CMIV Workshop on Matrix Computation

Abstracts: TBC

Abstracts: In this paper, we investigate the generalized eigenvalue problem $A{\bf x}=\lambda B {\bf x}$ arising from economic models. Under certain conditions, there is a simple generalized eigenvalue $\rho(A, B)$ in the interval $(0, 1)$ with a positive eigenvector. Based on the Noda iteration, a modified Noda iteration (MNI) and a generalized Noda iteration (GNI) are proposed for finding the generalized eigenvalue $\rho(A, B)$ and the associated unit positive eigenvector. It is proved that the GNI method always converges and has a quadratic asymptotic convergence rate. So GNI has a similar convergence behavior as MNI. The efficiency of these algorithms is illustrated by numerical examples.

Abstracts: TBC

Abstracts: Rank minimization problem has a wide range of applications in different areas. However, since this problem is NP-hard and non-convex, the frequently used method is to replace the matrix rank minimization with nuclear norm minimization. Nuclear norm is the convex envelope of the matrix rank and it is more computationally tractable. Matrix completion is a special case of rank minimization problem. In this paper, we consider directly using matrix rank as the regularization term instead of nuclear norm in the cost function for matrix completion problem. The solution is analyzed and obtained by a hard-thresholding operation on the singular values of the observed matrix. Then by exploiting patch-based nonlocal self-similarity scheme, we apply the proposed rank minimization algorithm to remove white Gaussian additive noise in images. Gamma multiplicative noise is also removed in logarithm domain. The experimental results illustrate that the proposed algorithm can remove noises in images more efficiently than nuclear norm can do. And the results are also competitive with those obtained by using the existing state-of-the-art noise removal methods in the literature.

Abstracts: In this talk, we study an image decomposition model for patterned fabric inspection. It is important to represent fabric patterns effectively so that fabric defects can be separated. One concern is that both patterned fabric (e.g., star- or box-patterned fabrics) and fabric defects contain mainly low frequency components. The main idea of this paper is to use the convolution of a lattice with a Dirac comb to characterize a patterned fabric image so that its repetitive components can be effectively represented in the image decomposition model. We formulate a model with total variation, sparsity and low-rank terms for patterned fabric inspection. The total variation term is used to regularize defective image, and the sparsity and the low-rank terms are employed to control Dirac comb function. The proposed model can be solved efficiently via a convex programming solver. Our experimental results for different types of patterned fabrics show that the proposed model can inspect defects at a higher accuracy compared with some classical methods in the literature.