Continuous Dictionaries Research Articles

Empi: GPU-Accelerated Matching Pursuit with Continuous Dictionaries

This article introduces an effective approach to performing matching pursuit calculations with continuous (quasi-infinite) dictionaries. Simulating continuous parameter space is accomplished by combining optimal dictionary construction as introduced previously with local parameter optimization. At the same time, the calculations can be performed not only with Gabor atoms, but with any type of oscillating atoms, or even multiple types of atoms at once. The proposed method is confirmed by introducing a first stable version of a high-performance implementation empi , supporting calculations with continuous as well as discrete dictionaries, further accelerated by the use of multi-threading and GPU support.

Joint sparse recovery for direction of arrival based on the generalized MUSIC criterion

The core of compressed sensing (CS) is to recover a set of sparse signals from incomplete measurements. In this paper, to solve the coherent direction-of-arrival (DOA) estimation problem, it is formulated as a joint-sparse recovery problem by exploiting the signal sparsity in the spatial domain. When the sparsity level of the signals satisfies certain conditions, the signals are recoverable even in the case of coherent sources. By further considering the structure of the sparse signals, a more general condition for the recoverability is theoretically derived. Then a method is proposed based on the CS theory and the generalized MUSIC criterion. This criterion improves on MUSIC so that the support is reliably recovered under the case of rank-defect. In our method, we modify the criterion to make it applicable for continuous dictionaries. When the partial DOAs has been estimated, the modified criterion can estimate the remaining DOAs with higher accuracy. By alternately using the modified criterion, the proposed method achieves super-resolution. Simulation results demonstrate that the proposed method has satisfactory performance even in the case of coherent sources.

When does OMP achieve exact recovery with continuous dictionaries?

This paper presents new theoretical results on sparse recovery guarantees for a greedy algorithm, Orthogonal Matching Pursuit (OMP), in the context of continuous parametric dictionaries. Here, the continuous setting means that the dictionary is made up of an infinite uncountable number of atoms. In this work, we rely on the Hilbert structure of the observation space to express our recovery results as a property of the kernel defined by the inner product between two atoms. Using a continuous extension of Tropp's Exact Recovery Condition, we identify key assumptions allowing to analyze OMP in the continuous setting. Under these assumptions, OMP unambiguously identifies in exactly $k$ steps the atom parameters from any observed linear combination of $k$ atoms. These parameters play the role of the so-called support of a sparse representation in traditional sparse recovery. In our paper, any kernel and set of parameters that satisfy these conditions are said to be admissible. In the one-dimensional setting, we exhibit a family of kernels relying on completely monotone functions for which admissibility holds for any set of atom parameters. For higher dimensional parameter spaces, the analysis turns out to be more subtle. An additional assumption, so-called axis admissibility, is imposed to ensure a form of delayed recovery (in at most $k^D$ steps, where $D$ is the dimension of the parameter space). Furthermore, guarantees for recovery in exactly $k$ steps are derived under an additional algebraic condition involving a finite subset of atoms (built as an extension of the set of atoms to be recovered). We show that the latter technical conditions simplify in the case of Laplacian kernels, allowing us to derive simple conditions for $k$-step exact recovery, and to carry out a coherence-based analysis in terms of a minimum separation assumption between the atoms to be recovered.

Inverse halftoning based on sparse representation

This Letter proposes a novel inverse halftoning algorithm that introduces a way of generating a pair of binary and continuous dictionaries optimized to a training image database composed of many pairs of halftoned patches and the corresponding continuous patches. The experiment results show that the two created binary and continuous dictionaries can be nicely used with the estimated sparse coefficients to represent an unknown continuous image with less noise and fine details from an input halftoned image.

Stochastic expansions using continuous dictionaries: Lévy adaptive regression kernels

This article describes a new class of prior distributions for nonparametric function estimation. The unknown function is modeled as a limit of weighted sums of kernels or generator functions indexed by continuous parameters that control local and global features such as their translation, dilation, modulation and shape. Lévy random fields and their stochastic integrals are employed to induce prior distributions for the unknown functions or, equivalently, for the number of kernels and for the parameters governing their features. Scaling, shape, and other features of the generating functions are location-specific to allow quite different function properties in different parts of the space, as with wavelet bases and other methods employing overcomplete dictionaries. We provide conditions under which the stochastic expansions converge in specified Besov or Sobolev norms. Under a Gaussian error model, this may be viewed as a sparse regression problem, with regularization induced via the Lévy random field prior distribution. Posterior inference for the unknown functions is based on a reversible jump Markov chain Monte Carlo algorithm. We compare the Lévy Adaptive Regression Kernel (LARK) method to wavelet-based methods using some of the standard test functions, and illustrate its flexibility and adaptability in nonstationary applications.