In this paper, we propose a reweighted low-rank matrix recovery method and demonstrate its application for robust image restoration. In the literature, principal component pursuit solves low-rank matrix recovery problem via a convex program of mixed nuclear norm and l1 norm. Inspired by reweighted l1 minimization for sparsity enhancement, we propose reweighting singular values to enhance low rank of a matrix. An efficient iterative reweighting scheme is proposed for enhancing low rank and sparsity simultaneously and the performance of low-rank matrix recovery is prompted greatly. We demonstrate the utility of the proposed method both on numerical simulations and real images/videos restoration, including single image restoration, hyperspectral image restoration, and background modeling from corrupted observations. All of these experiments give empirical evidence on significant improvements of the proposed algorithm over previous work on low-rank matrix recovery.

We consider the problem of super-resolution from unregistered aliased images with unknown spatial scaling factors and shifts. Due to the limitation of pixel size in the image sensor, the sampling rate for each image is lower than the Nyquist rate of the scene. Thus, we have aliasing in captured images, which makes it hard to register the low-resolution images and then generate a high-resolution image. To work out this problem, we formulate it as a multichannel sampling and reconstruction problem with unknown parameters, spatial scaling factors and shifts. We can estimate the unknown parameters and then reconstruct the high-resolution image by solving a nonlinear least square problem using the variable projection method. Experiments with synthesized 1-D signals and 2-D images show the effectiveness of the proposed algorithm.

This paper studies the problem of simultaneously aligning a batch of linearly correlated images despite gross corruption (such as occlusion). Our method seeks an optimal set of image domain transformations such that the matrix of transformed images can be decomposed as the sum of a sparse matrix of errors and a low-rank matrix of recovered aligned images. We reduce this extremely challenging optimization problem to a sequence of convex programs that minimize the sum of l1-norm and nuclear norm of the two component matrices, which can be efficiently solved by scalable convex optimization techniques. We verify the efficacy of the proposed robust alignment algorithm with extensive experiments on both controlled and uncontrolled real data, demonstrating higher accuracy and efficiency than existing methods over a wide range of realistic misalignments and corruptions.

This paper studies the problem of simultaneously aligning a batch of linearly correlated images despite gross corruption (such as occlusion). Our method seeks an optimal set of image domain transformations such that the matrix of transformed images can be decomposed as the sum of a sparse matrix of errors and a low-rank matrix of recovered aligned images. We reduce this extremely challenging optimization problem to a sequence of convex programs that minimize the sum of ℓ1-norm and nuclear norm of the two component matrices, which can be efficiently solved by scalable convex optimization techniques with guaranteed fast convergence. We verify the efficacy of the proposed robust alignment algorithm with extensive experiments with both controlled and uncontrolled real data, demonstrating higher accuracy and efficiency than existing methods over a wide range of realistic misalignments and corruptions.

Principal component analysis is a fundamental operation in computational data analysis, with myriad applications ranging from web search to bioinformatics to computer vision and image analysis. However, its performance and applicability in real scenarios are limited by a lack of robustness to outlying or corrupted observations. This paper considers the idealized “robust principal component analysis” problem of recovering a low rank matrix A from corrupted observations D = A + E. Here, the error entries E can be arbitrarily large (modeling grossly corrupted observations common in visual and bioinformatic data), but are assumed to be sparse. We prove that most matrices A can be efficiently and exactly recovered from most error sign-and-support patterns, by solving a simple convex program. Our result holds even when the rank of A grows nearly proportionally (up to a logarithmic factor) to the dimensionality of the observation space and the number of errors E grows in proportion to the total number of entries in the matrix. A by-product of our analysis is the first proportional growth results for the related problem of completing a low-rank matrix from a small fraction of its entries. Simulations and real-data examples corroborate the theoretical results, and suggest potential applications in computer vision.

In this paper, we propose a novel face recognition method based on anisotropic dual-tree complex wavelet packets(ADT-CWP). 2-D dual-tree complex wavelet transform(DT-CWT) provides a geometrically oriented decomposition for image representation as well as shift invariance. By applying anisotropic wavelet packet decomposition on DT-CWT further, ADT-CWP can be used to extract facial features better, which turns out to benefit for face recognition. With adaptively assigning different weights to different wavelet subbands, consistent best performances can be obtained based on different face databases which are under different conditions, such as varying illuminations and expressions, compared to PCA and other face recognition methods, especially Gabor-based method. Furthermore, in addition to the consistent and promising classification performances, our proposed ADT-CWP-based method has a really low computational complexity.