Total Variation Minimization with Separable Sensing Operator

Abstract

Abstract. Compressed Imaging is the theory that studies the problem of image recovery from an under-determined system of linear measurements. One of the most popular methods in this field is Total Variation (TV) Minimization, known for accuracy and computational efficiency. This paper applies a recently developed Separable Sensing Operator approach to TV Minimization, using the Split Bregman framework as the optimization approach. The internal cycle of the algorithm is performed by efficiently solving coupled Sylvester equations rather than by an iterative optimization procedure as it is done conventionally. Such an approach requires less computer memory and computational time than any other algorithm published to date. Numerical simulations show the improved -by an order of magnitude or more -time vs. image quality compared to two conventional algorithms.

Compressive imaging by generalized total variation minimization

“…Note that in (10) we have applied the Split Bregman technique [7] for the splitting D x = DV and D y = VD T , and introduced an additional split as U = V which guarantees that this is, in fact, the same one variable; however such a split allows us to decompose a most expensive step of the algorithm into two much simpler steps [3].…”

“…We stress that unlike other formulations presented in the literature, in problem (1) images are regarded as matrix variables instead of column-stacked vectors. Such formulation implicitly applies the separable sensing operator [1], [3] which facilitates efficient analysis, hence reduces storage complexity and makes fast computation possible. The matrix-based analysis of TV regularization model and its application to compressive imaging are the main contributions of this paper.…”

“…In [1], a two-dimensional separable sensing operator is used to reduce the storage of matrices of size m 2 × n 2 produced by the Kronecker product of two m × n matrices. In addition, TV minimization using a Split Bregman approach with a separable sensing operator is studied in [3].…”

Sparsity-inspired image reconstruction for electrical capacitance tomography

“…We extend the method by applying a separable sensing operator in [23] for further speed-up. The key idea is to split the L 1 and L 2 components of the optimization problem.…”

Edge-preserving nonlocal weighting scheme for total variation based compressive sensing recovery

“…Compressive sensing (CS) is an emerging sensing paradigm which allows to sample a signal at much lower rate than the Nyquist's sampling rate Total variation (TV) -based regularization is proved to have a tendency of preserving edge locations well [3], and it is being widely used in CS recovery [6][7][8][9][10]. The anisotropic TV optimization is given as:…”