Learning Image Dictionary using Non-negative Matrix Factorization and K-SVD

In this thesis, a theoretical background of algorithms called NLS-BB-NMF and K-SVD for computing the image dictionary have been introduced. The NLS-BB-NMF algorithm computes the matrix factorization V ≈ WH of the training data matrix V (in our case the set of image patches from training image) using gradient descent methods by applying non-negative constraint on matrices W and H. The K-SVD in turn computes the matrix factorization WH applying sparsity constraint on the coefficient matrix H using Orthogonal Matching Pursuit (OMP) and Singular Value Decomposition (SVD). In the factorization, matrix W is the so called dictionary and it contains features, also called atoms, of the data V . The atoms serve as a building blocks of the original data, and they are also assumed to represent data that is similar to the training data V .\n\nThe testing of the methods were carried in two phases. Initially, in the so called training phase, the dictionary was learned by the algorithms from a training image. The visual structure of the atoms learned by the algorithms were notably different although the approximations WH made by both dictionaries were visually very close to the original image. The visual difference between the learned dictionaries was seen as a consequence of the sparsity constraint that was forced for the coefficient matrix in K-SVD but not in NLS-BB-NMF.\n\nSecondly, in the test phase, a test image with various noise levels was approximated using the learned dictionary. The algorithms were able to produce approximations that were closer to the clean test image than the noisy test image. This was seen as the effect of dictionaries whose atoms were representing only the features of clean images.\n\nThis observation led to a second test where the algorithms were tested to compute the denoised reconstructions of the test image with varying noise levels by using an extended dictionary containing additionally atoms learned from a noise sample.\n\nThe qualities of the reconstructions were evaluated by using the Frobenius matrix norm and Structural Similarity (ssim) index that has been observed to adapt better the visual perception of human eyes.

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