Optimal Representation of Piecewise H older Smooth ...num.math.uni-goettingen.de/plonka/pdfs/epwt4ablack.pdf · pathways, to optimal N-term approximations for piecewise H older smooth

Optimal Representation of Piecewise Holder SmoothBivariate Functions by the Easy Path Wavelet Transform

Gerlind Plonka1, Stefanie Tenorth1, and Armin Iske2

1 Institute for Numerical and Applied Mathematics, University of Gottingen,Lotzestr. 16-18, 37083 Gottingen, Germany

plonka,[email protected] Department of Mathematics, University of Hamburg, 20146 Hamburg, Germany

[email protected]

Abstract

The Easy Path Wavelet Transform (EPWT) [22] has recently been proposed by oneof the authors as a tool for sparse representations of bivariate functions from discretedata, in particular from image data. The EPWT is a locally adaptive wavelet transform.It works along pathways through the array of function values and it exploits the localcorrelations of the given data in a simple appropriate manner. Using polyharmonic splineinterpolation, we show in this paper that the EPWT leads, for a suitable choice of thepathways, to optimal N -term approximations for piecewise Holder smooth functions withsingularities along curves.

Key words. sparse data representation, wavelet transform along pathways, N -term ap-proximation

AMS Subject classifications. 41A25, 42C40, 68U10, 94A08

1 Introduction

During the last few years, there has been an increasing interest in efficient representationsof large high-dimensional data, especially for signals. In the one-dimensional case, waveletsare particularly efficient to represent piecewise smooth signals with point singularities. Intwo dimensions, however, tensor product wavelet bases are no longer optimal for therepresentation of piecewise smooth functions with discontinuities along curves.

Just very recently, more sophisticated methods were developed to design approximationschemes for efficient representations of two-dimensional data, in particular for images,where correlations along curves are essentially taken into account to capture the geometryof the given data. Curvelets [2, 3], shearlets [11, 12] and directionlets [31] are examplesfor non-adaptive highly redundant function frames with strong anisotropic directionalselectivity.

For piecewise Holder smooth functions of second order with discontinuities alongC2-curves, Candes and Donoho [2] proved that a best approximation fN to a given functionf with N curvelets satisfies the asymptotic bound

‖f − fN‖22 ≤ C N−2 (log2N)3,

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whereas a (tensor product) wavelet expansion leads to asymptotically only O(N−1) [18].Up to the (log2N)3 factor, this curvelet approximation result is optimal (see [7, Sec-tion 7.4]). A similar estimate has been achieved by Guo and Labate [11] for shearletframes. These results, however, are not adaptive with respect to the assumed regularity ofthe target function, and so they cannot be applied to images of less regularity, i.e., imageswhich are not at least piecewise C2 with discontinuities along C2-curves.

In such relevant cases, one may rather adapt the approximation scheme to the imagegeometry instead of fixing a basis or a frame beforehand to approximate f . During the lastfew years, several different approaches were developed for doing so [1, 5, 6, 8, 9, 13, 16, 17,19, 22, 23, 24, 26, 27, 30]. In [17], for instance, bandelet orthogonal bases and frames areintroduced to adapt to the geometric regularity of the image. Due to their construction,the utilized bandelets are anisotropic wavelets that are warped along a geometrical flowto generate orthonormal bases in different bands. LePennec and Mallat [17] showed thattheir bandelet dictionary yields asymptotically optimal N -term approximations, even inmore general image models, where the edges may also be blurred.

Further examples for geometry-based image representations are the nonlinear edge-adapted (EA) multiscale decompositions in [1, 13] (and references therein) based on ENOreconstructions. We remark that the resulting ENO-EA schemes lead to an optimal N -term approximation, yielding ‖f − fN‖22 ≤ C N−2 for piecewise C2-functions with discon-tinuities along C2-curves. Moreover, unlike previous non-adaptive schemes, the ENO-EAmultiresolution techniques provide optimal approximation results also for BV -spaces andLp spaces, see [1].

In [22], a new locally adaptive discrete wavelet transform for sparse image representa-tions, termed Easy Path Wavelet Transform (EPWT), has been proposed by one of theauthors. The EPWT works along pathways through the array of function values, where itessentially exploits the local correlations of image values in a simple appropriate manner.We remark that the EPWT is not restricted to a regular (two-dimensional) grid of imagepixels, but it can be extended, in a more general setting, to scattered data approximationin higher dimensions. In [23], the EPWT has been applied to data representations on thesphere. In the implementation of the EPWT, one needs to work with suitable data struc-tures to efficiently store the path vectors that need to be accessed during the performanceof the EPWT reconstruction. To reduce the resulting adaptivity costs, we have proposeda hybrid method for smooth image approximations in [24], where an efficient edge repre-sentation by the EPWT is combined with favorable properties of the biorthogonal tensorproduct wavelet transform.

In this paper, we show that piecewise Holder smooth bivariate functions with singular-ities along smooth curves can optimally be represented by N -term approximations usinga suitable EPWT. More precisely, we prove optimal N -term approximations of the form

‖f − fN‖22 ≤ C N−α (1.1)

for the application of the EPWT to piecewise Holder smooth functions of order α > 0,with allowing discontinuities along smooth curves of finite length.

This paper essentially generalizes our results in [25], where we proved optimal N -termapproximation of the EPWT for piecewise Holder continuous functions of order α ∈ (0, 1].In that paper, the proofs were mainly based on the adaptive multiresolution analysisstructure, which is only available for piecewise constant Haar wavelets.

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But the EPWT needs not to be restricted to this simple wavelet transform. In fact,the numerical results in [22, 28] show a much higher efficiency for smoother wavelet basesas e.g. Daubechies D4 filters and biorthogonal 7-9 filters. These observations motivatedus to study the N -term approximation for Holder smooth functions of order α > 1 inthis paper. Interestingly, the detailed study of needed conditions for the path vectorsimplying best N -term approximation by the EPWT shows that the heuristically chosenpath constructions of the “relaxed EPWT” in [22] already yield a good compromise toproduce many small wavelet coefficients on the one hand and low costs for path codingon the other hand. We hope that our new results give rise to improve the known EPWTpath constructions.

With using piecewise constant functions for the approximation of a bivariate functionf , the EPWT yields an adaptive multiresolution analysis when relying on an adaptiveHaar wavelet basis (see [22, 25]). If, however, smoother wavelet bases are utilized in theEPWT approach, such an interpretation is not obvious. In fact, while Haar waveletsadmit a straight forward transfer from one-dimensional functions along pathways to bi-variate Haar-like functions, we cannot rely on such simple connections between smoothone-dimensional wavelets (used by the EPWT) and a bivariate approximation of the “low-pass” function. Therefore, in this paper we will apply a suitable interpolation method, byusing polyharmonic spline kernels, to represent the arising bivariate “low-pass” functionsafter each level of the EPWT. One key property of polyharmonic spline interpolation ispolynomial reproduction of arbitrary order. We will come back to relevant approximationproperties of polyharmonic splines in Section 2.

The outline of this paper is as follows. In Section 2, we first introduce the utilizedfunction model, some issues on polyharmonic spline interpolation, and the EPWT algo-rithm. Then, in Section 3, we study the decay of EPWT-wavelet coefficients, where wewill consider the highest level of the EPWT in detail. To achieve optimal decay results forthe EPWT wavelet coefficients at all levels, we require three specific side conditions forthe path vectors in the EPWT algorithm. Two of these conditions for the path vectors aresimilar to those used in [25]. Finally, Section 4 is devoted to the proof of asymptoticallyoptimal N -term error estimates of the form (1.1) for piecewise Holder smooth functions.

2 The EPWT and Polyharmonic Spline Interpolation

2.1 The Function Model

Suppose that F ∈ L2([0, 1)2) is a piecewise smooth bivariate function, being smoothover a finite set of regions Ωi1≤i≤K , where each region Ωi has a sufficiently smoothLipschitz boundary ∂Ωi of finite length, i.e., the boundary curves have bounded derivatives.Moreover, the set Ωi1≤i≤K is assumed to be a disjoint partition of [0, 1)2, so that

K⋃i=1

Ωi = [0, 1)2,

where each closure Ωi is a connected subset of [0, 1]2, for i = 1, . . . ,K.More precisely, we assume that F is Holder smooth of order α > 0 in each region Ωi,

1 ≤ i ≤ K, so that any µ-th derivative of F on Ωi with |µ| = bαc satisfies an estimate of

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the form|F (µ)(x)− F (µ)(y)| ≤ C ‖x− y‖α−|µ|2 for all x, y ∈ Ωi.

Note that this assumption for F is equivalent to the condition that for each x0 ∈ Ωi thereexists a bivariate polynomial qα of degree bαc (usually the Taylor polynomial of F ofdegree bαc at x0 ∈ Ωi) satisfying

|F (x)− qα(x− x0)| ≤ C‖x− x0‖α2 (2.1)

for every x ∈ Ωi in a neighborhood of x0, where the constant C > 0 does not depend on xor x0. But F may be discontinuous across the boundaries between adjacent regions. Notethat the Holder space Cα(Ωi) of order α > 0, being equipped with the norm

‖F‖Cα(Ωi) := ‖F‖Cbαc(Ωi) +∑|µ|=bαc

supx 6=y

|F (µ)(x)− F (µ)(y)|‖x− y‖α−bαc2

coincides with the Besov space Bα∞,∞(Ωi), when α is not an integer. Here, we use the

Cbαc(Ωi) norm

‖F‖Cbαc(Ωi) := supx∈Ωi

|F (x)|+∑|µ|=bαc

supx∈Ωi

|∂µF (x)|,

see e.g. [4, chapter 3.2]. Now by uniform sampling, the bivariate function F is assumedto be given by its function values taken at a finite rectangular grid. For a suitable integerJ > 1, let F (2−Jn)n∈I2J be the given samples of F , where

I2J := n = (n1, n2) : 0 ≤ n1 ≤ 2J − 1, 0 ≤ n2 ≤ 2J − 1,

and, moreover, let

Γ2Ji :=

n ∈ I2J :

n

2J∈ Ωi

for 1 ≤ i ≤ K

be the index sets of grid points that are contained in the regions Ωi, for 1 ≤ i ≤ K.Obviously,

K⋃i=1

Γ2Ji = I2J ,

and for the size #Γ2Ji of Γ2J

i we have #Γ2Ji ≤ #I2J = 22J for any 1 ≤ i ≤ K.

2.2 Polyharmonic Spline Interpolation

Next we compute a (piecewise) sufficiently smooth approximation to F from its givensamples. To this end, we construct a suitable interpolant F 2J satisfying the interpolationconditions

F 2J(n/2J) = F (n/2J) for all n ∈ I2J . (2.2)

This is accomplished by the application of polyharmonic spline interpolation separatelyin each individual region Ωi, in order to first obtain, for any index i, 1 ≤ i ≤ K, apolyharmonic spline interpolant F 2J

i satisfying

F 2Ji (n/2J) = F (n/2J) for all n ∈ I2J

i (2.3)

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at the interpolation points I2Ji ⊂ Ωi. For the required (global) interpolant F 2J , we then

let

F 2J(x) :=K∑i=1

F 2Ji (x)χΩi(x) for x ∈ [0, 1)2 (2.4)

to satisfy the interpolation conditions in (2.2), where χΩi is the characteristic function ofΩi.

Recall that polyharmonic splines, due to Duchon [10], are suitable tools for multivariateinterpolation from scattered data. For further details on polyharmonic splines, especiallyfor aspects on approximation, numerics, and efficient implementation, we refer the readerto [15]. We apply the polyharmonic spline interpolation scheme in two dimensions, wherethe interpolant F 2J

i in (2.3) is assumed to have the form

F 2Ji (x) =

∑n∈Γ2J

i

cin φm

(∥∥∥x− n

2J

∥∥∥2

)+ pim(x), (2.5)

where φm(r) = r2m log(r), for m := dαe, is a fixed polyharmonic spline kernel, and pim ∈Pm is a polynomial in the linear space Pm of all bivariate polynomials of degree at mostm. Note that, by the form of φm, the interpolant F 2J

i (x) is, for any i, in Cm. Moreover,by the above choice of m, F 2J

i (x) is Holder smooth of order α, and in this way, we canmaintain the local Holder regularity of F around each lattice point in Γ2J

i .Note that the interpolant F 2J

i in (2.5) has #Γ2Ji +dim(Pm) parameters, i.e., the #Γ2J

i

coefficients cin in its major part and another dim(Pm) = (m + 1)(m + 2)/2 parametersin its polynomial part. However, the interpolation problem (2.3) yields only #Γ2J

i linearconditions on the #Γ2J

i + (m + 1)(m + 2)/2 parameters of F 2Ji . To eliminate additional

degrees of freedom, we require another set of (m + 1)(m + 2)/2 linear constraints on thecoefficients cin, as given by the vanishing moment conditions∑

n∈Γ2Ji

cin p(n/2J) = 0 for all p ∈ Pm. (2.6)

To compute the coefficients of the polyharmonic spline interpolant F 2Ji , this then amounts

to solving a square linear system with #Γ2Ji + dim(Pm) equations, (2.3) and (2.6), for

#Γ2Ji + dim(Pm) unknowns, given by the parameters of F 2J

i .According to the seminal work of Michelli [20] on (conditionally) positive definite

functions, this square linear equation system has a unique solution, provided that the setof interpolation points are Pm-regular, i.e., for p ∈ Pm we have the implication

p(n/2J) = 0 for all p ∈ Pm =⇒ p ≡ 0, (2.7)

so that every polynomial in Pm can uniquely be reconstructed from its values on Γ2Ji . In

fact, this complies with earlier results in [10], where the well-posedness of the polyharmonicspline interpolation scheme is proven via a variational theory concerning (polyharmonic)splines minimizing rotation-invariant semi-norms in Sobolev spaces.

Therefore, we can conclude that the polyharmonic spline interpolant F 2Ji in (2.5),

satisfying (2.3) and (2.6), is unique provided that the interpolation points Γ2Ji satisfy the

(rather weak) regularity conditions in (2.7). From now we will tacitly assume that the

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conditions in (2.7) are fulfilled. In this case, we can further conclude that the polyharmonicspline interpolation scheme achieves to reconstruct polynomials of degree m.

Concerning the local approximation order of polyharmonic spline interpolation, werefer to [14]. As was proven in [14], the asymptotic estimate

|F 2Ji (hn/2J)− F (hn/2J)| = O(hm+1) for h 0

holds for F ∈ Cm+1, in which case the polyharmonic spline interpolation scheme is saidto have local approximation order m+ 1, see [14] or [15, Section 3.8] for details.

We remark that the concept in [14] concerning local approximation orders is in contrastto that of global approximation orders, where the latter is well-understood for polyhar-monic splines and related radial kernels since much longer. In the subsequent analysis inthis paper, we require one specific approximation result for polyharmonic spline interpo-lation concerning its global approximation behaviour. We can describe this as follows.

From the embedding theorem for Besov spaces we obtain Bα∞,∞(Ωi) ⊂ Bα

2,2(Ωi), seee.g. [29] or [4, page 163]. Since Bα

2,2(Ωi) is equivalent to the Sobolev space Hα(Ωi),see [4, 29], this allows us to use the estimate

‖F − F 2J‖L2(Ω) ≤ CFK∑i=1

hαΩi‖F‖Bα2,2(Ωi) (2.8)

for the interpolation error in Sobolev spaces, as shown in [21], where the fill distance

hΩi := supx∈Ωi

infn∈Γ2J

i

‖x− n

2J‖2 ≤ 2−J for 1 ≤ i ≤ K

measures the density of the interpolation points in Ωi.

2.3 The EPWT Algorithm

Now let us briefly recall the EPWT algorithm from our previous work [22]. To this end,let ϕ ∈ Cβ with β ≥ α be a sufficiently smooth, compactly supported, one-dimensionalscaling function, i.e., the integer translates of ϕ form a Riesz basis of the scaling spaceV0 := closL2span ϕ(· − k) : k ∈ Z. Further, let ϕ be a corresponding biorthogonal andsufficiently smooth scaling function with compact support, and let ψ and ψ be a corre-sponding pair of compactly supported wavelet functions. We refer to [4, Chapter 2] for acomprehensive survey on biorthogonal scaling functions and wavelet bases and summarizeonly the notation needed for the biorthogonal wavelet transform. For j, k ∈ Z, let

ϕj,k(t) := 2j/2 ϕ(2jt− k) and ψj,k(t) := 2j/2 ψ(2jt− k),

likewise for ϕ and ψ. The functions ϕ, ϕ and ψ, ψ are assumed to satisfy the refinementequations

ϕ(x) =√

2∑n

hnϕ(2x− n) ψ(x) =√

2∑n

qnϕ(2x− n)

ϕ(x) =√

2∑n

hnϕ(2x− n) ψ(x) =√

2∑n

qnϕ(2x− n)

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with finite sequences of filter coefficients (hn)n∈Z, (hn)n∈Z and (qn)n∈Z, (qn)n∈Z. Byassumption, the polynomial reproduction property∑

k

〈pm, ϕj,k〉ϕj,k = pm for all j ∈ Z,

is satisfied for any polynomial pm of degree less than or equal to m = dαe, and so,

〈pm, ψj,k〉 = 0 for all j, k ∈ Z.

With these assumptions, ψj,k : j, k ∈ Z and ψj,k : j, k ∈ Z form biorthogonalRiesz bases of L2(R), i.e., for each function f ∈ L2(R), we have

f =∑j,k∈Z〈f, ψj,k〉ψj,k =

∑j,k∈Z〈f, ψj,k〉ψj,k.

For any given univariate function f j , j ∈ Z, of the form f j(x) =∑

n∈Z cj(n)ϕj,n one

decomposition step of the discrete (biorthogonal) wavelet transform can be represented inthe form

f j(x) = f j−1(x) + gj−1(x),

wheref j−1(x) =

∑n∈Z

cj−1(n)ϕj−1,n and gj−1(x) =∑n∈Z

dj−1(n)ψj−1,n

withcj−1(n) = 〈f j , ϕj−1,n〉 and dj−1(n) = 〈f j , ψj−1,n〉. (2.9)

Conversely, one step of the inverse discrete wavelet transform yields for given functionsf j−1 and gj−1 the reconstruction

f j(x) =∑n∈Z

cj(n)ϕj,n with cj(n) = 〈f j−1, ϕj,n〉+ 〈gj−1, ϕj,n〉.

We recall that the EPWT is a wavelet transform that works along path vectors throughindex subsets of I2J . For the characterization of suitable path vectors we first need to intro-duce neighborhoods of indices. For any index n = (n1, n2) ∈ I2J , we define its neighborhoodby

N(n) := m = (m1,m2) ∈ I2J \ n : ‖n−m‖2 ≤√

2,

where ‖n−m‖22 = (n1 −m1)2 + (n2 −m2)2. Hence, an interior index, i.e., an index thatdoes not lie on the boundary of the index domain I2J , has eight neighbors.

Now the EPWT algorithm is performed as follows. For the application of the 2J-th

level of the EPWT we need to find a path vector p2J = (p2J(n))22J−1n=0 through the index

set I2J . This path vector is a suitable permutation of all indices in I2J , which can bedetermined by using the following strategy from [22]. Recall that I2J = ∪Ki=1Γ2J

i , whereΓ2Ji corresponds to lattice points in Ωi. Start with one index p2J(0) in Γ2J

1 . Now, for agiven n-th component p2J(n) being contained in the index set Γ2J

i , for some i ∈ 1, . . . ,K,we choose the next component p2J(n+ 1) of the path vector p2J , such that

p2J(n+ 1) ∈ (N(p2J(n)) ∩ Γ2Ji ) \ p2J(0), . . . , p2J(n), (2.10)

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i.e., p2J(n+ 1) should be a neighbor index of p2J(n), lying in the same index set Γ2Ji , that

has not been used in the path, yet.

In situations where (N(p2J(n))∩Γ2Ji )\p2J(0), . . . p2J(n) is an empty set, the path is

interrupted, and we need to start a new pathway by choosing the next index p2J(n+1) fromΓ2Ji \ p2J(0), . . . , p2J(n). If, however, this set is also empty, we choose p2J(n + 1) from

the set of remaining indices I2J \ p2J(0), . . . , p2J(n). For a more detailed description ofthe path vector construction we refer to [22].

In particular, for a suitably chosen path vector p2J , the number of interruptions canbe bounded by K = C1K, where K is the number of regions, and where the constant C1

does not depend on J but only on the shape of the regions Ωi, see [25]. The so obtainedvector p2J is composed of connected pathways, i.e., each pair of consecutive componentsin these pathways is neighboring.

Now, we consider the data vector

(c2J(`)

)22J−1

`=0:=

(F 2J

(p2J(`)

2J

))22J−1

`=0

and apply one level of a one-dimensional (periodic) wavelet transform to the function

values of F 2J along the path p2J . This yields the low-pass vector (c2J−1(`))22J−1−1`=0 and

the vector of wavelet coefficients (d2J−1(`))22J−1−1`=0 according to the formulae in (2.9).

Due to the piecewise smoothness of F 2J along the path vector p2J , it follows that mostof the wavelet coefficients in d2J−1 are small, where particularly the wavelet coefficientscorresponding to an interruption of the path (within one region or from one region toanother) may possess significant amplitudes. For the case of piecewise Holder smoothimages of order α > 1, we will introduce a smooth path function p2J in Section 3.1. Thispath function interpolates the components of the path vector p2J and is assumed to havesuitably bounded derivatives almost everywhere. Indeed for α > 1, significant waveletcoefficients may also occur, if the derivatives of the path function are not well bounded,as e.g. at tight turns of p2J .

As regards the next level of the EPWT, the path vector p2J determines a new subsetof indices

Γ2J−1 :=p2J(2`) : ` = 0, . . . , 22J−1 − 1

=

K⋃i=1

Γ2J−1i ,

where Γ2J−1i := p2J(2`) : p2J(2`) ∈ Γ2J

i . At level j = 2J − 1, we first locate a second

connected path vector p2J−1 = (p2J−1(`))22J−1−1`=0 through Γ2J−1, i.e., the entries of p2J−1

form a permutation of the indices in Γ2J−1. Similarly as before, we require that p2J−1(n)and p2J−1(n + 1) are neighbors lying in the same index set Γ2J−1

i . Here, p2J−1(n) andp2J−1(n+ 1) are said to be neighbors, i.e., p2J−1(n+ 1) ∈ N(p2J−1(n)), iff∥∥p2J−1(n)− p2J−1(n+ 1)

∥∥2≤ 2.

Again, the number of path interruptions can be bounded by C1K, where C1 does notdepend on J . Then we apply one level of the one-dimensional wavelet transform to the

permuted data vector (c2J−1(per2J−1(`)))22J−1−1`=0 , where the permutation per2J−1 is de-

fined by per2J−1(`) := k iff p2J−1(`) = p2J(2k) for `, k = 0, . . . 22J−1 − 1. Then we

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obtain the low-pass vector (c2J−2(`))22J−2−1`=0 and the vector (d2J−2(`))22J−2−1

`=0 of waveletcoefficients.

We continue by iteration over the remaining levels j for j = 2J − 2, . . . , 1, where atany level j we first construct a path pj = (pj(`))2j−1

`=0 through the index set

Γj := pj+1(2`) : ` = 0, . . . , 2j − 1 =K⋃i=0

Γji

with applying similar strategies as described above. Here, pj(n) and pj(n + 1) are calledneighbors, iff ∥∥pj(n)− pj(n+ 1)

∥∥2≤ D2J−j/2, (2.11)

where D ≥√

2 is a suitably determined constant (in the above description of p2J andp2J−1 we have chosen D =

√2). Then we apply the wavelet transform to the permuted

vector (cj(perj(`))2j−1`=0 , yielding cj−1 and dj−1.

Example. In this example, we explain the construction of the path vectors throughthe remaining data points with the low-pass values by a toy example. To this end, let[0, 1)2 be divided into only two regions, Ω1 and Ω2. The function F is assumed to be Holdersmooth in each of these regions, but may be discontinuous across the curve separating thetwo regions Ω1 and Ω2. In our toy example, we have J = 3, i.e., an 8 × 8 image with64 data values. At the highest level of the EPWT, we choose a path p6 through theunderlying index set I6 = Γ6

1∪Γ62, such that each two consecutive components in the path

are neighbors. We first pick all indices in Γ61, before jumping to Γ6

2, see Figure 1(a). Forthe path construction at the next level, we first determine the index set Γ5 = Γ5

1 ∪ Γ52

(containing only each second index of p6), see Figure 1(b), before we construct a pathaccording to the above description. Figures 1(c) and 1(d) show the index sets Γ4 and Γ3

along with their corresponding path vectors. Besides, we tried to find the path vectorsin a way such that the angles formed by the polygonal line of the path are as large aspossible. In fact, the polygonal lines for p6 and p5 do not contain angles being smallerthan 90 degrees, while the polygonal line of p4 possesses two such angles, one in eachregion. In fact, locally straight polygonal lines of the path vectors give rise to obtainbetter smoothness bounds for the path functions pj that we will introduce in Section 3.In this example, we have

‖p6(n+ 1)− p6(n)‖2 ≤√

5 ≤ D, ‖p5(n+ 1)− p5(n)‖2 ≤√

5 ≤√

2D,

‖p4(n+ 1)− p4(n)‖2 ≤ 4 ≤ 2D, ‖p3(n+ 1)− p3(n)‖2 ≤√

10 ≤√

8D

(with one path interruption at each level for the jump from one region to the other), sothat the path construction satisfies the above requirements with D =

√5. This simple

example also illustrates that the path construction leads at each level to index sets Γjiwith quasi-uniformly distributed indices. See Section 3.2 for more explanation about theused quasi-uniformity.

Remark 1. Considering the above strategy of the EPWT algorithm, it is heuristicallyclear that we are able to reduce the number of significant wavelet coefficients to a multipleof the number K of regions, where the target function F is smooth. Indeed, only when the

9

(a) (b)

(c) (d)

Figure 1: Path construction. (a) level six, (b) level five, (c) level four, (d) level three.

path skips from one region to another, a finite number of significant wavelet coefficientswill occur. This is in contrast to the usual tensor product wavelet transform, wherethe number of significant wavelet coefficients is usually related to the total length of the“smooth regions” boundaries and hence depends on the level j of the wavelet transform.

Remark 2. As shown in [22], the choice of the path vectors in each level is crucialfor the performance of the approximation by means of the EPWT. In particular, it is nota good idea to take simply the same path vector pj(`) = pj+1(2`) at each level, since oneonly exploits the data correlation along this one path and the EPWT is reduced to a one-dimensional transform. In Section 3.2, we will determine the so called diameter conditionthat requires the inequality (2.11) for neighboring indices in the path construction therebyavoiding the above mentioned simple path choice. Our numerical experiments with theEPWT imply that the constant D should be chosen rather small, e.g. D ≈ 2, see [28].On the other hand, a too small D induces a small number of neighborhood indices andhence yields a higher number of path interruptions. In Section 3.2, we will show that theinequality (2.11) for path construction implies a quasi-uniform distribution of indices inΓj at each level of the EPWT.

Remark 3. Observe that the EPWT algorithm as described above can be rathersimply implemented, where at each level, we first determine a path vector pj and thenapply a wavelet filter bank to the one-dimensional data vector that is ordered accordingto the path vector. In contrast to the above procedure, we will consider slightly differentdata vectors cjp(`) in the theoretical estimates of Sections 3 and 4, where we will be usingL2-projection operators determined by the dual scaling and wavelet functions, ϕ and ψ.Further, we will employ polyharmonic spline interpolation and smooth path functions foranalytical purpose.

Remark 4. Note that the components of the path vector pj lie in Γ2J with containing

10

2d integer entries. This is in contrast to the notation in [22]. Further, unlike in [22], wedo no longer consider index sets but define a neighborhood of pixels by the Euclideandistance between corresponding indices.

3 Decay of Wavelet Coefficients using the EPWT

Before we turn to the technical details, let us first sketch the basic ideas of the proof foroptimal N -term approximations by the EPWT.

As explained in the Subsection 2.2, we consider applying polyharmonic spline inter-polation, from given image values F (2−Jn), separately in the individual domains Ωi,i = 1, . . . ,K. We assume that F is Holder smooth of order α on each Ωi, so that thepolyharmonic spline interpolant F 2J in (2.4), being Holder smooth of order at least α,reconstructs bivariate polynomials of degree m = dαe.

At the 2J-th level of the EPWT, we define a path p2J through all indices of I2J

such that consecutive components of p2J are neighboring indices lying in the same indexset Γi (up to O(K) exceptions). The path p2J can be constructed in a way such thatonly C1K “interruptions” occur. Next, we consider a one-dimensional function f2J(t) =∑22J−1

k=0 c2Jp (k)ϕ2J,k(t), t ∈ [0, 1) that suitably approximates a smooth one-dimensional

and scaled restriction of F resp. F 2J “along the path p2J” with

|F (2−Jp2J(`))− f2J(2−2J`)| . 2−Jα,

and apply one level of a smooth wavelet transform to f2J .Significant wavelet coefficients will only occur at a finite number of locations on the

interval [0, 1) that correspond to interruptions red or to tight turns of the path. However,the number of such significant coefficients does not depend on J but only on the numberof regions, K. Therefore, with the performance of one level of the (periodic) wavelettransform, we will find that most of the wavelet coefficients of f2J = f2J−1 + g2J−1

occurring in the wavelet part g2J−1, are small.At the next levels of the EPWT, we consider the index sets Γj = ∪Ki=1Γji . By

construction, we have the inclusion Γj ⊂ Γj+1, for j = 1, . . . , 2J − 1, and, moreover,#Γj = 2j . To obtain sufficiently accurate polyharmonic spline interpolations F j to Fwith F j(2−Jn) = F (2−Jn) for n ∈ Γj , and to apply the same arguments as at the highestlevel, three specific conditions on the index sets Γj and the path vectors pj need to be sat-isfied. We can briefly explain these conditions, termed region condition, path smoothnesscondition and diameter condition, as follows (for more details on these conditions we referto Subsection 3.2).

Firstly, the region condition requires that the path should prefer to traverse the indicesbelonging to one region set Γji , before “jumping” to another region, where jumps within oneregion should be avoided. Assuming further that the path function is smooth with suitablybounded derivatives (path smoothness condition), we can optimally exploit the smoothnessof the function F along the path. Finally, the diameter condition ensures a quasi-uniformdistribution of remaining pixels in each Γji , and therefore leads to a sufficiently accuratepolyharmonic spline interpolation F j at each level of the EPWT.

We remark that the region condition and the diameter condition can be forced byusing the strategies for the path construction as proposed in Subsection 2.2. The path

11

smoothness condition is more difficult to implement and can be forced by avoiding “smallangles” in the path vector and by preferring “path snakes”, see e.g. examples in [22].The three path conditions allow us to estimate the EPWT wavelet coefficients similarlyas for one-dimensional piecewise smooth functions with a finite number of singularities tofinally obtain an optimal N -term approximation using only the N most significant EPWTwavelet coefficients for the image reconstruction.

3.1 The Highest Level of the EPWT

Let us now explain the 2J-th level of the EPWT in detail. The performance of the furtherlevels of the EPWT and the corresponding estimates are then derived in a similar manner.

We consider a sufficiently smooth parametric curve p2J(t), t ∈ [0, 1], through the planeapproximating the path p2J , i.e., we assume that

p2J(`/22J) = 2−J p2J(`), ` = 0, . . . , 22J − 1 (3.1)

with the convention that p2J(t) ∈ Ωi, for t ∈ [`/22J , (` + 1)/22J ], if 2−J p2J(`) and2−J p2J(` + 1) are in Ωi. Now, we regard the function f2J that is defined by the one-dimensional restriction of F 2J along the curve p2J ,

f2J (t) = F 2J(p2J(t)

)for t ∈ [0, 1].

If the pass function p2J crosses over from one region Ωi to another, there may occura discontinuity of f2J caused by a discontinuity of F 2J , i.e., there are indices ` and` + 1 in 0, 1, . . . , 22J − 1, where we have a discontinuity between F 2J(p2J(2−2J`)) andF 2J(p2J(2−2J(` + 1))), i.e., a discontinuity of f2J in the subinterval [`/22J , (` + 1)/22J ].For simplicity, we assume that we have only discontinuities of f2J caused by jumping fromone region to another.

Recall that the trace theorem for Holder resp. Besov spaces (see [29]) implies that forF 2J |Ωi ∈ Bα

∞,∞(Ωi), the (scaled) restriction f2J(t) along the curve p2J(t) is again Holdersmooth of order α in each subinterval of [0, 1], determined by

Ti = t ∈ [0, 1] : p2J(t) ∈ Ωi

with assuming that the corresponding path vector p2J has no interruptions in Ωi. Inparticular, we find for α ∈ (0, 1) the estimate

|f2J |Cα(Ti) = supt1 6=t2t1,t2∈Ti

|F 2J (p2J (t1))−F 2J (p2J (t2))||t1−t2|α

= supt1 6=t2t1,t2∈Ti

|F 2J (p2J (t1))−F 2J (p2J (t2))|‖p2J (t1)−p2J (t2)‖α2

‖p2J (t1)−p2J (t2)‖α2|t1−t2|α

≤ ‖F 2J‖Cα(Ωi)‖p2J‖αC1(Ti)

.

For the case α ∈ (1, 2), the chain rule yields

|f2J |Cα(Ti) = supt1 6=t2t1,t2∈Ti

|∇F 2J (p2J (t1))(p2J )′(t1)−∇F 2J (p2J (t2))(p2J )′(t2)||t1−t2|α−1

12

≤ supt1 6=t2t1,t2∈Ti

(‖∇F 2J (p2J (t1))−∇F 2J (p2J (t2))‖2

‖p2J (t1)−p2J (t2)‖α−12

‖p2J (t1)−p2J (t2)‖α−12

|t1−t2|α−1 ‖(p2J)‖C1(Ti)

+‖F 2J‖C1(Ωi)(p2J )′(t1)−(p2J )′(t2)

|t1−t2|α−1

)≤ ‖F 2J‖Cα(Ωi) ‖p

2J‖αC1(Ti)+ ‖F 2J‖C1(Ωi) ‖p

2J‖Cα(Ti).

Generally, we have

Lemma 3.1 Suppose that the parametric curve p2J : [0, 1] 7→ [0, 1]2 interpolates the pathvector, i.e.,

p2J(2−2J`) = 2−Jp2J(`), ` = 0, . . . , 22J − 1,

and satisfies in each subinterval Ti the smoothness condition

‖p2J‖Cβ(Ti) ≤ C2 2Jβ (3.2)

for all β ≤ maxα, 1 and with a constant C2 being independent of J , i.e., pj ∈ Cα(Ti) forα > 1 and pj ∈ C1(Ti) for α ≤ 1, i = 1, . . . ,K, with strictly bounded derivatives. Thenfor the one-dimensional restriction f2J(t) = F 2J(p2J(t)) along the curve p2J we have

‖f2J‖Cα(Ti) ≤ C2Jα‖F 2J‖Cα(Ωi). (3.3)

Proof. For α < 2 the assertion already follows from the previous observations. Letm := bαc. Further, we use the abbreviations F = F 2J , p = p2J . In the general case theformula by Faa di Bruno implies that Dmf2J(t) := dm

dtm f2J(t) = Dm(F (p(t))) is a finite

linear combination of terms

Sν,J(t) := DνF (x)|x=p(t) (Dm1 p1(t))b1 (Dm2 p2(t))b2 ,

where p = p2J = (p1, p2)T , m1,m2, b1, b2 ∈ N with m1 +m2 ≤ m, m1b1 +m2b2 = m, andwith ν ∈ N2

0, |ν| ≤ m. For t1, t2 ∈ Ti with t1 6= t2 we now obtain with (3.2) the estimate

Sν,J (t1)−Sν,J (t2)|t1−t2|α−m

≤(|(DνF )(p(t1))−(DνF )(p(t2))|

‖p(t1)−p(t2)‖α−m2

) (‖p(t1)−p(t2)‖α−m2|t1−t2|α−m

)|(Dm1 p1(t1))b1(Dm2 p2(t1))b2 |

+|(DνF )(p(t2))|(|(Dm1 p1(t1))b1 (Dm2 p2(t1))b2−(Dm1 p1(t2))b1 (Dm2 p2(t2))b2 |

|t1−t2|α−m

)≤ |F |Cα+|ν|−m(Ωi)

‖p‖α−mC1(Ti)

‖p‖b1Cm1 (Ti)‖p‖b2Cm2 (Ti)

+ ‖F‖C|ν|(Ωi)(b1‖p‖b2Cm2 (Ti)

∗‖p‖b1−1Cm1 (Ti)

‖p‖Cm1+α−m(Ti) + b2‖p‖b1Cm1 (Ti)‖p‖b2−1

Cm2 (Ti)‖p‖Cm2+α−m(Ti)

)≤ |F |Cα+|ν|−m(Ωi)

2J(α−m) 2J(m1b1) 2J(m2b2)

+‖F‖C|ν|(Ωi)(b12J(m2b2+m1b1−m1+m1+α−m) + b22J(m1b1+m2b2−m2+m2+α−m))

≤ Cν2Jα‖F‖Cα(Ωi),

where we have used the inequality |xp − yp| ≤ p|x− y||maxx, y|p−1 for x, y ∈ R, p ∈ N.Hence, the assertion follows.

13

Let us assume in the sequel that the path function p satisfies the smoothness condition(3.2) in Lemma 3.1. Thus, we obtain for the N -th order modulus of smoothness of f2J

the estimate

ωN (f2J , h)∞ := sup|h|≤h

‖∆Nhf2J‖∞ . hα‖f2J‖Cα(Ti) . (2Jh)α‖F 2J‖Cα(Ωi) (3.4)

within the subintervals, where f2J is smooth, i.e., for N = bα+ 1c and

Ti,h :=t : p2J(t+ kh) ∈ Ωi, k = 0, . . . , N

,

see [4]. Next, we consider the L2-projection f2J := P2J f2J of f2J onto the scaling space

V 2J := closL2[0,1)spanϕ2J,n : n = 0, . . . , 22J − 1,

where ϕ is assumed to be a sufficiently smooth scaling function, see Section 2.2. Then,

f2J = P2J f2J :=

∑22J−1n=0 〈f2J , ϕ2J,n〉ϕ2J,n also satisfies a Holder smoothness condition of

order α. Along the lines of [4, Theorem 3.3.3], we now have in the subintervals Ti,2−2J

‖f2J − f2J‖L∞(Ti,2−2J ) = ‖f2J − P2J f

2J‖L∞(Ti,2−2J ) . ωN (f2J , 2−2J)∞

. (2−2J)α‖f2J‖Cα(Ti) . (2−J)α‖F 2J‖Cα(Ωi). (3.5)

In particular,

|f2J(2−2J`)− f2J(2−2J`)| = |F 2J(p2J(2−2J`))− f2J(2−2J`)| . 2−Jα.

In the next step, we decompose the function f2J =∑c2Jp (`)ϕ2J,` with c2J

p (`) := 〈f2J , ϕ2J,`〉

into the low-pass part f2J−1 and the high-pass part g2J−1. Applying one level of the one-

dimensional wavelet transform to the data set (c2Jp (`))22J−1

`=0 , we obtain the decomposition

f2J = f2J−1 + g2J−1 with

f2J−1 =

22J−1−1∑n=0

c2J−1p (n)ϕ2J−1,n and g2J−1 =

22J−1−1∑n=0

d2J−1p (n)ψ2J−1,n,

where c2J−1p (n) := 〈f2J , ϕ2J−1,n〉 and d2J−1

p (n) := 〈f2J , ψ2J−1,n〉. From the Holder

smoothness of f2J , we find for t ∈ Ti the representation

f2J(t) = qα(t− t0) +R(t− t0)

for t0 ∈ 2−2Jk : k = 0, . . . , 22J − 1∩ Ti and |t− t0| ≤ 2−2J , where qα denotes the Taylorpolynomial of degree bαc of f2J at t0, and where the remainder R satisfies |R(t − t0)| ≤cϕ(t)2−Jα. Hence, if supp(ψ2J−1,n) ∈ Ti for some i, the wavelet coefficients satisfy

|d2J−1p (n)| = |〈qα(· − t0) +R(· − t0), ψ2J−1,n〉| = |〈R(· − t0), ψ2J−1,n〉|

≤ cϕ,n 2−Jα ‖ψ2J−1,n‖1 = cϕ,n 2(−J+1/2)(α+1),

where we have used ‖ψ2J−1,n‖1 = 2−J+1/2‖ψ‖1. Observe that the constant cϕ,n in thisinequality depends on the choice of the wavelet basis but also on the (local) smoothness

14

properties of f2J , and hence on the (local) boundedness properties of the derivatives ofthe chosen path function p2J in Lemma 3.1.

Now let Λ2J−1 be the set of all n ∈ 0, . . . , 22J−1 − 1, where the above estimate ford2J−1(n) is satisfied. Then, the number of the remaining wavelet coefficients 22J−1 −#Λ2J−1 corresponds to the number of positions, where f2J is discontinuous (e.g. causedby crossing over of the path function to another region) or where derivatives of f2J are notsuitably bounded, i.e., the constant cϕ,n is too large (caused e.g. by tight turns of the pathfunction p2J). We assume that this number is bounded by CK, where K is the numberof regions in the original image F , and where the constant C does not depend on J .

Now, we consider the low-pass function f2J−1 and reconstruct a bivariate functionF 2J−1 as follows. Taking only the path function values p2J(2−2J+1(n)) = 2−Jp2J(2n), weput

Γ2J−1i := p2J(2−2J+1(n)) : n = 0, . . . , 22J−1 − 1, p2J(2−2J+1n) ∈ Ωi

for each i = 1, . . . ,K and Γ2J−1 := ∪Ki=1Γ2J−1i . We compute the polyharmonic spline

interpolant

F 2J−1(x) :=K∑i=1

∑y∈Γ2J−1

i

ciy φm

(∥∥∥x− y

2J

∥∥∥2

)+ pim(x)

χΩi(x),

satisfying the interpolation conditions

F 2J−1(p2J(2−2J+1n)

)= f2J−1(2−2J+1n) for all n = 0, . . . , 22J−1 − 1.

Therefore,∣∣F 2J(p2J(2−2J+1n))− F 2J−1(p2J(2−2J+1n))∣∣ = |f2J(2−2J+1n)− f2J−1(2−2J+1n)|

= |f2J(2−2J+1n)− P2J−1f2J(2−2J+1n)|

. 2(−J+1)α = Dα(2−J+1/2)α,

where D =√

2, and where the last inequality again follows analogously as in (3.5) sincef2J−1 is the orthogonal projection of f2J to

V 2J−1 := closL2[0,1)spanϕ2J−1,n : n = 0, . . . 22J−1 − 1.

The last inequality implies that F 2J−1 is still a good approximation for F , since theinterpolation points have changed only slightly. However, only half of the interpolationpoints are left, which are irregularly distributed in [0, 1]2. Moreover, we have

maxx∈Ωi

miny∈Γ2J−1

i

|x− 2−Jy| ≤ 2−J+1 = D 2−J+1/2 and miny1,y2∈Γ2J−1

i

|y1 − y2| ≥ 2−J .

Together with (2.8) we observe

‖F 2J − F 2J−1‖L2(Ωi) . (2−J+1/2)α for all i = 1, . . . ,K.

Remark. In fact, we can even relax the condition (3.1) in the sense that we need tochoose a smooth path function p2J : [0, 1] 7→ [0, 1]2 with suitably bounded derivatives such

15

that the set of values p2J(2−2J`) : ` = 0, . . . , 22J − 1 is quasi-uniformly distributed in[0, 1]2 and that the number of crossings from one region Ωi to another is bounded by C1K,where K is the number of regions and C1 > 0 is a suitable constant being independent ofJ . In this case we redefine the sampling set as

Γ2J := 2J p(2−2Jn) : n = 0, . . . , 22J−1 − 1,

and this set coincides with Γ2J if p2J satisfies (3.1), i.e., if p2J interpolates 2−Jp2J at thesampling points 2−JΓ2J .

3.2 Conditions for the Path Vectors

Before we proceed with the error estimates for the further levels of the EPWT algorithm,we want to fix specific side conditions for the path functions that are required for our erroranalysis, and that have been implicitly used already in the estimates at the highest levelin Section 3.1. The side conditions are termed (a) region condition, (b) path smoothnesscondition, and (b) diameter condition, as stated below.

The region condition and the diameter condition have been similarly stated already in[25] to prove the N -term approximation estimates. The region condition ensures that ateach level of the EPWT, the path function “jumps” only C1K times from one region Ωi toanother region or inside a region. The diameter condition ensures that the remaining in-dices in Γj are quasi-uniformly distributed, such that there is a constant D, not dependingon J or j, satisfying

maxx∈Ωi

miny∈Γj‖x− 2−Jy‖2 ≤ (1 +

√2)D2−j/2. (3.6)

Further, at each level j of the EPWT, the path pj is chosen, such that we can finda smooth function pj : [0, 1] 7→ [0, 1]2 with suitably bounded derivatives satisfying theinterpolation condition pj(2−j`) = 2−Jpj(`) for ` = 0, . . . , 2j − 1 and with the conventionthat pj(t) ∈ Ωi, for t ∈ [`/2j , (`+ 1)/2j ], if 2−J pj(`) and 2−J pj(`+ 1) are in Ωi.

Let us introduce the three conditions more explicitly.

(a) Region condition. At each level j of the EPWT, the path pj is chosen, such that itcontains only at most C1K interruptions caused by pairs of components, pj(`) andpj(`+1), which are no (spatial) neighbors or are not belonging to the same index setΓji . Hence, pj possesses not more than C1K crossings from one region to another.

(b) Path smoothness condition. The path function pj satisfies in each subinterval Tithe smoothness condition

‖pj‖Cβ(Ti) ≤ C2 2jβ/2

for all β ≤ maxα, 1 and with a constant C2 being independent of j, i.e., pj ∈ Cα(Ti)for α > 1 and pj ∈ C1(Ti) for α ≤ 1, i = 1, . . . ,K, with strictly bounded derivatives.

(c) Diameter condition. At each level of the EPWT, we require for almost all com-ponents of the path the bound

‖pj(`)− pj(`+ 1)‖2 ≤ D 2J−j/2, (3.7)

16

where D is independent of J and j, and where the number of components of pj

which are not satisfying the diameter condition, is bounded by a constant C2 notdepending on J or j. Hence, at each level j, consecutive components in the pathshould be spatial neighbors.

The conditions (a) and (c) can be enforced by the proposed path construction inSubsection 2.2, see (2.10) for the region condition and (2.11) for the diameter condition.Particularly, the diameter condition ensures a quasi-uniform distribution of the indicesin Γj . This can be seen inductively as follows. For j = 2J , the assertion (3.6) is obvious.Generally, assuming (3.6) and (3.7) at the level j + 1, it follows that

maxx∈Ωi

miny∈Γj‖x− 2−Jy‖2 ≤ max

x∈Ωiminy∈Γj+1

‖x− 2−Jy‖2 + maxy∈Γj+1

minz∈Γj‖2−Jy − 2−Jz‖2

≤ (1 +√

2)D2−(j+1)/2 + 2−JD2J−(j+1)/2 = (1 +√

2)D 2−j/2.

3.3 The Further Levels of the EPWT

Let us now explain the further levels of the EPWT. These are performed following alongthe lines of the 2J-th level. We start with the polyharmonic spline interpolant

F j(x) :=K∑i=1

∑y∈Γji

ciy φm

(∥∥x− y2J

∥∥2

)+ pim(x)

χΩi(x)

that satisfies the interpolation conditions F j(2−Jpj+1(2n)) = F j(pj+1(2−jn)) = f j(2−jn)for all n = 0, . . . , 2j − 1. We fix a suitable path vector pj passing through the set Γj =

pj+1(2n) : n = 0, . . . , 2j − 1 with corresponding data values F j(pj+1(2n)

2J

): n =

0, . . . , 2j−1, such that the region condition and the diameter condition in Section 3.2 aresatisfied. Next, we consider a sufficiently smooth parametric curve pj(t), t ∈ [0, 1], throughthe plane interpolating pj , i.e., with pj(2−j`) = 2−J pj(`) for ` = 0, . . . , 2j − 1, such thatpj(t) ∈ Ωi for t ∈ [2−j`, 2−j(` + 1)], if 2−Jpj(`) and 2−Jpj(` + 1) are in the same regionΩi. Then we determine the one-dimensional restriction of F j along pj , f j(t) := F j

(pj(t)

),

t ∈ [0, 1]. As in the above discussion concerning the 2J-th level, we assume that pj satisfiesthe smoothness condition ‖pj‖Cβ(Ti) ≤ C22jβ/2. Then the piecewise Holder smoothness of

f j(t) ensures that

ωN (f j , h)∞ . (2Jh)α for Ti,h := t : pj(t+ kh) ∈ Ωi, k = 0, . . . , N.

Considering the L2-projection Pj fj :=

∑2j−1n=0 cjp(n)ϕj,n with cjp(n) := 〈f j , ϕj,n〉 of f j onto

the scaling spaceV j := closL2[0,1)spanϕj,n : n = 0, . . . , 2j − 1,

where ϕ is assumed to be a smooth scaling function as in Subsection 3.1, we have

‖f j − Pj f j‖L∞(Ti,2−j ) . wN (f j , 2−J−j/2) . 2−jα/2,

where we have used the diameter condition (3.7). We use the decomposition

Pj fj =

∑`

cjp(`)ϕj,` = f j−1 + gj−1 =∑n

cj−1p (n)ϕj−1,n +

∑n

dj−1p (n)ψj−1,n.

17

Then the Holder smoothness of Pj fj in the intervals Ti yields the Taylor expansion

Pj fj(t) = qα(t− t0) +R(t− t0) with |R(t− t0)| ≤ cϕ(t)Dα 2−jα/2

for t, t0 ∈ Ti, where D is the constant in the diameter condition (3.7), and this gives theestimate for the wavelet coefficients corresponding to the region Ωi,

|dj−1p (n)| = |〈R(t− t0), ψj−1,n〉| ≤ cϕ,nDα 2−jα/2 2−(j−1)/2 ‖ψ‖1

≤ cϕ,nDα 2−(j−1)(α+1)/2, (3.8)

where cϕ,n depends on the local smoothness of Pj fj and hence on the derivative bounds

of pj in (3.2). Again, let Λj−1 be the set of indices n from 0, . . . , 2j−1−1, where dj−1p (n)

satisfies the above estimate. Then the number of wavelet coefficients 2j−1−#Λj−1 whichare not satisfying this estimate (since supp ψj−1,n 6⊂ Ti for some i) is bounded by aconstant independent of J and j.

Finally, we obtain the polyharmonic spline interpolant

F j−1(x) :=K∑i=1

∑y∈Γj−1

i

ciy φm

(∥∥x− y2J

∥∥2

)+ pim(x)

χΩi(x),

where Γj−1i := pj(2n) : n = 0, . . . , 2j−1−1, 2−J pj(2n) ∈ Ωi, Γj−1 := ∪Ki=1Γj−1

i , throughthe interpolation conditions F j−1(2−Jpj(2n)) = f j−1(2−j+1n) for all n = 0, . . . , 2j−1 − 1.Hence, we obtain the estimate∣∣∣F j (pj(2n)

2J

)− F j−1

(pj(2n)

2J

)∣∣∣ = |f j(2−j+1n)− Pj−1fj(2−j+1n)| . 2−jα/2.

Particularly,∣∣∣F 2J(pj+1(2n)

2J

)− F j

(pj+1(2n)

2J

)∣∣∣ ≤ ∑2J−1ν=j

∣∣∣F ν+1(pj+1(2n)

2J

)− F ν

(pj+1(2n)

2J

)∣∣∣.

∑2J−1ν=j 2−(ν+1)α/2 ≤ 2−(j+1)α/2

1−2−α/2. (3.9)

Let us summarize the above findings on the decay of the EPWT wavelet coefficientsin the following theorem.

Theorem 3.2 For j = 2J − 1, . . . , 0, let djp(`) = 〈f j+1, ψj,`〉, ` = 0, . . . , 2j − 1, denotethe wavelet coefficients that are obtained by applying the EPWT algorithm to F (accord-ing to the above definitions), where we assume that F ∈ L2([0, 1)2) is piecewise Holdersmooth of order α as prescribed in Subsection 2.1. Further assume that the path vectors(pj+1(`))2j+1−1

`=0 and the corresponding path functions pj+1, j = 2J−1, . . . , 0, in the EPWTalgorithm satisfy the region condition (a), the path smoothness condition (b), and the di-ameter condition (c) of Subsection 3.2. Then, for all j = 2J − 1, . . . , 0 and ` ∈ Λj, theestimate

|djp(`)| ≤ C Dα 2−j(α+1)/2 (3.10)

holds, where D > 1 is the constant of the diameter condition (3.7), α is the Holderexponent of F , and C depends on the utilized wavelet basis and on the Holder constantin (2.1). Furthermore, for all ` ∈ 0, . . . , 2j − 1 \ Λj, we obtain the estimate

|djp(`)| ≤ C ′ 2−j/2 (3.11)

with some constant C ′ being independent of J and j.

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Proof. The proof of (3.10) follows directly from (3.8). Likewise, for all ` ∈ 0, . . . , 2j−1 \ Λj , i.e., for index sets that do not satisfy the diameter, the path smoothness, or theregion condition, we observe at least

|djp(`)| ≤ C ′ 2−j/2 = C ′ 2−j/2

since we can assume that F j is bounded, and hence the above estimate (3.10) holds forα = 0. Thus (3.11) follows.

4 N-term Approximation with the EPWT

Consider now the vector of all EPWT wavelet coefficients

dp = ((d2J−1p )T , . . . , d0

p, d−1p )T

with djp = (djp(`))2j−1`=0 for j = 0, . . . , 2J − 1, and with the mean value

d−1p = d−1

p (0) := f0(0) = 2−2J∑n∈IJ

F 2J(2−Jn),

together with the side information on the path vectors at each iteration step

p = ((p2J)T , . . . , (p1)T )T ∈ R2(22J−1).

With this information the reconstructed image F 2Jrec is uniquely recovered, where F 2J

rec isthe polyharmonic spline interpolation satisfying

F 2Jrec

(2−Jp2J(n)

)= f2J(2−2Jn), n = 0, . . . , 22J − 1.

Indeed, reconsidering the (j + 1)-th level of the EPWT procedure, we observe that thescaling coefficients cjp(n) = 〈f j+1, ϕj,n〉 and the wavelet coefficients djp = 〈f j+1, ψj,n〉determine f j and gj , and hence Pj+1f

j+1 uniquely. Further, the polyharmonic splineinterpolation F j+1 is entirely determined by the function values f j+1(2−(j+1)n).

By the choice of the wavelet basis, it further follows for n = 0, . . . , 2j+1 − 1 that

|F j+1(2−Jpj+1(n))− F j+1rec (2−Jpj+1(n))| = |f j+1(2−(j+1)n)− Pj+1f

j+1(2−(j+1)n)|. 2−(j+1)α/2,

where F j+1rec is uniquely determined by Pj+1f

j+1(2−(j+1)n), n = 0, . . . , 2j+1 − 1 and theside information about the path pj+1.

In order to find a sparse approximation of the digital image F resp. F 2J , we applya shrinkage procedure to the EPWT wavelet coefficients djp(`), using the hard thresholdfunction

sσ(x) =

x |x| ≥ σ,0 |x| < σ,

for some σ > 0.

We now study the error of a sparse representation using only the N wavelet coefficientswith largest absolute value for an approximative reconstruction of F 2J . For convenience,let S2J

N be the set of indices (j, `) of the N wavelet coefficients with largest absolute value.

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Moreover, let F 2JN,rec denote the polyharmonic spline interpolation determined by the

reconstructed function f2JN =

∑n c

2Jp,N (n)ϕ2J,n using only the N wavelet coefficients with

largest absolute value, satisfying the interpolation conditions

F 2JN,rec

(p2J (n)

2J

)= f2J

N (2−2Jn) for n = 0, . . . , 22J − 1.

While the wavelet basis used above is not orthonormal but stable, we can still estimatethe distance of F 2J and F 2J

N,rec by

εN = ‖F 2J − F 2JN,rec‖2L2(Ω) .

∑(j,`)6∈S2J

N

|djp(`)|2.

This estimate is a direct consequence of Theorem 3.2 and (3.9). Indeed, at each level ofthe EPWT, we observe that

‖F j+1 − F jrec‖L2(Ω) ≤ ‖F j+1 − F j‖L2(Ω) + ‖F j − F jrec‖L2(Ω)

.

(2j−1∑n=0|djp(n)|2

)1/2

. 2−jα/2,

where the number of wavelet coefficients satisfying (3.10) is 2j−C1K+C2, and where theconstants C1 and C2 do not depend on J or j, see Section 3.2.

Now we obtain the main result of this paper, showing the optimal N -term approxima-tion of the EPWT algorithm.

Theorem 4.1 Let F 2JN be the N -term approximation of F 2J as constructed above, and let

the assumptions of Theorem 3.2 be satisfied. Then the estimate

εN = ‖F 2J − F 2JN ‖22 ≤ C N−α (4.1)

holds for all J ∈ N, where the constant C <∞ does not depend on J .

Proof. The proof can be carried out by following along the lines of the proof ofTheorem 3.3 in [25].

Let us finally conclude by stating the following corollary.

Corollary 4.2 Let F ∈ L2([0, 1)2) be piecewise Holder continuous (as assumed in Sub-section 2.1). Then, for any ε > 0 there exists an integer J(ε), such that for all J ≥ J(ε)the N -term estimate

‖F − F 2JN ‖2L2 < CN−α + ε

holds, where C is the constant in (4.1).

Proof. The proof follows directly from Theorem 4.1 and (2.8).

Remark. Please observe that the above “construction” using polyharmonic spline inter-polations F j and smooth path functions f j has been done only for analytic purpose. Fora practical implementation of the EPWT we refer to the references [22], [24], and [28]. Inthese papers, the EPWT has been successively tested with different path constructions forimage approximation as sketched in Section 2. In particular, the so-called relaxed EPWT[22] already covers the path conditions heuristically, where in the numerical implementa-tion “path snakes” are preferred that lead to a better path smoothness.

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Acknowledgment

The authors would like to thank the anonymous referees for the detailed and helpful com-ments to improve the paper. In particular, they pointed out to us the strong dependenceof the smoothness properties of the restricted functions f j on the boundedness of deriva-tives of the path functions pj . This point had not been considered rigorously in the firstversion of the paper. A closer look at this problem led to a further path condition thatneeds to be satisfied in order to obtain best N-term approximation results.

This work is supported by the priority program SPP 1324 of the Deutsche Forschungs-gemeinschaft (DFG), projects PL 170/13-2 and IS 58/1-2.

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