases and frames are ntroduced to adapt to the geometrc regularty of the mage. Due to ther constructon, the utlzed bandelets are ansotropc wavelets that are warped along a geometrcal flow to generate orthonormal bases n dfferent bands. LePennec and Mallat [20] showed that ther bandelet dctonary yelds asymptotcally optmal N-term approxmatons, even n more general mage models, where the edges may also be blurred. Further examples for geometry-based mage representatons are the nonlnear edgeadapted (EA) multscale decompostons n [1, 14] (and references theren) based on ENO reconstructons. We remark that the resultng ENO-EA schemes lead to an optmal N- term approxmaton, yeldng f f N 2 2 C N 2 for pecewse C 2 -functons wth dscontnutes along C 2 -curves. Moreover, unlke prevous non-adaptve schemes, the ENO-EA multresoluton technques provde optmal approxmaton results also for BV -spaces and L p spaces, see [1]. In [25], a new locally adaptve dscrete wavelet transform for sparse mage representatons, termed Easy Path Wavelet Transform (EPWT), has been proposed by one of the authors. The EPWT works along pathways through the array of functon values, where t essentally explots the local correlatons of mage values n an approprate manner. We remark that the EPWT s not restrcted to a regular (two-dmensonal) grd of mage pxels, but t can be extended, n a more general settng, to scattered data approxmaton n hgher dmensons. In [26], the EPWT has been appled to data representatons on the sphere. In the mplementaton of the EPWT, one needs to work wth sutable data structures to effcently store the path vectors that need to be accessed durng the performance of the EPWT reconstructon. To reduce the resultng adaptvty costs, we have proposed a hybrd method for smooth mage approxmatons n [27], where an effcent edge representaton by the EPWT s combned wth favorable propertes of the borthogonal tensor product wavelet transform. 2

3 In [28], we have proven optmal N-term approxmaton rates for the EPWT for pecewse Hölder contnuous functons of order α (0, 1]. Our proof n [28] s manly based on an adaptve multresoluton analyss structure, whch s only avalable usng pecewse constant Haar wavelets along the pathways. But the EPWT does not necessarly need to be restrcted to Haar wavelets. In fact, the numercal results n [25, 31] show a much hgher effcency for smoother wavelet bases as e.g. Daubeches D4 flters and borthogonal 7-9 flters. These observatons motvate us to study the N-term approxmaton for Hölder smooth functons of order α > 1 n ths paper. In partcular, we wll derve condtons for the path vectors leadng to optmal N-term approxmaton by the EPWT. More precsely, we present three condtons for the optmal path vectors that mply optmal N-term approxmatons of the form f f N 2 2 C N α (1.1) for the applcaton of the EPWT to pecewse Hölder smooth functons of order α > 1, wth allowng dscontnutes along smooth curves of fnte length. Unfortunately, the three condtons on the path vectors are very dffcult to ensure. We regard these condtons as an dealzed settng whch heurstcally explans the performance of the EPWT wth hgher order wavelets, and whch gve us a hnt on how to construct sutable path vectors. The path constructons of the relaxed EPWT n [25] amed at a good compromse to produce many small wavelet coeffcents, on the one hand, and low costs for path codng, on the other hand. Interestngly enough, our results n ths paper show that the path constructons n [25] yeld already a farly good tradeoff to meet the optmal path vector condtons. In our prevous paper [15], we have appled the EPWT n mage denosng, wth path vectors that approxmatvely satsfy the optmal path vector condtons. Wth usng pecewse constant functons for the approxmaton of a bvarate functon f, the EPWT yelds an adaptve multresoluton analyss when relyng on an adaptve Haar wavelet bass (see [25, 28]). If, however, smoother wavelet bases are utlzed n the EPWT approach, such an nterpretaton s not obvous. In fact, whle Haar wavelets admt a straght forward transfer from one-dmensonal functons along pathways to bvarate Haar-lke functons, we cannot rely on such smple connectons between smooth one-dmensonal wavelets (used by the EPWT) and a bvarate approxmaton of the lowpass functon. Therefore, n ths paper we wll apply a sutable nterpolaton method, by usng polyharmonc splne kernels, to represent the arsng bvarate low-pass functons after each level of the EPWT. One key property of polyharmonc splne nterpolaton s polynomal reproducton of arbtrary order. We wll come back to relevant approxmaton propertes of polyharmonc splnes n Secton 2. The outlne of ths paper s as follows. In Secton 2, we frst ntroduce the utlzed functon model, some ssues on polyharmonc splne nterpolaton, and the EPWT algorthm. Then, n Secton 3, we study the decay of EPWT-wavelet coeffcents, where we wll consder the hghest level of the EPWT n detal. To acheve optmal decay results for the EPWT wavelet coeffcents at all levels, we requre three specfc sde condtons for the path vectors n the EPWT algorthm, the regon condton, the path smoothness condton and the dameter condton, see Secton 3.2. We also show, why the path smoothness condton s very restrctve. In fact, t cannot be met for the usual EPWT as descrbed n [25] for α > 1. Therefore, n ths paper we ntroduce sutable smooth path functons n 3

4 order to derve the path vectors for the EPWT. Fnally, Secton 4 s devoted to the proof of asymptotcally optmal N-term error estmates of the form (1.1) for pecewse Hölder smooth functons. 2 The EPWT and Polyharmonc Splne Interpolaton 2.1 The Functon Model Suppose that Ω [0, 1] 2 approxmates [0, 1] 2 and has suffcently smooth Lpschtz boundary. Further, let F L 2 (Ω) be a pecewse smooth bvarate functon, beng smooth over a fnte set of regons {Ω } 1 K, where each regon Ω has a suffcently smooth Lpschtz boundary Ω of fnte length,.e., the boundary curves have bounded dervatves. Moreover, the set {Ω } 1 K s assumed to be a dsjont partton of Ω, so that K Ω = Ω, =1 where each closure Ω s a connected subset of Ω, for = 1,..., K. More precsely, we assume that F s Hölder smooth of order α > 0 n each regon Ω, 1 K, so that any µ-th dervatve of F on Ω wth µ = α satsfes an estmate of the form F (µ) (x) F (µ) (y) C x y α µ 2 for all x, y Ω. Note that ths assumpton for F s equvalent to the condton that for each x 0 Ω there exsts a bvarate polynomal q α of degree α (usually the Taylor polynomal of F of degree α at x 0 Ω ) satsfyng F (x) q α (x x 0 ) C x x 0 α 2 (2.1) for every x Ω n a neghborhood of x 0, where the constant C > 0 does not depend on x or x 0. But F may be dscontnuous across the boundares between adjacent regons. Note that the Hölder space C α (Ω ) of order α > 0, beng equpped wth the norm F C α (Ω ) := F C α (Ω ) + sup x y µ = α F (µ) (x) F (µ) (y) x y α α 2 concdes wth the Besov space B, (Ω α ), when α s not an nteger. Here, we use the C α (Ω ) norm F C α (Ω ) := sup F (x) + sup µ F (x), x Ω x Ω µ = α see e.g. [4, Chapter 3.2]. Now by quas-unform samplng, the bvarate functon F s assumed to be gven by ts functon values taken at a fnte grd I 2J. For a sutable nteger J > 1, let {F (y)} y I2J be the gven samples of F, where I 2J Ω wth #I 2J = 2 2J, and mn y 2 y 2 2 κ 1 2 J, y 1,y 2 I 2J sup nf x y 2 2 (J+1/2) y I 2J x Ω 4

5 wth a postve constant κ 1 beng ndependent of J. Moreover, let Γ 2J := {y I 2J : y Ω } for 1 K (2.2) be the set of grd ponts that are contaned n the regons Ω, for 1 K. Obvously, K =1 Γ 2J = I 2J, and for the sze #Γ 2J of Γ 2J we have #Γ 2J #I 2J = 2 2J for any 1 K. 2.2 Polyharmonc Splne Interpolaton Next we compute a (pecewse) suffcently smooth approxmaton to F from ts gven samples. To ths end, we construct a sutable nterpolant F 2J satsfyng the nterpolaton condtons F 2J (y) = F (y) for all y I 2J. (2.3) Ths s accomplshed by the applcaton of polyharmonc splne nterpolaton separately n each ndvdual regon Ω, n order to frst obtan, for any ndex, 1 K, a polyharmonc splne nterpolant F 2J satsfyng F 2J (y) = F (y) for all y Γ 2J (2.4) at the nterpolaton ponts Γ 2J Ω. For the requred (global) nterpolant F 2J, we then let K F 2J (x) := F 2J (x)χ Ω (x) for x Ω (2.5) =1 to satsfy the nterpolaton condtons n (2.3), where χ Ω s the characterstc functon of Ω. Recall that polyharmonc splnes, due to Duchon [10], are sutable tools for multvarate nterpolaton from scattered data. For further detals on polyharmonc splnes, especally for relevant aspects concernng ther local approxmaton propertes, we refer to [17, Secton 3.8]. We apply the polyharmonc splne nterpolaton scheme n two dmensons, where the nterpolant F 2J n (2.4) s assumed to have the form F 2J (x) = y Γ 2J c n φ m ( x y 2 ) + p m(x), (2.6) where φ m (r) = r 2m log(r), for m := α, s a fxed polyharmonc splne kernel, and p m P m s a polynomal n the lnear space P m of all bvarate polynomals of degree at most m. Note that, by the form of φ m, the nterpolant F 2J (x) s, for any, n C m. Moreover, by the above choce of m, F 2J (x) s Hölder smooth of order α, and n ths way, we can mantan the local Hölder regularty of F around each lattce pont n Γ 2J. Note that the nterpolant F 2J n (2.6) has #Γ 2J + dm(p m ) parameters,.e., the #Γ 2J coeffcents c n n ts major part and another dm(p m ) = (m + 1)(m + 2)/2 parameters n ts polynomal part. However, the nterpolaton problem (2.4) yelds only #Γ 2J lnear 5

6 condtons on the #Γ 2J + (m + 1)(m + 2)/2 parameters of F 2J. To elmnate addtonal degrees of freedom, we requre another set of (m + 1)(m + 2)/2 lnear constrants on the coeffcents c n, as gven by the vanshng moment condtons c n p(y) = 0 for all p P m. (2.7) y Γ 2J To compute the coeffcents of the polyharmonc splne nterpolant F 2J, ths then amounts to solvng a square lnear system wth #Γ 2J + dm(p m ) equatons, (2.4) and (2.7), for #Γ 2J + dm(p m ) unknowns, gven by the parameters of F 2J. Accordng to the semnal work of Mchell [23] on (condtonally) postve defnte functons, ths square lnear equaton system has a unque soluton, provded that the set of nterpolaton ponts are P m -regular,.e., for p P m we have the mplcaton p(y) = 0 for all p P m = p 0, (2.8) so that every polynomal n P m can unquely be reconstructed from ts values on Γ 2J. In fact, ths comples wth earler results n [10], where the well-posedness of the polyharmonc splne nterpolaton scheme s proven va a varatonal theory concernng (polyharmonc) splnes mnmzng rotaton-nvarant sem-norms n Sobolev spaces. Therefore, we can conclude that the polyharmonc splne nterpolant F 2J n (2.6), satsfyng (2.4) and (2.7), s unque provded that the nterpolaton ponts Γ 2J satsfy the (rather weak) regularty condtons n (2.8). From now we wll tactly assume that the condtons n (2.8) are fulflled. In ths case, we can further conclude that the polyharmonc splne nterpolaton scheme acheves to reconstruct polynomals of degree m. Concernng the local approxmaton order of polyharmonc splne nterpolaton, we refer to [16]. As was proven n [16], the asymptotc estmate F 2J (hy) F (hy) = O(h m+1 ) for h 0 holds for F C m+1, n whch case the polyharmonc splne nterpolaton scheme s sad to have local approxmaton order m + 1, see [16] or [17, Secton 3.8] for detals. We remark that the concept n [16] concernng local approxmaton orders s n contrast to that of global approxmaton orders, where the latter s well-understood for polyharmonc splnes and related radal kernels snce much longer. In the subsequent analyss n ths paper, we requre one specfc approxmaton result for polyharmonc splne nterpolaton concernng ts global approxmaton behavour. We can descrbe ths as follows. From the embeddng theorem for Besov spaces we obtan B α, (Ω ) B α 2,2 (Ω ), see e.g. [32] or [4, page 163]. Snce B α 2,2 (Ω ) s equvalent to the Sobolev space H α (Ω ), see [4, 32], ths allows us to use the estmate F F 2J L 2 (Ω) C F K =1 h α Ω F B α 2,2 (Ω ) (2.9) for the nterpolaton error n Sobolev spaces, as shown n [24], where the fll dstance h Ω := sup x Ω nf y Γ 2J x y 2 2 J+1/2 measures the densty of the nterpolaton ponts n Ω. 6 for 1 K

7 2.3 The EPWT Algorthm Now let us brefly recall the EPWT algorthm from our prevous work [25]. To ths end, let ϕ C β, for β α, be a suffcently smooth, compactly supported, one-dmensonal scalng functon,.e., the nteger translates of ϕ form a Resz bass of the scalng space V 0 := clos L 2span {ϕ( k) : k Z}. Further, let ϕ be a correspondng borthogonal and suffcently smooth scalng functon wth compact support, and let ψ and ψ be a correspondng par of compactly supported wavelet functons. We refer to [4, Chapter 2] for a comprehensve survey on borthogonal scalng functons and wavelet bases and summarze only the notaton needed for the borthogonal wavelet transform. For j, k Z, let ϕ j,k (t) := 2 j/2 ϕ(2 j t k) and ψ j,k (t) := 2 j/2 ψ(2 j t k), lkewse for ϕ and ψ. The functons ϕ, ϕ and ψ, ψ are assumed to satsfy the refnement equatons ϕ(x) = 2 n ϕ(x) = 2 n h n ϕ(2x n) h n ϕ(2x n) ψ(x) = 2 q n ϕ(2x n) n ψ(x) = 2 q n ϕ(2x n) n wth fnte sequences of flter coeffcents (h n ) n Z, ( h n ) n Z and (q n ) n Z, ( q n ) n Z. assumpton, the polynomal reproducton property p m, ϕ j,k ϕ j,k = p m for all j Z, k By s satsfed for any polynomal p m of degree less than or equal to m = α, and so, p m, ψ j,k = 0 for all j, k Z. Wth these assumptons, {ψ j,k : j, k Z} and { ψ j,k : j, k Z} form borthogonal Resz bases of L 2 (R),.e., for each functon f L 2 (R), we have f = f, ψ j,k ψ j,k = f, ψ j,k ψ j,k. j,k Z For any gven unvarate functon f j, j Z, of the form f j (x) = n Z cj (n) ϕ j,n one decomposton step of the dscrete (borthogonal) wavelet transform can be represented n the form f j (x) = f j 1 (x) + g j 1 (x), where f j 1 (x) = n Z j,k Z c j 1 (n) ϕ j 1,n and g j 1 (x) = n Z d j 1 (n) ψ j 1,n wth c j 1 (n) = f j, ϕ j 1,n and d j 1 (n) = f j, ψ j 1,n. (2.10) 7

8 Conversely, one step of the nverse dscrete wavelet transform yelds for gven functons f j 1 and g j 1 the reconstructon f j (x) = n Z c j (n) ϕ j,n wth c j (n) = f j 1, ϕ j,n + g j 1, ϕ j,n. We recall that the EPWT s a wavelet transform that works along path vectors through pont subsets of I 2J. For the characterzaton of sutable path vectors we frst need to ntroduce neghborhoods of ponts n I 2J. For any pont y = (y 1, y 2 ) I 2J, we defne ts neghborhood by N(y) := {x = (x 1, x 2 ) I 2J \ {y} : x y 2 2 J+1/2 }, where x y 2 2 = (x 1 y 1 ) 2 + (x 2 y 2 ) 2. Now the EPWT algorthm s performed as follows. For the applcaton of the 2J-th level of the EPWT we need to fnd a path vector p 2J = (p 2J (n)) 22J 1 n=0 through the pont set I 2J. Ths path vector s a sutable permutaton of all ponts n I 2J, whch can be determned by usng the followng strategy from [25]. Recall that I 2J = K =1 Γ2J, where Γ 2J determnes the lattce ponts n Ω. Start wth one pont p 2J (0) n Γ 2J 1. Now, for a gven n-th component p 2J (n) beng contaned n the set Γ 2J n (2.2), for some {1,..., K}, we choose the next component p 2J (n + 1) of the path vector p 2J, such that p 2J (n + 1) (N(p 2J (n)) Γ 2J ) \ {p 2J (0),..., p 2J (n)}, (2.11).e., p 2J (n + 1) should be a neghbor pont of p 2J (n), lyng n the same set Γ 2J, that has not been used n the path, yet. In stuatons where (N(p 2J (n)) Γ 2J ) \ {p 2J (0),... p 2J (n)} s an empty set, the path s nterrupted, and we need to start a new pathway by choosng the next component p 2J (n+1) from Γ 2J \ {p 2J (0),..., p 2J (n)}. If, however, ths set s also empty, we choose p 2J (n + 1) from the set of remanng ponts I 2J \{p 2J (0),..., p 2J (n)}. For a more detaled descrpton of the path vector constructon we refer to [25]. In partcular, for a sutably chosen path vector p 2J, the number of nterruptons can be bounded by K = C 1 K, where K s the number of regons, and where the constant C 1 does not depend on J but only on the shape of the regons Ω, see [28]. The so obtaned vector p 2J s composed of connected pathways,.e., each par of consecutve components n these pathways s neghborng. Now, we consder the data vector ( c 2J (l) ) 2 2J 1 := ( F 2J ( p 2J (l) )) 2 2J 1 l=0 l=0 and apply one level of a one-dmensonal (perodc) wavelet transform to the functon values of F 2J along the path p 2J. Ths yelds the low-pass vector (c 2J 1 (l)) 22J 1 1 l=0 and the vector of wavelet coeffcents (d 2J 1 (l)) 22J 1 1 l=0 accordng to the formulae n (2.10). Due to the pecewse smoothness of F 2J along the path vector p 2J, t follows that most of the wavelet coeffcents n d 2J 1 are small, whereas the wavelet coeffcents correspondng to an nterrupton of the path (wthn one regon or from one regon to another) may possess sgnfcant ampltudes. We remark already here that, by contrast to the orgnal 8

9 EPWT algorthm descrbed n ths subsecton, for the case of pecewse Hölder smooth mages of order α > 1, we wll ntroduce a smooth path functon p 2J n Secton 3.1. Ths path functon wll determne the components of the path vector p 2J and s assumed to have bounded dervatves almost everywhere. Note that for α > 1, wavelet coeffcents of (arbtrary) large magntude may occur n stuatons where the dervatves of the path functon are not unformly bounded, as e.g. at tght turns of p 2J. As regards the next level of the EPWT, the path vector p 2J yelds a new subset of ponts where Γ 2J 1 Γ 2J 1 := { p 2J (2l) : l = 0,..., 2 2J 1 1 } = K =1 Γ 2J 1, := {p 2J (2l) : p 2J (2l) Γ 2J }. At level j = 2J 1, we frst locate a second connected path vector p 2J 1 = (p 2J 1 (l)) 22J 1 1 l=0 through Γ 2J 1,.e., the entres of p 2J 1 form a permutaton of the ponts n Γ 2J 1. Smlarly as before, we requre that p 2J 1 (n) and p 2J 1 (n + 1) are neghbors lyng n the same pont set Γ 2J 1. Here, p 2J 1 (n) and p 2J 1 (n + 1) are sad to be neghbors,.e., p 2J 1 (n + 1) N(p 2J 1 (n)), ff p 2J 1 (n) p 2J 1 (n + 1) 2 2 J+1. Agan, the number of path nterruptons can be bounded by C 1 K, where C 1 does not depend on J. Then we apply one level of the one-dmensonal wavelet transform to the permuted data vector (c 2J 1 (per 2J 1 (l))) 22J 1 1 l=0, where the permutaton per 2J 1 s defned by per 2J 1 (l) := k ff p 2J 1 (l) = p 2J (2k) for l, k = 0, J 1 1. We obtan the low-pass vector (c 2J 2 (l)) 22J 2 1 l=0 and the vector (d 2J 2 (l)) 22J 2 1 l=0 of wavelet coeffcents. We contnue by teraton over the remanng levels j for j = 2J 2,..., 1, where at any level j we frst construct a path p j = (p j (l)) 2j 1 l=0 through the set Γ j := {p j+1 (2l) : l = 0,..., 2 j 1} = wth applyng smlar strateges as descrbed above. Here, p j (n) and p j (n + 1) are called neghbors, ff p j (n) p j (n + 1) 2 D2 j/2, (2.12) where D 2 s a sutably determned constant (n the above descrpton of p 2J and p 2J 1 we have chosen D = 2). Then we apply the wavelet transform to the permuted vector (c j (per j (l)) 2j 1 l=0, yeldng cj 1 and d j 1. Example. In ths example, we explan the constructon of the path vectors through the remanng data ponts wth the low-pass values by a toy example. To ths end, let Ω = [0, 1) 2 be dvded nto only two regons, Ω 1 and Ω 2. The functon F s assumed to be Hölder contnuous n each of these regons, but may be dscontnuous across the curve separatng the two regons Ω 1 and Ω 2. In our toy example, we have J = 3,.e., an 8 8 mage wth 64 data values. At the hghest level of the EPWT, we choose a path p 6 through the underlyng pont set I 6 = {(2 3 n 1, 2 3 n 2 ) : n 1, n 2 = 0,..., 7} = Γ 6 1 Γ6 2, such that each two consecutve components n the path are neghbors. We frst pck all ponts n Γ 6 1, 9 K =0 Γ j

10 (a) (b) (c) (d) Fgure 1: Path constructon. (a) level sx, (b) level fve, (c) level four, (d) level three. before jumpng to Γ 6 2, see Fgure 1(a). For the path constructon at the next level, we frst determne the set Γ 5 = Γ 5 1 Γ5 2 (contanng only each second component of p6 ), see Fgure 1(b), before we construct a path accordng to the above descrpton. Fgures 1(c) and 1(d) show the sets Γ 4 and Γ 3 along wth ther correspondng path vectors. Besdes, we tred to fnd the path vectors n a way such that the angles formed by the polygonal lne of the path are as large as possble. In fact, the polygonal lnes for p 6 and p 5 do not contan angles beng smaller than 90 degrees, whle the polygonal lne of p 4 possesses two such angles, one n each regon. In fact, locally straght polygonal lnes of the path vectors gve rse to obtan better smoothness bounds for the path functons p j that we wll ntroduce n Secton 3. In ths example, we have p 6 (n + 1) p 6 (n) 2 5 D, p 5 (n + 1) p 5 (n) 2 5 2D, p 4 (n + 1) p 4 (n) 2 4 2D, p 3 (n + 1) p 3 (n) D (wth one path nterrupton at each level for the jump from one regon to the other), so that the path constructon satsfes the above requrements wth D = 5. Ths smple example also llustrates that the path constructon leads at each level to pont sets Γ j wth quas-unformly dstrbuted ponts. See Secton 3.2 for more explanaton about the used quas-unformty. Remark 1. Consderng the above strategy of the EPWT algorthm, t s heurstcally clear that we are able to reduce the number of sgnfcant wavelet coeffcents to a multple of the number K of regons, where the target functon F s smooth. Indeed, only when the path skps from one regon to another, a fnte number of sgnfcant wavelet coeffcents wll occur. Ths s n contrast to the usual tensor product wavelet transform, where the number of sgnfcant wavelet coeffcents s usually related to the total length of the smooth regons boundares and hence depends on the level j of the wavelet transform. 10

11 Remark 2. As shown n [25], the choce of the path vectors n each level s crucal for the performance of the approxmaton by means of the EPWT. In partcular, t s not a good dea to take smply the same path vector p j (l) = p j+1 (2l) at each level, snce one only explots the data correlaton along ths one path and the EPWT s reduced to a onedmensonal transform. In Secton 3.2, we wll determne the so called dameter condton that requres the nequalty (2.12) for neghborng components n the path constructon thereby avodng the above mentoned smple path choce. Our numercal experments wth the EPWT mply that the constant D should be chosen rather small, e.g. D 2, see [31]. On the other hand, a too small D nduces a small number of neghborhood ponts and hence yelds a hgher number of path nterruptons. In Secton 3.2, we wll show that the nequalty (2.12) for path constructon mples a quas-unform dstrbuton of values n Γ j at each level of the EPWT. Remark 3. Note that the mplementaton of the EPWT algorthm, as descrbed above, s rather straghtforward. In the orgnal verson of the EPWT, we frst determne, at each level, a path vector p j, before we apply a wavelet flter bank to the one-dmensonal data vector that s ordered accordng to the path vector. In contrast to that strategy (from our prevous work), we now consder usng a slghtly dfferent approach to obtan the theoretcal estmates of our present paper. To ths end, we derve the path vector p j by frst determnng a pecewse smooth path functon p j, whose constructon s detaled n the followng of ths paper. We also consder slghtly dfferent data vectors c j p(l) n the theoretcal estmates of Sectons 3 and 4, where we wll use L 2 -projecton operators determned by the dual scalng and wavelet functons, ϕ and ψ. Further, we wll employ polyharmonc splne nterpolaton for analytcal purpose. Remark 4. Note that the components of the path vector p j le n I 2J wth contanng 2d entres. Ths s n contrast to the notaton n [25]. Further, unlke n [25], we do no longer consder ndex sets but defne a neghborhood of ponts by the Eucldean dstance. 3 Decay of Wavelet Coeffcents generated by the EPWT Before we turn to the techncal detals, let us frst sketch the basc deas of the proof for optmal N-term approxmatons by the EPWT, where we use slghtly changed notatons. As explaned n the Subsecton 2.2, we consder applyng polyharmonc splne nterpolaton, from gven mage values F (y), separately n the ndvdual domans Ω, = 1,..., K. We assume that F s Hölder smooth of order α on each Ω, so that the polyharmonc splne nterpolant F 2J n (2.5), beng Hölder smooth of order at least α, reconstructs bvarate polynomals of degree m = α. Further, we apply a generalzed noton of a path p gven as a suffcently smooth planar parameter curve. At the 2J-th level of the EPWT, we frst determne a (pecewse) smooth path functon p 2J : [0, 1] Ω, such that each value y I 2J can be approxmated at suffcent accuracy by p 2J ( l ), for some l {0,..., 2 2J 1}. Consecutve values p 2J ( l ) and p 2J ( l+1 ) 2 2J 2 2J 2 2J should approxmate neghborng values n Γ 2J := I 2J lyng n the same set Ω (up to O(K) exceptons). Now, a path vector p 2J R 22J 2 through all values of I 2J s determned by the permutaton gven by the order n whch p 2J (l/2 2J ), l = 0,..., 2 2J 1, approxmates the values n I 2J. 11

12 The path functon p 2J should be constructed such that only at most C 1 K dscontnutes (ncurred by possble transtons from one regon to another) may occur. In ths way, we can bound the number of nterruptons n the path vector p 2J by C 1 K. Next, we consder a one-dmensonal functon f 2J (t) = 22J 1 k=0 c 2J p (k) ϕ 2J,k (t), t [0, 1), whch approxmates the pecewse smooth one-dmensonal and scaled restrcton of F resp. F 2J along the path p 2J satsfyng F ( p 2J (2 2J l) f 2J (2 2J l) 2 Jα. Then, we apply one level of a smooth wavelet transform to f 2J. In ths way, sgnfcant wavelet coeffcents may only occur at a fnte number of locatons on the nterval [0, 1) that correspond to dscontnutes or to tght turns of the path functon p 2J. However, the number of such sgnfcant coeffcents does not depend on 2 J but on the number of regons, K. Therefore, wth the performance of one level of the (perodc) wavelet transform, we wll fnd that most of the wavelet coeffcents of f 2J = f 2J 1 + g 2J 1 occurrng n the wavelet part g 2J 1, are small. At the next j-th level of the EPWT, we consder the set Γ j := { p j+1 (2 j l) : l = 0,..., 2 j 1} = K =1 Γ j. By constructon, we have # Γ j = 2 j. Observe that Γ j are pont sets n Ω. Agan, we then construct a smooth path functon p j that approxmates the values n Γ j n a sutable order and thus determnes the path vector p j whose components are the values n Γ j n the order n whch they are approxmated by p j. To obtan suffcently accurate polyharmonc splne nterpolatons F j to F wth F j (y) = F (y) for y Γ j, and to apply the same arguments as at the hghest level, three specfc condtons on the path functons p j need to be satsfed. We can brefly explan these condtons, termed regon condton, path smoothness condton and dameter condton, as follows (for more detals on these condtons we refer to Subsecton 3.2). Frstly, the regon condton requres that the path functon should prefer to approxmate all values of one regon set Γ j, before proceedng wth the values of the next regon. In other words, the path vector should prefer to traverse the ponts belongng to one regon set Γ j, before jumpng to another regon, where jumps wthn one regon should be avoded. Assumng further that the path functon s smooth wth sutably bounded dervatves (path smoothness condton), we can optmally explot the smoothness of the functon F along the path. Fnally, the dameter condton ensures a quas-unform dstrbuton of remanng ponts n each Γ j, and therefore leads to a suffcently accurate polyharmonc splne nterpolaton F j at each level of the EPWT. We remark that the regon condton and the dameter condton can be forced by usng the strateges for the path constructon as proposed n Subsecton 2.2. The path smoothness condton cannot be satsfed by the orgnal path constructon methods n [25]. But snce the numercal evaluatons contan only a few levels of the EPWT, the path smoothness condton can be forced by avodng small angles n the path vector and by preferrng path snakes, see e.g. examples n [25]. The three path condtons wll allow us to estmate the EPWT wavelet coeffcents smlarly as for one-dmensonal pecewse smooth functons wth a fnte number of sngulartes to fnally obtan an optmal N-term 12

13 approxmaton usng only the N most sgnfcant EPWT wavelet coeffcents for the mage reconstructon. 3.1 The Hghest Level of the EPWT Let us now explan the 2J-th level of the EPWT n detal. The performance of the subsequent levels of the EPWT and the correspondng estmates are then derved n a smlar manner. We consder a suffcently smooth parameter curve p 2J (t), t [0, 1], through Ω approxmatng the values n Γ 2J n a certan order and the correspondng path vector p 2J of length 2 2J wth values n I 2J, where we assume that p 2J (2 2J l) p 2J (l) µ2 J l = 0,..., 2 2J 1, (3.1) and where µ s a fxed small constant (ndependent of J) wth assumng p 2J (t) Ω, for t [l/2 2J, (l + 1)/2 2J ], f p 2J (2 2J l) and p 2J (2 2J (l + 1)) are n Ω. Now, we regard the functon f 2J that s defned by the one-dmensonal restrcton of F 2J along the curve p 2J, f 2J (t) = F 2J ( p 2J (t) ) for t [0, 1]. If the path functon p 2J crosses over from one regon Ω to another, there may occur a dscontnuty of f 2J caused by a dscontnuty of F 2J,.e., there are ndces l and l + 1 n {0, 1,..., 2 2J 1}, where we have a dscontnuty between F 2J ( p 2J (2 2J l)) and F 2J ( p 2J (2 2J (l + 1))),.e., a dscontnuty of f 2J n the subnterval [l/2 2J, (l + 1)/2 2J ]. Wthout loss of generalty, we assume that we have only dscontnutes of f 2J caused by jumps from one regon to another. Recall that the trace theorem for Hölder resp. Besov spaces (see [32]) mples that for F 2J Ω B α, (Ω ), the (scaled) restrcton f 2J (t) along the curve p 2J (t) s agan Hölder smooth of order α n each subnterval of [0, 1], determned by T = {t [0, 1] : p 2J (t) Ω } and wth assumng that the correspondng path vector p 2J has no nterruptons n Ω. In partcular, for any α (0, 1), we fnd the estmate f 2J C α (T ) = sup t 1 t 2 t 1,t 2 T = sup t 1 t 2 t 1,t 2 T F 2J ( p 2J (t 1 )) F 2J ( p 2J (t 2 )) t 1 t 2 α F 2J ( p 2J (t 1 )) F 2J ( p 2J (t 2 )) p 2J (t 1 ) p 2J (t 2 ) α p 2J (t 1 ) p 2J (t 2 ) α 2 2 t 1 t 2 α F 2J C α (Ω ) p 2J α C 1 (T ). For the case α (1, 2), the chan rule yelds f 2J C α (T ) = sup t 1 t 2 t 1,t 2 T sup t 1 t 2 t 1,t 2 T F 2J ( p 2J (t 1 ))( p 2J ) (t 1 ) F 2J ( p 2J (t 2 ))( p 2J ) (t 2 ) t 1 t 2 α 1 ( F 2J ( p 2J (t 1 )) F 2J ( p 2J (t 2 )) 2 p 2J (t 1 ) p 2J (t 2 ) α 1 2 ) + F 2J C 1 (Ω ) ( p2j ) (t 1 ) ( p 2J ) (t 2 ) t 1 t 2 α 1 F 2J C α (Ω ) p 2J α C 1 (T ) + F 2J C 1 (Ω ) p 2J C α (T ). 13 p 2J (t 1 ) p 2J (t 2 ) α 1 2 t 1 t 2 α 1 ( p 2J ) C 1 (T )

14 In general, we have Lemma 3.1 Suppose that the parameter curve p 2J : [0, 1] Ω satsfes n each subnterval T the smoothness condton p 2J C β (T ) C 2 2 Jβ (3.2) for all β max{α, 1} and wth a constant C 2 beng ndependent of J,.e., p j C α (T ) for α > 1 and p j C 1 (T ) for α 1, = 1,..., K, wth strctly bounded dervatves. Then for the one-dmensonal restrcton f 2J (t) = F 2J ( p 2J (t)) along the curve p 2J we have f 2J C α (T ) C2 Jα F 2J C α (Ω ). (3.3) Proof. For α < 2 the asserton already follows from the prevous observatons. Let m := α. Further, we use the abbrevatons F = F 2J, p = p 2J. In the general case the formula by Faà d Bruno mples that D m f 2J (t) := dm f dt 2J (t) = D m (F ( p(t))) s a fnte m lnear combnaton of terms S ν,j (t) := D ν F (x) x= p(t) (D m1 p 1 (t)) b 1 (D m2 p 2 (t)) b 2, where p = p 2J = ( p 1, p 2 ) T, m 1, m 2, b 1, b 2 N wth m 1 + m 2 m, m 1 b 1 + m 2 b 2 = m, and wth ν N 2 0, ν m. For t 1, t 2 T wth t 1 t 2 we now obtan wth (3.2) the estmate S ν,j (t 1 ) S ν,j (t 2 ) t 1 t 2 ( α m ) ( ) (D ν F )( p(t 1 )) (D ν F )( p(t 2 )) p(t1 ) p(t 2 ) α m p(t 1 ) p(t 2 ) α m 2 t 2 1 t 2 ( α m ) + (D ν F )( p(t 2 )) (D m 1 p1 (t 1 )) b 1 (D m 2 p 2 (t 1 )) b 2 (D m 1 p 1 (t 2 )) b 1 (D m 2 p 2 (t 2 )) b 2 t 1 t 2 α m F C α+ ν m (Ω ) p α m C 1 (T ) p b 1 C m 1 (T ) p b 2 C m 2 (T ) + F C ν (Ω ) (D m1 p 1 (t 1 )) b 1 (D m2 p 2 (t 1 )) b 2 ( b 1 p b 2 p b 1 1 C m 1 (T ) p C m 1 +α m (T ) + b 2 p b 1 C m 1 (T ) p b 2 1 C m 2 (T ) p C m 2 +α m (T ) F C α+ ν m (Ω ) 2J(α m) 2 J(m 1b 1 ) 2 J(m 2b 2 ) C m 2 (T ) + F C ν (Ω ) (b 12 J(m 2b 2 +m 1 b 1 m 1 +m 1 +α m) + b 2 2 J(m 1b 1 +m 2 b 2 m 2 +m 2 +α m) ) C ν 2 Jα F C α (Ω ), where we have used the nequalty x k y k k x y max{x, y} k 1 for x, y R, k N. Hence, the asserton follows. Let us assume n the sequel that the path functon p satsfes the smoothness condton (3.2) n Lemma 3.1. Thus, we obtan for the N-th order modulus of smoothness of f 2J the estmate ω N ( f 2J, h) := sup Ñ f 2J h h α f 2J C α (T ) (2 J h) α F 2J C α (Ω ) (3.4) h h wthn the subntervals, where f 2J s smooth,.e., for N = α + 1 and T,h := { t : p 2J (t + kh) Ω, k = 0,..., N }, ) 14

15 see [4]. Next, we consder the L 2 -projecton f 2J := P 2J f 2J of f 2J onto the scalng space V 2J := clos L 2 [0,1)span{ϕ 2J,n : n = 0,..., 2 2J 1}, where ϕ s assumed to be a suffcently smooth scalng functon, see Secton 2.2. Then, f 2J = P 2J f 2J := 2 2J 1 n=0 f 2J, ϕ 2J,n ϕ 2J,n also satsfes a Hölder smoothness condton of order α. Followng along the lnes of [4, Theorem 3.3.3], we obtan n the subntervals T,2 2J the estmate In partcular, f 2J f 2J L (T,2 2J ) = f 2J P 2J f 2J L (T,2 2J ) ω N ( f 2J, 2 2J ) (2 2J ) α f 2J C α (T ) (2 J ) α F 2J C α (Ω ). (3.5) f 2J (2 2J l) f 2J (2 2J l) = F 2J ( p 2J (2 2J l)) f 2J (2 2J l) 2 Jα. In the next step, we decompose the functon f 2J = c 2J p (l)ϕ 2J,l wth c 2J p (l) := f 2J, ϕ 2J,l l nto the low-pass part f 2J 1 and the hgh-pass part g 2J 1. Applyng one level of the onedmensonal wavelet transform to the data set (c 2J p (l)) 22J 1 l=0, we obtan the decomposton f 2J = f 2J 1 + g 2J 1 wth f 2J 1 = 2 2J 1 1 n=0 2 2J 1 1 c 2J 1 p (n) ϕ 2J 1,n and g 2J 1 = n=0 d 2J 1 p (n) ψ 2J 1,n, where c 2J 1 p (n) := f 2J, ϕ 2J 1,n and d 2J 1 p (n) := f 2J, ψ 2J 1,n. From the Hölder smoothness of f 2J, we fnd for t T the representaton f 2J (t) = q α (t t 0 ) + R(t t 0 ) for t 0 {2 2J k : k = 0,..., 2 2J 1} T and t t 0 2 2J, where q α denotes the Taylor polynomal of degree α of f 2J at t 0, and where the remander R satsfes the bound R(t t 0 ) c ϕ (t)2 Jα. Hence, f supp( ψ 2J 1,n ) T for some, the wavelet coeffcents satsfy d 2J 1 p (n) = q α ( t 0 ) + R( t 0 ), ψ 2J 1,n = R( t 0 ), ψ 2J 1,n c ϕ,n 2 Jα ψ 2J 1,n 1 = c ϕ,n 2 ( J+1/2)(α+1), where we have used ψ 2J 1,n 1 = 2 J+1/2 ψ 1. Observe that the constant c ϕ,n n ths nequalty depends on the choce of the wavelet bass but also on the (local) smoothness propertes of f 2J, and hence on the (local) boundedness propertes of the dervatves of the chosen path functon p 2J n Lemma 3.1. Now let Λ 2J 1 be the set of all n {0,..., 2 2J 1 1}, where the above estmate for d 2J 1 (n) s satsfed. Then, the number of the remanng 2 2J 1 #Λ 2J 1 wavelet coeffcents corresponds to the number of postons, where f 2J s dscontnuous (e.g. caused by crossng over of the path functon to another regon) or where dervatves of f 2J are not sutably bounded,.e., the constant c ϕ,n s too large (caused e.g. by tght turns of the path 15

16 functon p 2J ). We assume that ths number s bounded by CK, where K s the number of regons n the orgnal mage F, and where the constant C does not depend on 2 J. Now, we consder the low-pass functon f 2J 1 and reconstruct a bvarate functon F 2J 1 as follows. Takng only the path functon values p 2J (2 2J+1 n), we put Γ 2J 1 := { p 2J (2 2J+1 n) : n = 0,..., 2 2J 1 1, p 2J (2 2J+1 n) Ω } for each = 1,..., K and Γ 2J 1 := K =1 Γ 2J 1. We compute the polyharmonc splne nterpolant K F 2J 1 (x) := c y φ m ( x y 2 ) + p m(x) χ Ω (x), =1 y Γ 2J 1 satsfyng the nterpolaton condtons F 2J 1 ( p 2J (2 2J+1 n) ) = f 2J 1 (2 2J+1 n) for all n = 0,..., 2 2J 1 1. Therefore, F 2J ( p 2J (2 2J+1 n)) F 2J 1 ( p 2J (2 2J+1 n)) = f 2J (2 2J+1 n) f 2J 1 (2 2J+1 n) = f 2J (2 2J+1 n) P 2J 1 f 2J (2 2J+1 n) 2 ( J+1)α = D α (2 J+1/2 ) α, where D = 2, and where the last nequalty agan follows analogously as n (3.5) snce f 2J 1 s the orthogonal projecton of f 2J to V 2J 1 := clos L 2 [0,1)span{ϕ 2J 1,n : n = 0, J 1 1}. The last nequalty mples that F 2J 1 s stll a good approxmaton to F, snce the nterpolaton ponts have changed only slghtly. However, only half of the nterpolaton ponts are left, whch are rregularly dstrbuted n Ω. Moreover, by (3.1) we have max x Ω mn y Γ 2J 1 x y 2 J+1 + µ2 J = D 2 J+1/2. (3.6) Assumng mn y1,y 2 Γ 2J 1 y 1 y 2 D 1 2 J, wth a sutable constant D 1 > 0 ndependent of J, we observe wth (2.9) F 2J F 2J 1 L 2 (Ω ) (2 J+1/2 ) α for all = 1,..., K. Remark. Compared wth the orgnal EPWT algorthm n [25], the condton (3.1) s an mportant relaxaton. Here the scalng and wavelet coeffcents are computed from the functon f 2J (t) that approxmates F 2J ( p 2J (t)), but snce p 2J (2 2J l), l = 0,..., 2 2J 1, do not nterpolate but only approxmate the orgnal grd ponts I 2J, we work wth new functon values F 2J ( p 2J (2 2J l)) of the polyharmonc splne functon F 2J nstead of the gven values F (y), y I 2J. The orgnal EPWT relates to the specal case µ = 0, where we have nterpolaton n (3.1). 16

17 3.2 Condtons for the Path Vectors Before we proceed wth the error estmates for the further levels of the EPWT algorthm, we want to fx specfc sde condtons for the path functons that are requred for our error analyss, and that have been mplctly used already n the estmates at the hghest level n Secton 3.1. The sde condtons are termed (a) regon condton, (b) path smoothness condton, and (b) dameter condton, as stated below. The regon condton and the dameter condton have been smlarly stated already n [28] to prove the N-term approxmaton estmates. The regon condton ensures that at each level of the EPWT, the path functon jumps only C 1 K tmes from one regon Ω to another regon or nsde a regon. The dameter condton ensures that the remanng ponts n Γ j are quas-unformly dstrbuted, such that there s a constant D, not dependng on J or j, satsfyng max mn x y 2 (1 + 2)D2 j/2. (3.7) x Ω y Γ j Further, at each level j of the EPWT, we assume that we can fnd a smooth functon p j : [0, 1] Ω wth unformly bounded dervatves that approxmates the values n Γ j. Ths leads us to the vector p j, whose components from Γ j are sutably ordered, and we have p j (2 j l) p j (l) µ2 j for l = 0,..., 2 j 1. Further, we assume that p j (t) Ω, for t [l/2 j, (l + 1)/2 j ], f p j (2 2J l) and p j (2 2J (l + 1)) are n Ω. Let us ntroduce the three condtons more explctly. (a) Regon condton. At each level j of the EPWT, the path functon p j s chosen, such that t contans only at most C 1 K dscontnutes caused by crossng over from one regon Ω to another regon or by jumpng wthn one regon Ω. (b) Path smoothness condton. The path functon p j satsfes n each subnterval T the smoothness condton p j C β (T ) C 2 2 jβ/2 for all β max{α, 1} and wth a constant C 2 beng ndependent of j,.e., p j C α (T ) for α > 1 and p j C 1 (T ) for α 1, = 1,..., K, wth bounded dervatves. (c) Dameter condton. At each level of the EPWT, we requre for almost all values 2 j l, l = 0,..., 2 j 2, the condton p j (2 j l) p j (2 j (l + 1)) 2 D 2 j/2, (3.8) where D s ndependent of J and j, and where the number of values of p j (2 j l) whch do not satsfy the dameter condton, s bounded by a constant C 3 not dependng on j. Hence, at each level j, consecutve components p j (2 j l) n the path functon should be spatal neghbors. The condtons (a) and (c) can be enforced by the proposed path constructon n Subsecton 2.2, see (2.11) for the regon condton and (2.12) for the dameter condton. Partcularly, the dameter condton ensures a quas-unform dstrbuton of the ponts n Γ j. Ths can be seen nductvely as follows. For j = 2J, the asserton (3.7) s obvous, 17

18 Fgure 2: Smooth path functon n a regon Ω wth smooth boundary. see also (3.6). Generally, assumng (3.7) and (3.8) at level j + 1, t follows that max mn x y 2 max mn x y 2 + max mn y z 2 x Ω y Γ j x Ω y Γ j+1 y Γ j+1 z Γ j (1 + 2)D2 (j+1)/2 + D2 (j+1)/2 = (1 + 2)D 2 j/2. Unfortunately, the gven path smoothness condton s rather restrctve. In Fgure 2, we sketch an example for a choce of a path functon p 2J for a convex doman Ω = Ω 1 wth a smooth boundary wth Hölder smoothness α (1, 2). The constructon of p 2J reles on smooth spral curves that are composed of straght lnes and smooth Bezer curves (or arcs) wth bounded frst dervatve. Observe that n the example of Fgure 2, the curvature of the path functon p 2J does not depend on 2 J but on the curvature of the boundary of Ω 1, whle all ponts n Γ 2J 1 can be well approxmated (e.g. by ther projectons onto the curve p 2J ). Moreover, assumng that the straght lne segments as shown n Fgure 2 are separated at dstance 2 J, the complete path functon p 2J has a length bounded by κ2 J. For a next path functon p 2J 1 one may now thnk about a smooth spral curve composed of straght lne segments n a horzontal drecton (nstead of the vertcal drecton n Fgure 2), and the dstance between two neghborng straght lne segments s now 2 J+1/2, etc. Remarks. 1. To understand the strength of the path smoothness condton (b), we follow the arguments of one of the revewers and show that the path vector p j can only be taken straght along a lne f µ < 1/4. In partcular, f the path functon p 2J s forced to nterpolate the grd ponts of the form (2 J n 1, 2 J n 2 ), n 1, n 2 Z (regular grd), as n the stuaton of the EPWT n [25], the path smoothness condton cannot be satsfed. Indeed, wth (3.1) we fnd the estmate p 2J (l + 1) 2p 2J (l) + p 2J (l 1) p 2J (2 2J (l + 1) 2 p 2J (2 2J (l) + p 2J (2 2J (l 1) + 2 J 4µ. 18

19 But the smoothness condton (3.2) mples that p(2 2J (l + 1) 2 p(2 2J (l) + p(2 2J (l 1) = ( p(2 2J (l + 1) p(2 2J (l)) ( p(2 2J (l) + p(2 2J (l 1)) = (( p 2J ) (t) ( p 2J ) (t 2 2J ) dt I 0 2 2J 2 2J(α 1) p 2J C α (I) = 2 2Jα p 2J C α (I) C 2 2 Jα, where I 0 := [2 2J l, 2 2J (l + 1)] I, and where I denotes an nterval where p 2J s C α smooth. Hence, p 2J (l + 1) 2p 2J (l) + p 2J (l 1) 2 J (C 2 2 J(1 α) + 4µ). For α > 1 the term C2 J(1 α) tends to zero, as J. On the other hand, f the ponts p 2J (l + 1), p 2J (l), p 2J (l 1) (beng samples from the regular grd (2 J n 1, 2 J n 2 ), n 1, n 2 Z) do not le on the same lne, then the norm on the left hand sde s at least 2 J. Hence, for µ < 1/4 the path condtons are ncompatble. 2. The condtons (a) and (c) can slghtly be relaxed the sense that the constants C 1 and C 3 may be allowed to depend polynomally on j (but not exponentally). In fact, consderng the proof of Theorem 4.1 (resp. Theorem 3.3 n [28]), one needs to ensure that a weghted sum of all wavelet coeffcents that are only satsfyng the estmate (3.11) s stll bounded. 3.3 The Further Levels of the EPWT Let us now explan the further levels of the EPWT. These are performed followng along the lnes of the 2J-th level. We start wth the polyharmonc splne nterpolant F j (x) := K c y φ m ( x y 2 ) + p m(x) χ Ω (x) =1 y Γ j that satsfes the nterpolaton condtons F j ( p j+1 (2 j n)) = f j (2 j n) for all n = 0,..., 2 j 1. We fx a sutable path functon p j approxmatng the set Γ j = { p j+1 (2 j n) : n = 0,..., 2 j 1} wth correspondng data values {F j ( p j+1 (2 j n) ) : n = 0,..., 2 j 1}, such that the three path condtons n Secton 3.2 are satsfed. Then we determne the onedmensonal restrcton of F j along p j, f j (t) := F j ( p j (t) ), t [0, 1]. The pecewse Hölder smoothness of f j (t) ensures that ω N ( f j, h) (2 J h) α for T,h := {t : p j (t + kh) Ω, k = 0,..., N}. Consderng the L 2 -projecton P j f j := 2 j 1 n=0 cj p(n) ϕ j,n wth c j p(n) := f j, ϕ j,n of f j onto the scalng space V j := clos L 2 [0,1)span{ϕ j,n : n = 0,..., 2 j 1}, where ϕ s assumed to be a smooth scalng functon as n Subsecton 3.1, we have f j P j f j L (T,2 j ) w N ( f j, 2 J j/2 ) 2 jα/2, 19

20 where we have appled the dameter condton (3.8). We use the decomposton P j f j = l c j p(l) ϕ j,l = f j 1 + g j 1 = n c j 1 p (n) ϕ j 1,n + n d j 1 p (n) ψ j 1,n. Then the Hölder smoothness of P j f j n the ntervals T yelds the Taylor expanson P j f j (t) = q α (t t 0 ) + R(t t 0 ) wth R(t t 0 ) c ϕ (t) D α 2 jα/2 for t, t 0 T, where D s the constant n the dameter condton (3.8), and ths gves the estmate for the wavelet coeffcents correspondng to the regon Ω, d j 1 p (n) = R(t t 0 ), ψ j 1,n c ϕ,n D α 2 jα/2 2 (j 1)/2 ψ 1 c ϕ,n D α 2 (j 1)(α+1)/2, (3.9) where c ϕ,n depends on the local smoothness of P j f j and hence on the dervatve bounds of p j n (3.2). Agan, let Λ j 1 be the set of ndces n from {0,..., 2 j 1 1}, where d j 1 p (n) satsfes the above estmate. Then the number of wavelet coeffcents 2 j 1 #Λ j 1 whch are not satsfyng ths estmate (snce supp ψ j 1,n T for some ) s bounded by a constant ndependent of J and j. Fnally, we obtan the polyharmonc splne nterpolant F j 1 (x) := K c y φ m ( x y 2 ) + p m(x) χ Ω (x), j 1 Γ =1 y Γ j 1 where := { p j (2 j+1 n) : n = 0,..., 2 j 1 1, p j (2 j+1 n) Ω }, Γ j 1 := K =1 Γ j 1, through the nterpolaton condtons F j 1 ( p j (2 j+1 n)) = f j 1 (2 j+1 n) for all n = 0,..., 2 j 1 1. Hence, we obtan the estmate F j ( p j (2 j+1 n) ) F j 1 ( p j (2 j+1 n) ) = f j (2 j+1 n) P j 1 f j (2 j+1 n) 2 jα/2. Let us summarze the above fndngs on the decay of the EPWT wavelet coeffcents n the followng theorem. Theorem 3.2 For j = 2J 1,..., 0, let d j p(l) = f j+1, ψ j,l, l = 0,..., 2 j 1, denote the wavelet coeffcents that are obtaned by applyng the EPWT algorthm to F (accordng to the above defntons n Secton 3), where we assume that F L 2 (Ω) s pecewse Hölder smooth of order α as prescrbed n Subsecton 2.1. Further assume that the path functons p j+1, j = 2J 1,..., 0, n the EPWT algorthm satsfy the regon condton (a), the path smoothness condton (b), and the dameter condton (c) of Subsecton 3.2. Then, for all j = 2J 1,..., 0 and l Λ j, the estmate d j p(l) C D α 2 j(α+1)/2 (3.10) holds, where D > 1 s the constant of the dameter condton (3.8), α s the Hölder exponent of F, and C depends on the utlzed wavelet bass and on the Hölder constant n (2.1). Furthermore, for all l {0,..., 2 j 1} \ Λ j, we obtan the estmate wth some constant C beng ndependent of J and j. d j p(l) C 2 j/2 (3.11) 20

21 Proof. The proof of (3.10) follows drectly from (3.9). Lkewse, for all l {0,..., 2 j 1} \ Λ j,.e., for pont sets that do not satsfy the dameter, the path smoothness, or the regon condton, we observe at least d j p(l) C 2 j/2 = C 2 j/2 snce we can assume that F j s bounded, and hence the above estmate (3.10) holds for α = 0. Thus (3.11) follows. 4 Optmal N-term Approxmaton obtaned from the EPWT Consder now the vector of all EPWT wavelet coeffcents d p = ((d 2J 1 p ) T,..., d 0 p, d 1 wth d j p = (d j p(l)) 2j 1 l=0 for j = 0,..., 2J 1, and wth the mean value d 1 p p ) T = d 1 p (0) := f 0 (0) = 2 2J F 2J (y), y I J together wth the sde nformaton on the path functons p 2J,..., p 1 at each teraton step. Wth ths nformaton the reconstructed mage Frec 2J s unquely recovered, where Frec 2J s the polyharmonc splne nterpolaton satsfyng F 2J rec ( p 2J (2 2J n) ) = f 2J (2 2J n), n = 0,..., 2 2J 1. Indeed, reconsderng the (j + 1)-th level of the EPWT procedure, we observe that the scalng coeffcents c j p(n) = f j+1, ϕ j,n and the wavelet coeffcents d j p = f j+1, ψ j,n determne f j and g j, and hence P j+1 f j+1 unquely. Further, the polyharmonc splne nterpolaton F j+1 s entrely determned by f j+1 (2 (j+1) n) and the sde nformaton about the path functon p j+1 (2 (j+1) n). By the choce of the wavelet bass, t further follows for n = 0,..., 2 j+1 1 that F j+1 ( p j+1 (2 (j+1) n)) F j+1 rec ( p j+1 (2 (j+1) n)) = f j+1 (2 (j+1) n) P j+1 f j+1 (2 (j+1) n) 2 (j+1)α/2, where Frec j+1 s unquely determned by P j+1 f j+1 (2 (j+1) n), n = 0,..., 2 j+1 1 and the sde nformaton about the path p j+1. In order to fnd a sparse approxmaton of the dgtal mage F resp. F 2J, we apply a shrnkage procedure to the EPWT wavelet coeffcents d j p(l), usng the hard threshold functon { x x σ, s σ (x) = for some σ > 0. 0 x < σ, We now study the error of a sparse representaton usng only the N wavelet coeffcents wth largest absolute value for an approxmatve reconstructon of F 2J. For convenence, let SN 2J be the set of ndces (j, l) of the N wavelet coeffcents wth largest absolute value. 21