Shearlets and Optimally Sparse Approximations

Multivariate functions are typically governed by anisotropic features such as edges in images or shock fronts in solutions of transport-dominated equations. One major goal both for the purpose of compression as well as for an efficient analysis is the provision of optimally sparse approximations of such functions. Recently, cartoon-like images were introduced in 2D and 3D as a suitable model class, and approximation properties were measured by considering the decay rate of the $L^2$ error of the best $N$-term approximation. Shearlet systems are to date the only representation system, which provide optimally sparse approximations of this model class in 2D as well as 3D. Even more, in contrast to all other directional representation systems, a theory for compactly supported shearlet frames was derived which moreover also satisfy this optimality benchmark. This chapter shall serve as an introduction to and a survey about sparse approximations of cartoon-like images by band-limited and also compactly supported shearlet frames as well as a reference for the state-of-the-art of this research field.Comment: in "Shearlets: Multiscale Analysis for Multivariate Data", Birkh\"auser-Springe

Image registration with sparse approximations in parametric dictionaries

We examine in this paper the problem of image registration from the new perspective where images are given by sparse approximations in parametric dictionaries of geometric functions. We propose a registration algorithm that looks for an estimate of the global transformation between sparse images by examining the set of relative geometrical transformations between the respective features. We propose a theoretical analysis of our registration algorithm and we derive performance guarantees based on two novel important properties of redundant dictionaries, namely the robust linear independence and the transformation inconsistency. We propose several illustrations and insights about the importance of these dictionary properties and show that common properties such as coherence or restricted isometry property fail to provide sufficient information in registration problems. We finally show with illustrative experiments on simple visual objects and handwritten digits images that our algorithm outperforms baseline competitor methods in terms of transformation-invariant distance computation and classification

Understanding and Comparing Scalable Gaussian Process Regression for Big Data

As a non-parametric Bayesian model which produces informative predictive distribution, Gaussian process (GP) has been widely used in various fields, like regression, classification and optimization. The cubic complexity of standard GP however leads to poor scalability, which poses challenges in the era of big data. Hence, various scalable GPs have been developed in the literature in order to improve the scalability while retaining desirable prediction accuracy. This paper devotes to investigating the methodological characteristics and performance of representative global and local scalable GPs including sparse approximations and local aggregations from four main perspectives: scalability, capability, controllability and robustness. The numerical experiments on two toy examples and five real-world datasets with up to 250K points offer the following findings. In terms of scalability, most of the scalable GPs own a time complexity that is linear to the training size. In terms of capability, the sparse approximations capture the long-term spatial correlations, the local aggregations capture the local patterns but suffer from over-fitting in some scenarios. In terms of controllability, we could improve the performance of sparse approximations by simply increasing the inducing size. But this is not the case for local aggregations. In terms of robustness, local aggregations are robust to various initializations of hyperparameters due to the local attention mechanism. Finally, we highlight that the proper hybrid of global and local scalable GPs may be a promising way to improve both the model capability and scalability for big data.Comment: 25 pages, 15 figures, preprint submitted to KB

Flexible Multi-layer Sparse Approximations of Matrices and Applications

The computational cost of many signal processing and machine learning techniques is often dominated by the cost of applying certain linear operators to high-dimensional vectors. This paper introduces an algorithm aimed at reducing the complexity of applying linear operators in high dimension by approximately factorizing the corresponding matrix into few sparse factors. The approach relies on recent advances in non-convex optimization. It is first explained and analyzed in details and then demonstrated experimentally on various problems including dictionary learning for image denoising, and the approximation of large matrices arising in inverse problems

Sparse approximations of protein structure from noisy random projections

Single-particle electron microscopy is a modern technique that biophysicists employ to learn the structure of proteins. It yields data that consist of noisy random projections of the protein structure in random directions, with the added complication that the projection angles cannot be observed. In order to reconstruct a three-dimensional model, the projection directions need to be estimated by use of an ad-hoc starting estimate of the unknown particle. In this paper we propose a methodology that does not rely on knowledge of the projection angles, to construct an objective data-dependent low-resolution approximation of the unknown structure that can serve as such a starting estimate. The approach assumes that the protein admits a suitable sparse representation, and employs discrete $L^1$-regularization (LASSO) as well as notions from shape theory to tackle the peculiar challenges involved in the associated inverse problem. We illustrate the approach by application to the reconstruction of an E. coli protein component called the Klenow fragment.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS479 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sparse image representation with encryption

In this thesis we present an overview of sparse approximations of grey level images. The sparse representations are realized by classic, Matching Pursuit (MP) based, greedy selection strategies. One such technique, termed Orthogonal Matching Pursuit (OMP), is shown to be suitable for producing sparse approximations of images, if they are processed in small blocks. When the blocks are enlarged, the proposed Self Projected Matching Pursuit (SPMP) algorithm, successfully renders equivalent results to OMP. A simple coding algorithm is then proposed to store these sparse approximations. This is shown, under certain conditions, to be competitive with JPEG2000 image compression standard. An application termed image folding, which partially secures the approximated images is then proposed. This is extended to produce a self contained folded image, containing all the information required to perform image recovery. Finally a modified OMP selection technique is applied to produce sparse approximations of Red Green Blue (RGB) images. These RGB approximations are then folded with the self contained approach

Regression with Sparse Approximations of Data

Publication in the conference proceedings of EUSIPCO, Bucharest, Romania, 201