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Tensor.m
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classdef Tensor
%
% To ensure compatibility, this class contains (almost) verbatim copies of the functions in https://github.com/markus-nilsson/md-dmri/tree/master/tools/tensor_maths,
% which is part of the open-source multi-dimensional diffusion MRI toolbox (Nilsson et al., Proc. Intl. Soc. Mag. Reson. Med. (26), Paris, France, 2018).
%
methods(Static)
function alpha = rot_6x6(R)
alpha = [
R(1,1).^2 R(1,2).^2 R(1,3).^2 sqrt(2)*R(1,1)*R(1,2) sqrt(2)*R(1,3)*R(1,1) sqrt(2)*R(1,2)*R(1,3) ;
R(2,1).^2 R(2,2).^2 R(2,3).^2 sqrt(2)*R(2,1)*R(2,2) sqrt(2)*R(2,3)*R(2,1) sqrt(2)*R(2,2)*R(2,3) ;
R(3,1).^2 R(3,2).^2 R(3,3).^2 sqrt(2)*R(3,1)*R(3,2) sqrt(2)*R(3,3)*R(3,1) sqrt(2)*R(3,2)*R(3,3) ;
sqrt(2)*R(1,1)*R(2,1) sqrt(2)*R(1,2)*R(2,2) sqrt(2)*R(1,3)*R(2,3) R(1,2)*R(2,1)+R(2,2)*R(1,1) R(1,3)*R(2,1)+R(2,3)*R(1,1) R(1,2)*R(2,3)+R(2,2)*R(1,3);
sqrt(2)*R(3,1)*R(1,1) sqrt(2)*R(3,2)*R(1,2) sqrt(2)*R(3,3)*R(1,3) R(3,2)*R(1,1)+R(1,2)*R(3,1) R(3,3)*R(1,1)+R(1,3)*R(3,1) R(3,2)*R(1,3)+R(1,2)*R(3,3);
sqrt(2)*R(2,1)*R(3,1) sqrt(2)*R(2,2)*R(3,2) sqrt(2)*R(2,3)*R(3,3) R(2,2)*R(3,1)+R(3,2)*R(2,1) R(2,3)*R(3,1)+R(3,3)*R(2,1) R(2,2)*R(3,3)+R(3,2)*R(2,3);
];
end
function E_iso = iso_3x3()
E_iso = eye(3)/3;
end
function [E_bulk, E_shear, E2_iso] = iso_6x6()
E_iso = Tensor.t_3x3_to_1x6(Tensor.iso_3x3());
E_bulk = Tensor.outer(E_iso, E_iso);
E_shear = eye(6)/3 - E_bulk;
E2_iso = eye(6)/3;
end
function [E_bulk, E_shear, E2_iso] = iso_1x21()
[E_bulk, E_shear, E2_iso] = Tensor.iso_6x6();
E_bulk = Tensor.t_6x6_to_1x21(E_bulk);
E_shear = Tensor.t_6x6_to_1x21(E_shear);
E2_iso = Tensor.t_6x6_to_1x21(E2_iso);
end
function t_1x6 = t_3x3_to_1x6(t_3x3)
t_1x6 = t_3x3([1 5 9 2 3 6]) .* [1 1 1 sqrt(2) sqrt(2) sqrt(2)];
end
function t_3x3 = t_1x6_to_3x3(t_1x6)
assert(size(t_1x6,2) == 6)
t = t_1x6 .* [1 1 1 sqrt(1/2) sqrt(1/2) sqrt(1/2)];
t_3x3 = t([1 4 5; 4 2 6; 5 6 3]);
end
function t_1x6 = v_1x3_to_1x6(v_1x3,ad,rd)
assert(size(v_1x3,2) == 3)
if nargin < 3
rd = 0;
end
if nargin < 2
ad = 1;
end
x = v_1x3(:,1);
y = v_1x3(:,2);
z = v_1x3(:,3);
o = zeros([1 3]);
e = ones([1 3]);
c = sqrt(2);
t_1x6 = [x.*x y.*y z.*z c.*x.*y c.*x.*z c.*y.*z] .* (ad - rd) + [e o] .* rd;
end
function t_1x15 = v_1x3_to_1x15(v_1x3)
assert(size(v_1x3,2) == 3)
t_1x15 = [...
v_1x3(:,1) .* v_1x3(:,1) .* v_1x3(:,1) .* v_1x3(:,1) * sqrt(1) ...
v_1x3(:,1) .* v_1x3(:,1) .* v_1x3(:,1) .* v_1x3(:,2) * sqrt(4) ...
v_1x3(:,1) .* v_1x3(:,1) .* v_1x3(:,1) .* v_1x3(:,3) * sqrt(4) ...
v_1x3(:,1) .* v_1x3(:,1) .* v_1x3(:,2) .* v_1x3(:,2) * sqrt(6) ...
v_1x3(:,1) .* v_1x3(:,1) .* v_1x3(:,2) .* v_1x3(:,3) * sqrt(12) ...
v_1x3(:,1) .* v_1x3(:,1) .* v_1x3(:,3) .* v_1x3(:,3) * sqrt(6) ...
v_1x3(:,1) .* v_1x3(:,2) .* v_1x3(:,2) .* v_1x3(:,2) * sqrt(4) ...
v_1x3(:,1) .* v_1x3(:,2) .* v_1x3(:,2) .* v_1x3(:,3) * sqrt(12) ...
v_1x3(:,1) .* v_1x3(:,2) .* v_1x3(:,3) .* v_1x3(:,3) * sqrt(12) ...
v_1x3(:,1) .* v_1x3(:,3) .* v_1x3(:,3) .* v_1x3(:,3) * sqrt(4) ...
v_1x3(:,2) .* v_1x3(:,2) .* v_1x3(:,2) .* v_1x3(:,2) * sqrt(1) ...
v_1x3(:,2) .* v_1x3(:,2) .* v_1x3(:,2) .* v_1x3(:,3) * sqrt(4) ...
v_1x3(:,2) .* v_1x3(:,2) .* v_1x3(:,3) .* v_1x3(:,3) * sqrt(6) ...
v_1x3(:,2) .* v_1x3(:,3) .* v_1x3(:,3) .* v_1x3(:,3) * sqrt(4) ...
v_1x3(:,3) .* v_1x3(:,3) .* v_1x3(:,3) .* v_1x3(:,3) * sqrt(1) ];
end
function t = t_1x15_to_1x21(N)
N = N ./ repmat(sqrt([1 4 4 6 12 6 4 12 12 4 1 4 6 4 1]), size(N,1), 1);
t = [...
[N(:,1) N(:,11) N(:,15)] * sqrt(1) ...
[N(:,4) N(:,6) N(:,13)] * sqrt(2) ...
[N(:,2) N(:,8) N(:,14)] * sqrt(4) ...
[N(:,5) N(:,3) N(:,12)] * sqrt(4) ...
[N(:,7) N(:,10) N(:,9)] * sqrt(4) ...
[N(:,13) N(:,6) N(:,4)] * sqrt(4) ...
[N(:,9) N(:,8) N(:,5)] * sqrt(8) ];
end
function d_1x6 = tpars_to_1x6(trace, delta, uvec)
assert(size(uvec,2) == 3)
t_stick = Tensor.v_1x3_to_1x6(uvec, 1, 0);
t_sphere = Tensor.t_3x3_to_1x6(Tensor.iso_3x3());
c2 = trace .* delta;
c1 = trace - c2;
d_1x6 = c1 .* t_sphere + c2 .* t_stick;
end
function t_1x21 = t_1x6_to_1x21(t_1x6)
assert(size(t_1x6,2) == 6)
xx = t_1x6(:,1);
yy = t_1x6(:,2);
zz = t_1x6(:,3);
xy = t_1x6(:,4);
xz = t_1x6(:,5);
yz = t_1x6(:,6);
t_1x21 = [...
[xx.*xx yy.*yy zz.*zz] * sqrt(1) ...
[xx.*yy xx.*zz yy.*zz] * sqrt(2) ...
[xx.*yz yy.*xz zz.*xy] * sqrt(2) ...
[xx.*xy xx.*xz] * sqrt(2) ...
[yy.*xy yy.*yz] * sqrt(2) ...
[zz.*xz zz.*yz] * sqrt(2) ...
[xy.*xy xz.*xz yz.*yz] * sqrt(1) ...
[xy.*xz xy.*yz xz.*yz] * sqrt(2) ];
% % equivalent:
% t_1x21 = zeros([size(t_1x6,1) 21]);
% for i = 1:size(t_1x6,1)
% t_1x21(i,:) = Tensor.t_6x6_to_1x21(Tensor.outer(t_1x6(i,:),t_1x6(i,:)));
% end
end
function t_1x21 = t_6x6_to_1x21(t_6x6)
assert(all(size(t_6x6) == [6 6]))
t_1x21 = [...
[t_6x6(1,1) t_6x6(2,2) t_6x6(3,3)] * sqrt(1) ...
[t_6x6(1,2) t_6x6(1,3) t_6x6(2,3)] * sqrt(2) ...
[t_6x6(1,6) t_6x6(5,2) t_6x6(4,3)] * sqrt(2) ... % xxyz yyxz zzxy
[t_6x6(1,4) t_6x6(1,5) t_6x6(2,4)] * sqrt(2) ...
[t_6x6(2,6) t_6x6(3,5) t_6x6(3,6)] * sqrt(2) ...
[t_6x6(4,4) t_6x6(5,5) t_6x6(6,6)] * sqrt(1) ...
[t_6x6(4,5) t_6x6(4,6) t_6x6(5,6)] * sqrt(2)];
end
function t_6x6 = t_1x21_to_6x6(t_1x21)
% function t = tm_1x21_to_6x6(t)
%
% Convert fourth-order tensor in 1x21 format to 6x6 format
assert(size(t_1x21,2) == 21)
xxxx = t_1x21(1); % all same
yyyy = t_1x21(2);
zzzz = t_1x21(3);
c = sqrt( 2 );
xxyy = t_1x21(4) / c;
xxzz = t_1x21(5) / c;
yyzz = t_1x21(6) / c;
c = sqrt( 2 );
xxyz = t_1x21(7) / c;
yyxz = t_1x21(8) / c;
zzxy = t_1x21(9) / c;
c = sqrt( 2 );
xxxy = t_1x21(10) / c;
xxxz = t_1x21(11) / c;
yyxy = t_1x21(12) / c;
yyyz = t_1x21(13) / c;
zzxz = t_1x21(14) / c;
zzyz = t_1x21(15) / c;
c = 1;
xyxy = t_1x21(16) / c;
xzxz = t_1x21(17) / c;
yzyz = t_1x21(18) / c;
c = sqrt(2);
xyxz = t_1x21(19) / c;
xyyz = t_1x21(20) / c;
xzyz = t_1x21(21) / c;
A = [...
xxxx xxyy xxzz;
xxyy yyyy yyzz;
xxzz yyzz zzzz]; % 6 unique
B = [...
xxxy xxxz xxyz;
yyxy yyxz yyyz;
zzxy zzxz zzyz]; % 9 unique
C = [...
xyxy xyxz xyyz;
xyxz xzxz xzyz;
xyyz xzyz yzyz];
t_6x6 = [A B; B' C];
end
function t_3x3x3x3 = t_1x21_to_3x3x3x3(t_1x21)
assert(size(t_1x21,2)==21)
xxxx = t_1x21(1); % all same
yyyy = t_1x21(2);
zzzz = t_1x21(3);
c = sqrt(2);
xxyy = t_1x21(4) / c;
xxzz = t_1x21(5) / c;
yyzz = t_1x21(6) / c;
c = sqrt(4);
xxyz = t_1x21(7) / c;
yyxz = t_1x21(8) / c;
zzxy = t_1x21(9) / c;
c = sqrt(4);
xxxy = t_1x21(10) / c;
xxxz = t_1x21(11) / c;
yyxy = t_1x21(12) / c;
yyyz = t_1x21(13) / c;
zzxz = t_1x21(14) / c;
zzyz = t_1x21(15) / c;
c = sqrt(4);
xyxy = t_1x21(16) / c;
xzxz = t_1x21(17) / c;
yzyz = t_1x21(18) / c;
c = sqrt(8);
xyxz = t_1x21(19) / c;
xyyz = t_1x21(20) / c;
xzyz = t_1x21(21) / c;
t_3x3x3x3 = zeros(3,3,3,3);
% 9
t_3x3x3x3(1,1,1,1) = xxxx;
t_3x3x3x3(1,1,1,2) = xxxy;
t_3x3x3x3(1,1,1,3) = xxxz;
t_3x3x3x3(1,1,2,1) = xxxy;
t_3x3x3x3(1,1,2,2) = xxyy;
t_3x3x3x3(1,1,2,3) = xxyz;
t_3x3x3x3(1,1,3,1) = xxxz;
t_3x3x3x3(1,1,3,2) = xxyz;
t_3x3x3x3(1,1,3,3) = xxzz;
% 9+9
t_3x3x3x3(1,2,1,1) = xxxy;
t_3x3x3x3(1,2,1,2) = xyxy;
t_3x3x3x3(1,2,1,3) = xyxz;
t_3x3x3x3(1,2,2,1) = xyxy;
t_3x3x3x3(1,2,2,2) = yyxy;
t_3x3x3x3(1,2,2,3) = xyyz;
t_3x3x3x3(1,2,3,1) = xyxz;
t_3x3x3x3(1,2,3,2) = xyyz;
t_3x3x3x3(1,2,3,3) = zzxy;
% 27
t_3x3x3x3(1,3,1,1) = xxxz;
t_3x3x3x3(1,3,1,2) = xyxz;
t_3x3x3x3(1,3,1,3) = xzxz;
t_3x3x3x3(1,3,2,1) = xyxz;
t_3x3x3x3(1,3,2,2) = yyxz;
t_3x3x3x3(1,3,2,3) = xzyz;
t_3x3x3x3(1,3,3,1) = xzxz;
t_3x3x3x3(1,3,3,2) = xzyz;
t_3x3x3x3(1,3,3,3) = zzxz;
% 36
t_3x3x3x3(2,1,1,1) = xxxy;
t_3x3x3x3(2,1,1,2) = xyxy;
t_3x3x3x3(2,1,1,3) = xyxz;
t_3x3x3x3(2,1,2,1) = xyxy;
t_3x3x3x3(2,1,2,2) = yyxy;
t_3x3x3x3(2,1,2,3) = xyyz;
t_3x3x3x3(2,1,3,1) = xyxz;
t_3x3x3x3(2,1,3,2) = xyyz;
t_3x3x3x3(2,1,3,3) = zzxy;
% 45
t_3x3x3x3(2,2,1,1) = xxyy;
t_3x3x3x3(2,2,1,2) = yyxy;
t_3x3x3x3(2,2,1,3) = yyxz;
t_3x3x3x3(2,2,2,1) = yyxy;
t_3x3x3x3(2,2,2,2) = yyyy;
t_3x3x3x3(2,2,2,3) = yyyz;
t_3x3x3x3(2,2,3,1) = yyxz;
t_3x3x3x3(2,2,3,2) = yyyz;
t_3x3x3x3(2,2,3,3) = yyzz;
% 54
t_3x3x3x3(2,3,1,1) = xxyz;
t_3x3x3x3(2,3,1,2) = xyyz;
t_3x3x3x3(2,3,1,3) = xzyz;
t_3x3x3x3(2,3,2,1) = xyyz;
t_3x3x3x3(2,3,2,2) = yyyz;
t_3x3x3x3(2,3,2,3) = yzyz;
t_3x3x3x3(2,3,3,1) = xzyz;
t_3x3x3x3(2,3,3,2) = yzyz;
t_3x3x3x3(2,3,3,3) = zzyz;
% 63
t_3x3x3x3(3,1,1,1) = xxxz;
t_3x3x3x3(3,1,1,2) = xyxz;
t_3x3x3x3(3,1,1,3) = xzxz;
t_3x3x3x3(3,1,2,1) = xyxz;
t_3x3x3x3(3,1,2,2) = yyxz;
t_3x3x3x3(3,1,2,3) = xzyz;
t_3x3x3x3(3,1,3,1) = xzxz;
t_3x3x3x3(3,1,3,2) = xzyz;
t_3x3x3x3(3,1,3,3) = zzxz;
% 72
t_3x3x3x3(3,2,1,1) = xxyz;
t_3x3x3x3(3,2,1,2) = xyyz;
t_3x3x3x3(3,2,1,3) = xzyz;
t_3x3x3x3(3,2,2,1) = xyyz;
t_3x3x3x3(3,2,2,2) = yyyz;
t_3x3x3x3(3,2,2,3) = yzyz;
t_3x3x3x3(3,2,3,1) = xzyz;
t_3x3x3x3(3,2,3,2) = yzyz;
t_3x3x3x3(3,2,3,3) = zzyz;
% 81
t_3x3x3x3(3,3,1,1) = xxzz;
t_3x3x3x3(3,3,1,2) = zzxy;
t_3x3x3x3(3,3,1,3) = zzxz;
t_3x3x3x3(3,3,2,1) = zzxy;
t_3x3x3x3(3,3,2,2) = yyzz;
t_3x3x3x3(3,3,2,3) = zzyz;
t_3x3x3x3(3,3,3,1) = zzxz;
t_3x3x3x3(3,3,3,2) = zzyz;
t_3x3x3x3(3,3,3,3) = zzzz;
end
function x = inner(t1,t2)
x = t1 * t2';
end
function x = outer(t1,t2)
x = t1' * t2;
end
function L = eigval(t_1x6)
assert(size(t_1x6,2) == 6)
L = zeros(size(t_1x6,1), 3);
for i = 1:size(t_1x6,1)
t_3x3 = Tensor.t_1x6_to_3x3(t_1x6(i,:));
if all(~isnan(t_3x3(:)))
L(i,:) = eigs(t_3x3);
else
L(i,:) = NaN(1,3);
end
end
end
function md = md(t_1x6)
assert(size(t_1x6,2) == 6)
md = Tensor.inner(t_1x6, Tensor.t_3x3_to_1x6(Tensor.iso_3x3()));
end
function v_lambda = v_lambda(t_1x6)
assert(size(t_1x6,2) == 6)
[~,E_shear] = Tensor.iso_6x6();
v_lambda = Tensor.inner(Tensor.t_1x6_to_1x21(t_1x6), Tensor.t_6x6_to_1x21(E_shear));
end
function fa = fa(a,b)
if nargin > 1
md = a;
vl = b;
else
assert(size(a,2) == 6)
md = Tensor.md(a);
vl = Tensor.v_lambda(a);
end
fa = sqrt((3/2) * (1 + md.^2./vl).^(-1));
fa = real(fa);
end
end
end