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Matrix Completion, and Some of its Applications in Image Processing

Ghasemi, Hooshang | 2012

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 43903 (05)
  4. University: Sharif University of Technology
  5. Department: Electrical Engineering
  6. Advisor(s): Babaei-Zadeh, Masoud; Ashtiani, Farid
  7. Abstract:
  8. Consider a matrix that only few of its entries are known and we would like to exactly recover the matrix based on the revealed entries. More preciesly, a set of entities which had been selected uniformally are revealed. Is it possible to recover the whole of the matrix from few revealed entities? If the answer is yes, which algorithm could recover the matrix? It is clear that solving this problem is not generally possible because for any unknown entity we could map the matrix to infinity matrix which coinside the constrain of the revealed entities. But in most case of the applications, we involve with the matrix which are low-rank. Now, what about low-rank matrix? Recovery of the matrix from sufficient number of known entries is possible if the matrix is low-rank and entries are selected uniformly. Matrix completion arises in wide range of applications including: collaborative filtering, triangulation from incomplete data and linear system identification. Matrix completion can be treated as a generalization of compressive sensing of vectors which has been attracting a lot of researchers during the last decade. In this thesis, we examin the matrix completion problem from theory and application point of view. We envastigate the problem in noiseless and sparse noise cases. Most of the pioneer in compressive sensing area belive that low-rank-driven modeling will become extremely important in the next decade (think of sparsity in the last decade)
  9. Keywords:
  10. Compressive Sensing ; Matrix Completion ; Nuclear Norm ; Sparse Corrupted Matrix Decomposition ; Real Field Coding

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