Multidimensional Data Recovery via Iterative Regularization based on Higher Order Singular Value Decomposition

With the recent advances of networking, sensors, and storage technologies, many mul- tidimentional datasets are being generated in various fields. These datasets are often incomplete or contaminated during the acquisition process. Recovering the missing or noise-free data from degraded observations thus becomes crucial to obtaining precise information to refer to. The aim of this thesis is towards restoration of multidimen- sional (tensor) data. Specifically, we consider three problems: 1) tensor inpainting, 2) magnetic resonance image denoising and, 3) hyperspectral image denoising. The theory of tensors has become popular in dealing with multidimensional data, due to the capability of tensors in exploiting additional structure in comparison with matrix based alternatives. The most commonly used decomposition of multidimensional data to date is higher order singular value decomposition (HOSVD). The HOSVD is an ef- ficient way for eliciting intrinsic structure of multidimensional data. It offers a simple, adaptive and natural way to exploit sparsity among all dimensions of multidimensional data. The HOSVD decomposes a particular tensor data into the product of a sparse ten- sor and a few orthogonal matrices, each of which captures the subspace information corresponding to one dimension. In this work, we solve the restoration problems by employing the HOSVD transform and by exploiting the sparsity of the multidimen- sional signals. We enforce the sparsity using iterative regularization technique, which is shown to be very effective for our problems. The first contribution of this work is employing the iterative regularization scheme for tensor inpainting. The rationale of this approach is based on an enhanced sparse rep- resentation in HOSVD domain and it uses the iterative regularization procedure for inpainting. Improved performances of this algorithm are demonstrated in our exper- iments on three dimensional tensors, taken from multi-channel (color) images, video sequences, and magnetic resonance images. The evaluation is made quantitatively in terms of peak signal-to-noise ratio and structural similarity index, and qualitatively via visual comparisons. Despite the success of magnetic resonance imaging techniques in many applications, acquisition noise is still a limiting factor for the quality and hence the usefulness of the techniques. Our second contribution is improving the application of the iterative higher order singular value decomposition framework to denoising the magnetic res- onance images. The proposed algorithm forms a single tensor from the noisy data. This tensor undergoes an HOSVD, where its sparse representation coefficients are cal- culated with respect to a set of directional orthogonal basis matrices. Denoising is achieved by iteratively applying soft thresholding on the calculated sparse representa- tion coefficients. The proposed algorithm is further enhanced with a post-process of Wiener filtering. The performance of the proposed method is evaluated using synthetic and real magnetic resonance images. Validation results and quantitative comparisons with the state-of-the-art in magnetic resonance image denoising clearly demonstrate the advantages of the proposed method. The hyperspectral data cube is considered as a three-order tensor that is able to jointly treat both the spatial and spectral dimensions. Noise in hyperspectral image can de- grade the visual quality and limit the applicability of computerized analysis processes. Hence, toward the third contribution we consider the denoising of the hyperspectral images to improve the performance of the subsequent applications. In this work, not only we use the proposed iterative higher order singular value decomposition, but also we go one step further and propose a new iterative denoising method which utilizes the advantages of the patch-based HOSVD sparse model and the iterative regularization technique. The experiments with both synthetic noisy data and real hyperspectral data reveal that the proposed iterative algorithm improves the hyperspectral data quality in terms of both quality metrics and visual inspection. The subsequent classification re- sults further validate the effectiveness of the proposed hyperspectral noise reduction algorithm. In conclusion, extensive experiments on synthetic and real world datasets have shown the competitive performance of the proposed algorithms for inpainting, magnetic reso- nance image denoising, and hyperspectral image denoising over existing state-of-the- art ones. Keywords: Denoising, higher order singular value decomposition, hyperspectral, iter- ative regularization, MR images, patch-based, soft thresholding, sparse representation, Tucker decomposition.