Efficient Algorithms for Convolutional Sparse Representations

Abstract

When applying sparse representation techniques to images, the standard approach is to independently compute the representations for a set of overlapping image patches. This method performs very well in a variety of applications, but results in a representation that is multi-valued and not optimized with respect to the entire image. An alternative representation structure is provided by a convolutional sparse representation, in which a sparse representation of an entire image is computed by replacing the linear combination of a set of dictionary vectors by the sum of a set of convolutions with dictionary filters. The resulting representation is both single-valued and jointly optimized over the entire image. While this form of a sparse representation has been applied to a variety of problems in signal and image processing and computer vision, the computational expense of the corresponding optimization problems has restricted application to relatively small signals and images. This paper presents new, efficient algorithms that substantially improve on the performance of other recent methods, contributing to the development of this type of representation as a practical tool for a wider range of problems.

SPORCO: A Python package for standard and convolutional sparse representations

“…Convolutional sparse coding (CSC) is based on a convolutional form of BPDN, referred to as Convolutional BPDN (CBPDN) [Woh16d] argmin…”

A two-term penalty function for inverse problems with sparsity constrains

“…For the latter case, see [4,Section 3]; moreover, recent ADMM-based approaches compute the convolutions in the frequency domain [21], [22], [23], [4], [24].…”

“…is defined in (9). Observing that each component of g(·) is independent 6 then we only need 4 prox p,λ = minx 1 2 x − y + p(x, λ) 5 Scalar operations applied to a vector are considered to apply element-wise. 6 …”

“…The former, which is distributed with the SPORCO library, computes the convolutions in the frequency domain in order to improve its computational performance (see [4] for details). The latter implementation computes the convolutions in the spatial domain, by first pre-computing a separable approximation of the original non-separable FB (see [12] for details).…”

“…Moreover, a convolutional sparse representation (CSR) [2], [3] models an entire image as a sum over a set of convolutions of coefficient maps, of the same size as the image, with their corresponding dictionary filters. Given a set of dictionary filters, the most widely used formulation of the convolutional sparse coding (CSC) problem is Convolutional BPDN (CBPDN) [4], defined as arg min…”