ForwardDXT.m

% -------------------------------------------------------------------------
% Copyright (C) 2019 by D. di Serafino, G. Landi, M. Viola.
%
%                           COPYRIGHT NOTIFICATION
%
% This file is part of ACQUIRE.
%
% ACQUIRE is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% ACQUIRE is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with ACQUIRE. If not, see <http://www.gnu.org/licenses/>.
% -------------------------------------------------------------------------

function DXTU = ForwardDXT(U)

%==========================================================================
%
% Authors:
%   Daniela di Serafino (daniela.diserafino [at] unicampania.it)
%   Germana Landi       (germana.landi [at] unibo.it )
%   Marco Viola         (marco.viola [at] unicampania.it)
%
% Version: 1.0
% Last Update: 1 November 2019
%
%==========================================================================
%
% Given a 2-dimensional array U, this function returns the result of
%                           DX'*u,
% where u is the column version of U and DX is the linear operator
% representing the forward first-order column differences. The result is
% reshaped to have the same size as U.
%
%==========================================================================
%
% 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
%     https://arxiv.org/abs/1807.10832
% and Optimization Online
%     http://www.optimization-online.org/DB_HTML/2018/07/6745.html
%
%==========================================================================
%
% INPUT ARGUMENTS
% 
% U    = double matrix.
%
% OUTPUT ARGUMENTS
% 
% DXTU = double matrix, each entry (i,j) contains
%              DXTU(i,j) = U(i,j-1) - U(i,j),
%        we assume periodic boundary conditions, i.e.,
%              DXTU(i,1) = U(i,end) - U(i,1).
%
%==========================================================================

DXTU = -[U(:,1) - U(:,end), diff(U,1,2)];

end