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Two-Dimensional Dictionary Learning and its Application in Image Denoising

Shahriari Mehr, Firooz | 2020

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 52859 (05)
  4. University: Sharif University of Technology
  5. Department: Electrical Engineering
  6. Advisor(s): Babaiezadeh, Masoud
  7. Abstract:
  8. Sparse representation and consequently, dictionary learning have been two of the great importance topics in signal processing problems for the last two decades. In sparse representation, each signal has to be represented as a linear combination of some basic signals, which are called atoms, and their collection is called a dictionary. To put it in other words, if complete dictionaries such as Fourier or Wavelet dictionaries are used for the representation of signals, the representation will be unique, but not sparse. On the other hand, if overcomplete dictionaries are used, we will confront with too many representations, and the goal of sparse representation is to find the sparsest one. Without any shadow of a doubt, the dictionary caused to sparse representations for signals plays an important role in the field of sparse signal processing. The dictionary may be chosen from fixed dictionaries; however, to have a sparser representation for a class of signals, one may opt to learn a dictionary for that class of signals, which is called dictionary learning.By growing the size of signals in one-dimensional (1D) dictionary learning for sparse representation, memory consumption and complex computations restrict the learning procedure. In applications of sparse representation and dictionary learning in two-dimensional (2D) signals (e.g., in image processing), if one opts to convert 2D signals to 1D signals and use the existing 1D dictionary learning and sparse representation techniques, too huge signals and dictionaries will be encountered. 2D dictionary learning and sparse representation have been proposed to avoid these problems by using the separable structure of atoms.In this thesis, four new algorithms are proposed for the 2D dictionary learning problem. The first one, 2D-MOD, is based on the MOD algorithm for 1D dictionary learning, which uses alternating minimization and gradient projection approach. The 2D dictionary learning problem is non-convex over its variables. In this thesis, a new jointly convex objective function is achieved for 2D dictionary learning, and the second one, 2D-CMOD, is proposed to solve it. The third algorithm is based on random permutation for updating variables in each iteration and can be applied to the previously proposed methods. Finally, the DL2D algorithm is proposed to change the 2D dictionary learning to two 1D dictionary learning problems. The proposed methods not only do reduce the computational complexity and the required memory of 1D methods but also need much fewer training signals and have a higher rate of convergence than 1D methods. We should consider if the training signals be sufficient, we should pay for the cost of a lower recovery percentage for these benefits
  9. Keywords:
  10. Sparse Representation ; Dictionary Learning ; Two Dimensional Signals ; Image Denoising ; Separable Structure ; Sparse Signal Processing

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