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function [s,TA]=framemulappr(Fa,Fs,T)
%-*- texinfo -*-
%@deftypefn {Function} framemulappr
%@verbatim
%FRAMEMULAPPR Best Approximation of a matrix by a frame multiplier
% Usage: s=framemulappr(Fa,Fs,T);
% [s,TA]=framemulappr(Fa,Fs,T);
%
% Input parameters:
% Fa : Analysis frame
% Fs : Synthesis frame
% T : The operator represented as a matrix
%
% Output parameters:
% s : Symbol of best approximation
% TA : The best approximation of the matrix T
%
% s=FRAMEMULAPPR(Fa,Fs,T) computes the symbol s of the frame
% multiplier that best approximates the matrix T in the Frobenious norm
% of the matrix (the Hilbert-Schmidt norm of the operator). The frame
% multiplier uses Fa for analysis and Fs for synthesis.
%
% Examples:
%
% T = eye(2,2);
% D = [0 1/sqrt(2) -1/sqrt(2); 1 -1/sqrt(2) -1/sqrt(2)];
% F = frame('gen',D);
% [coeff,TA] = framemulappr(F,F,T)
%
%
%@end verbatim
%@strong{Url}: @url{http://ltfat.github.io/doc/operators/framemulappr.html}
%@end deftypefn
% Copyright (C) 2005-2016 Peter L. Soendergaard <peter@sonderport.dk>.
% This file is part of LTFAT version 2.3.1
%
% This program 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.
%
% This program 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 this program. If not, see <http://www.gnu.org/licenses/>.
% Literature : [1] P. Balazs; Irregular And Regular Gabor frame multipliers
% with application to psychoacoustical masking
% (Ph.D. thesis 2005)
% [2] P. Balazs; Hilbert- Schmidt Operators and Frames -
% Classification, Best Approximation by Multipliers and
% Algorithms;
% International Journal of Wavelets, Multiresolution and
% Information Processing}, to appear,
% http://arxiv.org/abs/math.FA/0611634
% Author: Peter Balazs and Peter L. Soendergaard
if nargin < 3
error('%s: Too few input parameters.',upper(mfilename));
end;
[N M] = size(T);
Mfix=M;
% Bootstrap the code
D=frsynmatrix(Fa,Mfix);
Ds=frsynmatrix(Fs,Mfix);
[Nd Kd] = size(D);
% TODO: Check for for correct framelengths
% TODO: Check this error('The frames must have the same number of
% elements.');
% TODO: Possible optimization for Fa=Fs
% TODO: Express the pinv as an iterative algorithm
% Compute the lower symbol.
% The more elegant code
%
% is slower, O(k(n^2+n^2)))
% see [Xxl]
if 1
% Original expression
%lowsym = diag(D'*T*D);
% New expression
lowsym = conj(diag(frana(Fa,frana(Fa,T)')));
else
lowsym = zeros(Kd,1); %lower symbol
for ii=1:Kd
lowsym(ii) = D(:,ii)'*(T*D(:,ii));
end;
end;
Gram = (Ds'*Ds).*((D'*D).');
% upper symbol:
s = Gram\lowsym;
% synthesis
if nargout>1
TA = zeros(N,M);
for ii = 1:Kd
P = Ds(:,ii)*D(:,ii)';
TA = TA + s(ii)*P;
end;
end;
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