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spm_matcomp.m
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function varargout = spm_matcomp(varargin)
%__________________________________________________________________________
% Collection of tools for matrices and vector/tensor fields.
%
% FORMAT out = spm_matcomp('name', input)
%
% FORMAT A = spm_matcomp('LoadDiag', A)
% FORMAT ld = spm_matcomp('LogDet', A)
% FORMAT s = spm_matcomp('logsumexp',b,dm)
% FORMAT iA = spm_matcomp('Inv', A)
% FORMAT c = spm_matcomp('Pointwise3', a, (b), (op))
% FORMAT c = spm_matcomp('Pointwise', a, (b), (op), (sym))
% FORMAT [Q,ll] = spm_matcomp('softmax',Q,scl)
% FORMAT ind = spm_matcomp('SymIndices', k)
% FORMAT ind = spm_matcomp('SymIndices', k)
% FORMAT M = spm_matcomp('threshold_eig',M)
%
% FORMAT help spm_matcomp>function
% Returns the help file of the selected function.
%__________________________________________________________________________
% Copyright (C) 2017 Wellcome Trust Centre for Neuroimaging
if nargin == 0
help spm_matcomp
error('Not enough argument. Type ''help spm_matcomp'' for help.');
end
id = varargin{1};
varargin = varargin(2:end);
switch lower(id)
case 'pointwise3'
[varargout{1:nargout}] = pointwise3(varargin{:});
case 'pointwise'
[varargout{1:nargout}] = pointwise(varargin{:});
case 'logdet'
[varargout{1:nargout}] = logdet(varargin{:});
case 'symindices'
[varargout{1:nargout}] = symIndices(varargin{:});
case 'loaddiag'
[varargout{1:nargout}] = loaddiag(varargin{:});
case {'inv', 'inverse'}
[varargout{1:nargout}] = inv(varargin{:});
case 'threshold_eig'
[varargout{1:nargout}] = threshold_eig(varargin{:});
case 'logsumexp'
[varargout{1:nargout}] = logsumexp(varargin{:});
case 'softmax'
[varargout{1:nargout}] = softmax(varargin{:});
otherwise
help spm_matcomp
error('Unknown function %s. Type ''help spm_matcomp'' for help.', id)
end
end
%% === logdet =============================================================
function ld = logdet(A)
% FORMAT ld = spm_matcomp('LogDet', A)
% A - A postive-definite square matrix
% ld - Logarithm of determinant of A
%
% Log-determinant of a positive-definite matrix.
% Cholesky factorisation is used to compute a more stable log-determinant.
%__________________________________________________________________________
% Copyright (C) 2017 Wellcome Trust Centre for Neuroimaging
% John Ashburner
% $Id$
% Cholseki decomposition of A (A = C' * C, with C upper-triangular)
[C, p] = chol(numeric(A));
if p > 0
% A should usually be positive definite, but check anyway.
warning(['Attempting to compute log determinant of matrix ' ...
'that is not positive definite (p=%d).'], p);
end
% Because C is triangular, |C| = prod(diag(C))
% Hence: log|C| = sum(log(diag(C)))
% And: log|A| = log|C'*C| = log(|C|^2) = 2 * sum(log(diag(C)))
ld = 2 * sum(log(diag(C)));
end
%% === loaddiag ===========================================================
function A = loaddiag(A)
% FORMAT A = spm_matcomp('LoadDiag', A)
% A - A square matrix
%
% Load A's diagonal until it is well conditioned for inversion.
%__________________________________________________________________________
% Copyright (C) 2017 Wellcome Trust Centre for Neuroimaging
factor = 1e-7;
while rcond(A) < 1e-5
A = A + factor * max([diag(A); eps]) * eye(size(A));
factor = 10 * factor;
end
end
%% === inv ================================================================
function A = inv(A)
% FORMAT iA = spm_matcomp('Inv', A)
% A - A positive-definite square matrix
% iA - Its inverse
%
% Stable inverse of a positive-definite matrix.
% Eigendecomposition is used to compute a more stable inverse.
%__________________________________________________________________________
% Copyright (C) 2017 Wellcome Trust Centre for Neuroimaging
[V,D] = eig(A);
if any(diag(D) <= 0)
warning('[spm_matcomp::inv] Matrix has negative eigenvalues')
D(D <= 0) = eps; % Threshold negative eigenvalues
end
D = loaddiag(D);
A = real(V * (D \ V'));
% A = 0.5*(A + A.'); % Ensure symetric inverse
end
%% === symindices =========================================================
function [ind, n] = symIndices(k, side)
% FORMAT [ind, n] = spm_matcomp('SymIndices', k, ('k'))
% FORMAT [ind, k] = spm_matcomp('SymIndices', n, 'n')
% k - Length of the linearized and sparse symmetric matrix
% n - Side size of the corresponding square matrix
%
% Returns a converter between row/col and linear indices for sparse
% symmetric matrices.
%
% 1) If needed, finds n the size of the corresponding square matrix
% 2) Returns a n*n matrix which links subscripts (ex: (1,2) or (2,3)) to a
% corresponding linear index when the data is stored "sparsely" to save
% redunduncies.
%__________________________________________________________________________
% Copyright (C) 2017 Wellcome Trust Centre for Neuroimaging
if nargin == 2
side = strcmpi(side, 'n');
else
side = false;
end
% 1) Compute n
if ~side
n = round(max(roots([1 1 -2*k])));
else
n = k;
k = n*(n+1)/2;
end
% 2) Build the correspondance matrix
ind = diag(1:n);
l = n;
for i1=1:n
for i2=(i1+1):n
l = l + 1;
ind(i1, i2) = l;
ind(i2, i1) = l;
end
end
if side
n = k;
end
end
%% === pointwise3 =========================================================
function c = pointwise3(a, b, op)
% FORMAT c = spm_matcomp('Pointwise3', a, (b), (op))
% a - [nx nz nz 3 (3)] ND-array of vectors/matrices
% b - [nx nz nz 3 (3)] ND-array of vectors/matrices
% op - Operation(s) to apply to an element of a before multiplying it
% with an element of b:
% 't' (transpose), 'i' (invert), 'd' (determinant).
% (they can be concatenated: 'itd' means det(inv(a).')
%
% Performs matrix operations at each point of a tensor field.
% If b is provided, at each point:
% c = op(a) * b
% Else, at each point:
% c = op(a)
%
% Note that
% - if one (or both) entry is a vector, the adequate transpose
% operation is usually automatically detected.
% - a and b can be fields of sparse symmetric matrices. In this case, they
% have the form of a 6 element vector, which is automatically detected.
% - Implicit expansion of the lattice is performed when one of the input
% dimensions is singular.
%__________________________________________________________________________
% Copyright (C) 2017 Wellcome Trust Centre for Neuroimaging
% --- Default arguments
if nargin < 3
if nargin < 2
op = '';
binary = false;
elseif ischar(b)
op = b;
binary = false;
else
op = '';
binary = true;
end
else
binary = true;
end
% --- Check dimensions
ismat = struct;
dim = struct;
sym = struct;
dim.a = size(a);
sym.a = size(a, 4) > 3;
ismat.a = sym.a || size(a, 5) > 1;
if length(dim.a) < 4
error('Inputs must be vector or tensor fields')
end
if binary
dim.b = size(b);
sym.b = size(b, 4) > 3;
ismat.b = sym.b || size(b, 5) > 1;
if length(dim.b) < 4
error('Inputs must be vector or tensor fields')
end
end
% The corresponding dimensions of A and B must be equal to each other,
% or equal to one.
if binary
dima = dim.a(1:3);
dimb = dim.b(1:3);
same = dima == dimb;
diff = min(dima(~same), dimb(~same));
if any(diff ~= 1)
error(['The corresponding lattice dimensions of A and B ' ...
'must be equal to each other, or equal to one.']);
end
end
% --- Deal with cases
switch lower(op)
case ''
if binary && ismat.a && ismat.b
if sym.a && sym.b
c = pw3_sasb(a, b);
elseif sym.a
c = pw3_samb(a, b);
elseif sym.b
c = pw3_masb(a, b);
else
c = pw3_mamb(a, b);
end
elseif binary && ismat.a && ~ismat.b
if sym.a
c = pw3_savb(a, b);
else
c = pw3_mavb(a, b);
end
elseif binary && ~ismat.a && ismat.b
if sym.b
c = pw3_vasb(a, b);
else
c = pw3_vamb(a, b);
end
elseif binary && ~ismat.a && ~ismat.b
c = pw3_vavb(a, b);
else
c = a;
end
case 't'
if binary && ismat.a && ismat.b
if sym.a && sym.b
c = pw3_sasb(a, b);
elseif sym.b
c = pw3_masb(a, b);
elseif sym.a
c = pw3_samb(a, b);
else
c = pw3_tmamb(a, b);
end
elseif binary && ismat.a && ~ismat.b
if syma
c = pw3_savb(a, b);
else
c = pw3_tmavb(a, b);
end
elseif binary && ~ismat.a && ismat.b
if sym.b
c = pw3_vasb(a, b);
else
c = pw3_vamb(a, b);
end
elseif binary && ~ismat.a && ~ismat.b
c = pw3_vavb(a, b);
elseif ismat.a
if sym.a
c = a;
else
c = pw3_tma(a);
end
else
c = a;
end
case 'i'
if binary && ismat.a && ismat.b
if sym.a && sym.b
c = pw3_sasb(pw3_isa(a), b);
elseif sym.a
c = pw3_samb(pw3_isa(a), b);
elseif sym.b
c = pw3_masb(pw3_ima(a), b);
else
c = pw3_mamb(pw3_ima(a), b);
end
elseif binary && ismat.a && ~ismat.b
if sym.a
c = pw3_savb(pw3_isa(a), b);
else
c = pw3_mavb(pw3_ima(a), b);
end
elseif ~binary && ismat.a
if sym.a
c = pw3_isa(a);
else
c = pw3_ima(a);
end
else
error('This case does not make sense')
end
case 'd'
if binary && ismat.a
if sym.a
c = bsxfun(@times, pw3_dsa(a), b);
else
c = bsxfun(@times, pw3_dma(a), b);
end
elseif ~binary && ismat.a
if sym.a
c = pw3_dsa(a);
else
c = pw3_dma(a);
end
else
error('This case does not make sense')
end
case {'it', 'ti'}
if binary && ismat.a && ismat.b
if sym.a && sym.b
c = pw3_sasb(pw3_isa(a), b);
elseif sym.a
c = pw3_samb(pw3_isa(a), b);
elseif sym.b
c = pw3_tmasb(pw3_ima(a), b);
else
c = pw3_tmamb(pw3_ima(a), b);
end
elseif binary && ismat.a && ~ismat.b
if sym.a
c = pw3_savb(pw3_isa(a), b);
else
c = pw3_tmavb(pw3_ima(a), b);
end
elseif ~binary && ismat.a
if sym.a
c = pw3_isa(a);
else
c = pw3_tma(pw3_ima(a));
end
else
error('This case does not make sense')
end
case {'itd', 'tid'}
if binary && ismat.a
if sym.a
c = bsxfun(@times, b, pw3_dsa(a));
else
c = bsxfun(@ldivide, b, pw3_dma(a));
end
elseif ~binary && ismat.a
if sym.a
c = pw3_dsa(a);
else
c = 1 ./ pw3_dma(a);
end
else
error('This case does not make sense')
end
otherwise
error('The provided operation does not make sense')
end
end
function c = pw3_mamb(a, b)
% a - 3x3 tensor field
% b - 3x3 tensor field
% c - 3x3 tensor field
% Returns pointwise a * b
dima = [size(a) 1 1];
dimb = [size(b) 1 1];
dim = max(dima(1:3), dimb(1:3));
c = zeros([dim 3 3], 'like', a);
c(:,:,:,1,1) = bsxfun(@times, a(:,:,:,1,1), b(:,:,:,1,1)) + bsxfun(@times, a(:,:,:,1,2), b(:,:,:,2,1)) + bsxfun(@times, a(:,:,:,1,3), b(:,:,:,3,1));
c(:,:,:,1,2) = bsxfun(@times, a(:,:,:,1,1), b(:,:,:,1,2)) + bsxfun(@times, a(:,:,:,1,2), b(:,:,:,2,2)) + bsxfun(@times, a(:,:,:,1,3), b(:,:,:,3,2));
c(:,:,:,1,3) = bsxfun(@times, a(:,:,:,1,1), b(:,:,:,1,3)) + bsxfun(@times, a(:,:,:,1,2), b(:,:,:,2,3)) + bsxfun(@times, a(:,:,:,1,3), b(:,:,:,3,3));
c(:,:,:,2,1) = bsxfun(@times, a(:,:,:,2,1), b(:,:,:,1,1)) + bsxfun(@times, a(:,:,:,2,2), b(:,:,:,2,1)) + bsxfun(@times, a(:,:,:,2,3), b(:,:,:,3,1));
c(:,:,:,2,2) = bsxfun(@times, a(:,:,:,2,1), b(:,:,:,1,2)) + bsxfun(@times, a(:,:,:,2,2), b(:,:,:,2,2)) + bsxfun(@times, a(:,:,:,2,3), b(:,:,:,3,2));
c(:,:,:,2,3) = bsxfun(@times, a(:,:,:,2,1), b(:,:,:,1,3)) + bsxfun(@times, a(:,:,:,2,2), b(:,:,:,2,3)) + bsxfun(@times, a(:,:,:,2,3), b(:,:,:,3,3));
c(:,:,:,3,1) = bsxfun(@times, a(:,:,:,3,1), b(:,:,:,1,1)) + bsxfun(@times, a(:,:,:,3,2), b(:,:,:,2,1)) + bsxfun(@times, a(:,:,:,3,3), b(:,:,:,3,1));
c(:,:,:,3,2) = bsxfun(@times, a(:,:,:,3,1), b(:,:,:,1,2)) + bsxfun(@times, a(:,:,:,3,2), b(:,:,:,2,2)) + bsxfun(@times, a(:,:,:,3,3), b(:,:,:,3,2));
c(:,:,:,3,3) = bsxfun(@times, a(:,:,:,3,1), b(:,:,:,1,3)) + bsxfun(@times, a(:,:,:,3,2), b(:,:,:,2,3)) + bsxfun(@times, a(:,:,:,3,3), b(:,:,:,3,3));
end
function c = pw3_masb(a, b)
% a - 3x3 tensor field
% b - 3x3 symmetric tensor field = 6d vector field
% c - 3x3 tensor field
% Returns pointwise a * b
dima = [size(a) 1 1];
dimb = [size(b) 1 1];
dim = max(dima(1:3), dimb(1:3));
c = zeros([dim 3 3], 'like', a);
ind = symIndices(size(b, 4));
c(:,:,:,1,1) = bsxfun(@times, a(:,:,:,1,1), b(:,:,:,ind(1,1))) + bsxfun(@times, a(:,:,:,1,2), b(:,:,:,ind(2,1))) + bsxfun(@times, a(:,:,:,1,3), b(:,:,:,ind(3,1)));
c(:,:,:,1,2) = bsxfun(@times, a(:,:,:,1,1), b(:,:,:,ind(1,2))) + bsxfun(@times, a(:,:,:,1,2), b(:,:,:,ind(2,2))) + bsxfun(@times, a(:,:,:,1,3), b(:,:,:,ind(3,2)));
c(:,:,:,1,3) = bsxfun(@times, a(:,:,:,1,1), b(:,:,:,ind(1,3))) + bsxfun(@times, a(:,:,:,1,2), b(:,:,:,ind(2,3))) + bsxfun(@times, a(:,:,:,1,3), b(:,:,:,ind(3,3)));
c(:,:,:,2,1) = bsxfun(@times, a(:,:,:,2,1), b(:,:,:,ind(1,1))) + bsxfun(@times, a(:,:,:,2,2), b(:,:,:,ind(2,1))) + bsxfun(@times, a(:,:,:,2,3), b(:,:,:,ind(3,1)));
c(:,:,:,2,2) = bsxfun(@times, a(:,:,:,2,1), b(:,:,:,ind(1,2))) + bsxfun(@times, a(:,:,:,2,2), b(:,:,:,ind(2,2))) + bsxfun(@times, a(:,:,:,2,3), b(:,:,:,ind(3,2)));
c(:,:,:,2,3) = bsxfun(@times, a(:,:,:,2,1), b(:,:,:,ind(1,3))) + bsxfun(@times, a(:,:,:,2,2), b(:,:,:,ind(2,3))) + bsxfun(@times, a(:,:,:,2,3), b(:,:,:,ind(3,3)));
c(:,:,:,3,1) = bsxfun(@times, a(:,:,:,3,1), b(:,:,:,ind(1,1))) + bsxfun(@times, a(:,:,:,3,2), b(:,:,:,ind(2,1))) + bsxfun(@times, a(:,:,:,3,3), b(:,:,:,ind(3,1)));
c(:,:,:,3,2) = bsxfun(@times, a(:,:,:,3,1), b(:,:,:,ind(1,2))) + bsxfun(@times, a(:,:,:,3,2), b(:,:,:,ind(2,2))) + bsxfun(@times, a(:,:,:,3,3), b(:,:,:,ind(3,2)));
c(:,:,:,3,3) = bsxfun(@times, a(:,:,:,3,1), b(:,:,:,ind(1,3))) + bsxfun(@times, a(:,:,:,3,2), b(:,:,:,ind(2,3))) + bsxfun(@times, a(:,:,:,3,3), b(:,:,:,ind(3,3)));
end
function c = pw3_samb(a, b)
% a - 3x3 symmetric tensor field = 6d vector field
% b - 3x3 tensor field
% c - 3x3 tensor field
% Returns pointwise a * b
dima = [size(a) 1 1];
dimb = [size(b) 1 1];
dim = max(dima(1:3), dimb(1:3));
c = zeros([dim 3 3], 'like', a);
ind = symIndices(size(a, 4));
c(:,:,:,1,1) = bsxfun(@times, a(:,:,:,ind(1,1)), b(:,:,:,1,1)) + bsxfun(@times, a(:,:,:,ind(1,2)), b(:,:,:,2,1)) + bsxfun(@times, a(:,:,:,ind(1,3)), b(:,:,:,3,1));
c(:,:,:,1,2) = bsxfun(@times, a(:,:,:,ind(1,1)), b(:,:,:,1,2)) + bsxfun(@times, a(:,:,:,ind(1,2)), b(:,:,:,2,2)) + bsxfun(@times, a(:,:,:,ind(1,3)), b(:,:,:,3,2));
c(:,:,:,1,3) = bsxfun(@times, a(:,:,:,ind(1,1)), b(:,:,:,1,3)) + bsxfun(@times, a(:,:,:,ind(1,2)), b(:,:,:,2,3)) + bsxfun(@times, a(:,:,:,ind(1,3)), b(:,:,:,3,3));
c(:,:,:,2,1) = bsxfun(@times, a(:,:,:,ind(2,1)), b(:,:,:,1,1)) + bsxfun(@times, a(:,:,:,ind(2,2)), b(:,:,:,2,1)) + bsxfun(@times, a(:,:,:,ind(2,3)), b(:,:,:,3,1));
c(:,:,:,2,2) = bsxfun(@times, a(:,:,:,ind(2,1)), b(:,:,:,1,2)) + bsxfun(@times, a(:,:,:,ind(2,2)), b(:,:,:,2,2)) + bsxfun(@times, a(:,:,:,ind(2,3)), b(:,:,:,3,2));
c(:,:,:,2,3) = bsxfun(@times, a(:,:,:,ind(2,1)), b(:,:,:,1,3)) + bsxfun(@times, a(:,:,:,ind(2,2)), b(:,:,:,2,3)) + bsxfun(@times, a(:,:,:,ind(2,3)), b(:,:,:,3,3));
c(:,:,:,3,1) = bsxfun(@times, a(:,:,:,ind(3,1)), b(:,:,:,1,1)) + bsxfun(@times, a(:,:,:,ind(3,2)), b(:,:,:,2,1)) + bsxfun(@times, a(:,:,:,ind(3,3)), b(:,:,:,3,1));
c(:,:,:,3,2) = bsxfun(@times, a(:,:,:,ind(3,1)), b(:,:,:,1,2)) + bsxfun(@times, a(:,:,:,ind(3,2)), b(:,:,:,2,2)) + bsxfun(@times, a(:,:,:,ind(3,3)), b(:,:,:,3,2));
c(:,:,:,3,3) = bsxfun(@times, a(:,:,:,ind(3,1)), b(:,:,:,1,3)) + bsxfun(@times, a(:,:,:,ind(3,2)), b(:,:,:,2,3)) + bsxfun(@times, a(:,:,:,ind(3,3)), b(:,:,:,3,3));
end
function c = pw3_sasb(a, b)
% a - 3x3 symmetric tensor field = 6d vector field
% b - 3x3 symmetric tensor field = 6d vector field
% c - 3x3 tensor field
% Returns pointwise a * b
dima = [size(a) 1 1];
dimb = [size(b) 1 1];
dim = max(dima(1:3), dimb(1:3));
c = zeros([dim 3 3], 'like', a);
ind = symIndices(size(a, 4));
c(:,:,:,1,1) = bsxfun(@times, a(:,:,:,ind(1,1)), b(:,:,:,ind(1,1))) + bsxfun(@times, a(:,:,:,ind(1,2)), b(:,:,:,ind(2,1))) + bsxfun(@times, a(:,:,:,ind(1,3)), b(:,:,:,ind(3,1)));
c(:,:,:,1,2) = bsxfun(@times, a(:,:,:,ind(1,1)), b(:,:,:,ind(1,2))) + bsxfun(@times, a(:,:,:,ind(1,2)), b(:,:,:,ind(2,2))) + bsxfun(@times, a(:,:,:,ind(1,3)), b(:,:,:,ind(3,2)));
c(:,:,:,1,3) = bsxfun(@times, a(:,:,:,ind(1,1)), b(:,:,:,ind(1,3))) + bsxfun(@times, a(:,:,:,ind(1,2)), b(:,:,:,ind(2,3))) + bsxfun(@times, a(:,:,:,ind(1,3)), b(:,:,:,ind(3,3)));
c(:,:,:,2,1) = bsxfun(@times, a(:,:,:,ind(2,1)), b(:,:,:,ind(1,1))) + bsxfun(@times, a(:,:,:,ind(2,2)), b(:,:,:,ind(2,1))) + bsxfun(@times, a(:,:,:,ind(2,3)), b(:,:,:,ind(3,1)));
c(:,:,:,2,2) = bsxfun(@times, a(:,:,:,ind(2,1)), b(:,:,:,ind(1,2))) + bsxfun(@times, a(:,:,:,ind(2,2)), b(:,:,:,ind(2,2))) + bsxfun(@times, a(:,:,:,ind(2,3)), b(:,:,:,ind(3,2)));
c(:,:,:,2,3) = bsxfun(@times, a(:,:,:,ind(2,1)), b(:,:,:,ind(1,3))) + bsxfun(@times, a(:,:,:,ind(2,2)), b(:,:,:,ind(2,3))) + bsxfun(@times, a(:,:,:,ind(2,3)), b(:,:,:,ind(3,3)));
c(:,:,:,3,1) = bsxfun(@times, a(:,:,:,ind(3,1)), b(:,:,:,ind(1,1))) + bsxfun(@times, a(:,:,:,ind(3,2)), b(:,:,:,ind(2,1))) + bsxfun(@times, a(:,:,:,ind(3,3)), b(:,:,:,ind(3,1)));
c(:,:,:,3,2) = bsxfun(@times, a(:,:,:,ind(3,1)), b(:,:,:,ind(1,2))) + bsxfun(@times, a(:,:,:,ind(3,2)), b(:,:,:,ind(2,2))) + bsxfun(@times, a(:,:,:,ind(3,3)), b(:,:,:,ind(3,2)));
c(:,:,:,3,3) = bsxfun(@times, a(:,:,:,ind(3,1)), b(:,:,:,ind(1,3))) + bsxfun(@times, a(:,:,:,ind(3,2)), b(:,:,:,ind(2,3))) + bsxfun(@times, a(:,:,:,ind(3,3)), b(:,:,:,ind(3,3)));
end
function c = pw3_mavb(a, b)
% a - 3x3 tensor field
% b - 3d vector field
% b - 3d vector field
% Returns pointwise a * b
dima = [size(a) 1 1];
dimb = [size(b) 1 1];
dim = max(dima(1:3), dimb(1:3));
c = zeros([dim 3], 'like', a);
c(:,:,:,1) = bsxfun(@times, a(:,:,:,1,1), b(:,:,:,1)) + bsxfun(@times, a(:,:,:,1,2), b(:,:,:,2)) + bsxfun(@times, a(:,:,:,1,3), b(:,:,:,3));
c(:,:,:,2) = bsxfun(@times, a(:,:,:,2,1), b(:,:,:,1)) + bsxfun(@times, a(:,:,:,2,2), b(:,:,:,2)) + bsxfun(@times, a(:,:,:,2,3), b(:,:,:,3));
c(:,:,:,3) = bsxfun(@times, a(:,:,:,3,1), b(:,:,:,1)) + bsxfun(@times, a(:,:,:,3,2), b(:,:,:,2)) + bsxfun(@times, a(:,:,:,3,3), b(:,:,:,3));
end
function c = pw3_savb(a, b)
% a - 3x3 symmetric tensor field = 6d vector field
% b - 3d vector field
% b - 3d vector field
% Returns pointwise a * b
dima = [size(a) 1 1];
dimb = [size(b) 1 1];
dim = max(dima(1:3), dimb(1:3));
c = zeros([dim 3], 'like', a);
ind = symIndices(size(a, 4));
c(:,:,:,1) = bsxfun(@times, a(:,:,:,ind(1,1)), b(:,:,:,1)) + bsxfun(@times, a(:,:,:,ind(1,2)), b(:,:,:,2)) + bsxfun(@times, a(:,:,:,ind(1,3)), b(:,:,:,3));
c(:,:,:,2) = bsxfun(@times, a(:,:,:,ind(2,1)), b(:,:,:,1)) + bsxfun(@times, a(:,:,:,ind(2,2)), b(:,:,:,2)) + bsxfun(@times, a(:,:,:,ind(2,3)), b(:,:,:,3));
c(:,:,:,3) = bsxfun(@times, a(:,:,:,ind(3,1)), b(:,:,:,1)) + bsxfun(@times, a(:,:,:,ind(3,2)), b(:,:,:,2)) + bsxfun(@times, a(:,:,:,ind(3,3)), b(:,:,:,3));
end
function c = pw3_vamb(a, b)
% a - 3d vector field
% b - 3x3 tensor field
% b - 3d vector field
% Returns pointwise a.' * b
dima = [size(a) 1 1];
dimb = [size(b) 1 1];
dim = max(dima(1:3), dimb(1:3));
c = zeros([dim 3], 'like', a);
c(:,:,:,1) = bsxfun(@times, a(:,:,:,1), b(:,:,:,1,1)) + bsxfun(@times, a(:,:,:,2), b(:,:,:,2,1)) + bsxfun(@times, a(:,:,:,3), b(:,:,:,3,1));
c(:,:,:,2) = bsxfun(@times, a(:,:,:,1), b(:,:,:,1,2)) + bsxfun(@times, a(:,:,:,2), b(:,:,:,2,2)) + bsxfun(@times, a(:,:,:,3), b(:,:,:,3,2));
c(:,:,:,3) = bsxfun(@times, a(:,:,:,1), b(:,:,:,1,3)) + bsxfun(@times, a(:,:,:,2), b(:,:,:,2,3)) + bsxfun(@times, a(:,:,:,3), b(:,:,:,3,3));
end
function c = pw3_vasb(a, b)
% a - 3d vector field
% b - 3x3 symmetric tensor field = 6d vector field
% b - 3d vector field
% Returns pointwise a.' * b
dima = [size(a) 1 1];
dimb = [size(b) 1 1];
dim = max(dima(1:3), dimb(1:3));
c = zeros([dim 3], 'like', a);
ind = symIndices(size(b, 4));
c(:,:,:,1) = bsxfun(@times, a(:,:,:,1), b(:,:,:,ind(1,1))) + bsxfun(@times, a(:,:,:,2), b(:,:,:,ind(2,1))) + bsxfun(@times, a(:,:,:,3), b(:,:,:,ind(3,1)));
c(:,:,:,2) = bsxfun(@times, a(:,:,:,1), b(:,:,:,ind(1,2))) + bsxfun(@times, a(:,:,:,2), b(:,:,:,ind(2,2))) + bsxfun(@times, a(:,:,:,3), b(:,:,:,ind(3,2)));
c(:,:,:,3) = bsxfun(@times, a(:,:,:,1), b(:,:,:,ind(1,3))) + bsxfun(@times, a(:,:,:,2), b(:,:,:,ind(2,3))) + bsxfun(@times, a(:,:,:,3), b(:,:,:,ind(3,3)));
end
function c = pw3_vavb(a, b)
% a - 3d vector field
% b - 3d vector field
% b - 3d vector field
% Returns pointwise a.' * b
c = bsxfun(@times, a(:,:,:,1), b(:,:,:,1)) + bsxfun(@times, a(:,:,:,2), b(:,:,:,2)) + bsxfun(@times, a(:,:,:,3), b(:,:,:,3));
end
function c = pw3_tmamb(a, b)
% a - 3x3 tensor field
% b - 3x3 tensor field
% c - 3x3 tensor field
% Returns pointwise a.' * b
dima = [size(a) 1 1];
dimb = [size(b) 1 1];
dim = max(dima(1:3), dimb(1:3));
c = zeros([dim 3 3], 'like', a);
c(:,:,:,1,1) = bsxfun(@times, a(:,:,:,1,1), b(:,:,:,1,1)) + bsxfun(@times, a(:,:,:,2,1), b(:,:,:,2,1)) + bsxfun(@times, a(:,:,:,3,1), b(:,:,:,3,1));
c(:,:,:,1,2) = bsxfun(@times, a(:,:,:,1,1), b(:,:,:,1,2)) + bsxfun(@times, a(:,:,:,2,1), b(:,:,:,2,2)) + bsxfun(@times, a(:,:,:,3,1), b(:,:,:,3,2));
c(:,:,:,1,3) = bsxfun(@times, a(:,:,:,1,1), b(:,:,:,1,3)) + bsxfun(@times, a(:,:,:,2,1), b(:,:,:,2,3)) + bsxfun(@times, a(:,:,:,3,1), b(:,:,:,3,3));
c(:,:,:,2,1) = bsxfun(@times, a(:,:,:,1,2), b(:,:,:,1,1)) + bsxfun(@times, a(:,:,:,2,2), b(:,:,:,2,1)) + bsxfun(@times, a(:,:,:,3,2), b(:,:,:,3,1));
c(:,:,:,2,2) = bsxfun(@times, a(:,:,:,1,2), b(:,:,:,1,2)) + bsxfun(@times, a(:,:,:,2,2), b(:,:,:,2,2)) + bsxfun(@times, a(:,:,:,3,2), b(:,:,:,3,2));
c(:,:,:,2,3) = bsxfun(@times, a(:,:,:,1,2), b(:,:,:,1,3)) + bsxfun(@times, a(:,:,:,2,2), b(:,:,:,2,3)) + bsxfun(@times, a(:,:,:,3,2), b(:,:,:,3,3));
c(:,:,:,3,1) = bsxfun(@times, a(:,:,:,1,3), b(:,:,:,1,1)) + bsxfun(@times, a(:,:,:,2,3), b(:,:,:,2,1)) + bsxfun(@times, a(:,:,:,3,3), b(:,:,:,3,1));
c(:,:,:,3,2) = bsxfun(@times, a(:,:,:,1,3), b(:,:,:,1,2)) + bsxfun(@times, a(:,:,:,2,3), b(:,:,:,2,2)) + bsxfun(@times, a(:,:,:,3,3), b(:,:,:,3,2));
c(:,:,:,3,3) = bsxfun(@times, a(:,:,:,1,3), b(:,:,:,1,3)) + bsxfun(@times, a(:,:,:,2,3), b(:,:,:,2,3)) + bsxfun(@times, a(:,:,:,3,3), b(:,:,:,3,3));
end
function c = pw3_tma(a)
% a - 3x3 tensor field
% c - 3x3 tensor field
% Returns pointwise a.'
dim = [size(a) 1 1];
c = zeros([dim(1:3) 3 3], 'like', a);
c(:,:,:,1,1) = a(:,:,:,1,1);
c(:,:,:,1,2) = a(:,:,:,2,1);
c(:,:,:,1,3) = a(:,:,:,3,1);
c(:,:,:,2,1) = a(:,:,:,1,2);
c(:,:,:,2,2) = a(:,:,:,2,2);
c(:,:,:,2,3) = a(:,:,:,3,2);
c(:,:,:,3,1) = a(:,:,:,1,3);
c(:,:,:,3,2) = a(:,:,:,2,3);
c(:,:,:,3,3) = a(:,:,:,3,3);
end
function c = pw3_ima(a)
% a - 3x3 tensor field
% c - 3x3 tensor field
% Returns pointwise inv(a)
dim = [size(a) 1 1];
c = zeros([dim(1:3) 3 3], 'like', a);
% - Cofactors
c(:,:,:,1,1) = a(:,:,:,2,2) .* a(:,:,:,3,3) - a(:,:,:,2,3) .* a(:,:,:,3,2);
c(:,:,:,1,2) = a(:,:,:,1,3) .* a(:,:,:,3,2) - a(:,:,:,1,2) .* a(:,:,:,3,3);
c(:,:,:,1,3) = a(:,:,:,1,2) .* a(:,:,:,2,3) - a(:,:,:,1,3) .* a(:,:,:,2,2);
c(:,:,:,2,1) = a(:,:,:,2,3) .* a(:,:,:,3,1) - a(:,:,:,2,1) .* a(:,:,:,3,3);
c(:,:,:,2,2) = a(:,:,:,1,1) .* a(:,:,:,3,3) - a(:,:,:,1,3) .* a(:,:,:,3,1);
c(:,:,:,2,3) = a(:,:,:,1,3) .* a(:,:,:,2,1) - a(:,:,:,1,1) .* a(:,:,:,2,3);
c(:,:,:,3,1) = a(:,:,:,2,1) .* a(:,:,:,3,2) - a(:,:,:,2,2) .* a(:,:,:,3,1);
c(:,:,:,3,2) = a(:,:,:,1,2) .* a(:,:,:,3,1) - a(:,:,:,1,1) .* a(:,:,:,3,2);
c(:,:,:,3,3) = a(:,:,:,1,1) .* a(:,:,:,2,2) - a(:,:,:,1,2) .* a(:,:,:,2,1);
% - Determinant
c = bsxfun(@rdivide, c, a(:,:,:,1,1) .* c(:,:,:,1,1) + ...
a(:,:,:,1,2) .* c(:,:,:,2,1) + ...
a(:,:,:,1,3) .* c(:,:,:,3,1) );
end
function c = pw3_isa(a)
% a - 3x3 symmetric tensor field = 6d vector field
% c - 3x3 symmetric tensor field = 6d vector field
% Returns pointwise inv(a)
dim = [size(a) 1 1];
c = zeros([dim(1:3) 6], 'like', a);
ind = symIndices(size(a, 4));
% - Cofactors
c(:,:,:,ind(1,1)) = a(:,:,:,ind(2,2)) .* a(:,:,:,ind(3,3)) - a(:,:,:,ind(2,3)) .* a(:,:,:,ind(3,2));
c(:,:,:,ind(1,2)) = a(:,:,:,ind(1,3)) .* a(:,:,:,ind(3,2)) - a(:,:,:,ind(1,2)) .* a(:,:,:,ind(3,3));
c(:,:,:,ind(1,3)) = a(:,:,:,ind(1,2)) .* a(:,:,:,ind(2,3)) - a(:,:,:,ind(1,3)) .* a(:,:,:,ind(2,2));
c(:,:,:,ind(2,2)) = a(:,:,:,ind(1,1)) .* a(:,:,:,ind(3,3)) - a(:,:,:,ind(1,3)) .* a(:,:,:,ind(3,1));
c(:,:,:,ind(3,2)) = a(:,:,:,ind(1,2)) .* a(:,:,:,ind(3,1)) - a(:,:,:,ind(1,1)) .* a(:,:,:,ind(3,2));
% - Determinant
c = bsxfun(@rdivide, c, a(:,:,:,ind(1,1)) .* c(:,:,:,ind(1,1)) + ...
a(:,:,:,ind(1,2)) .* c(:,:,:,ind(2,1)) + ...
a(:,:,:,ind(1,3)) .* c(:,:,:,ind(3,1)) );
end
function c = pw3_dma(a)
% a - 3x3 tensor field
% c - scalar field
% Returns pointwise det(a)
c = a(:,:,:,1,1) .* ( a(:,:,:,2,2) .* a(:,:,:,3,3) - a(:,:,:,2,3) .* a(:,:,:,3,2) ) + ...
a(:,:,:,1,2) .* ( a(:,:,:,2,3) .* a(:,:,:,3,1) - a(:,:,:,2,1) .* a(:,:,:,3,3) ) + ...
a(:,:,:,1,3) .* ( a(:,:,:,2,1) .* a(:,:,:,3,2) - a(:,:,:,2,2) .* a(:,:,:,3,1) );
end
function c = pw3_dsa(a)
% a - 3x3 symmetric tensor field = 6d vector field
% c - scalar field
% Returns pointwise det(a)
ind = symIndices(size(a, 4));
c = a(:,:,:,ind(1,1)) .* ( a(:,:,:,ind(2,2)) .* a(:,:,:,ind(3,3)) - a(:,:,:,ind(2,3)) .* a(:,:,:,ind(3,2)) ) + ...
a(:,:,:,ind(1,2)) .* ( a(:,:,:,ind(2,3)) .* a(:,:,:,ind(3,1)) - a(:,:,:,ind(2,1)) .* a(:,:,:,ind(3,3)) ) + ...
a(:,:,:,ind(1,3)) .* ( a(:,:,:,ind(2,1)) .* a(:,:,:,ind(3,2)) - a(:,:,:,ind(2,2)) .* a(:,:,:,ind(3,1)) );
end
%% === pointwise ==========================================================
function c = pointwise(a, varargin)
% FORMAT c = spm_matcomp('Pointwise', a, (b), (op), (sym))
% a - [nx nz nz A (B)] ND-array of vectors/matrices
% b - [nx nz nz C (D)] ND-array of vectors/matrices
% op - Operation(s) to apply to an element of a before multiplying it
% with an element of b:
% 't' (transpose).
% sym - If true, at least one of a and b is sparse symmetric.
%
% Performs matrix operations at each point of a tensor field.
% If b is provided, at each point:
% c = op(a) * b
% Else, at each point:
% c = op(a)
%
% Note that
% - if one (or both) entry is a vector, the adequate transpose
% operation is usually automatically detected.
% - a and b can be fields of sparse symmetric matrices. In this case, they
% have the form of a vector field, and sym must be set to true.
% - Implicit expansion of the lattice is performed when one of the input
% dimensions is singular.
%__________________________________________________________________________
% Copyright (C) 2017 Wellcome Trust Centre for Neuroimaging
% --- Default arguments
b = [];
issym = false;
op = '';
while ~isempty(varargin)
if ischar(varargin{1})
op = varargin{1};
elseif isscalar(varargin{1})
issym = varargin{1};
else
b = varargin{1};
end
varargin = varargin(2:end);
end
binary = ~isempty(b);
% --- Check dimensions
ismat = struct;
dim = struct;
sym = struct;
dim.a = [size(a) 1 1];
if ~binary
sym.a = issym;
end
if binary
dim.b = [size(b) 1 1];
if issym
if size(a,5) == 1 && size(b,5) > 1
sym.a = true;
sym.b = false;
elseif size(a,5) > 1 && size(b,5) == 1
sym.b = true;
sym.a = false;
else
sym.a = false;
sym.b = false;
if size(a,4) >= size(b,4)
sym.a = true;
end
if size(b,4) >= size(a,4)
sym.b = true;
end
end
else
sym.a = false;
sym.b = false;
end
end
% The corresponding dimensions of A and B must be equal to each other,
% or equal to one.
if binary
dima = dim.a(1:3);
dimb = dim.b(1:3);
same = dima == dimb;
diff = min(dima(~same), dimb(~same));
if any(diff ~= 1)
error(['The corresponding lattice dimensions of A and B ' ...
'must be equal to each other, or equal to one.']);
end
end
ismat.a = sym.a || size(a, 5) > 1;
ismat.b = sym.b || size(b, 5) > 1;
% --- Deal with cases
switch lower(op)
case ''
if binary && ismat.a && ismat.b
if sym.a && sym.b
c = pw_sasb(a, b);
elseif sym.a
c = pw_samb(a, b);
elseif sym.b
c = pw_masb(a, b);
else
c = pw_mamb(a, b);
end
elseif binary && ismat.a && ~ismat.b
if sym.a
c = pw_savb(a, b);
else
c = pw_mavb(a, b);
end
elseif binary && ~ismat.a && ismat.b
if sym.b
c = pw_vasb(a, b);
else
c = pw_vamb(a, b);
end
elseif binary && ~ismat.a && ~ismat.b
c = pw_vavb(a, b);
else
c = a;
end
case 't'
if binary && ismat.a && ismat.b
if sym.a && sym.b
c = pw_sasb(a, b);
elseif sym.b
c = pw_masb(a, b);
elseif sym.a
c = pw_samb(a, b);
else
c = pw_tmamb(a, b);
end
elseif binary && ismat.a && ~ismat.b
if sym.a
c = pw_savb(a, b);
else
c = pw_tmavb(a, b);
end
elseif binary && ~ismat.a && ismat.b
if sym.b
c = pw_vasb(a, b);
else
c = pw_vamb(a, b);
end
elseif binary && ~ismat.a && ~ismat.b
c = pw_vavb(a, b);
elseif ismat.a
if sym.a
c = a;
else
c = pw_tma(a);
end
else
c = a;
end
otherwise
error('The provided operation is not supported')
end
end
function c = pw_mamb(a, b)
% a - NxM tensor field
% b - MxK tensor field
% c - NxK tensor field
% Returns pointwise a * b
if size(a,5) ~= size(b,4)
error('Matrix dimensions not consistant: %d and %d', size(a,5), size(b,4))
end
dima = [size(a) 1 1];
dimb = [size(b) 1 1];
dim = max(dima(1:3), dimb(1:3));
c = zeros([dim size(a,4) size(b,5)], 'like', a);
for i=1:size(a,4)
for j=1:size(b,5)
for k=1:size(a,5)
c(:,:,:,i,j) = c(:,:,:,i,j) + bsxfun(@times, a(:,:,:,i,k), b(:,:,:,k,j));
end
end
end
end
function c = pw_masb(a, b)
% a - NxM tensor field
% b - MxM symmetric tensor field = 6d vector field
% c - NxM tensor field
% Returns pointwise a * b
[ind, n] = symIndices(size(b, 4));
if size(a,5) ~= n
error('Matrix dimensions not consistant: %d and %d', size(a,5), n)
end
dima = [size(a) 1 1];
dimb = [size(b) 1 1];
dim = max(dima(1:3), dimb(1:3));
c = zeros([dim size(a,4) n], 'like', a);
for i=1:size(a,4)
for j=1:n
for k=1:n
c(:,:,:,i,j) = c(:,:,:,i,j) + bsxfun(@times, a(:,:,:,i,k), b(:,:,:,ind(k,j)));
end
end
end
end
function c = pw_samb(a, b)
% a - NxN symmetric tensor field = 6d vector field
% b - NxM tensor field
% c - NxM tensor field
% Returns pointwise a * b
ind = symIndices(size(a, 4));
if n ~= size(b,4)
error('Matrix dimensions not consistant: %d and %d', n, size(b,4))
end
dima = [size(a) 1 1];
dimb = [size(b) 1 1];
dim = max(dima(1:3), dimb(1:3));
c = zeros([dim n size(b,5)], 'like', a);
for i=1:n
for j=1:size(b,5)
for k=1:n
c(:,:,:,i,j) = bsxfun(@times, a(:,:,:,ind(i,k)), b(:,:,:,k,j));
end
end
end
end
function c = pw_sasb(a, b)
% a - NxN symmetric tensor field = 6d vector field
% b - NxN symmetric tensor field = 6d vector field
% c - NxN tensor field
% Returns pointwise a * b
[inda, na] = symIndices(size(a, 4));
[indb, nb] = symIndices(size(b, 4));
if na ~= nb
error('Matrix dimensions not consistant: %d and %d', na, nb)
end
dima = [size(a) 1 1];
dimb = [size(b) 1 1];
dim = max(dima(1:3), dimb(1:3));
c = zeros([dim na na], 'like', a);
for i=1:na
for j=1:na
for k=1:na
c(:,:,:,i,j) = c(:,:,:,i,j) + bsxfun(@times, a(:,:,:,inda(i,k)), b(:,:,:,indb(k,j)));
end
end
end
end
function c = pw_mavb(a, b)
% a - NxM tensor field
% b - Md vector field
% c - Nd vector field
% Returns pointwise a * b
transpose = false;
n = size(a,4);
if size(a,5) ~= size(b,4)
if size(a,4) ~= size(b,4)
error('Matrix dimensions not consistant: %d and %d', size(a,5), size(b,4))
else
n = size(a,5);
transpose = true;
end
end
dima = [size(a) 1 1];
dimb = [size(b) 1 1];
dim = max(dima(1:3), dimb(1:3));
c = zeros([dim n], 'like', a);
if ~transpose
for i=1:size(a,4)
for k=1:size(b,4)
c(:,:,:,i) = c(:,:,:,i) + bsxfun(@times, a(:,:,:,i,k), b(:,:,:,k));
end
end
else
for i=1:size(a,5)
for k=1:size(b,4)
c(:,:,:,i) = c(:,:,:,i) + bsxfun(@times, a(:,:,:,k,i), b(:,:,:,k));
end
end
end
end
function c = pw_savb(a, b)
% a - NxN symmetric tensor field = 6d vector field
% b - Nd vector field
% c - Nd vector field
% Returns pointwise a * b
[ind, n] = symIndices(size(a, 4));
if n ~= size(b,4)
error('Matrix dimensions not consistant: %d and %d', n, size(b,4))
end
dima = [size(a) 1 1];
dimb = [size(b) 1 1];
dim = max(dima(1:3), dimb(1:3));
c = zeros([dim n], 'like', a);