DFG-Schwerpunktprogramm 1324

”Extraktion quantifizierbarer Information aus komplexen Systemen”

Optimal Representation of Piecewise HolderSmooth Bivariate Functions by the Easy Path

Wavelet Transform

G. Plonka, S. Tenorth, A. Iske

Preprint 78

Edited by

AG Numerik/OptimierungFachbereich 12 - Mathematik und InformatikPhilipps-Universitat MarburgHans-Meerwein-Str.35032 Marburg

DFG-Schwerpunktprogramm 1324

”Extraktion quantifizierbarer Information aus komplexen Systemen”

Optimal Representation of Piecewise HolderSmooth Bivariate Functions by the Easy Path

Wavelet Transform

G. Plonka, S. Tenorth, A. Iske

Preprint 78

The consecutive numbering of the publications is determined by theirchronological order.

The aim of this preprint series is to make new research rapidly availablefor scientific discussion. Therefore, the responsibility for the contents issolely due to the authors. The publications will be distributed by theauthors.

Optimal Representation of Piecewise Holder Smooth

Bivariate 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,s.tenorth@math.uni-goettingen.de2 Department of Mathematics, University of Hamburg, 20146 Hamburg, Germany

iske@math.uni-hamburg.de

Abstract

The Easy Path Wavelet Transform (EPWT) [20] 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, image datacompression, adaptive wavelet bases, N -term approximation

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. Inhigher 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 [10, 11] and directionlets [28] 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,

1

whereas a (tensor product) wavelet expansion leads to asymptotically only O(N−1) [17].Up to the (log2N)3 factor, this curvelet approximation result is asymptotically optimal(see [7, Section 7.4]). A similar estimate has been achieved by Guo and Labate [10]for shearlet frames. These results, however, are not adaptive with respect to the assumedregularity of the target function, and so they cannot be applied to images of less regularity,i.e., images which are not at least piecewise C2 with discontinuities along C2-curves.

In such relevant cases, one should 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, 12, 15, 16,18, 20, 21, 22, 24, 25, 27]. In [16], 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 [16] 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, 12] (and references therein), being basedon ENO reconstructions. We remark that the resulting ENO-EA schemes lead to anoptimal N -term approximation, yielding ‖f − fN‖2

2 ≤ C N−2 for piecewise C2-functionswith discontinuities along C2-curves. Moreover, unlike previous non-adaptive schemes,the ENO-EA multiresolution techniques provide optimal approximation results also forBV -spaces and Lp spaces, see [1].

In [20], 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 [21], 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 [22], 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 usingthe 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.

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 [20, 23]). If, however, smoother wavelet bases are utilized in theEPWT approach, such an interpretation is not obvious. In fact, while Haar wavelets

2

admit 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, leading to a corresponding local approxima-tion order [13]. We will come back to relevant approximation properties of polyharmonicsplines in Section 2.

This paper essentially generalizes our results in [23], 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.

The outline of this paper is as follows. In Section 2, we first introduce the utilizedfunction model and the EPWT algorithm. Then, in Section 3, we study the decay ofEPWT-wavelet coefficients, where we will consider the highest level of the EPWT indetail. To achieve optimal decay results for the EPWT wavelet coefficients at the furtherlevels, we require specific side conditions for the path vectors in the EPWT algorithm.These side conditions can be ensured by a suitable path vector construction as proposed inSection 2. Similar conditions for the path vectors have been used already in [23]. Finally,Section 4 is devoted to the proof of asymptotically optimal N -term error estimates of theform (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 smooth overa finite set of regions Ωi1≤i≤K , where each region Ωi has a sufficiently smooth Lipschitzboundary ∂Ωi of finite length. Moreover, the set Ωi1≤i≤K is assumed to be a disjointpartition 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| = ⌊α⌋ satisfies an estimate ofthe 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 ⌊α⌋ (usually the Taylor polynomial of F ofdegree ⌊α⌋ 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. Note

3

that the Holder space Cα(Ωi) of order α > 0, being equipped with the norm

‖F‖Cα(Ωi) := ‖F‖C⌊α⌋(Ωi)+

∑

|µ|=m

supx 6=y

|F (µ)(x) − F (µ)(y)|‖x− y‖α−m

2

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

Cm(Ωi) norm

‖F‖Cm(Ωi) := supx∈Ωi

|F (x)| +∑

|α|=m

supx∈Ωi

|∂αF (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.

Next we compute a (piecewise) sufficiently smooth approximation to F from its givensamples. To this end, we apply polyharmonic spline interpolation to obtain an interpola-tion to F of the form

F 2J(x) :=K

∑

i=1

∑

n∈Γ2Ji

cin φm

(∥

∥

∥x− n

2J

∥

∥

∥

2

)

+ pim(x)

χΩi(x) for x ∈ [0, 1)2 (2.2)

satisfying the interpolation conditions

F 2J(n/2J) = F (n/2J) for all n ∈ I2J . (2.3)

Here, χΩiis the characteristic function of Ωi, φm(r) = r2m log(r), for m := max(⌊α⌋, 2), is

a fixed polyharmonic spline kernel, and pim denotes a bivariate polynomial of degree at most

m. It is well-known that polyharmonic spline interpolation leads, on given interpolationconditions (2.3), to a unique interpolant of the form (2.2). In particular, for the specificchoice of a polyharmonic spline kernel φm, the interpolation scheme achieves to reconstructpolynomials of degree m. Consequently, the local approximation order of polyharmonicspline interpolation is m + 1, see [13]. Further relevant details on polyharmonic splineinterpolation and their approximation properties can be found in [14, Section 3.8].

4

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

2,2(Ωi), seee.g. [26] or [4, page 163]. Since Bα

2,2(Ωi) is equivalent to the Sobolev space Hα(Ωi),see [4, 26], this allows us to use the estimate

‖F − F 2J‖L2(Ω) ≤ CF

K∑

i=1

hαΩi‖F‖Bα

2,2(Ωi) (2.4)

for the interpolation error in Sobolev spaces, as shown in [19], where the fill distance

hΩi:= sup

x∈Ωi

infn∈Γ2J

i

‖x− n

2J‖2 ≤ 2−J for 1 ≤ i ≤ K

measures the density of the interpolation points in Ωi.

Remark. Note that by the above representation (2.2), we apply polyharmonic splineinterpolation separately in each individual region Ωi. We prefer interpolation (rather thanany other projection method), since we essentially require to maintain the local Holderregularity around each lattice point in I2J .

2.2 The EPWT Algorithm

Now let us briefly recall the EPWT algorithm from our previous work [20]. 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, we use thenotation

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)

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 m = max(⌊α⌋, 2), and so,

〈pm, ψj,k〉 = 0 for all j, k ∈ Z.

5

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∈Zcj(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),

where

f j−1(x) =∑

n∈Z

cj−1(n)ϕj−1,n and gj−1(x) =∑

n∈Z

dj−1(n)ψj−1,n

with

cj−1(n) = 〈f j , ϕj−1,n〉 and dj−1(n) = 〈f j , ψj−1,n〉. (2.5)

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 that

does 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−1

n=0 through the indexset I2J . This path vector is a suitable permutation of all indices in I2J , which can bedetermined by using the following strategy from [20]. Recall that I2J = ∪K

i=1Γ2Ji , where

Γ2Ji 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),

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) fromΓ2J

i \ p2J(0), . . . , p2J(n). If, however, this set is also empty, we choose p2J(n + 1) fromthe set of remaining indices I2J \ p2J(0), . . . , p2J(n). For a more detailed description ofthe path vector construction we refer to [20].

6

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 constantC1 does not depend on J , see [23]. The so obtained vector p2J is composed of connectedpathways, i.e., each pair of consecutive components in these pathways are neighboring.

Now, we consider the data vector

(

c2J(ℓ))22J−1

ℓ=0:=

(

F 2J

(

p2J(ℓ)

2J

))22J−1

ℓ=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(ℓ))22J−1−1

ℓ=0 and

the vector of wavelet coefficients (d2J−1(ℓ))22J−1−1

ℓ=0 according to the formulae in (2.5). Dueto the piecewise smoothness of F 2J along the path vector p2J , it follows that most of thewavelet coefficients in d2J−1 are small, where only the wavelet coefficients correspondingto an interruption (from one region to another) may possess significant amplitudes.

The path vector p2J determines a new subset of indices

Γ2J−1 :=

p2J(2ℓ) : ℓ = 0, . . . , 22J−1 − 1

=K⋃

i=1

Γ2J−1i ,

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

i .As regards the next level of the EPWT, where j = 2J − 1, we first locate a second

connected path vector p2J−1 = (p2J−1(ℓ))22J−1−1

ℓ=0 through Γ2J−1, i.e., the entries of p2J−1

form a permutation of the indices in Γ2J−1. Similar 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 not de-pend on J . Then we apply one level of the one-dimensional wavelet transform to the per-

muted data vector (c2J−1(p2J−1(ℓ)))22J−1−1

ℓ=0 to obtain the low-pass vector (c2J−2(ℓ))22J−2−1

ℓ=0

and the vector (d2J−2(ℓ))22J−2−1

ℓ=0 of wavelet coefficients.We continue by iteration over the remaining levels j + 1 for j = 2J − 3, . . . , 0, where

at any level j + 1 we first construct a path pj+1 = (pj+1(ℓ))2j+1−1

ℓ=0 through the index set

Γj+1 := pj+2(2ℓ) : ℓ = 0, . . . , 2j+1 − 1 =

K⋃

i=0

Γj+1i

with applying similar strategies as described above. Here, pj+1(n) and pj+1(n + 1) arecalled neighbors, iff

∥

∥pj+1(n) − pj+1(n+ 1)∥

∥

2≤ D2J−(j+1)/2,

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+1(pj+1(ℓ))2j+1−1

ℓ=0 , yielding cj and dj .

7

(a) (b)

(c) (d)

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

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 in theunderlying index set I6 = Γ6

1 ∪ Γ62, such that each pair of consecutive components in the

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

2, see Figure 1(a).For the 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.

In this example, we have

‖p6(n+ 1) − p6(n)‖2 ≤√

2 ≤ D, ‖p5(n+ 1) − p5(n)‖2 ≤ 2 ≤√

2D,

‖p4(n+ 1) − p4(n)‖2 ≤√

10 ≤ 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 =

√10/2 ≈ 1.5811.

This simple example also illustrates that the path construction leads at each level to indexsets Γj

i with quasi-uniformly distributed indices. Further details concerning the requiredquasi-uniformity of the index distributions are explained in Section 3.2.

8

Remark 1. Note that the components of the path vector pj lie in Γ2J with containing2d integer entries. This is in contrast to the notation in [20]. Further, unlike in [20], wedo no longer consider index sets but define a neighborhood of pixels by the Euclideandistance between corresponding indices. The constant D should be chosen rather small,e.g. D ∈ (

√2, 2), as in the previous example. We remark that the choice of a sufficiently

small constant D leads to a sequence of paths, where the distribution of the indices in Γj

is quasi-uniform at each level of the EPWT, see Section 3.2.

Remark 2. 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 thepath 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.

To show this fact also theoretically, we have proven in [23] that for piecewise Holdercontinuous functions of order α ∈ (0, 1], we obtain optimal convergence rates for N -termapproximations when using univariate Haar wavelets. However, our numerical experimentsshow that the EPWT is much more effective when smoother wavelet bases are used, e.g. theDaubechies wavelets with two vanishing moments or the 7 − 9 biorthogonal transform.

Therefore, the goal of this paper is to show that a higher Holder smoothness of thetarget function within the different regions leads to optimal N -term approximations, whenusing the EPWT in combination with smooth wavelet bases. In this sense, the results ofthis paper can be viewed as a generalization of our previous results in [23].

Remark 3. In contrast to the above procedure, where we have used the data vectors(

cj(ℓ))2j−1

ℓ=0=

(

F j(2−Jpj(ℓ))2j−1

ℓ=0, we will consider slightly different data vectors cjp(ℓ) in the

theoretical estimates of Sections 3 and 4, where we will be using L2-projection operatorsdetermined by the dual scaling and wavelet functions, ϕ and ψ.

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 already explained in the previous section, we consider applying polyharmonic splineinterpolation, from given image values F (2−Jn), separately in the individual domainsΩi, i = 1, . . . ,K. We assume that F is Holder smooth of order α on each Ωi, andthat the polyharmonic spline interpolant F 2J in (2.2), being Holder smooth of order atleast α, reconstructs bivariate polynomials of degree m = max(⌊α⌋, 2), thus yielding localapproximation order m+ 1.

At the 2J-th level of the EPWT, we define a path p2J through all indices of I2J suchthat consecutive components of p2J are neighboring indices lying in the same index setΓi. The path p2J can be constructed in a way such that only 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

9

“along the path p2J” with

|F (2−Jp2J(ℓ)) − f2J(2−2Jℓ)| . 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 theinterval [0, 1) that correspond to interruptions of the path. However, the number of suchinterruptions does not depend on J but only on the number of regions, K. Therefore, withthe performance of one level of the (periodic) wavelet transform, we will find that most ofthe wavelet coefficients of f2J = f2J−1 + g2J−1 occurring in the wavelet part g2J−1, aresmall.

At the next levels of the EPWT, we consider the index sets Γj = ∪Ki=1Γ

ji . 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 , two specific conditions on the index sets Γj and thepath vectors pj need to be satisfied. We can briefly explain these two conditions, termedregion condition and diameter condition, as follows (for more details on these conditionswe refer to Subsection 3.2).

Firstly, the region condition requires that the path should prefer to traverse the indicesbelonging to one region set Γj

i , before “jumping” to another region. In this way, theregion condition ensures that we can optimally exploit the smoothness of the function Falong the path. Secondly, the diameter condition requires a quasi-uniform distribution ofremaining pixels in each Γj

i , and so the diameter condition leads to a sufficiently accuratepolyharmonic spline interpolation F j at each level of the EPWT.

We remark that the two conditions can be satisfied by using the strategies for the pathconstruction as proposed in Subsection 2.2. This then allows us to estimate the EPWTwavelet coefficients similarly as for one-dimensional piecewise smooth functions with afinite number of singularities to finally obtain an optimal N -term approximation usingonly the N most significant EPWT wavelet 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 theplane interpolating the path p2J , i.e., with p2J(ℓ/22J) = 2−J p2J(ℓ), for ℓ = 0, . . . , 22J − 1,and with the convention that p2J(t) ∈ Ωi, for t ∈ [ℓ/22J , (ℓ + 1)/22J ], if 2−J p2J(ℓ) and2−J p2J(ℓ + 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).

For each interruption in the path vector p2J , originated within one region Ωi or by a jumpfrom one region to another, there are indices ℓ and ℓ+1, where p2J(ℓ) and p2J(ℓ+1) are notneighbors or where there is a discontinuity between F 2J(2−J p2J(ℓ)) and F 2J(2−J p2J(ℓ+1)). These interruptions correspond to small subintervals of [0, 1) of length 2−2J , wherethe univariate function f2J may also have discontinuities. More precisely, an interruption

10

between the path components p2J(ℓ) and p2J(ℓ + 1) generates such a “jump interval”at [ℓ/22J , (ℓ + 1)/22J ]. For simplicity, we assume that we only have path interruptionsfrom one region to another. Note that this is only a mild restriction, since we mayotherwise decide to further subdivide a region into several smaller subregions according tothe regularity of F along the paths. Finally, we remark that for any convex region Ωi wecan show that there is at least one path without any interruptions, see [23].

Recall that the trace theorem for Holder resp. Besov spaces (see [26]) implies that forF 2J |Ωi

∈ Bα∞,∞(Ωi), the (scaled) restriction f2J(t) along the curve p2J(t) is again Holder

smooth of order α in each subinterval of [0, 1), determined by t ∈ [0, 1) : p2J(t) ∈ Ωiwith assuming that the corresponding path vector p2J has no interruptions. In particular,we obtain for the N -th order modulus of smoothness the estimate

ωN (f2J , h)∞ := sup|h|≤h

‖∆Nhf2J‖∞ . (2Jh)α‖f2J‖Bα

∞,∞(3.1)

within the subintervals, where f2J is smooth, i.e., for N = ⌊α+ 1⌋ and

Ti,h :=

t : p2J(t+ kh) ∈ Ωi, k = 0, . . . , N

,

see [4]. Observe that the factor 2Jα in (3.1) is due to the scaling of f2J on [0, 1] while thelength of the complete curve p2J is c 2J , where the constant c does not depend on J .

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−J)α‖f2J‖Bα∞,∞

. (3.2)

In particular,

|f2J(2−2Jℓ) − f2J(2−2Jℓ)| = |F 2J(2−Jp2J(ℓ)) − f2J(2−2Jℓ)| . 2−Jα.

In the next step, we decompose the function f2J =∑

ℓ

c2Jp (ℓ)ϕ2J,ℓ with c2J

p (ℓ) := 〈f2J , ϕ2J,ℓ〉

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 (ℓ))2

2J−1ℓ=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 in Ti := t ∈ [0, 1) : p2J(t) ∈ Ωi, we find for t ∈ Ti the represen-tation

f2J(t) = qα(t− t0) +R(t− t0)

11

for t0 ∈ 2−2Jk : k = 0, . . . , 22J − 1∩ Ti and |t− t0| ≤ 2−2J , where qα denotes the Taylorpolynomial of degree ⌊α⌋ of f2J at t0, and where the remainder R satisfies |R(t − t0)| ≤cϕ2−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 2−Jα ‖ψ2J−1,n‖1 = cϕ 2(−J+1/2)(α+1),

where we have used ‖ψ2J−1,n‖1 = 2−J+1/2‖ψ‖1.Now let Λ2J−1 be the set of all n ∈ 0, . . . , 22J−1 − 1, where the above estimate for

d2J−1(n) is satisfied. Then, the number of the remaining wavelet coefficients 22J−1 −#Λ2J−1 corresponds to the number of discontinuities of f2J and is hence bounded by CK,where K is the number of regions in the original image F , and where the constant C doesnot 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 components of p2J with even indices, we put

Γ2J−1i := p2J(2n) : n = 0, . . . , 22J−1 − 1, 2−Jp2J(2n) ∈ Ωi

for each i = 1, . . . ,K and Γ2J−1 := ∪Ki=1Γ

2J−1i . We compute the polyharmonic spline

interpolant

F 2J−1(x) :=K

∑

i=1

∑

y∈Γ2J−1i

ciy φm

(∥

∥

∥x− y

2J

∥

∥

∥

2

)

+ pim(x)

χΩi

(x),

satisfying the interpolation conditions

F 2J−1

(

p2J(2n)

2J

)

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

Therefore,∣

∣

∣

∣

F 2J

(

p2J(2n)

2J

)

− F 2J−1

(

p2J(2n)

2J

)∣

∣

∣

∣

= |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 (3.2) 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 (2.4) we observe

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

12

3.2 Conditions for the Path Vectors

Before we proceed with the error estimates for the further levels of the EPWT algorithm,we need to fix two specific side conditions for the path vectors that are required for ourerror analysis. The two side conditions are termed (a) region condition and (b) diametercondition, as stated below.

Similar conditions have already been applied to the N -term estimates in [23]. Theregion condition ensures that at each level of the EPWT, the path vector should be chosenin a way such that all indices belonging to one region Ωi, i ∈ 1, . . . ,K, should be takenfirst before jumping to another region, and that one should have only a small numberof index pairs pj(ℓ), pj(ℓ + 1) belonging to different regions Γj

i and Γjk. The diameter

condition ensures that the remaining indices in Γj are quasi-uniformly distributed, suchthat there is a constant D, not depending on J or j, satisfying

maxx∈Ωi

miny∈Γj

‖x− 2−Jy‖2 ≤ D2−j/2.

Let us introduce the two 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(ℓ) andpj(ℓ+ 1), which are no neighbors or are not belonging to the same index set Γj

i .

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

‖pj(ℓ) − pj(ℓ+ 1)‖2 ≤ D 2J−j/2, (3.3)

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.

The two conditions, (a) and (b), are already ensured by the proposed path constructionin Subsection 2.2. Particularly, the diameter condition ensures a quasi-uniform distributionof the indices in Γj .

3.3 The Further Levels of the EPWT

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

F j+1(x) :=K

∑

i=1

∑

y∈Γj+1i

ciy φm

(∥

∥

∥x− y

2J

∥

∥

∥

2

)

+ pim(x)

χΩi

(x)

satisfying the interpolation conditions

F j+1

(

pj+2(2n)

2J

)

= f j+1(2−(j+1)n) for all n = 0, . . . , 2j+1 − 1.

13

We first fix a suitable path vector pj+1 passing through the set

Γj+1 = pj+2(2n) : n = 0, . . . , 2j+1 − 1

with corresponding data values F j+1(

pj+2(2n)2J

)

: n = 0, . . . , 2j+1 − 1, such that the

region condition and the diameter condition in Section 3.2 are satisfied.

Next, we consider a sufficiently smooth parametric curve pj+1(t), t ∈ [0, 1), throughthe plane interpolating pj+1, i.e., with pj+1(2−j−1ℓ) = 2−J pj+1(ℓ) for ℓ = 0, . . . , 2j+1 − 1,such that pj+1(t) ∈ Ωi for t ∈ [2−(j+1)ℓ, 2−(j+1)(ℓ+ 1)], if 2−Jpj+1(ℓ) and 2−Jpj+1(ℓ+ 1)are in the same region Ωi.

Then we determine the one-dimensional restriction of F j+1 along pj+1,

f j+1(t) := F j+1(

pj+1(t))

t ∈ [0, 1).

As in the above discussion concerning the 2J-th level, the piecewise Holder smoothnessof f j+1(t) ensures that

ωN (f j+1, h)∞ ≤ c (2Jh)α for Ti,h := t : pj+1(t+ kh) ∈ Ωi, k = 0, . . . , N.

Considering the L2-projection

f j+1 = Pj+1fj+1 :=

2j+1−1∑

n=0

cj+1p (n)ϕj+1,n

with cj+1p (n) := 〈f j+1, ϕj+1,n〉 of f j+1 onto the scaling space

V j+1 := closL2[0,1)spanϕj+1,n : n = 0, . . . , 2j+1 − 1,

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

‖f j+1 − f j+1‖L∞(Ti,2−(j+1)) = ‖f j+1 − Pj+1f

j+1‖L∞(Ti,2−(j+1))

. wN (f j+1, 2−J−(j+1)/2) . 2−(j+1)α/2,

where we have used the diameter condition (3.3), so that

‖pj+1(ℓ) − pj+1(ℓ+ 1)‖2 ≤ D 2J−(j+1)/2.

We use the decomposition

f j+1 =∑

ℓ

cj+1p (ℓ)ϕj+1,ℓ

= f j + gj =∑

n

cjp(n)ϕj,n +∑

n

djp(n)ψj,n.

Then the Holder smoothness of f j+1 in the intervals Ti yields the Taylor expansion

f j+1(t) = qα(t− t0) +R(t− t0) with |R(t− t0)| ≤ cϕDα 2−(j+1)α/2

14

for t, t0 ∈ Ti, and this gives the following estimate for the wavelet coefficients correspondingto the region Ωi:

|djp(n)| = |〈R(t− t0), ψj,n〉| ≤ cϕD

α 2−(j+1)α/2 2−j/2 ‖ψ‖1 ≤ cϕDα 2−j(α+1)/2.

Again, let Λj be the set of indices n from 0, . . . , 2j − 1, where djp(n) satisfies the above

estimate. Then the number of wavelet coefficients 2j − #Λj which are not satisfying thisestimate (since supp ψj,n 6⊂ Ti for some i) is bounded by a constant independent of Jand j.

Finally, we obtain the polyharmonic spline interpolant

F j(x) :=K

∑

i=1

∑

y∈Γji

ciy φm

(∥

∥

∥x− y

2J

∥

∥

∥

2

)

+ pim(x)

χΩi

(x),

where Γji := pj+1(2n) : n = 0, . . . , 2j − 1, 2−J pj+1(2n) ∈ Ωi, Γj := ∪K

i=1Γji , through the

interpolation conditions

F j

(

pj+1(2n)

2J

)

= f j(2−jn) for all n = 0, . . . , 2j − 1.

Hence, we obtain the estimate∣

∣

∣

∣

F j+1

(

pj+1(2n)

2J

)

− F j

(

pj+1(2n)

2J

)∣

∣

∣

∣

= |f j+1(2−jn) − Pj fj+1(2−jn)| . 2−(j+1)α/2.

Particularly,

∣

∣

∣

∣

F 2J

(

pj+1(2n)

2J

)

− F j

(

pj+1(2n)

2J

)∣

∣

∣

∣

≤2J−1∑

ν=j

∣

∣

∣

∣

F ν+1

(

pj+1(2n)

2J

)

− F ν

(

pj+1(2n)

2J

)∣

∣

∣

∣

.

2J−1∑

ν=j

2−(ν+1)α/2 ≤ 2−(j+1)α/2

1 − 2−α/2. (3.4)

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

Theorem 3.1 For j = 2J −1, . . . , 0, let djp(ℓ) = 〈f j+1, ψj,ℓ〉, ℓ = 0, . . . , 2j −1, denote the

wavelet coefficients that are obtained by applying the EPWT algorithm to F (according toSubsection 2.2), where we assume that F ∈ L2([0, 1)2) is piecewise Holder smooth of order

α as prescribed in Subsection 2.1. Further assume that the path vectors (pj+1(ℓ))2j+1−1

ℓ=0 ,j = 2J−1, . . . , 0, in the EPWT algorithm satisfy the region condition (a) and the diametercondition (b) of Subsection 3.2. Then, for all j = 2J − 1, . . . , 0 and ℓ ∈ Λj, the estimate

|djp(ℓ)| ≤ C Dα 2−j(α+1)/2 (3.5)

holds, where D > 1 is the constant of the diameter condition (3.3), α is the Holderexponent of F , and C depends on the utilized wavelet basis and on the Holder constantin (2.1). Furthermore, for all ℓ ∈ 0, . . . , 2j − 1 \ Λj, we obtain the estimate

|djp(ℓ))| ≤ C ′ 2−j/2 (3.6)

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

15

Proof. The proof of (3.5) follows directly from the above considerations. Likewise,for all ℓ ∈ 0, . . . , 2j − 1 \ Λj , i.e., for index sets that do not satisfy the diameter or theregion condition, we observe at least

|djp(ℓ)| ≤ C ′ 2−j/2 = C ′ 2−j/2

since we can assume that F j is bounded, and hence the above estimate (3.5) holds forα = 0. Hence (3.6) 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 = (dj

p(ℓ))2j−1ℓ=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 in each iteration step

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

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

rec is the polyharmonicspline 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 dj

p = 〈f j+1, ψj,n〉determine f j and gj , and hence f j+1 uniquely. Further, the polyharmonic spline interpo-lation 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) − f j+1(2−(j+1)n)|

= |f j+1(2−(j+1)n) − P j+1f j+1(2−(j+1)n)|. 2−(j+1)α/2,

i.e., F j+1rec is uniquely determined by f j+1(2−(j+1)n), n = 0, . . . , 2j+1 − 1 and the side

information about the path pj+1.In order to find a sparse approximation of the digital image F resp. F 2J , we apply

a shrinkage procedure to the EPWT wavelet coefficients djp(ℓ), using the hard threshold

function

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, ℓ) of the N wavelet coefficients with largest absolute value.

16

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

scaling function f2JN =

∑

n c2Jp,N (n)ϕ2J,n using only the N wavelet coefficients with largest

absolute value, satisfying the interpolation conditions

F 2JN,rec

(

p2J(n)

2J

)

= f2JN (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

ǫN = ‖F 2J − F 2JN,rec‖2

L2(Ω) .∑

(j,ℓ) 6∈S2JN

|djp(ℓ)|2.

This estimate is a direct consequence of Theorem 3.1 and (3.4). 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 j

rec‖L2(Ω)

.

2j−1∑

n=0

|djp(n)|2

1/2

. 2−(j+1)α/2,

where the number of wavelet coefficients satisfying (3.5) 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 3.1 be satisfied. Then the estimate

ǫN = ‖F 2J − F 2JN ‖2

2 ≤ 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 [23].

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 ǫ > 0 there exists an integer J(ǫ), such that for all J ≥ J(ǫ)the N -term estimate

‖F − F 2JN ‖2

L2 < CN−α + ǫ

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

Proof. The proof follows directly from Theorem 4.1 and (2.4).

Acknowledgment

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

17

References

[1] F. Arandiga, A. Cohen, R. Donat, N. Dyn, and B. Matei, Approximation of piecewisesmooth functions and images by edge-adapted (ENO-EA) nonlinear multiresolutiontechniques, Appl. Comput. Harmon. Anal. 24 (2008), 225–250.

[2] E.J. Candes and D.L. Donoho, New tight frames of curvelets and optimal represen-tations of objects with piecewise singularities, Comm. Pure Appl. Math. 57 (2004),219–266.

[3] E.J. Candes, L. Demanet, D.L. Donoho, and L. Ying, Fast discrete curvelet trans-forms, Multiscale Model. Simul. 5 (2006), 861–899.

[4] A. Cohen, Numerical Analysis of Wavelet Methods. Studies in Mathematics and itsApplications 32, Elsevier, Amsterdam, 2003.

[5] S. Dekel and D. Leviatan, Adaptive multivariate approximation using binary spacepartitions and geometric wavelets, SIAM J. Numer. Anal. 43 (2006), 707–732.

[6] L. Demaret, N. Dyn, and A. Iske, Image compression by linear splines over adaptivetriangulations, Signal Processing 86 (2006), 1604–1616.

[7] R.A. DeVore, Nonlinear approximation, Acta Numerica, 1998, 51–150.

[8] M.N. Do and M. Vetterli, The contourlet transform: an efficient directional multires-olution image representation, IEEE Trans. Image Process. 14 (2005), 2091–2106.

[9] D.L. Donoho, Wedgelets: Nearly minimax estimation of edges, Ann. Stat. 27 (1999),859–897.

[10] K. Guo and D. Labate, Optimally sparse multidimensional representation using shear-lets, SIAM J. Math. Anal. 39 (2007), 298–318.

[11] K. Guo, W.-Q. Lim, D. Labate, G. Weiss, and E. Wilson, Wavelets with compositedilations, Electr. res. Announc. of AMS 10 (2004), 78–87.

[12] A. Harten, Multiresolution representation of data: general framework, SIAM J. Nu-mer. Anal. 33 (1996), 1205–1256.

[13] A. Iske, On the approximation order and numerical stability of local Lagrange inter-polation by polyharmonic splines, Modern Developments in Multivariate Approxima-

tion, W. Haußmann, K. Jetter, M. Reimer, J. Stockler (eds.), ISNM 145, Birkhauser,Basel, 2003, 153–165.

[14] A. Iske, Multiresolution Methods in Scattered Data Modelling. Lecture Notes in Com-putational Science and Engineering 37, Springer, Berlin, 2004.

[15] L. Jaques and J.-P. Antoine, Multiselective pyramidal decomposition of images:wavelets with adaptive angular selectivity, Int. J. Wavelets Multiresolut. Inf. Pro-cess. 5 (2007), 785–814.

[16] E. Le Pennec and S. Mallat, Bandelet image approximation and compression, Multi-scale Model. Simul. 4 (2005), 992–1039.

[17] S. Mallat, A Wavelet Tour of Signal Processing. Academic Press, San Diego, 1999.

[18] S. Mallat, Geometrical grouplets, Appl. Comput. Harmon. Anal. 26 (2009), 161–180.

[19] F.J. Narcowich, J.D. Ward, and H. Wendland, Sobolev error estimates and a Bern-stein inequality for scattered data interpolation via radial basis functions, Constr.Approx. 24 (2006), 175–186.

18

[20] G. Plonka, The easy path wavelet transform: a new adaptive wavelet transform forsparse representation of two-dimensional data, Multiscale Modelling Simul. 7 (2009),1474–1496.

[21] G. Plonka and D. Rosca, Easy path wavelet transform on triangulations of the sphere,Mathematical Geosciences 42(7) (2010), 839–855.

[22] G. Plonka, S. Tenorth, D. Rosca. A hybrid method for image approximation usingthe easy path wavelet transform. IEEE Trans. Image Process., 2011, to appear, DOI10.1109/TIP.2010.2061861.

[23] G. Plonka, S. Tenorth, and A. Iske, Optimally sparse image representation by theeasy path wavelet transform. Preprint, 2010.

[24] D.D. Po and M.N. Do, Directional multiscale modeling of images using the contourlettransform, IEEE Trans. Image Process. 15 (2006), 1610–1620.

[25] R. Shukla, P.L. Dragotti, M.N. Do, and M. Vetterli, Rate-distortion optimized treestructured compression algorithms for piecewise smooth images, IEEE Trans. ImageProcess. 14 (2005), 343–359.

[26] H. Triebel, Theory of Function Spaces. Birkhauser, Basel 1983.

[27] M.B. Wakin, J.K. Romberg, H. Choi, and R.G. Baraniuk, Wavelet-domain approxi-mation and compression of piecewise smooth images, IEEE Trans. Image Process. 15(2006), 1071–108.

[28] V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, and P.L. Dragotti, Directionlets:anisotropic multidirectional representation with separable filtering, IEEE Trans. Im-age Process. 15(7) (2006), 1916–1933.

19

Preprint Series DFG-SPP 1324

http://www.dfg-spp1324.de

Reports

[1] R. Ramlau, G. Teschke, and M. Zhariy. A Compressive Landweber Iteration forSolving Ill-Posed Inverse Problems. Preprint 1, DFG-SPP 1324, September 2008.

[2] G. Plonka. The Easy Path Wavelet Transform: A New Adaptive Wavelet Transformfor Sparse Representation of Two-dimensional Data. Preprint 2, DFG-SPP 1324,September 2008.

[3] E. Novak and H. Wozniakowski. Optimal Order of Convergence and (In-) Tractabil-ity of Multivariate Approximation of Smooth Functions. Preprint 3, DFG-SPP1324, October 2008.

[4] M. Espig, L. Grasedyck, and W. Hackbusch. Black Box Low Tensor Rank Approx-imation Using Fibre-Crosses. Preprint 4, DFG-SPP 1324, October 2008.

[5] T. Bonesky, S. Dahlke, P. Maass, and T. Raasch. Adaptive Wavelet Methods andSparsity Reconstruction for Inverse Heat Conduction Problems. Preprint 5, DFG-SPP 1324, January 2009.

[6] E. Novak and H. Wozniakowski. Approximation of Infinitely Differentiable Multi-variate Functions Is Intractable. Preprint 6, DFG-SPP 1324, January 2009.

[7] J. Ma and G. Plonka. A Review of Curvelets and Recent Applications. Preprint 7,DFG-SPP 1324, February 2009.

[8] L. Denis, D. A. Lorenz, and D. Trede. Greedy Solution of Ill-Posed Problems: ErrorBounds and Exact Inversion. Preprint 8, DFG-SPP 1324, April 2009.

[9] U. Friedrich. A Two Parameter Generalization of Lions’ Nonoverlapping DomainDecomposition Method for Linear Elliptic PDEs. Preprint 9, DFG-SPP 1324, April2009.

[10] K. Bredies and D. A. Lorenz. Minimization of Non-smooth, Non-convex Functionalsby Iterative Thresholding. Preprint 10, DFG-SPP 1324, April 2009.

[11] K. Bredies and D. A. Lorenz. Regularization with Non-convex Separable Con-straints. Preprint 11, DFG-SPP 1324, April 2009.

[12] M. Dohler, S. Kunis, and D. Potts. Nonequispaced Hyperbolic Cross Fast FourierTransform. Preprint 12, DFG-SPP 1324, April 2009.

[13] C. Bender. Dual Pricing of Multi-Exercise Options under Volume Constraints.Preprint 13, DFG-SPP 1324, April 2009.

[14] T. Muller-Gronbach and K. Ritter. Variable Subspace Sampling and Multi-levelAlgorithms. Preprint 14, DFG-SPP 1324, May 2009.

[15] G. Plonka, S. Tenorth, and A. Iske. Optimally Sparse Image Representation by theEasy Path Wavelet Transform. Preprint 15, DFG-SPP 1324, May 2009.

[16] S. Dahlke, E. Novak, and W. Sickel. Optimal Approximation of Elliptic Problemsby Linear and Nonlinear Mappings IV: Errors in L2 and Other Norms. Preprint 16,DFG-SPP 1324, June 2009.

[17] B. Jin, T. Khan, P. Maass, and M. Pidcock. Function Spaces and Optimal Currentsin Impedance Tomography. Preprint 17, DFG-SPP 1324, June 2009.

[18] G. Plonka and J. Ma. Curvelet-Wavelet Regularized Split Bregman Iteration forCompressed Sensing. Preprint 18, DFG-SPP 1324, June 2009.

[19] G. Teschke and C. Borries. Accelerated Projected Steepest Descent Method forNonlinear Inverse Problems with Sparsity Constraints. Preprint 19, DFG-SPP1324, July 2009.

[20] L. Grasedyck. Hierarchical Singular Value Decomposition of Tensors. Preprint 20,DFG-SPP 1324, July 2009.

[21] D. Rudolf. Error Bounds for Computing the Expectation by Markov Chain MonteCarlo. Preprint 21, DFG-SPP 1324, July 2009.

[22] M. Hansen and W. Sickel. Best m-term Approximation and Lizorkin-Triebel Spaces.Preprint 22, DFG-SPP 1324, August 2009.

[23] F.J. Hickernell, T. Muller-Gronbach, B. Niu, and K. Ritter. Multi-level MonteCarlo Algorithms for Infinite-dimensional Integration on RN. Preprint 23, DFG-SPP 1324, August 2009.

[24] S. Dereich and F. Heidenreich. A Multilevel Monte Carlo Algorithm for Levy DrivenStochastic Differential Equations. Preprint 24, DFG-SPP 1324, August 2009.

[25] S. Dahlke, M. Fornasier, and T. Raasch. Multilevel Preconditioning for AdaptiveSparse Optimization. Preprint 25, DFG-SPP 1324, August 2009.

[26] S. Dereich. Multilevel Monte Carlo Algorithms for Levy-driven SDEs with GaussianCorrection. Preprint 26, DFG-SPP 1324, August 2009.

[27] G. Plonka, S. Tenorth, and D. Rosca. A New Hybrid Method for Image Approx-imation using the Easy Path Wavelet Transform. Preprint 27, DFG-SPP 1324,October 2009.

[28] O. Koch and C. Lubich. Dynamical Low-rank Approximation of Tensors.Preprint 28, DFG-SPP 1324, November 2009.

[29] E. Faou, V. Gradinaru, and C. Lubich. Computing Semi-classical Quantum Dy-namics with Hagedorn Wavepackets. Preprint 29, DFG-SPP 1324, November 2009.

[30] D. Conte and C. Lubich. An Error Analysis of the Multi-configuration Time-dependent Hartree Method of Quantum Dynamics. Preprint 30, DFG-SPP 1324,November 2009.

[31] C. E. Powell and E. Ullmann. Preconditioning Stochastic Galerkin Saddle PointProblems. Preprint 31, DFG-SPP 1324, November 2009.

[32] O. G. Ernst and E. Ullmann. Stochastic Galerkin Matrices. Preprint 32, DFG-SPP1324, November 2009.

[33] F. Lindner and R. L. Schilling. Weak Order for the Discretization of the StochasticHeat Equation Driven by Impulsive Noise. Preprint 33, DFG-SPP 1324, November2009.

[34] L. Kammerer and S. Kunis. On the Stability of the Hyperbolic Cross DiscreteFourier Transform. Preprint 34, DFG-SPP 1324, December 2009.

[35] P. Cerejeiras, M. Ferreira, U. Kahler, and G. Teschke. Inversion of the noisy Radontransform on SO(3) by Gabor frames and sparse recovery principles. Preprint 35,DFG-SPP 1324, January 2010.

[36] T. Jahnke and T. Udrescu. Solving Chemical Master Equations by AdaptiveWavelet Compression. Preprint 36, DFG-SPP 1324, January 2010.

[37] P. Kittipoom, G. Kutyniok, and W.-Q Lim. Irregular Shearlet Frames: Geometryand Approximation Properties. Preprint 37, DFG-SPP 1324, February 2010.

[38] G. Kutyniok and W.-Q Lim. Compactly Supported Shearlets are Optimally Sparse.Preprint 38, DFG-SPP 1324, February 2010.

[39] M. Hansen and W. Sickel. Best m-Term Approximation and Tensor Products ofSobolev and Besov Spaces – the Case of Non-compact Embeddings. Preprint 39,DFG-SPP 1324, March 2010.

[40] B. Niu, F.J. Hickernell, T. Muller-Gronbach, and K. Ritter. Deterministic Multi-level Algorithms for Infinite-dimensional Integration on RN. Preprint 40, DFG-SPP1324, March 2010.

[41] P. Kittipoom, G. Kutyniok, and W.-Q Lim. Construction of Compactly SupportedShearlet Frames. Preprint 41, DFG-SPP 1324, March 2010.

[42] C. Bender and J. Steiner. Error Criteria for Numerical Solutions ofBackward SDEs. Preprint 42, DFG-SPP 1324, April 2010.

[43] L. Grasedyck. Polynomial Approximation in Hierarchical Tucker Format by Vector-Tensorization. Preprint 43, DFG-SPP 1324, April 2010.

[44] M. Hansen und W. Sickel. Best m-Term Approximation and Sobolev-Besov Spacesof Dominating Mixed Smoothness - the Case of Compact Embeddings. Preprint 44,DFG-SPP 1324, April 2010.

[45] P. Binev, W. Dahmen, and P. Lamby. Fast High-Dimensional Approximation withSparse Occupancy Trees. Preprint 45, DFG-SPP 1324, May 2010.

[46] J. Ballani and L. Grasedyck. A Projection Method to Solve Linear Systems inTensor Format. Preprint 46, DFG-SPP 1324, May 2010.

[47] P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk.Convergence Rates for Greedy Algorithms in Reduced Basis Methods. Preprint 47,DFG-SPP 1324, May 2010.

[48] S. Kestler and K. Urban. Adaptive Wavelet Methods on Unbounded Domains.Preprint 48, DFG-SPP 1324, June 2010.

[49] H. Yserentant. The Mixed Regularity of Electronic Wave Functions Multiplied byExplicit Correlation Factors. Preprint 49, DFG-SPP 1324, June 2010.

[50] H. Yserentant. On the Complexity of the Electronic Schrodinger Equation.Preprint 50, DFG-SPP 1324, June 2010.

[51] M. Guillemard and A. Iske. Curvature Analysis of Frequency Modulated Manifoldsin Dimensionality Reduction. Preprint 51, DFG-SPP 1324, June 2010.

[52] E. Herrholz and G. Teschke. Compressive Sensing Principles and Iterative SparseRecovery for Inverse and Ill-Posed Problems. Preprint 52, DFG-SPP 1324, July2010.

[53] L. Kammerer, S. Kunis, and D. Potts. Interpolation Lattices for Hyperbolic CrossTrigonometric Polynomials. Preprint 53, DFG-SPP 1324, July 2010.

[54] G. Kutyniok and W.-Q Lim. Shearlets on Bounded Domains. Preprint 54, DFG-SPP 1324, July 2010.

[55] A. Zeiser. Wavelet Approximation in Weighted Sobolev Spaces of Mixed Orderwith Applications to the Electronic Schrodinger Equation. Preprint 55, DFG-SPP1324, July 2010.

[56] G. Kutyniok, J. Lemvig, and W.-Q Lim. Compactly Supported Shearlets.Preprint 56, DFG-SPP 1324, July 2010.

[57] A. Zeiser. On the Optimality of the Inexact Inverse Iteration Coupled with AdaptiveFinite Element Methods. Preprint 57, DFG-SPP 1324, July 2010.

[58] S. Jokar. Sparse Recovery and Kronecker Products. Preprint 58, DFG-SPP 1324,August 2010.

[59] T. Aboiyar, E. H. Georgoulis, and A. Iske. Adaptive ADER Methods Using Kernel-Based Polyharmonic Spline WENO Reconstruction. Preprint 59, DFG-SPP 1324,August 2010.

[60] O. G. Ernst, A. Mugler, H.-J. Starkloff, and E. Ullmann. On the Convergence ofGeneralized Polynomial Chaos Expansions. Preprint 60, DFG-SPP 1324, August2010.

[61] S. Holtz, T. Rohwedder, and R. Schneider. On Manifolds of Tensors of FixedTT-Rank. Preprint 61, DFG-SPP 1324, September 2010.

[62] J. Ballani, L. Grasedyck, and M. Kluge. Black Box Approximation of Tensors inHierarchical Tucker Format. Preprint 62, DFG-SPP 1324, October 2010.

[63] M. Hansen. On Tensor Products of Quasi-Banach Spaces. Preprint 63, DFG-SPP1324, October 2010.

[64] S. Dahlke, G. Steidl, and G. Teschke. Shearlet Coorbit Spaces: Compactly Sup-ported Analyzing Shearlets, Traces and Embeddings. Preprint 64, DFG-SPP 1324,October 2010.

[65] W. Hackbusch. Tensorisation of Vectors and their Efficient Convolution.Preprint 65, DFG-SPP 1324, November 2010.

[66] P. A. Cioica, S. Dahlke, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, and R. L.Schilling. Spatial Besov Regularity for Stochastic Partial Differential Equations onLipschitz Domains. Preprint 66, DFG-SPP 1324, November 2010.

[67] E. Novak and H. Wozniakowski. On the Power of Function Values for the Ap-proximation Problem in Various Settings. Preprint 67, DFG-SPP 1324, November2010.

[68] A. Hinrichs, E. Novak, and H. Wozniakowski. The Curse of Dimensionality forMonotone and Convex Functions of Many Variables. Preprint 68, DFG-SPP 1324,November 2010.

[69] G. Kutyniok and W.-Q Lim. Image Separation Using Shearlets. Preprint 69, DFG-SPP 1324, November 2010.

[70] B. Jin and P. Maass. An Analysis of Electrical Impedance Tomography with Ap-plications to Tikhonov Regularization. Preprint 70, DFG-SPP 1324, December2010.

[71] S. Holtz, T. Rohwedder, and R. Schneider. The Alternating Linear Scheme forTensor Optimisation in the TT Format. Preprint 71, DFG-SPP 1324, December2010.

[72] T. Muller-Gronbach and K. Ritter. A Local Refinement Strategy for ConstructiveQuantization of Scalar SDEs. Preprint 72, DFG-SPP 1324, December 2010.

[73] T. Rohwedder and R. Schneider. An Analysis for the DIIS Acceleration Methodused in Quantum Chemistry Calculations. Preprint 73, DFG-SPP 1324, December2010.

[74] C. Bender and J. Steiner. Least-Squares Monte Carlo for Backward SDEs.Preprint 74, DFG-SPP 1324, December 2010.

[75] C. Bender. Primal and Dual Pricing of Multiple Exercise Options in ContinuousTime. Preprint 75, DFG-SPP 1324, December 2010.

[76] H. Harbrecht, M. Peters, and R. Schneider. On the Low-rank Approximation by thePivoted Cholesky Decomposition. Preprint 76, DFG-SPP 1324, December 2010.

[77] P. A. Cioica, S. Dahlke, N. Dohring, S. Kinzel, F. Lindner, T. Raasch, K. Ritter,and R. L. Schilling. Adaptive Wavelet Methods for Elliptic Stochastic PartialDifferential Equations. Preprint 77, DFG-SPP 1324, January 2011.

[78] G. Plonka, S. Tenorth, and A. Iske. Optimal Representation of Piecewise HolderSmooth Bivariate Functions by the Easy Path Wavelet Transform. Preprint 78,DFG-SPP 1324, January 2011.