Applied and Computational Harmonic Analysis

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1 Appl. Comput. Harmon. Anal. 7 9) 5 34 Contents lists available at ScienceDirect Applied and Computational Harmonic Analysis Directional Haar wavelet frames on triangles Jens Krommweh Gerlind Plonka Department of Mathematics University of Duisburg-Essen Campus Duisburg 4748 Duisburg Germany article info abstract Article history: Received 6 September 7 Revised 8 March 9 Accepted 9 March 9 Available online 6 March 9 Communicated by Peter Oswald Keywords: Haar wavelet frames Non-separable wavelets Composite dilation wavelets Dual frames Sparse representation Image denoising raditional wavelets are not very effective in dealing with images that contain orientated discontinuities edges). o achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. In recent years several approaches like curvelets and shearlets have been studied providing essentially optimal approximation properties for images that are piecewise smooth and have discontinuities along C -curves. While curvelets and shearlets have compact support in frequency domain we construct directional wavelet frames generated by functions with compact support in time domain. Our Haar wavelet constructions can be seen as special composite dilation wavelets being based on a generalized multiresolution analysis MRA) associated with a dilation matrix and a finite collection of shear matrices. he complete system of constructed wavelet functions forms a Parseval frame. Based on this MRA structure we provide an efficient filter bank algorithm. he freedom obtained by the redundancy of the applied Haar functions will be used for an efficient sparse representation of piecewise constant images as well as for image denoising. 9 Elsevier Inc. All rights reserved.. Introduction Over the past few years there has been a great interest in improved methods for sparse representations of higher dimensional data sets. Multiscale methods based on wavelets have been shown to provide successful schemes for data compression and denoising. Indeed wavelets are optimally efficient in representing functions with point singularities [3]. In addition the multiresolution analysis MRA) associated with wavelets results in fast algorithms for computing the wavelet coefficients [93]. However due to the missing rotation invariance of tensor product wavelets the wavelet representation of D-functions is not longer optimal. herefore in recent years several attempts for improvement of wavelet systems in higher dimensions have been made including complex wavelets [7] contourlets [4] brushlets [8] curvelets [45] bandelets [] shearlets [45] and directionlets [7]. Curvelets [45] and shearlets [45] are examples of non-adaptive highly redundant function frames with strong anisotropic directional selectivity. For piecewise Hölder continuous functions of order with discontinuities along C -curves Candès and Donoho [5] proved that a best approximation f M of a given function f with M curvelets satisfies f f M CM log M) 3 * Corresponding author. addresses: jens.krommweh@uni-due.de J. Krommweh) gerlind.plonka@uni-due.de G. Plonka) /$ see front matter 9 Elsevier Inc. All rights reserved. doi:.6/j.acha.9.3.

2 6 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) 5 34 while a tensor product) wavelet expansion only leads to an approximation error OM ) [3]. Up to the log M) 3 factor this curvelet approximation result is asymptotically optimal. A similar estimation has been achieved by Guo and Labate [4] for shearlet frames. Instead of choosing a priori a basis or a frame to approximate f one can rather adapt the approximation scheme to the image geometry. For example one can construct an approximation f M which is piecewise linear over an optimized triangulation including M triangles and satisfies f f M CM. his requires adapting the triangulation to the edge geometry see e.g. []). In [] bandelet orthogonal bases and frames are introduced that adapt the geometric regularity of the image. Further we want to mention the nonlinear edge adapted multiscale decompositions based on ENO schemes in [7] and the multidirectional edge adapted compression algorithm [] based on an edge detection procedure. In this paper we are especially interested in non-adaptive directional wavelet frames being compactly supported in time domain. he curvelet and shearlet systems constructed so far are tight frames of well-localized functions at different scales positions and directions. he corresponding generating functions have compact support on triangles parabolic wedges or sheared wedges in frequency domain. In particular for curvelet frames there is no underlying multiresolution analysis supporting the efficient computation of curvelet representations. hese circumstances can be seen as a certain drawback for the application of efficient wavelet filter banks based on these frames. herefore we are strongly interested in directional wavelet frames generated by functions with compact support in time domain and providing an efficient algorithm based on the MRA structure. he construction of directional Haar wavelet frames on triangles introduced in this paper is a first attempt in this direction. A different approach can also be found in [8]. Further constructions of piecewise constant wavelets in L R n ) with n for other purposes are due to [365]. As in [8] our Haar wavelet constructions can be seen as special composite dilation wavelets see [5]) being based on a generalized MRA associated with a dilation matrix and a collection of shear matrices. But in contrast to [8] we shall use the dilation matrix A = I leading to simple decomposition and reconstruction formulas. Since we use a finite collection of shear matrices the considered MRA can also be understood as generated by a refinable function vector and the corresponding wavelet functions form a multiwavelet vector. Our results presented in this paper go far beyond the ideas given in [8] where Haar wavelet frames based on the quincunx matrix as dilation matrix are constructed but without any consideration of frame properties redundancy and applications. Furthermore we achieve a higher directional sensitivity considering eight directions instead of four which produces an essential improvement in applications. Due to their support our scaling functions and wavelets are able to detect different directions. he complete system of constructed wavelet functions forms a Parseval frame. he freedom obtained by the redundancy of the applied Haar functions will be used for an efficient sparse representation of piecewise constant) images as well as for image denoising. In order to find a sparse representation of the image we will apply a minimization of the vector of wavelet coefficients in the l -seminorm in each decomposition step. he considered l -minimization problem is related to the construction of M-term approximations of functions in a redundant dictionary by greedy algorithms see e.g. [66]). We shall present a new and simple but efficient algorithm for finding frame representations with small l -seminorm that uses the known dependence relations in the frame and provides an optimal solution for piecewise constant images. he piecewise constant directional Haar wavelet frame presented in this paper possesses some limitations yet. On the one hand a construction of directional wavelet systems with small support in time domain and with higher smoothness is desirable. On the other hand for image analysis one wishes to have as many different directions as possible. Unfortunately this desire conflicts with a small redundancy of the wavelet system. We will discuss these issues and possible extensions of our approach in Section 7. he paper is organized as follows. In Section we introduce the space of scaling functions with compact support on triangles. Further we present the canonical dual frames for the scaling spaces V j. Section 3 is devoted to the construction of the directional Haar wavelet frame and corresponding decomposition and reconstruction formulas. In Section 4 we present the directional Haar wavelet filter bank based on the new wavelet frames on triangles. Further we study the question of how to find a suitable orthogonal projection of a given digital image into the scaling space V as well as the back projection after application of the filter bank algorithm. In Section 5 a new algorithm for sparse representation of images is presented where we apply an l -minimization to the wavelet coefficients of the constructed frames. Finally Section 6 is devoted to the application of the redundant directional Haar wavelet filter bank to image denoising and sparse image approximation. In particular we shall compare its performance with curvelets and contourlets.. he space of scaling functions We consider the domain Ω := [ ] and divide it into 6 triangles with the same area see Fig.. We want to introduce a vector of characteristic functions on these 6 triangles. Let the first scaling function φ be a characteristic function on the triangle { U = conv ) ) / )} { := x R : x x x }

3 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) Fig.. Construction of scaling functions a) coarsest level V b) refinements). i.e. ) x φ x) = φ x x ) = χ U x x ) = χ [] χ [] x ). x he second scaling function φ is given by ) x φ x) = φ x x ) = χ U x x ) = χ [] χ [] x ) x where U = { conv ) ) / )}. Observe that introducing the shear matrix S := / ) wehaveφ x) = φ S x). Letus apply the group B := {B i : i =...7} of isometries of the square [ ] with ) ) ) ) B = B = B = B 3 = ) ) ) ) B 4 = B 5 = B 6 = B 7 =. hen for i =...7wehave U i = { } B x: x U i = B U i U i+ = { } B x: x U i = B U i and we define the further scaling functions φ i by φ i x) := φ B i x) = χ U B i x) = χ B U i x) = χ Ui x) φ i+ x) := φ B i x) = χ U B i x) = χ B U i x) = χ Ui+ x) i =...7. In the following we consider the translated versions of φ i with support in [ ] and put them into a vector Φ of length 6 )) Φ := φ...φ 3 φ 4 ))...φ 7 ))) φ 8 ))...φ ))φ ))...φ 5..) Wedefinenowthesequenceofspaces{V j } j Z given by with V j := clos L R ) span{ φ i jk φ i+ jk : i =...7; k Z }.) φ i jk x) := j φ Bi j x )) k φ i+ jk x) := j φ Bi j x )) k i =...7 k Z i.e. j denotes the scale k the translation and i the rotation/shearing. Note that these functions can be understood as scaling functions with composite dilations see [5]). We show that {V j } j Z forms a generalized stationary MRA of L R ) that can also be interpreted as a so-called AB-MRA with A = I and B B as introduced in [5].

4 8 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) 5 34 Fig.. Redundancies of scaling functions. Lemma.. he sequence {V j } j Z of subspaces of L R ) satisfies the following properties:. V j V j+ j Z.. clos L R ) j Z V j = L R ). 3. j Z V j ={}. 4. {φ i k) φ i+ k): i =...7; k Z } forms a frame of V. Proof. First observe that the scaling functions are refinable compare Fig. b). In particular we have )) )) )) φ = φ ) + φ + φ + φ 9 = φ ) + φ ) + φ ) + φ 9 )) )) )) )) φ = φ ) + φ + φ + φ 8 = φ ) + φ ) + φ ) + φ 8 )). he two-scale relations for the other scaling functions now simply follow as as well as φ i = φ B i ) = φ B i ) + φ B i = φ i ) + φ ib i φ i+ = φ i+ ) + φ i+b i )) + φ B i )) + φ B 4 B i ) + φ i+b i ) + φ i+9) mod 6B )) i ) + φ ib i ) + φ i+8) mod 6B )). i ))) Hence V V holds. For j Z and k Z we get the general refinement equations φ i jk = φ Bi j k )) = φ i j+k + φ i j+k+b i ) + φ i+ j+k+b i ) + φ i+9) mod 6 j+k+b )) i φ i+ jk = φ )) Bi j k = φ i+ j+k + φ i+ j+k+b i ) + φ i j+k+b i ) + φ i+8) mod 6 j+k+b )). i hus we have V j V j+ for all j Z. Secondly since the spaces V j defined in.) contain the subspaces of Haar scaling functions i.e. V H V j j with V H j := clos L R ) span{ j χ [) ) j k : k Z } we find clos L R ) j Z V j = L R ). Further the condition j Z V j ={} easily follows for a stationary sequence {V j } j Z see e.g. [3 Corollary 4.4]). Now the last property remains to proved. he family of functions {φ i k) φ i+ k): i =...7; k Z } does not generate a basis of V. Obviously we have the following dependencies see Fig. ):

5 φ + φ = φ φ + φ 3 = φ 8 φ 4 + φ 5 = φ 4 φ 6 + φ 7 = φ φ + φ + φ + φ 3 = φ 4 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) )) + φ )) + φ 9 )) + φ 5 )) + φ 3 )) )) )) + φ 5 )) )) )) + φ 6 )) + φ 7 ))..3) Indeed the space V is already generated by the set of functions {φ i : i =...7} {φ φ 3 φ 5 }. he Gram matrix G := ΦΦ R 6 6 with Φ in.) is given by I 4 G G G G = G I 4 G G 4 G G.4) I 4 G G G G I 4 with the identity matrix I 4 of size 4 4 and with /5 /5 / /6 /3 /3 7/5 /5 /6 /6 /3 /3 G = G /3 /5 7/5 =. /3 /3 /3 /3 /5 /5 /3 /3 We observe that rankg) =. he nonzero eigenvalues of G provide us with the frame constants of {φ i k): i =...5; k Z }; i.e. the inequality 5 A f L R ) f φ i k) B f L R ) i= k Z is satisfied for all f V with A.745 and B =. Now we look for a dual frame { φ i k): i =...5; k Z } of V such that f = 5 f φ i k) φ i k) = i= k Z 5 i= k Z f φi k) φ i k) f V..5) he dual frame Φ of Φ can be computed by Φ = G Φ.6) where G is the well-defined Moore Penrose generalized inverse of the Gram matrix G. Indeed the first part of.5) can be seen as follows. Since G is symmetric and positive semidefinite there exists an orthogonal matrix P R 6 6 and a diagonal matrix D = diagλ...λ...) R 6 6 such that G = P DP. Hence the pseudo-inverse G is given by G = P D P where D = diag/λ.../λ...). InparticularV is also generated by PΦ where PΦ PΦ =P ΦΦ P = PGP = D i.e. the last five functions in the vector PΦ are zero functions. Now for an arbitrary function g = c PΦ V restricted to [ ] ) it follows that 5 i= g φ i φ i = c PΦ G Φ Φ = c P ΦΦ G Φ = c PGG Φ = c DD PΦ = c PΦ = g. he second part of.5) follows similarly. he pseudo-inverse G has again block structure Ĝ Ĝ Ĝ Ĝ G Ĝ Ĝ Ĝ Ĝ = Ĝ Ĝ Ĝ Ĝ Ĝ Ĝ Ĝ Ĝ

6 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) 5 34 Fig. 3. Construction of directional wavelets ψ ψ andψ 3. where Ĝ = Ĝ = and Ĝ = Here we computed G with the common Maple procedure and its rational entries are rounded to two digits.) 3. Construction of a tight directional wavelet frame and reconstruction formulas Let us now consider the wavelet spaces W j satisfying the condition V j + W j = V j+ for all j Z. he locality and refinability of generating functions φ i i = imply to consider the wavelet functions for φ ψ := φ ) + φ ) φ ) φ 9 )) ψ := φ ) φ ) φ ) + φ 9 )) ψ 3 := φ ) φ ) + φ ) φ 9 )) see Fig. 3 and for φ ψ := φ ) φ ) + φ ) φ 8 )) ψ := φ ) φ ) φ ) + φ 8 )) ψ 3 := φ ) + φ ) φ ) φ 8 )). he wavelet functions ψ i ψ i ψ 3 i have the same support as φ i i =. All further wavelet functions can be obtained by rotation/reflection of these six functions namely ψ r i := ψr B i ) and ψ r i+ := ψr B i ) for i =...7 r = 3. Now we are able to define the wavelet spaces W j := clos L R ) span{ ψi r jk : i =...5; r = 3; k Z} where ψi r jk := j ψ r B i j k)) ψi+ r jk := j ψ r B i j k)). he above refinement equations for ψ r and ψr r = 3) directly imply the relations ψi jk = φ i j+k + φ i j+k+b i ψi+ jk = φ i+ j+k φ i+ j+k+b i ) φ i+ j+k+b i ) φ i+9) mod 6 j+k+b i )) ) + φ i j+k+b ) φ i+8) mod 6 j+k+b )) 3.) i i

7 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) 5 34 and analogous relations for ψi jk and ψ i 3 jk.huswehavew j V j+ for all j Z. Obviously the wavelet functions ψi r jk r = 3 possess the same compact support as the corresponding scaling functions φ i jk for all i j k. Reconstruction formulas can now be derived as follows see also Fig. 3) as well as φ j+k = φ jk + ψ jk + ψ jk + ψ jk) 3 φ j+k+ ) = φ jk + ψ jk ψ jk ψ ) 3 jk φ j+k+ ) = φ9 jk+ ) ψ + ψ ψ ) 3 9 jk+ ) 9 jk+ ) 9 jk+ ) φ j+k+ ) = φ jk + ψ jk ψ jk ψ ) 3 jk φ j+k = φ jk + ψ jk + ψ jk + ψ jk) 3 φ j+k+ ) = φ8 jk+ ) ψ + ψ ψ ) 3 8 jk+ ) 8 jk+ ) 8 jk+ ) φ j+k+ ) = φ jk ψ jk ψ jk + ψ ) 3 jk φ j+k+ ) = φ jk ψ jk ψ jk + ψ ) 3 jk. he reconstruction formulas for the rotated and reflected functions follow accordingly. Hence we indeed have V j + W j = V j+. Now we can prove the essential tight frame property of the system Ψ D := { ψi r jk : i =...5; r = 3; j Z; k Z} generating L R ). heorem 3.. he directional Haar wavelet system Ψ D forms a Parseval frame of L R ) i.e. f L R ) = f ψ f L R ). ψ Ψ D Proof. Firstly we consider the following subspaces V j V j V j and V 3 j of V j given by V ν j := clos L R ) span{ φ ν jk φ ν+ jk φ ν+8 jk φ ν+9 jk : k Z } ν = 3. 3.) From the observations in Section it follows that the sequences {V ν j } j Z themselves already form a multiresolution of L R ) and moreover the generating functions form an orthogonal basis of V ν j where φ i jk L R ) = 4 i =...5; j Z; k Z. Now taking the subspaces W j W j W j and W 3 j of W j in the same manner i.e. W ν j := clos L R ) span{ ψν r jk ψr ν+ jk ψr ν+8 jk ψr ν+9 jk : r = 3; k Z} for arbitrary for ν = 3 we find that W ν j V ν j and for each ν = 3 this generating system is even an orthogonal basis of W ν j. Hence each of the function sets Ψ ν := { ψν r jk ψr ν+ jk ψr ν+8 jk ψr ν+9 jk : r = 3; j Z; k Z} ν = 3 forms an orthogonal basis of L R ) and the Parseval identity implies f L R ) = f ψ = 4 f ψ ψ Ψ ν ψψ ψ Ψ ν for all f L R ) ν = 3 such that the complete system Ψ D forms a tight frame of L R ) with f L R ) = f ψ. ψ Ψ D Remark 3... An alternative proof of the tight frame property is given by one of the authors in [9] with arguments in the frequency domain.

8 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) Note the important fact that the directional wavelet frame consists of four orthogonal bases i.e. it can be interpreted as a redundant dictionary. In our applications we will exploit this fact in order to get an efficient implementation. 4. Directional Haar wavelet filter bank Let J := {...N } {...N } be the index set of a digital image a = a k ) k J function f L Ω) with Ω =[ N ] [ N ] with N N pixels. he f x x ) = N k = N k = a k k χ [) x k x k ) = k J a k χ [) x k) 4.) can be seen as the corresponding L -version of the discrete image a. Here χ [) denotes the characteristic function on [ ) and we assume that N = n j N = n j with some n n N and a fixed j N. We want to apply our redundant Haar wavelet frame constructed above for an efficient analysis of f. Let us shortly describe the procedure before going into a detailed analysis of the single steps. First we compute an orthogonal projection f resp. f j )of f into the space V defined in.) or into a coarser space V j with j < see Section 4.). hen we apply the directional Haar wavelet filter bank generated by the decomposition and reconstruction formulas for φ i ψ i ψ i ψ3 i i =...5 in order to decompose f j into f j V j and g j W j as usual. Using the fact that our constructed frame can be split into four bases the decomposition can be done by a fourfold application of the fast wavelet transform FW) see Section 4.. If we use the directional wavelet frame for image denoising we do not reduce the redundancies because redundant information is desirable with denoising. By contrast if we apply the filter bank algorithm to find a sparse image representation we have to reduce redundancies. his issue will be considered in detail in Section Orthogonal projection of f into V In order to apply the directional Haar wavelet frames constructed in the preceding sections we need a suitable projection f of a given function f of the form 4.) into the scaling space V defined in.). For this projection we require two conditions. Firstly the redundancy introduced by this projection i.e. the ratio between the number of coefficients determining f and the 4N N coefficients determining f should be as small as possible. Secondly there should be no loss of information i.e. we desire that f can be perfectly reconstructed from f. We are interested in the orthogonal projection of f in 4.) into the space V of the form f = k J c k) Φ k) 4.) where J := {...N } {...N } c k = c k...c 5k ) R 6 and where the support of all functions in Φ is contained in [ ] see.)). Since the basis functions χ [) x k) in 4.) as well as the scaling functions in Φ k) have small compact support we can look at the projection problem locally. We restrict ourselves to the case k = and consider the area [ ). Hence we need a projection of )) )) )) f [) x) = a )χ [) x) + a )χ [) x + a )χ [) x + a )χ [) x to f [) x) = c ) Φx). Obviously such a projection provides the redundancy factor 4. Using the dual canonical frame Φ = G Φ defined in.6) the coefficient vector c R6 is now given by c = Φ f =G Φ f = a )G Φχ [) ) + a )G + a )G Φχ [) Φχ [) )) + a )G )) Φχ [) )). he vectors Φχ[) l) = ) y + l Φ dy 4 [) l { ) ) ) )} in R 6 can now easily be computed and we find

9 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) Fig. 4. Computation of the coefficients c k c k c 8k c from four image pixels. 9k )) Φχ [) )) Φχ[) M := ) Φχ [) Φχ [) 3 3 = Hence the coefficient vector c is found by c = G Ma ) a ) a ) a. )) Generally the coefficient vectors c k in 4.) are obtained for all k J by. )) ) c k = G Ma k a k+ ) a k+ ) a k+. )) Next we will show that this projection f in 4.) contains the full information of f. In other words f can be perfectly reconstructed from f. his can be seen as follows. Consider the subspace V of V containing only the Z -translates of φ φ φ 8 and φ 9 we obtain an orthogonal projection of f in 4.) into the subspace V by f = k J c k φ k + c k φ k + c 8k φ 8k+ ) + c 9k φ 9k+ ) where with the same arguments as above c k a 3 k c k c = a k+ ) 4 3 a 8k k+ c ). 4.3) a k+ 9k ) Here the coefficient matrix contains the th st 8th and 9th rows of M because these rows of M provide the coefficients of φ φ φ 8 ) and φ 9 ) see Fig. 4. Further since φ φ φ 8 ) φ 9 are orthogonal the corresponding Gramian ) matrix has the form 4 I 4. Since the coefficient matrix in 4.3) is invertible we can reconstruct f from f.butv is a subspace of V and in particular it follows that f is also found as the orthogonal projection of f into V.Hencetaking ) Φ k) φrk = c c rk = f φ rk = c k ) Φ k) φr+8k+ ) k) gr r = = c k) gr+8 r = c r+8k = f φ r+8k+ ) = c k where g r is the rth column vector of the Gramian matrix G we obtain for all k J c k c k c = g g g 8 g 9 ) c k = G c k 8k c 9k where G R 6 4 contains the th st 8th and 9th columns of G and with 4.3) we have the reconstruction formula a k a k+ ) a k+ ) = 4 a k+ ) G c k.

10 4 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) 5 34 Fig. 5. Order of components in the function vectors Φ ν for ν = left)ν = middle left) ν = middle right) and ν = 3 right). he figures indicate the supports of the ith component of Φ ν for i = Directional wavelet filter bank algorithm Let now f V be given as in 4.). We want to derive an efficient algorithm for the decomposition of f into f V and g W and for the reconstruction f + g.with J := {... N } {... N } we can write f = ))) ) c ) k Φ k) + c Φ k+ k ) k + J + ))) ) c Φ k+ ) k + + ))) c k+ ) Φ ) k +. Again we derive the decomposition of f locally. On [ ) there are 64 basis functions of V namely the 6 components of Φ as given in.)) with a support inside [ ] and the components of Φ )) Φ )) andφ )).Forthe filter bank algorithm we want to apply the knowledge that V j is composed by the subspaces V j V j V j and V 3 as given j in the proof of heorem 3. see 3.)) i.e. V j = V j + V j + V j + V 3 j. herefore we reorder the frame functions in Φ Φ )) Φ )) Φ )) ) according to the subspaces V ν ν = 3. In [ ) we hence consider the function vectors Φ := φ )φ 9 )φ )φ )φ )φ 8 )φ )φ ) ) φ 8 )φ )φ 8 )φ 9 )φ 9 )φ )φ 8 )φ 9 ) as well as Φ ν := Φ B ν ) ν = 3. he order of functions in Φ ν is taken for simplifying the refinement relations and the application of the corresponding wavelet filter bank. See Fig. 5 for the new order of frame functions in the four directions related to the four subspaces V ν. By corresponding reordering of the coefficients of f in the coefficient vectors c k+l l { ) ) ) )} weobtain f [) = ) d Φ + ) d Φ + ) d Φ + ) d 3 Φ 3. Since Φ ν contains the 6 frame functions in [ ) that correspond only to the direction ν we can separately apply the decomposition formulas 3.). For every ν = 3 we find with and the relation Ψ := φ )ψ )ψ )ψ3 )φ )ψ )ψ )ψ3 ) ) φ 8 )ψ 8 )ψ 8 )ψ3 8 )φ 9 )ψ 9 )ψ 9 )ψ3 9 ) Ψ ν := Ψ B ν ) ν = 3 Ψ ν = AΦν ν = 3 where the orthogonal matrix A R 6 6 is a tensor product matrix of the form B B A = I 4 B) = with B := B. B

11 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) Observe that Ψ ν generates the directional subspace V ν W ν ν = 3. Now we denote with Φ := ) Φ Φ Φ Φ3 4.4) the reordered) vector of functions generating the frame in V restricted to [ ) ) and with Ψ := Ψ Ψ Ψ Ψ3 ) the generating frame in V W restricted to [ ) ). Both Φ and Ψ are now function vectors of length 64. Using A = A = A the representation of f with generating functions from V resp. V W can be given in the form f [) = d ) d ) d ) d 3 ) ) Φ = D Φ = D I 4 A)Ψ where D := d ) d ) d ) d 3 ) ) R 64. For the complete image f it follows the decomposition f = k J D k Φ k) = k J D k I 4 A)Ψ k) 4.5) with D k := d k ) d k ) d k ) d 3 k ) ) R 64. Summing up the decomposition algorithm for the constructed directional wavelets on triangles has the following form. Algorithm a Decomposition by directional Haar wavelet filter bank).. Input: Initial image obtained by orthogonal projection of f into V ) c k Φ k) c k = f Φ k). f = k J. Reorder the frame functions resp. corresponding coefficients) by directions. Let P be the permutation matrix for reordering of generating functions in V Φ Φ )) Φ )) Φ )) ) P = Φ. Now for each k J reorder the coefficient vectors D k = ) d ) k d ) k d ) k d 3 ) ) k = c ) k c ) c ) c ) P k+ ) k+ ) k+ ) such that f = k J D k Φ k). 3. Decompose f V into f V and g W using the relation Φ = I 4 A)Ψ. Compute the corresponding coefficients by D k = D k I 4 A). 4. Using the definition of Ψ reorder the coefficients in the vectors D k = d k) d k ) d k ) d 3 k ) ) k J in order to obtain f and g. 5. Iterative application: Apply the same procedure to the low pass part f while the high pass part g is stored. As previously described the decomposition algorithm of f V into f V and g W only involves some permutations and some additions/subtractions. Since the transformation matrix I 4 A) is orthogonal the algorithm is numerically stable. he reconstruction procedure then easily follows by reversing the steps of Algorithm a. 5. Sparse image representations in wavelet spaces At present a usual approach to find a sparse representation of f in a redundant dictionary is the orthogonal matching pursuit OMP) see e.g. [6] and references therein). OMP is an iterative greedy algorithm that selects at each step the dictionary element best correlated with the residual part of the signal. hen a new approximation of the signal is produced by a projection on the dictionary elements that have already been selected. Unfortunately because of the large coherence of the considered dictionary of Haar wavelet functions the OMP algorithm does not provide satisfying sparse representations of f V in our case.

12 6 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) 5 34 In order to get a sparse representation of images we need to reduce the existing fourfold) redundancy by exploiting our explicit knowledge about it. he idea is as follows. After decomposing a given image f j V j into f j V j and g j W j we aim to exploit the redundancy of the frames generating V j and W j and try to find a representation of f j and g j that contains as many zero coefficients as possible. he procedure will be applied after each decomposition step. Finally using a threshold procedure to remove remaining small frame coefficients we obtain a suitable sparse approximation of the image where due to the frame construction different directions of the image are well adapted. Again we use the decomposition V = V + V + V + V 3 and recall that V is spanned by {Φ k): k Z } where Φ is the function vector of length 64 defined in 4.4) and V ν = span{φν k): k Z } for ν = 3 see Fig. 5. Considering 4.5) we note that the representations of f in V as well as in V + W are not uniquely determined. he dependence relations.3) in V imply with unit vectors e k := δ kl ) 5 the equations l= e + e 4 ) Φ ν e 4 + e 5 ) Φ ν+ = e + e 3 ) Φ ν e + e ) Φ ν+ = e 9 + e 3 ) Φ ν e + e 5 ) Φ ν+ = e 6 + e 7 ) Φ ν e 8 + e ) Φ ν+ = e + e 4 ) Φ ν+ e 4 + e 5 ) Φ ν = e + e 3 ) Φ ν+ e + e ) Φ ν = e 9 + e 3 ) Φ ν+ e + e 5 ) Φ ν = e 6 + e 7 ) Φ ν+ e 8 + e ) Φ ν = for ν = and e + e 4 ) Φ + ) Φ e9 + e 3 ) Φ e + e 3 ) Φ 3 = e 9 + e 3 ) Φ + e + e 3 ) Φ e 6 + e 7 ) Φ + ) Φ3 = e + e 3 ) Φ + e 9 + e 3 ) Φ e + e 4 ) Φ + ) Φ3 = e 6 + e 7 ) Φ + ) Φ e + e 3 ) Φ e 9 + e 3 ) Φ 3 =. hese relations directly provide a matrix U R 64 containing these dependencies such that ) U Φ ) Φ ) Φ ) Φ 3 ) = UΦ = on [ ). 5.) Hence we obtain from 4.5) for arbitrary vectors g k R k J a redundant representation of f V of the form f = D k Φ k) = D k + g k U) Φ k) k J k J = k J D k + g k U) I 4 A)Ψ k). We aim to represent f in V + W with the smallest possible number of nonzero wavelet coefficients. Observe that because of the local supports of Φ resp. Ψ this problem can be considered separately for each k J. hus for each k J we have to determine a vector g k R such that the l -seminorm D k + g k U) I 4 A) is minimized where the l -seminorm of a vector simply counts the number of its nonzero components. his minimization leads to a large amount of vanishing wavelet frame coefficients. he modified decomposition algorithm has the following form. Algorithm b Decomposition with redundancy reduction).. Input: Initial image in V by orthogonal projection of f into V ) c k Φ k) c k = f Φ k). f = k J. Reorder the basis functions resp. corresponding coefficients) by directions see step of Algorithm a) f = k J D k Φ k).

13 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) Add redundancies in V and apply the transform to V + W : D k + g k U) Φ k) = D k + g k U) I 4 A)Ψ k). k J f = k J 4. For each k J compute g k R such that the l -seminorm D k + g k U) I 4 A) = D k + g k UI 4 A) with D k = D k I 4 A) becomes minimal. 5. For each k J let D k = D k + g k UI 4 A) be this minimized coefficient vector where g k := arg min g k R D k + g k UI 4 A). Compute the sparse representation f = k J D k Ψ k) and determine f V and g W from this representation. 6. Iterative application: Apply the same procedure to the low pass part f while the high pass part g is stored. Let us now focus on the local minimization problem in step 4 arg min g k R { D k + I 4 A)U g k } = arg min g k R { D k + Rg k } 5.) that has to be solved for each k J and where D k R 64 as well as the matrix R := I 4 A)U R 64 are given. Observe that R has full rank. A naive approach to the problem is to consider all possibilities to take linearly independent rows of R to build a matrix C k R and to solve the system C k g k = D k ) q where the vector D k ) q R is a subvector of D k obtained by taking the components of D k that correspond to the rows of R generating C k. hen the vector D k + Rg k = D k RC D k k ) q contains at least zeros. All vectors D k + Rg k obtained in this manner need to be compared with respect to their l -seminorm. Obviously such a procedure is inefficient for our purposes. Unfortunately the idea of replacing the l -seminorm by the l -norm does not work in our case. For example for constant parts of the image i.e. f = c on [ ) the coefficient vectors in the two representations and f = c ) Φ = c w 6 6 6) Ψ f = c ) 6 Φ = c w w w ) w Ψ 4 of f have the same l -norm while their l -seminorm strongly differs. Here 6 :=...) R 6 6 denotes the zero vector of length 6 and w := ) R ) We are interested in a simple algorithm to find a nearly optimal local representation of the image in the sense of 5.) that also uses our explicit knowledge about the special properties of the redundant function system and especially on the dependence relations of the system collected in the matrix U. herefore we propose a new method that can be proved to be optimal for piecewise constant images. Further we will show that the sparse representation found by our algorithm does not depend on the representation of the signal to start with. he algorithm is based on the following idea. We consider the local orthogonal projections of the image f into the subspaces V ν W ν ν = 3 that represent four different directions see Fig. 5. For a fixed k = k k ) J we recall that for each ν { 3} the generating functions in Ψ ν k) form an orthogonal basis of V ν W ν restricted to Q k := [k k + ) [k k + )). We consider the 64 coefficients obtained altogether in the four projections and select the smallest coefficients first. hose functions in our frame being connected with the smallest coefficients are not so important for the representation of f and will be pushed to be zero by using a suitable vector g k. he complete algorithm is described as follows. It needs to be applied for all k J.

14 8 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) 5 34 Algorithm Sparse local representation in V + W ). Input: D k R 64 as in Algorithm b such that f Q k = D k Φ k) and R := I 4 A)U R 64.. Compute the local orthogonal projections of f Q k = D k Φ k) into the subspaces V ν W ν ν = 3). Start with the ansatz f ν := hν k ) Ψ ν k) in Q k for the local orthogonal projection of f into V ν W ν.he orthogonality of basis functions in Ψ ν implies h ν k := 4 Ψ ν k) f R 6. Introducing the vector h k := h k ) h k ) h k ) h 3 k ) ) R 64 weobtain h k = 4 Ψ k) f = 4 I4 A)Φ k) D k Φ k) = 4I 4 A) Φ Φ D k. Let again P be the permutation matrix for reordering of generating functions in V )) )) )) ) Φ Φ Φ Φ P = Φ then with the Gram matrix G of Φ givenin.4)weobtain Φ Φ =P I 4 G)P.hus h k = 4I 4 A)P I 4 G)P D k. he vector h k now contains all local coefficients of the four orthogonal projections of f into V ν W ν ν = 3).. Arrange the components of h k R 64 from lowest absolute value to highest value and compute the corresponding permutation p...p 64 ) of indices...64). Ifsomevaluesinh k have the same absolute value then take that with the smallest index first. 3. Compute an invertible matrix C k R by choosing rows of R as follows. a) he first row of C k is the p th row of R. b) he second row of C k is the p th row of R if it is linearly independent of the p th row of R. Otherwise consider the p 3 th row of R etc. In general proceed as follows for i =...:ifthep i th row of R is linearly independent of the rows being already chosen in C k then take this row as a further row of C k. Otherwise go further to the p i+ th one. c) Stop this procedure if linearly independent rows of R are found and C k is completely determined. Since rankr) = the procedure comes to an end. 4. Let q = q...q ) be the vector of indices of rows from R taken in C k.solvethelinearsystem C k g k = D k ) q where D k ) q contains the components with indices q...q of D k = I 4 A)D k in this order. With the resulting vector g k we determine the desired sparse coefficient vector D k = D k + g k R and find the new sparse local representation D k Ψ of f in V + W. Step 4 of Algorithm implies that in the new representation of f V given by D k at least wavelet coefficients vanish namely those corresponding to the indices q...q ). Finally we show two important properties of the proposed algorithm. Lemma 5.. he sparse local representation of f V in V + W obtained by Algorithm is uniquely determined i.e. it does not depend on the initial redundant representation of f in V. Proof. Since the components of Ψ ν form a basis of V ν W ν for each ν = 3 we observe that the local projections of f into V ν W ν are uniquely determined and do not depend on the initial representation of f in V.Hencethe matrix C k computed in step 3 of Algorithm is uniquely determined too. aking the parameter vector g k = C D k k ) q as given in step 4 of Algorithm the obtained new local representation of f in V + W f = [ D k ) D k) ] q C k R Ψ contains by construction zero coefficients corresponding to the indices q...q ) i.e. the components ψ q...ψ q of the function vector Ψ = ψ μ ) 64 μ= are not longer involved in the representation of f.let f = ˆD k Ψ be a second representation of f in V + W then there exists a vector b R with ˆD k = D k + b R. Algorithm provides now the representation

15 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) f = [ ) ˆD k ˆD k ) ] q C k R Ψ = [ D k + b R [ D k ) q + ] ) ] b Rq C k R Ψ = [ D k ) D k) ] q C k R Ψ since b Rq C ) R = b C C ) R = b R by construction. Here b R k k k q R denotes the subvector of b R R 64 with components indexed by q...q ). We can show that the procedure in Algorithm provides optimal results if the function f is locally constant. Lemma 5.. Let f be constant on the square Q k =[k k + ) [k k + ) for some k = k k ) J. hen Algorithm provides an optimal representation of f in V + W. Proof. If f c on Q k with some constant c R it can be represented in V by f = c ) Φ. According to Lemma 5. we can reduce our considerations to this representation of f. We apply Algorithm and show that the resulting coefficient vector is a sparsest possible one. Computing the orthogonal projections of f into V ν W ν ν = 3 as in step of Algorithm the vector h k = cw w w w ) with w R 6 from 5.3) is obtained. Applying step of Algorithm the vector h k yields the permutation p = ). Without loss of generality let the rows of U be determined by the dependence relations given in Section 5. in the order as mentioned there. By a simple computation according to step 3 of Algorithm it can be observed that the 48 rows with indices ) of R = I 4 A)U contain exactly 5 linearly independent rows that will be used to determine the first 5 rows of C k.helast5rowsofc k are determined by the indices ) of R. Now we a apply the next step of Algorithm step 4 and we obtain with D k = ci 4 A) ) = c w 6 6 ) 6 the vector D k ) q = ce 6 + e 7 ) since only the indices q 6 = and q 7 = 9 determining the 6th and the 7th rows of C k correspond to nonzero values in D k.heree μ := δ μν ) ν= denote the unit vectors of length. Using w := ) 8 R 6 w := w w the 6th and 7th rows of the matrix RC ) have the form k w w 6 6 ) and w w 6 w ) and we obtain ) ] C R Ψ f = [ D k D k) q = c [ w = c w ) Ψ. k ) w w 6 ) 6 w w 6 )] w Ψ his is an optimal sparse representation of f. 6. Numerical results In this section we use the described algorithms for image denoising and image approximation. Both applications are based on the efficient multiscale decomposition using the proposed directional Haar wavelet filter bank and a suitable wavelet shrinkage. 6.. Image denoising We consider a Gaussian noise with standard deviation σ = that is added to the synthetic image Fig. 6a)) and to the pepper image of the same size Fig. 7a)). We apply the directional Haar wavelet filter bank algorithm Algorithm a) with a global hard-thresholding after a complete decomposition of the image. Choosing the shrinkage parameter λ = σ logn)/ where N denotes the number of pixels we obtain good denoising results see Fig. 6b)) because the directional edges of the geometrical figures are well detected. Figs. 6c) and 6d) show that our method outperforms curvelets [5] as well as contourlets [] for piecewise constant images. Even for natural images the denoising result is acceptable albeit

16 3 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) 5 34 Fig. 6. Piecewise constant image denoising. a) Noisy image PSNR.8 b) denoised image by our method PSNR 33. c) denoised image by curvelets PSNR 3.8 d) denoised image by contourlets PSNR 9.9. not excellent see Fig. 7. However it is of the same PSNR scale as with curvelets and it outperforms contourlets again. For the computation of the curvelet and the contourlet transform we have used the toolboxes from and Image approximation While for image denoising redundancy information is helpful for image approximation it is not. herefore in order to get a sparse image representation we reduce the redundancies by applying Algorithm b. By doing this the redundancy decreases from 4 to.74 i.e. for the pepper image of size we get nonzero coefficients after decomposition. he same scale of redundancy occurs with the discrete curvelet transform. here we get curvelet coefficients what corresponds with a redundancy of.83. In comparison the contourlet transform achieves a remarkable low redundancy of.3 i.e. the decomposition leads to 86 6 coefficients. Now in Figs. 8 and 9 we keep the 3 resp. 6554) largest coefficients and reconstruct the images.

17 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) Fig. 7. Pepper image denoising. a) Noisy image PSNR.8 b) denoised image by our method PSNR 8. c) denoised image by curvelets PSNR 8.4 d) denoised image by contourlets PSNR Conclusions Certain drawbacks of the existing non-adaptive directional wavelet constructions like curvelets are the global support of curvelet elements in time domain and the missing MRA structure leading to rather complex algorithms for the digital curvelet transform. herefore we desire to construct wavelet frame functions with small support in time domain simple structure based on a multiresolution analysis and leading to efficient filter bank algorithms high directionality low redundancy and suitable smoothness. he proposed directional Haar wavelet frames on triangles can be seen as a first step in this direction. However our approach has some limitations. First one would like to have continuous or smooth) frame functions instead of piecewise constants. Unfortunately the usage of box splines on a multi-directional mesh seems not to be advantageous due to their fast increasing support. his results in large corresponding filters and filter bank algorithms with high complexity. herefore we are currently investigating a multiwavelet approach using piecewise linear wavelet functions with small support on directional triangles.

18 3 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) 5 34 Fig. 8. Piecewise constant image approximation with % coefficients. a) Original image b) approximation by our method PSNR 3.8 c) approximation by curvelets PSNR 6.77 d) approximation by contourlets PSNR One may also ask for an extension of the proposed Haar wavelet filter bank to more directions using thinner support triangles. But a further splitting of the considered triangles leads to the problem that vertices of these triangles may not lie in the set j Z. One way out of this limitation of directionality may be the choice of different dilation matrices. An important question is how to relate high directionality with small redundancy of the wavelet dictionary. his problem may be only solvable by a locally adaptive choice of directional frame functions. In particular for the proposed Haar wavelet dictionary one may think about a local optimization procedure in order to activate only frame functions that correspond to certain locally important directions. his topic is also subject of further research. Acknowledgments he research in this paper is funded by the project PL 7/- of the Deutsche Forschungsgemeinschaft DFG). his is gratefully acknowledged. he authors thank the referees for the extensive comments and the helpful suggestions which have essentially improved this paper.

19 J. Krommweh G. Plonka / Appl. Comput. Harmon. Anal. 7 9) Fig. 9. Pepper image approximation with % coefficients. a) Original image b) approximation by our method PSNR 7.8 c) approximation by curvelets PSNR 5.5 d) approximation by contourlets PSNR 3.. References [] F. Arandiga J. Baccou M. Doblas J. Liandrat Image compression based on a multi-directional map-dependent algorithm Appl. Comput. Harmon. Anal. 3 7) [] F. Arandiga A. Cohen R. Donat N. Dyn B. Matei Approximation of piecewise smooth functions and images by edge-adapted ENO-EA) nonlinear multiresolution techniques Appl. Comput. Harmon. Anal. 4 8) 5 5. [3] C. de Boor R.A. DeVore A. Ron On the construction of multivariate pre)wavelets Constr. Approx ) [4] E.J. Candès D.L. Donoho Curvelets a surprisingly effective nonadaptive representation for objects with edges in: C. Rabut A. Cohen L.L. Schumaker Eds.) Curves and Surfaces Vanderbilt University Press Nashville pp. 5. [5] E.J. Candès D.L. Donoho New tight frames of curvelets and optimal representations of objects with piecewise C singularities Comm. Pure Appl. Math. 56 4) [6] S.S. Chen Basis pursuit PhD thesis Stanford University 995. [7] A. Cohen B. Matei Compact representation of images by edge adapted multiscale transforms in: Proc. IEEE Int. Conf. on Image Proc. ICIP) hessaloniki Greece pp. 8. [8] R.R. Coifman F.G. Meyer Brushlets: A tool for directional image analysis and image compression Appl. Comput. Harmon. Anal ) [9] I. Daubechies en Lectures on Wavelets SIAM Philadelphia 99. [] L. Demaret N. Dyn A. Iske Image compression by linear splines over adaptive triangulations Signal Process. 86 6)

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