% GETIPSPH - Compute "in place" (m by n) sphering or quasi-sphering matrix for an (n by t) 
%              input data matrix. Quasi-sphering reduces dimensionality of the data, while
%              maintaining approximately the "original" positions of the axes. That is, 
%              quasi-sphering "rotates back" as much as possible into the original channel 
%              axes, versus simple PCA reduction to the principal subspace (i.e., projecting 
%              the data onto a largest eigenvector basis).
% Usage:
%           >> S = getipsph(x,m)
%
% Input:
%           x  size [n,t] input n-channel data matrix
%
% Optional input:
%           m  (int <= n) Reduce output dimensions to [n,m]. If m is not present, or m==n, 
%              then the usual sphering matrix (the symmetric square root of the data 
%              covariance matrix) is returned. {default: [], return usual sphering matrix).
% Output:
%
%           S  size [m,n] (if m==n) sphering matrix, or (if m < n) quasi-sphering matrix 
% Example:
%           >> x = nrand(10,1000); % random (10,1000) matrix
%           >> S = getipsph(x,8);  % return quasi-sphering matrix reducing dimension to 8
%           >> d = S*x;            % d is the quasi-sphered 8-dimensional data
%           %
%           % If ICA decomposition is performed on d, as >> [w] = runica(d,'sphering','off');
%           % then the mixing matrix containing the 8 component maps in original channel 
%           % coordinates is >> A = pinv(W*S); % where A is size [n,m]
%

% Author: Jason Palmer, SCCN / INC / UCSD, 2008

% Copyright (C) Jason Palmer, SCCN / INC / UCSD , La Jolla 2008
%
% This file is part of EEGLAB, see http://www.eeglab.org
% for the documentation and details.
%
% Redistribution and use in source and binary forms, with or without
% modification, are permitted provided that the following conditions are met:
%
% 1. Redistributions of source code must retain the above copyright notice,
% this list of conditions and the following disclaimer.
%
% 2. Redistributions in binary form must reproduce the above copyright notice,
% this list of conditions and the following disclaimer in the documentation
% and/or other materials provided with the distribution.
%
% THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
% AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
% IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
% ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
% LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
% CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
% SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
% INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
% CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
% ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF
% THE POSSIBILITY OF SUCH DAMAGE.

function S = getipsph(x,m)

[n,N] = size(x);
if nargin < 2
    m = n;
end

if m>n
   help getipsph
   return
end

mn = mean(x,2);
for i = 1:n
    x(i,:) = x(i,:) - mn(i);
end
Sxx = x*x'/N; Sxx = (Sxx+Sxx')/2;
[U,D,V] = svd(Sxx);
ds = diag(D);

if m == n
    S = U * pinv(diag(sqrt(ds))) * U';
elseif m < n
    [sd,so] = sort(diag(Sxx),1,'descend');
    [uv,sv,vv] = svd(U(so(1:m),1:m));
    S = uv * vv' * pinv(diag(sqrt(ds(1:m)))) * U(:,1:m)';
else
    error('m must be less than or equal to n');
end