DATA-DRIVEN TIGHT FRAMES OF KRONECKER TYPE AND APPLICATIONS TO SEISMIC DATA INTERPOLATION AND DENOISING

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1 DATA-DRIVEN TIGHT FRAMES OF KRONECKER TYPE AND APPLICATIONS TO SEISMIC DATA INTERPOLATION AND DENOISING LINA LIU, GERLIND PLONKA, AND JIANWEI MA Abstract. Seismic data interpolation and denoising plays a key role in seismic data processing. These problems can be understood as sparse inverse problems, where the desired data are assumed to be sparsely representable within a suitable dictionary. In this paper, we present a new data-driven tight frame of Kronecker type (KronTF) method that avoids the vectorization step and considers the multidimensional structure of data in a tensor-product way. It takes advantage of the structure contained in all different modes (dimensions) simultaneously. In order to overcome the limitations of a usual tensor-product approach we also incorporate data-driven directionality (KronTFD). The complete method is formulated as a sparsity-promoting minimization problem. It includes two main steps. In the first step, a hard thresholding algorithm is used to update the frame coefficients of the data in the dictionary. In the second step, an iterative alternating method is used to update the tight frame (dictionary) in each different mode. The dictionary that is learned in this way contains the principal components in each mode. Furthermore, we apply the proposed tight frames of Kronecker type to seismic interpolation and denoising. Examples with synthetic and real seismic data show that the proposed method achieves better results in comparison with the traditional projection onto convex sets (POCS) method based the Fourier transform and the previous vectorized data-driven tight frame (DDTF) methods. In particular, the simple structure of the new frame construction makes it essentially more efficient. Key words. Seismic data, Interpolation and denoising, Dictionary learning, Tight frame AMS subject classifications. 65T, 68U, 86A Introduction. Seismic data processing is the essential bridge connecting the seismic data acquisition and interpretation. Many processes benefit from fully sampled seismic volumes. Examples of the latter are multiple suppression, migration, amplitude versus offset analysis, and shear wave splitting analysis. However, sampling is often limited by the high cost of the acquisition, obstacles in the field, and sampled data contain missing traces. Interpolating 3D data on a regular grid, but with an irregular pattern of missing traces, is more delicate. In that case undesired attenuate artifacts can arise from improper wave field sampling. Because of its practical importance, interpolation and denoising of irregular missing seismic data have become an important topic for the seismic data-processing community. Various interpolation methods have been proposed to handle the arising aliasing effects. We especially refer to the f-x domain prediction method by Spitz [4] that is based on predictability of linear events without a priori knowledge of the directions of lateral coherence of the events. This approach can be also extended to 3D regular data. Seismic data interpolation and data denoising can also be regarded as an inverse problem where we employ sparsity constraints. Using the special structure of seismic data, we want to exploit that it can be represented sparsely in a suitable basis or frame, i.e., the original signal can be well approximated using only a small number of significant frame coefficients. In this paper, we restrict ourselves to finite-dimensional Submitted to the editors July, 6. Department of Mathematics, Harbin Institute of Technology, Harbin; Institute for Numerical and Applied Mathematics, University of Göttingen, Göttingen, Germany (liulina lx@63.com) Institute for Numerical and Applied Mathematics, University of Göttingen, Göttingen, Germany (plonka@math.uni-goettingen.de) Department of Mathematics, Harbin Institute of Technology, Harbin, China (jma@hit.edu.cn )

2 L. LIU, G. PLONKA, AND J. MA tight frames. Generalizing the idea of orthogonal transforms, a frame transform is determined by a transform matrix D C M N with M N, such that for given data y C N the vector u = Dy C N contains the frame coefficients. For tight frames, D possesses a left inverse of the form c D = c DT C N M, i.e., D D = ci, where I is the identity operator. The constant c is the frame bound that can be set to by suitable normalization. In this case the tight frame is called Parseval frame. The idea to employ sparsity of the data in a certain basis or frame has been already used in seismic data processing. A fundamental question is here the choice of the transform D. Transforms can be mainly divided into two categories: fixed-basis (basis also can be called frame) transforms and adaptive learned transforms. Transforms in the first category are data independent and include e.g. Radon transform [7, 4], Fourier transform [, 9, 3,, 6], curvelet transform [, 3, 3, ], shearlet transform [], and others. In recent years, adaptive learning transform methods came up for data processing. The corresponding transforms are data dependent and therefore usually more expensive. On the other hand they often achieve essentially better results. Transforms of adaptive learning type include principal component analysis (PCA) [9] and generalized PCA [8], the method of optimal directions (MOD) [8, 4], the K-SVD method (K-mean singular value decomposition) [], and others. Particularly, K-SVD is a very popular tool for training the dictionary. It trains a dictionary from patches of the input image. When all of the patches of the input image are used, K- SVD is actually an adaptive frame learning method in the image space, where elements of the learned adaptive frame are translations of atoms in the patch dictionary. In this case, the adaptive frame transform can be understood as a filtering with filters being the atoms the learned patch dictionary, or a concatenation of block Toeplitz matrices in matrix form. However, the filters in the K-SVD frame are highly correlated and highly redundant, and in each step each filter is updated by one SVD, which causes the demand of a large computational effort. In order to reduce the computational complexity, [6] proposed a new method, named data-driven tight frame (DDTF), which in the image space learns a tight frame with a set of orthogonal filters. Due to the orthogonality, the DDTF method is able to update the complete set of orthonormal filters by one SVD in one iteration step, in contrast to the K-SVD that employs an SVD to update each filters consecutively. Therefore, the DDTF method is significantly faster than K-SVD, while achieving similar results in terms of image quality. The DDTF method has been applied to seismic data interpolation and denoising of twodimensional and high-dimensional seismic data [8, 3]. Fixed dictionary methods enjoy the high efficiency advantage, while adaptive learning dictionary methods can take use of the information of data itself. By combining the seislets [9] and DDTF, a double sparsity method was proposed for seismic data processing [7], which can benefit from both fixed and learned dictionary methods. However, a vast majority of existing dictionary learning methods vectorize the patches and deal with vectors and therefore do not fully exploit the data structure, i.e., spatial correlation of the data pixels. To overcome this problem, Kronecker-based DDTFs has been applied to dynamic computed tomography [5, 3] treating the patches as tensors instead of vectors and applying a learned filters with tensor structures. The tensor decomposition has been also used for seismic interpolation [6]. Generalizing these ideas, in this paper we present a data-driven tight frame construction of Kronecker type (KronTF), that inherits the simple tensor-product structure of the filters to ensure fast computation also for 3D tensors. This approach is

3 DATA-DRIVEN TIGHT FRAMES OF KRONECKER TYPE further generalized by incorporating adaptive directionality (KronTFD). The separability of the learned filters makes the resulting dictionaries highly scalable. The orthonormality among filters leads to very efficient sparse coding computation, as each sub-problem encountered during the iterations for solving the resulting optimization problem has a simple closed-form solution. These two characteristics, i.e., the computational efficiency and scalability, make the proposed method very suitable for processing tensor data. The main contribution of our work includes the following aspects: ) To avoid the vectorization step in previous DDTF method, we unfold the seismic data in each mode and use one SVD of small size per iteration to learn the filters of the learned tight frame. Therefore, the learned tight frame contains structures of data sets in different modes. ) Further, we introduce a simple method to incorporate significant directions of the data structure into the filters. 3) Finally the proposed new data-driven dictionaries KronTF and KronTFD are applied to seismic data interpolation and denoising. This paper is organized as follows. We first introduce the notation borrowed from tensor algebra and definitions pertaining tensors that are used throughout our paper. In Section, we shortly recall the basic idea of construction a data-driven frame and present an iterative scheme that is based on alternating updates of the frame coefficients for sparse representation of the given training data on the one hand and of the data-dependent filters of the tight frame on the other hand. Section 3 is devoted to the construction of a new data-driven frame that is based on the tensor-product structure, and is therefore essentially cheaper to learn. The employed idea can be simply transferred to the case of 3-tensors. In order to make the construction more flexible, we extend the data-driven Kronecker frame construction by incorporating also directionality in Section 4. The two data-driven tight frames are applied to data interpolation/reconstruction and to denoising in Section 5, and the corresponding numerical tests for synthetic D and 3D data and real D and 3D data are presented in Section 6. The paper finishes with a short conclusion on the achievements in this paper and further open problems... Notations. In this paper, we denote vectors by lowercase letters and ma- ( trices by uppercase letters. For x = (x i ) N i= N ) / CN let x := i= x i be 3 the Euclidean norm, and for X = (x i,j) N,N i=,j= we denote by X CN N F = ( N ) / 4 N i= j= x i,j the Frobenius norm. 5 For a third order tensor X = (x i,j,k) N,N,N3 i=,j=,k=, we introduce the CN N N3 6 three matrix unfoldings that involve the tensor dimensions N, N and N 3 in a cyclic 7 way, ( ) X () := (x i,j, ) N,N i=,j=, (x i,j,) N,N i=,j=,..., (x i,j,n 3 ) N,N i=,j= C N NN3, ( ) X () := (x,j,k ) N,N3 j=,k=, (x,j,k) N,N3 j=,k=,..., (x N,j,k) N,N3 j=,k= C N NN3, ( ) X (3) := (x i,,k ) N3,N k=,i=, (x i,,k) N3,N k=,i=,..., (x i,n,k) N3,N k=,i= C N3 NN. The multiplication of a tensor X with a matrix is defined by employing the ν- mode product, ν =,, 3, as defined in [7]. For X C N N N3 and matrices V () = (v () i,i )M,N i =,i= CM N, V () = (v () j,j )M,N j =,j= CM N, and V (3) =

4 4 L. LIU, G. PLONKA, AND J. MA (v (3) k,k )M3,N3 k =,k= CM3 N3 we define X V () := ( N i= x i,j,k v () i,i N X V () := x i,j,k v () X 3 V (3) := j= ( N3 k= j,j x i,j,k v (3) k,k ) M,N,N 3 i =,j=,k= N,M,N 3 i=,j =,k= ) N,N,M 3 i=,j=,k = C M N N3, C N M N3, C N N M3. Generalizing the usual singular value decomposition for matrices it has been shown in [7] that every (N N N 3 )-tensor X can be written as a product X = S V () V () 3 V (3) with unitary matrices V () C N N, V () C N N, and V (3) C N3 N3 and with a sparse core tensor S C N N N3. By unfolding, this factorization can be represented as () X () = V () S () (V (3) V () ) T, X () = V () S () (V () V (3) ) T, X (3) = V (3) S (3) (V () V () ) T, where the S (ν) denote the corresponding ν-modes of the tensor S. Further, denotes the Kronecker product, see [5], which is for two matrices A = (a i,j ) M,N i=,j= CM N and B C L K defined as A B = (a i,j B) M,N i=,j= CML NK In order to compute a generalized SVD for the tensor X one may first consider the SVDs of the unfolded matrices X (), X () and X (3). Interpreting the equations in () as SVD decompositions, we obtain the ν-mode orthogonal matrices V (ν), ν =,, 3, directly from the left singular matrices of these SVDs, see [7].. Data-driven Tight frames. In this section, we describe the construction of DDTF for two-dimensional data or images, as proposed in [6, 3]. Generalizing the approach of unitary matrices, we say that a matrix D C M N represents a (Parseval) tight frame of C N, if D D = I N, i.e., D = D T is the Moore Penrose inverse of D. Obviously, each vector a C N can be represented as a linear combination of the M columns of D, i.e., there exists a vector c C M with a = D c, and for M > N, the coefficient vector is usually not longer uniquely defined. One canonical choice for c would be c = Da, since obviously D c = D Da = a is true. In frame theory, D is called synthesis operator while D is the analysis operator. Within the last time, many frame transforms have been developed for signal and image processing, as e.g. wavelet framelets, curvelet frames, shearlet frames etc. In [6] a DDTF construction has been proposed for image denoising, where the analysis operator D is composed of filtering with a set of orthogonal filters.

5 DATA-DRIVEN TIGHT FRAMES OF KRONECKER TYPE Let X be an input image. We now aim at finding a tight frame transform D under which X has a sparse representation. It is impossible to achieve this gaol without imposing any additional structure of D, as we have only one input image for training. In [6], D is assumed to be a filtering by a set of -D filters D i for i =,..., K. It is proved in [6] that D is a tight frame if the filters D i for i =,..., K are orthogonal and K = S, where the support of the filters are all S S. Then, the DDTF can be constructed by solving () K K D i X C i F +λ C i s.t. D i, D j = δ ij /K, i, j K, min D i C K K,C C M K i= i= where C i denotes either C i, the number of nonzero entries of C i, or C i, the sum of all absolute values of entries in C i. The parameter λ > balances the approximation term and the sparsity term. The DDTF construction () can be reformulated as an orthogonal dictionary learning in the patch space [3]. Let Y,..., Y K are patches of X that are vectorized to y,..., y K C N. Define D = [d, d,..., d K ], where d i is the vectorization of D i. Then () is equivalent to () min D C M N,C C M K DY C F + λ C s.t. D D = I N. Observe that in the above model not only C, the set of sparse vectors c k representing y k in the dictionary domain, but also the filters D has to be learned, where the side condition D D = I N ensures that D is a Parseval tight frame. To solve this optimization problem, we employ an alternating iterative scheme that in turns updates the coefficient matrix C and the frame matrix D, where we start with a suitable initial frame D = D. Step. For a fixed frame D C M N we have to solve the problem (3) min C C M K DY C F + λ C. We obtain for the solution C = S λ/ (DY), applied componentwisely, with the soft threshold operator { (sign x)( x λ/) for x > λ/, (4) S λ/ (x) = for x λ/. For for we obtain C = T γ (DY), applied componentwisely, with the hard threshold operator { x for x > γ, T γ (x) = for x γ, where the threshold parameter γ depends on λ and on the data DY. Step. For given C we now want to update the frame matrix D by solving the optimization problem (5) min DY C D C M N F s.t. D D = I N. Similarly as in [33] we can show

6 6 L. LIU, G. PLONKA, AND J. MA Theorem. The above optimization problem (5) is equivalent to the problem max Re (tr (DYC )) s.t. D D = I N, D C M N where tr denotes the trace of a matrix, i.e., the sum of its diagonal entries, and Re the real part of a complex number. If YC has full rank N, this optimization problem is solved by D opt = V N U, where UΛV denotes the singular value decomposition of YC C N M with the unitary matrices U C N N, V C M M, and V N C M N is the restriction of V to its first N columns. Proof. We give the short proof for the complex case that is not contained in [33]. The objective function in (5) can be rewritten as DY C F = tr(((dy) C )(DY C)) = tr(y D DY + C C C DY Y D C) = tr(y Y) + tr(c C) Re(tr(DYC )), where we have used that tr(dyc ) = tr(c DY) = tr(y D C). Thus (5) is equivalent to max D C M N Re (tr (DYC )) s.t. D D = I N. Let now the singular value decomposition of YC be given by YC = UΛV with unitary matrices U C N N, V C M M, and a diagonal matrix of (real nonnegative) singular values Λ = (diag(λ,..., λ N ), ) R N M. Choosing D opt := V N U, where V N is the restriction of V to its first N columns, we simply observe that tr(d opt YC ) = tr(v N U UΛV ) = tr(v N ΛV ) = tr(v V N Λ) = tr Λ N R, = diag(λ,..., λ N ) is the restriction of Λ to its first N columns. More- where Λ N over, (D opt ) D opt = UV N V N U = I N. We show now that the value tr Λ N is indeed maximal. Let D C M N be an arbitrary matrix satisfying D D = I N. Let U be again the unitary matrix from the SVD YC = UΛV, and let W N := DU. Then the constraint implies that WN W N = U D DU = I N. Thus tr(dyc ) = tr(w N U UΛV ) = tr(w N ΛV ) = tr(v W N Λ). Since V is unitary and since W N contains N orthonormal columns of length M also V W N consists of N orthonormal columns. Particularly, each diagonal entry of V W N has modulus smaller than or equal to. Thus, Re (tr(v W N Λ)) trλ N, 6 and equality only occurs if W N = V N, i.e., if D = D opt.

7 DATA-DRIVEN TIGHT FRAMES OF KRONECKER TYPE Data-driven Kronecker tight frame. In the patch space, the DDTF transforms the patches in C n n to vectors of length C N with N = n n. However, the vectorization ignores the spatial correlation of the patch pixels hence not fully exploit the data structure. Moreover, the number of filters becomes huge as the dimension increases. To overcome these, we present the data-driven Kronecker tight frame, where each -D filter is a Kronecker product (tensor product) of -D vectors. Similar to DDTF construction which is equivalent to finding an orthogonal transform in the patch space, the data-driven Kronecker tight frame construction can be reformulated as a tensor-product structured orthogonal transform learning from the patches. We omit the details. A usual (tensor-product) two-dimensional orthogonal transform for a given patch Y C n n is of the form (6) C = D Y D T, where D C m n, D C m n are orthogonal matrices. Many existing orthogonal transform takes this structure, such as DCT, DFT, and discrete wavelet transform. Employing the properties of the Kronecker tensor product, we obtain by vectorization of (6) c := vec(c) = vec(d Y D T ) = (D D ) vec(y ) = (D D ) y, where the operator vec reshapes the matrix into a vector, and y := vec(y ). Without vectorizing the patches, and by imposing the tensor-product structure of the orthogonal transform, Eq. () in Section is revised to (7) min D,D,C,...,C K k= K ( D Y k D T C k ) F + λ C k s.t. D νd ν = I nν, ν =,. where Y,..., Y K C n n are patches of the input image X, and C,..., C K C m m are the corresponding coefficient matrices. Equivalently, using the notations y k := vec(y k ) C n n, c k := vec(c k ) C m m as well as Y := (y,..., y K ) C nn K and C := (c,..., c K ) C mm K, we can write the model in the form (8) min D,D,C (D D )Y C F + λ C s.t. D νd ν = I nν, ν =,. Here, as before denotes either or. Due to the orthogonality, Proposition 3 in [6] implies that the filtering with filters being columns of D D forms a tight frame transform in the image space. Observe that, compared to (), the matrix (D D ) C mm nn of filters has now a Kronecker structure that we did not impose before. Thus, learning the filter matrix (D D ) requires to determine only n m + n m instead of n n m m components. Surely, we need to capture the important structures of the images in a sparse manner. In order to solve the minimization problem (7) resp. (8), we adopt the alternating minimization scheme proposed in the last section. Now, each iteration step consists of three independent steps: Step. First, for fixed matrices D, D, we minimize only with respect to C by applying either a hard or a soft threshold analogously as in Step for the model (). Step. We fix C and D, and minimize (7) resp. (8) with respect to D. For this purpose, we rewrite (7) as (9) min D C m n K D Y k D T C k F s.t. DD = I n. k=

8 8 L. LIU, G. PLONKA, AND J. MA As in (5), this problem is equivalent to ( ( K )) max D C m n Re tr D (Y k D T )Ck k= s.t. D D = I n. We take the singular value decomposition K k= Y k D T C k = U Λ V and obtain unitary matrices ( U C n n, V) C m m and a diagonal matrix of singular values Λ = diag(λ (),..., λ() n ), R n m. Now, similarly as shown in Theorem, the optimal dictionary matrix is obtained by D,opt = V,n U, where V,n denotes the restriction of V to its first n columns. Since K k= Y kd T Ck is a matrix of size n m, we only need to apply the singular value decomposition of this size here to obtain an update for D. Step 3. Analogously, in the third step we fix C and D and minimize (7) resp. (8) with respect to D. Here, we observe that is equivalent to min D C m n min D C m n K D Y k D T C k F s.t. DD = I n k= K k= D Y T k D T C T k F s.t. D D = I n. 5 5 The SVD K k= Yk T D T C k = U Λ V with unitary matrices U C n n, V C m m, and Λ = R n m yields the update D = V,n U, ( ) diag(λ (),..., λ() n ), where V,n again denotes the restriction of V to its first n columns. We outline the pseudo code for learning the tight frame with Kronecker structure (KronTF) in the following Algorithm. This approach to construct a data-driven tight frame can now easily be extended to third order tensors. Using tensor notion (for -tensors) the product in (6) reads C = Y D D. Generalizing the concept above to 3-tensors, we want to find matrices D ν C mν nν, ν =,, 3, with m ν n ν and D νd ν = I nν, ν =,, 3, such that for the tensor patches Y k C n n n3, k =,..., K, of a given big tensor X, the core tensors S k = Y k D D 3 D 3 C m m m3

9 DATA-DRIVEN TIGHT FRAMES OF KRONECKER TYPE 9 Algorithm KronTF Algorithm Input: Image X, number of iterations T Take patches of X, denoted by Y, Y Y K C n n. Initialize the filter matrices D C m n and D C m n with D D = I n, D D = I n. for k =,,..., T do Use the hard/soft thresholding to update the coefficient matrix C = (C,..., C K ) as given in Step for n = to do do Use the SVD method to update the filter matrices D n as given in Steps and 3 end for end for return D, D 53 are simultaneously sparse. This is done by solving the minimization problem () min D,D,D 3,S,...,S K k= K ( Yk D D 3 D 3 S k ) F + λ S k s.t. D νd ν = I nν, ν =,, Here, the Frobenius norm of X is defined by X F := ( n i= j= k=3 x i,j,k ) /, and S k denotes in the case = the number of nonzero entries of S k, and for = the sum of the moduli of all entries of S k. In the space of the input tensor X, we are training a tight frame whose filters are tensor products of three sets of orthogonal -D filters. The minimization problem () can be solved by four steps at each iteration level. In step, for fixed D, D, D 3, one minimizes S k, k =,..., K, by applying a threshold procedure as before. In step, for fixed D, D 3 and S k, k =,..., K, we use the unfolding (S k ) () = D (Y k ) () (D 3 D ) T and have to solve min D C m n n n3 K (S k ) () D (Y k ) () (D 3 D ) T F s.t. DD = I n. k= This problem has exactly the same structure as (9) and is solved by choosing D = V,n U, where U Λ V is the singular value decomposition of the (n m )-matrix K (Y k ) () (D 3 D ) T (S k ) (). k= Analogously, we find in step 3 and step 4 the updates D = V,n U and D 3 = V 3,n3 U 3 from the singular decompositions K (Y k ) () (D D 3 ) T (S k ) () = U Λ V k=

10 L. LIU, G. PLONKA, AND J. MA 63 resp. K (Y k ) (3) (D D ) T (S k ) (3) = U 3Λ 3 V k= Remark 3... This dictionary learning approach requires at each iteration step three SVDs but for matrices of moderate size n ν m ν, ν =,, 3.. One may also connect the two approaches considered in Section and in Section 3 by e.g. enforcing a tensor product structure of the filters only in the third direction while employing a more general filter for the first and second direction. In this case we can use the unfolding (S k ) (3) = D 3 (Y k ) (3) D T, where the dictionary (D D ) is replaced by the matrix D C mm nn that not necessarily has the Kronecker product structure. The dictionary learning procedure is then applied as for two-dimensional tensors. 4. Directional DDTF with Kronecker structure. Though the data-driven Kronecker tight frame considered in Section 3 is of simple structure and therefore is faster to implement, the Kronecker structure is limited in representing directional features, as usual for tensor product approaches. Therefore, we want to propose now a data-driven frame construction whose filters contain both a Kronecker structure and a directional structure. We first focus on the -D case. It is well-known that tensor-product filters are especially well suited for representing vertical or horizontal features in an image. Our idea is now to incorporate other favorite directions by mimicking rotation of the image. While an exact rotation is not possible when we want to stay with the original grid, we apply the following simple procedure. Let V : C n C n be the cyclic shift operator, i.e., for some x = (x j ) n j= we have V x := (x (j+) mod n ) n j=. For a given image X = (x,..., x n ) C n n with columns x,..., x n in C n, we consider e.g. the new image X = (x, V x, V x,..., V n x n ), where the j-th column is cyclically shifted by j steps. This procedure for example yields X = X = , 3 3 such that diagonal structures of X turn to horizontal structures of X. Vectorizing the image X into x = vec(x), it follows that I V x = vecx = diag(v j ) n j= x = V x,... V n

11 DATA-DRIVEN TIGHT FRAMES OF KRONECKER TYPE Fig.. Illustration of different angles in the range [, π/4]. where diag(v j ) n j= C nn nn denotes the block diagonal matrix with blocks V j. Other rotations of X can be mimicked for example by multiplying x = vecx with diag (V jl/n ) n j= for capturing the directions in [ π/4, ] and by, l =,..., n diag (V jl/n ) n, l = n +, n +,..., j= for capturing directions in [, π/4]. This range is sufficient in order to bring essential edges into horizontal or vertical form. If a priori information about favorite directions (or other structures) in images is available, we may suitably adopt the method to transfer this structure into a linear vertical or horizontal structure. Possible column shifts in the data matrices mimicking directions are illustrated in Figure. Using this idea to incorporate directionality, we generalize the model (7) resp. (8) for filters learning as follows. nstead of considering only one set of orthogonal Kronecker filters D D, we employ R sets of orthogonal Kronecker filters () (D D )(diag ( V jα ) n j= ). (D D )(diag ( V jα R ) n j= ) CRmm nn with R constants α,..., α R { + n, + n,... n ) n, } capturing R favorite directions. Then the model reads () min D,D,C R, α...,α R R r= ( (D D )(diagv jαr ) n j= Y C r F + λ C r ) s.t. D νd ν = I nν, ν =,. where C r C mmr K contains the blocks of transform coefficients for each direction, and where. denotes either the -norm or the -subnorm as before. Similar to [6, Proposition 3], one can prove that the filtering with filters being rows of () form a tight frame transform for the input image. In other words, through (), we are finding a data-driven tight frame with filters containing both Kronecker and directional

12 L. LIU, G. PLONKA, AND J. MA 98 structures. Compared to (8), the filter matrix is now composed of a Kronecker matrix 99 (D D ) C mm nn and block diagonal matrices (diagv jαr ) n j=, r =,..., R 3 capturing different directions of the image features. Particularly, learning the filter 3 matrix () only requires to determine n m + n m components of D, D and R 3 components α r to fix the directions. 33 The minimization problem in () can again be solved by alternating minimiza- 34 tion, where we determine the favorite directions already in a preprocessing step. Preprocessing step. For fixed matrices D and D we solve the problem min (D D )(diagv jαr ) n j= Y C α r F + λ C αr C αr 35 for each direction α r from a predetermined set of possible directions by applying ei- 36 ther a hard or a soft threshold operator to the transformed patches Y = (y,..., y K ). 37 We emphasize, that computing (diagv jα ) n j= Y for some fixed α (, ] is very 38 cheap, it just means a cyclic shifting of columns in the matrices Y k, such that 39 (D D )(diagv jα ) n j= Y corresponds to applying the two-dimensional dictionary 3 transform to the images that are obtained from Y k, k =,..., K, by taking cyclic 3 shifts. 3 We a priori fix the number R of considered directions (in practice often just R = 33 or R = ) and find the R favorite directions, e.g. by comparing the energies of C αr 34 for fixed thresholds or by comparing the PSNR values of the reconstructions of Y 35 after thresholding. 36 Once the directions α,..., α R are fixed, we start the iteration process as before Step. At each iteration level, we first minimize C αr, r =,..., R, while the complete set of filters (directions and D, D ) is fixed. This is done by applying the thresholding procedure. This step is not necessary at the first level since we can use here C αr obtained in the preprocessing step. Step. For fixed directions α,..., α R, corresponding C αr, r =,..., R, and fixed D we consider the minimization problem for D. We recall that C αr consists of K columns, where the k-th column is the vector of transform coefficients of Y k. We reshape C αr C mm K (C αr,,... C αr,k) C m mk and diag (V jαr ) n j= Y C nn K (Y αr,,..., Y αr,k) C n nk, where the k-th column of C αr of length m m is reshaped back to an (m m )- matrix C αr,k by inverting the vec operator, and analogously, the k-th column of diag ( V jαr ) n j= y k C nn is reshaped to a matrix Y αr,k C n n. Now we have to solve min D C m n R r= k= K D (Y αr,kd T C αr,k F s.t. DD = I n with similar structure as in (9). As before, we apply the singular value decomposition R K r= k= Y lr,kd T C l r,k = U Λ V

13 DATA-DRIVEN TIGHT FRAMES OF KRONECKER TYPE with unitary ( matrices U C n n,) V C m m and a diagonal matrix of singular values Λ = diag(λ (),..., λ() n ), R n m. Now, as shown in Theorem, the optimal dictionary matrix is obtained by D,opt = V,n U, where V,n denotes the restriction of V to its first n columns. Step 3. For fixed directions α,..., α R, corresponding C αr, r =,..., R, and fixed D we consider the minimization problem for D. With the same notations as above, we can write min D C m n R r= k= K D Yα T r,kd T Cα T r,k F s.t. DD = I n and obtain the optimal solution D,opt = V,n U from the singular value decomposition R K Yα T r,kd T C αr,k = U Λ V, r= k= where V,n denotes the restriction of V to its first n columns. We outline the pseudo code for learning the tight frame with Kronecker structure and one optimal direction in Algorithm. Algorithm KronTFD Algorithm Input: Training set of data, number of iterations T Let Y, Y Y K C n n be patches of X Initialize the filter matrices D C m n and D C m n with DD = I n, DD = I n. First: Find optimal angle direction of patches. for k =,,..., K do for angle = 45,,..., 45 do Adjust Y k by cyclic shifting of columns by the angle. Soft thresholding to update the coefficient matrix C = (C,..., C K ). Reconstruct the image by the tight frame synthesis operator and record the achieved SNR value. end for The largest SNR value yields the optimal angle direction of training data Adjust Y k by cyclic shifting of columns by the optimal angle. end for Second: Learn filters. for k =,,..., T do Use the hard/soft thresholding to update the coefficient matrix C = (C,..., C K ) for n = to do do Use the SVD method to update the filter matrices D n as given in Steps and 3 end for end for return D, D, optimal angle 38

14 4 L. LIU, G. PLONKA, AND J. MA Application to seismic data reconstruction and denoising. We want to apply the new data-driven tight frame constructions to D and 3D data reconstruction (resp. data interpolation) and to data denoising. Let X denote the complete correct data, and let Y be the observed data. We assume that these data are connected by the following relation (3) Y = A X + γξ. Here A X denotes the pointwise product (Hadamard product) of the two matrices A and X, ξ denotes an array of normalized Gaussian noise with expectation, and γ determines the noise level. The matrix A contains only the entries or and is called trace sampling operator. If γ = then the above relation models a seismic interpolation problem, and the task ist to reconstruct the missing data. If A = I, where I is the matrix containing only ones, and γ >, it models a denoising problem. The two problems can be solved by a sparsity-promoting minimization method, see e.g. [6]. We assume that our desired data X can be sparsely represented by the datadriven tight frame that has been learned beforehand, either by KronTF with kronecker product filters presented in Section 3 or by KronTFD with directional kronecker product filters presented in Section 4. We use D to denote the analysis operator of the learned tight frame. In geophysical community, alternating projection algorithms as POCS (projection onto convex sets) are very popular. The approach in [] for Fourier bases (instead of frames) can be interpreted as follows. We may try to formulate the interpolation problem as a feasibility problem. We look for a solution X of (3) that on the one hand satisfies the interpolation condition A X = Y, i.e., is contained in the set of all data M := {Z : A Z = Y } possessing the observed data Y. projection operator onto M, This constraint can be enforced by applying the P M X = (I A) X + Y that leaves the unobserved data unchanged and projects the observed traces to Y. On the other hand, we want to ensure that the solution X has a sparse representation in the constructed data-driven frame. The sparsity constraint is enforced by applying a (soft) thresholding to the transformed data, i.e., we compute P Dλ X := D S λ DX, where D denotes the tight frame analysis operator, and S λ is the soft threshold operator as in (4). We observe however, that P Dλ is not longer a projector and therefore this approach already generalizes a usual alternating projection method. The complete iteration scheme can be obtained by alternating application of P M and P Dλk, (4) X k+ = P M (P Dλk X k ) = (I A) (P Dλk X k ) + Y, where λ k is the threshold value that can vary at each iteration step. We recall that all elements of the matrix I are one. To show convergence of this scheme in the finitedimensional setting one can transfer ideas from [], where an iterative projection

15 DATA-DRIVEN TIGHT FRAMES OF KRONECKER TYPE scheme had been applied to a phase retrieval problem incorporating sparsity in a shearlet frame. In the context of image inpainting, (4) was proposed in [5] with framelet transform and the convergence was also proved there. To improve the convergence of this iteration scheme in numerical experiments, we adopt the following exponentially decreasing thresholding parameters, see [], λ k = λ max e b(k ), k =,,..., iter, where λ = λ max represents the maximum parameter, λ iter = λ min the minimum λmax parameter, and b is chosen as b = ( iter ) ln( λ min ), where iter is the fixed number of iterations in the scheme. For data denoising, we only apply the simple one-step (non-iterative) soft thresholding procedure given by P Dλ X, where λ is the threshold parameter related to the noise level γ. 6. Numerical Simulations. In a first illustration, we show the results of the dictionary learning method, where we have used (64 64) blocks of the seismic image in Figure 5(a). In Figure (b) we present the learned dictionary using the DDTF method. In comparison, the dictionaries learned by KronTF are shown in Figure (c) and (d). For the DDTF method, a spline wavelet is applied as the initial filter and the filter size is set to be 7 7. The KronTF method chooses the Fourier basis of size as initial filter. The Figures (e) and (f) show, how well the data can be sparsified by the learned filters compared to the initial filters. DDTF works here better than KronTF, but requires a much higher computational effort for dictionary learning. The comparison of time costs is illustrated in Figure 3. Here, we also show the comparison to KSVD that is even more expensive since it incorporates many SVDs, see Remark.3. We use the the new data-driven tight frames of Kronecker type (KronTF) and the data-driven directional frames of Kronecker type (KronTFD) for interpolation and denoising of D and 3D seismic data, and compare the performance with the results using the POCS algorithm based on the Fourier transform [], curvelet frames and data-driven tight frames (DDTF) method [8]. For the POCS method, seismic data are cropped into overlapping patches to suppress the artifacts resulting from the global Fourier transform. For the DDTF method, again a spline wavelet is applied as the initial dictionary and the filter size is set to be 7 7, see [6] The quality of the reconstructed images is compared using the PSNR (peak signalto-noise ratio) value, given by (5) P SNR = log( MN max X min X i,j (X i,j X i,j ) ), where X C M N denotes the original seismic data and X C M N is the recovered seismic data. In a first test we consider synthesis data, see Figure 4. Figure 4(a)-(b) show the original real data and sub-sampled data with % random traces missing.we compare the reconstructions (interpolation) using the DDFT method in Figure 4(c), the KronTF in Figure 4(d) and the KronTFD method in Figure 4(e). In a next example, we consider real seismic data. In Figure 5 and Figure 6 we present the interpolation results using different reconstruction methods. Figure 5(c)- (f) show the interpolation results by the POCS method and the curvelet transform as

16 6 L. LIU, G. PLONKA, AND J. MA well as the difference images. In Figure 6, we show the corresponding interpolation results for DDTF, and KronTF and KronTFD together with the error between the recovery and original data. For a further comparison of the recovery results, we compare a single trace in Figure 7. In Table, we list the comparisons of reconstruction results obtained from incomplete data with different sampling ratios. Table PSNR comparison of five methods for different sampling ratio. Sampling KronTFD KronTF DDTF Curvelet POCS

17 DATA-DRIVEN TIGHT FRAMES OF KRONECKER TYPE 7 (a) (b) (c) (d) 4 x Learned Initial Learned Initial 3.5 abs(u) abs(u) k (e) k (f) Fig.. (a) Initial filters. (b) Learned filter via DDTF. (c) The matrix D learned via KronTF. (d) The matrix D learned via KronTF. (e) (f) Absolute values of frame coefficients using KronTF and DDTF. Solid lines denote absolute values of frame coefficients using initial filters.

18 8 L. LIU, G. PLONKA, AND J. MA 7 6 DDTF KronTF KSVD DDTF 5 Training time(s) 4 3 Training time(s) Data size Data size (a) (b) Fig. 3. Comparison of time costs for filter learning Figures 8 and 9 show denoising results for the data in Figure 8(b) where the noise level is. We compare the results of POCS, the curvelet transform, DDTF, and our new frames KronTF and KronTFD, respectively. With our new data-driven frames, we can achieve better results than with the other methods, because the dictionary learning methods contain the information on the special seismic data. For better comparison, we also display the single trace comparisons in Figure. In Table, we list the comparisons of the achieved PSNR value with different noise levels. KronTF and KronTFD achieve competitive results both for interpolation and denoising. Finally, we test the interpolation of 3D data with % randomly missing traces. In Figure, we applied the KronTF and the KronTFD tensor technique to the synthesis 3D seismic data (size of ). Here, patches are used for training. The results of the DDTF method for 3D data are shown in Figures (c). In Figure (d)-(e) we present the results obtained by the KronTF and KronTFD method, respectively. The single trace comparisons are also shown in Figure (f). Finally, Figure and Figure 3 show interpolation comparison of the DDTF method and KronTF method for a real 3D marine data. Table PSNR comparison of five methods for different noise levels. Noise KronTFD KronTF DDTF Curvelet POCS Conclusion. This paper aims at exploiting sparse representation of seismic data for interpolation and denoising. We have proposed a new method to construct data-driven tensor-product tight frames, which learn a separable dictionaries. In order to enlarge the flexibility of the filters, we have also proposed a simple method to incorporate favorite local directions that are learned from the data. In the numerical experiments, we employed the new dictionaries to improve both seismic data interpolation and denoising. The main advantage of the proposed tight frame construction is its computational

19 DATA-DRIVEN TIGHT FRAMES OF KRONECKER TYPE efficiency which is due to the imposed tensor-product structure. The tight fame learning method is applied here to rather small blocks of data, such that local features are well recovered. However tensor-based method has its limitation, The algorithm need to store the each unfold-mode matrix is used to learn the dictionary, we need more storage space. In future, we will extend the idea to learn more global features (e.g. by applying a multiscale approach) and incorporate other direction and slope detection methods for the KronTFD. Furthermore, we are interested in applications to five-dimensional seismic data, where efficiency of the method is of even greater importance.

20 L. LIU, G. PLONKA, AND J. MA (a) (b) (c) (d) (e) Original KronTFD Original KronTF Original DDTF (f) Fig. 4. Interpolation results using DDTF and the new KronTF methods for synthesis data with irregular sampling ratio of.5. (a) Original seismic data. (b) Seismic data with % traces missing. (c) Interpolation by DDTF method. (d) Interpolation by KronTF method. (e) Interpolation by KronTFD method. (f) Single trace comparisons of the reconstructions with the original data.

21 DATA-DRIVEN TIGHT FRAMES OF KRONECKER TYPE (a) (b) (c) (d) (e) (f) Fig. 5. Interpolation results on real data with an irregular sampling ratio of.5. (a) Original seismic data. (b) Seismic data with % trace missing. (c) Interpolation using the POCS method. (d) Difference between (c) and (a). (e) Interpolation using the curvelet method. (f) Difference between (e) and (a).

22 L. LIU, G. PLONKA, AND J. MA (a) (b) (c) (d) (e) (f) Fig. 6. Interpolation results (continued) of Figure 5(b). (a) Interpolation using the DDTF. (b) Difference between (a) and Figure 5(a). (c) Interpolation using the KronTF. (d) Difference between (c) and Figure 5(a). (e) Interpolation using the KronTFD. (f) Difference between (e) and Figure 5 (a).

23 DATA-DRIVEN TIGHT FRAMES OF KRONECKER TYPE 3 3 original recover Time sampling data (a) 3 3 original recover Time sampling data (b) (c) 3 original recover 3 original recover (d) (e) Fig. 7. (a)-(e) Single trace comparison of the five interpolation results in Figure 5 and 6 with the original data as shown Figure 5(a).

24 4 L. LIU, G. PLONKA, AND J. MA (a) (b) (c) (d) (e) (f) Fig. 8. Denoising results of methods on real data with noise level of. (a) Original seismic data. (b) Noisy data. (c) Denoising result by POCS method. (d) Difference between (c) and (a). (e) Denoising result by curvelet method. (f) Difference between (e) and (a).

25 DATA-DRIVEN TIGHT FRAMES OF KRONECKER TYPE 5 (a) (b) (c) (d) (e) (f) Fig. 9. Denoising results for data in Figure 8(b). (a) Denoising by DDTF method. (b) Difference between (a) and 8(a). (c) Denoising by KronTF method. (d) Difference between (c) and 8(a). (e) Denoising by KronTFD method. (f) Difference between (e) and 8(a).

26 6 L. LIU, G. PLONKA, AND J. MA 3 original denoise (a) 3 Original Curvelet 3 original denoise Time sampling data (b) (c) 3 original denoise 3 original denoise Time sampling data (d) (e) Fig.. Single trace comparisons for the recovery results in Figure 8 and Figure 9 and the original Figure 8 (a). Here we have (a) application of POCS method, (b) curvelet method, (c)-(e) application of the data-driven frames DDTF, KronTF and KronTFD.

27 DATA-DRIVEN TIGHT FRAMES OF KRONECKER TYPE 7 Time (s) Receiver (km)..4 Shot (km) Receiver (km) Time (s) Receiver (km)..4 Shot (km) Receiver (km) (a) (b) Time (s) Receiver (km) Time (s) Receiver (km) Shot (km) Receiver (km) Shot (km) Receiver (km) (c) (d) Time (s) Receiver (km) Shot (km) Receiver (km) (e) Original KronTFD Time sampling data (f) Original DDTF Original KronTF Fig.. Interpolation of a synthetic seismic 3D data with data size (a) Original data. (b) Data with % randomly missing traces. (c)-(e) Interpolation by DDTF, KronTF and KronTFD.(f) Single trace comparisons of their reconstructions with the original data.

28 8 L. LIU, G. PLONKA, AND J. MA Time (s) Receiver (km)..4 Shot (km) Receiver (km).... Time (s) Receiver (km)..4 Shot (km) Receiver (km).... (a) (b) Time (s) Receiver (km)..4 Shot (km) Receiver (km).... Time (s) Receiver (km)..4 Shot (km) Receiver (km).... (c) (d) Time (s) Receiver (km)..4 Shot (km) Receiver (km).... Time (s) Receiver (km)..4 Shot (km) Receiver (km).... (e) (f) Time (s) Receiver (km)..4 Shot (km) Receiver (km).... (g) Fig.. Interpolation of a real 3D marine data with data size 5. (a) Original data; (b) Data with % randomly missing traces. (c)-(g) Interpolation by POCS method, Curvelet method, DDTF, KronTF and KronTFD.

29 DATA-DRIVEN TIGHT FRAMES OF KRONECKER TYPE 9.6 Original POCS (a).8.6 Original Curvelet.8.6 Original DDTF (b) (c) Original KronTF.6 Original KronTFD (d) (e) Fig. 3. Single trace comparisons for the recovery results in Figure and the original Figure (a). Here we have (a) POCS method, (b) curvelet denoising, (c)-(e) application of the data-driven frames DDTF, KronTF and KronTFD.

30 3 L. LIU, G. PLONKA, AND J. MA Acknowledgments. The authors thank Jian-Feng Cai from Hongkong University of Science and Technology for insightful technical discussions and helpful edit on the DDTF. This work is supported by NSFC (grant number: NSFC 9338, 4374, 6373), and the Fundamental Research Funds for the Central Universities (grant number: HIT.PIRS.A) REFERENCES [] R. Abma and N. Kabir, 3d interpolation of irregular data with a pocs algorithm, Geophysics, 7 (6), pp. E9 E97, doi:.9/ [] M. Aharon, M. Elad, and A. Bruckstein, The k-svd: An algorithm for designing of overcomplete dictionaries for sparse representation, IEEE Transactions On Signal Processing, 54 (6), pp , doi:.9/tsp [3] C. Bao, J.-F. Cai, and H. Ji, Fast sparsity-based orthogonal dictionary learning for image restoration, in Proceedings of the IEEE International Conference on Computer Vision, 3, pp [4] S. Beckouche and J. Ma, Simultaneously dictionary learning and denoising for seismic data, Geophysics, 79 (4), pp. A7 A3, doi:.9/geo [5] J.-F. Cai, R. H. Chan, and Z. Shen, A framelet-based image inpainting algorithm, Applied and Computational Harmonic Analysis, 4 (8), pp [6] J.-F. Cai, H. Ji, Z. Shen, and G.-B. Ye, Data-driven tight frame construction and image denoising, Applied and Computational Harmonic Analysis, 37 (4), pp. 89 5, doi:.6/j.acha.3... [7] Y. Chen, J. Ma, and S. Fomel, Double sparsity dictionary for seismic noise attenuation, Geophysics, 8 (6), pp. V3 V6, doi:.3997/ [8] K. Engan, S. O. Aase, and J. H. Husoy, Method of optimal directions for frame design, IEEE International Conference on., 5 (999), pp , doi:.9/icassp [9] S. Fomel and Y. Liu, Seislet transform and seislet frame, Geophysics, 75 (), pp. V5 V38, doi:.9/ [] J. J. Gao, X. H. Chen, J. Y. Li, G. C. Liu, and J. Ma, Irregular seismic data reconstruction based on exponential threshold model of pocs method, Applied Geophysics, 7 (), pp. 9 38, doi:.7/s [] S. Häuser and J. Ma, Seismic data reconstruction via shearlet-regularized directional inpainting, (). [] G. Hennenfent and F. J. Herrmann, Simply denoise: wavefield reconstruction via jittered undersampling, Geophysics, 73 (8), pp. V9 V8, doi:.9/.838. [3] F. J. Herrmann and G. Hennenfent, Non-parametric seismic data recovery with curvelet frames, Geophysical Journal International, 73 (8), pp , doi:./j x x. [4] M. M. N. Kabir and D. J. Verschuur, Restoration of missing offsets by parabolic radon transform, Geophysical Prospecting, 43 (995), pp , doi:./j tb57.x. [5] T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM review, 5 (9), pp. 455, doi:.37/77x. [6] N. Kreimer and M. D. Sacchi, A tensor higher-order singular value decomposition for prestack seismic data noise reduction and interpolation, Geophysics, 77 (), pp. V3 V, doi:.9/geo [7] L. D. Lathauwer, B. D. Moor, and J. Vandewalle, A multilinear singular value decomposition, SIAM journal on Matrix Analysis and Applications, (), pp , doi:.37/s [8] J. Liang, J. Ma, and X. Zhang, Seismic data restoration via data-driven tight frame, Geophysics, 79 (4), pp. V65 V74, doi:.9/geo3-5.. [9] B. Liu and M. D. Sacchi, Minimum weighted norm interpolation of seismic records, Geophysics, 69 (4), pp , doi:.9/ [] S. Loock and G. Plonka, Phase retrieval for fresnel measurements using a shearlet sparsity constraint, Inverse Problems, 3 (4), p. 55. [] M. Naghizadeh and M. D. Sacchi, Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data, Geophysics, 75 (), pp. WB89 WB, doi:.9/ [] M. D. Sacchi, T. J. Ulrych, and C. J. Walker, Interpolation and extrapolation using a highresolution discrete fourier transform, IEEE Transactions on Signal Processing, 46 (998),