Solving a non linear equation using the Least squares method

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B 2022 年 5 月 22 日
This concern MRI data and diffusion tensor imaging
I am trying to estimate D within each voxel using the data in data1.mat and i am not sure if i have done it correctly
Any help would appreciated
syms d_xx d_yy d_zz d_xy d_xz d_yz real
D = [d_xx, d_xy, d_xz; d_xy, d_yy, d_yz; d_xz, d_yz, d_zz]
syms x y z real
g = [x; y; z]
syms b S_0
S = S_0*exp(-b*g'*D*g)
logS = expand(1/2* g'*D*g)
Load data
load data1.mat
b_val = 1000;
xvec = g(:,1);
yvec = g(:,2);
zvec = g(:,3);
Svec = reshape(S, [], 64);
S0vec = reshape(S0, [], 1);
Construct A and b
A = [-xvec.^2/2, -xvec.*yvec, -xvec.*zvec, -yvec.*2/2, -yvec.*zvec, -zvec.^2/2];
b = log(Svec./S0vec);
c = zeros(6, size(S0vec, 1));
for i = 1:size(S0vec, 1)
c(:,i) = A \ b(i,:)';
% Tensor for each voxel
D = zeros(3, 3, size(S0vec, 1));
for i = 1:size(S0vec, 1)
D(:,:,i) = [c(1,i), c(2,i), c(3,i); c(2,i), c(4,i), c(5,i); c(3,i), c(5,i), c(6,i)];

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