ACQUIRE

A MATLAB package for the solution of optimization problems modeling the restoration of images corrupted by Poisson noise.

Authors

Daniela di Serafino, University of Campania "Luigi Vanvitelli", Caserta, Italy, daniela.diserafino[at]unicampania.it
Germana Landi, University of Bologna, Bologna, Italy, germana.landi[at]unibo.it
Marco Viola, University of Campania "Luigi Vanvitelli", Caserta, Italy, marco.viola[at]unicampania.it

Last Update

Version 1.0 - November 1, 2019

Description

ACQUIRE is a MATLAB implementation of the "Algorithm based on Consecutive QUadratic and Iteratively REweighted norm approximations" for the solution of constrained optimization problems of the form

          min  F(x) := KL(A*x + b, y) + lambda*TV(x)
          s.t. x in Omega,

modeling, e.g., the restoration of images corrupted by Poisson noise.

The objective function is the sum of a data-fidelity term consisting of the generalized Kullback-Leibler (KL) divergence of the blurred image (A*x + b) from the observed image (y) (see eq. (2) in [1]) and a regularization term consisting of the discrete isotropic Total Variation (TV) (see eq. (3) in [1]) with weight lambda. Here A is a linear blurring operator and b is the background noise.

The feasible set Omega represents either nonnegativity constraints, i.e.,

  Omega = {x : x_i>=0 for all i}

or nonnegativity contraints plus a linear constraint imposing total flux conservation, i.e.,

  Omega = {x : (x_i>=0, for all i) & (sum(x) = sum_i (y_i - b)}.

ACQUIRE is a line-search method that considers a smoothed version of TV, based on a Huber-like function, and computes the search directions by minimizing quadratic approximations of the problem, built by exploiting some second-order information. The KL divergence is approximated by a classical second-order Taylor expansion plus a strong convexity term. An Iteratively Reweighted Norm (IRN) approach is used to approximate the smoothed TV. See Section 3 in [1] for details.

The minimization of the quadratic approximations is performed by using a Scaled Gradient Projection (SGP) method [S. Bonettini, R. Zanella and L. Zanni, Inverse Problems 25 (2009), 015002].

References

[1] D. di Serafino, G. Landi and M. Viola, ACQUIRE: an inexact iteratively reweighted norm approach for TV-based Poisson image restoration, Applied Mathematics and Computation, volume 364, 2020, article 124678, DOI: 10.1016/j.amc.2019.124678. Preprint available from ArXiv and Optimization Online.

Software requirements

ACQUIRE runs under MATLAB. It has been tested under MATLAB 2018b.

Contents of the package

Here's the list of ACQUIRE files in alphabetical order:

• acquire.m : main function;
• DFproj_pg.m : Dai-Fletcher algorithm for the projection onto simplex-like constraints used for the computation of projected gradients;
• DFproj_sgp.m : Dai-Fletcher algorithm for the projection onto simplex-like constraints used in the SGP algorithm to project points onto the feasible set;
• ForwardD.m : function computing forward first-order finite differences with periodic boundary conditions;
• ForwardDXT.m : function evaluating the adjoint operator of the forward first-order column-difference operator;
• ForwardDYT.m : function evaluating the adjoint operator of the forward first-order row-difference operator;
• KLfunction.m : function evaluationg the Kullback-Leibler (KL) divergence (see eq (2) in [1]);
• KLhessprod.m : function computing the product between the Hessian of the quadratic approximation of the KL function and a vector;
• KLTVscaling.m : function computing the scaling matrix for the SGP algorithm by means of the splitting technique described, e.g., in [Zanella et al., Inverse Problems 25 (2009), 045010];
• line_search.m : function performing the backtracking Armijo line search;
• projgrad.m : function computing the projected gradient at a given point by using DFproj_pg;
• sgp_subp.m : SGP algorithm for the minimization of quadratic subproblems;
• TVapprox.m : function computing the IRN approximation of the smoothed TV;
• TVfunction.m : function evaluating the smoothed TV approximation.

See the documentation inside each file for further details.

Example of use

• example.m : example of use of ACQUIRE;
• TEST_phantom_gauss.mat : "phantom" image with Gaussian blur and SNR 40;
• TEST_cameraman_motion.mat : "cameraman" image with motion blur and SNR 40.

License