Constrained Overcomplete Analysis Operator Learning for Cosparse Signal Modelling

We consider the problem of learning a low-dimensional signal model from a collection of training samples. The mainstream approach would be to learn an overcomplete dictionary to provide good approximations of the training samples using sparse synthesis coefficients. This famous sparse model has a less well known counterpart, in analysis form, called the cosparse analysis model. In this new model, signals are characterised by their parsimony in a transformed domain using an overcomplete (linear) analysis operator. We propose to learn an analysis operator from a training corpus using a constrained optimisation framework based on L1 optimisation. The reason for introducing a constraint in the optimisation framework is to exclude trivial solutions. Although there is no final answer here for which constraint is the most relevant constraint, we investigate some conventional constraints in the model adaptation field and use the uniformly normalised tight frame (UNTF) for this purpose. We then derive a practical learning algorithm, based on projected subgradients and Douglas-Rachford splitting technique, and demonstrate its ability to robustly recover a ground truth analysis operator, when provided with a clean training set, of sufficient size. We also find an analysis operator for images, using some noisy cosparse signals, which is indeed a more realistic experiment. As the derived optimisation problem is not a convex program, we often find a local minimum using such variational methods. Some local optimality conditions are derived for two different settings, providing preliminary theoretical support for the well-posedness of the learning problem under appropriate conditions.Comment: 29 pages, 13 figures, accepted to be published in TS

Sparse representation-based synthetic aperture radar imaging

There is increasing interest in using synthetic aperture radar (SAR) images in automated target recognition and decision-making tasks. The success of such tasks depends on how well the reconstructed SAR images exhibit certain features of the underlying scene. Based on the observation that typical underlying scenes usually exhibit sparsity in terms of such features, we develop an image formation method which formulates the SAR imaging problem as a sparse signal representation problem. Sparse signal representation, which has mostly been exploited in real-valued problems, has many capabilities such as superresolution and feature enhancement for various reconstruction and recognition tasks. However, for problems of complex-valued nature, such as SAR, a key challenge is how to choose the dictionary and the representation scheme for effective sparse representation. Since we are usually interested in features of the magnitude of the SAR reflectivity field, our new approach is designed to sparsely represent the magnitude of the complex-valued scattered field. This turns the image reconstruction problem into a joint optimization problem over the representation of magnitude and phase of the underlying field reflectivities. We develop the mathematical framework for this method and propose an iterative solution for the corresponding joint optimization problem. Our experimental results demonstrate the superiority of this method over previous approaches in terms of both producing high quality SAR images as well as exhibiting robustness to uncertain or limited data

Sparse representation-based SAR imaging

There is increasing interest in using synthetic aperture radar (SAR) images in automated target recognition and decision-making tasks. The success of such tasks depends on how well the reconstructed SAR images exhibit certain features of the underlying scene. Based on the observation that typical underlying scenes usually exhibit sparsity in terms of such features, we develop an image formation method which formulates the SAR imaging problem as a sparse signal representation problem. Sparse signal representation, which has mostly been exploited in real-valued problems, has many capabilities such as superresolution and feature enhancement for various reconstruction and recognition tasks. However, for problems of complex-valued nature, such as SAR, a key challenge is how to choose the dictionary and the representation scheme for effective sparse representation. Since we are usually interested in features of the magnitude of the SAR reflectivity field, our new approach is designed to sparsely represent the magnitude of the complex-valued scattered field. This turns the image reconstruction problem into a joint optimization problem over the representation of magnitude and phase of the underlying field reflectivities. We develop the mathematical framework for this method and propose an iterative solution for the corresponding joint optimization problem. Our experimental results demonstrate the superiority of this method over previous approaches in terms of both producing high quality SAR images as well as exhibiting robustness to uncertain or limited data