Contents

introduction

% s_SDnDTI_designBvec4AcquiredData.m
%
%   A script for selecting diffusion encoding directions for SDnDTI from
%   acquired data with given diffusion encoding directions (e.g., 30 out of
%   90 directions).
%
%   Source code:
%       https://github.com/qiyuantian/SDnDTI/blob/main/s_SDnDTI_designBvec4AcquiredData.m
%
%   HTML file can be automatically generaged using command:
%       publish('s_SDnDTI_designBvec4AcquiredData.m', 'html');
%
%   Reference:
%       [1] Tian Q, Li Z, Fan Q, Polimeni JR, Bilgic B, Salat DH, Huang SY.
%       SDnDTI: Self-supervised deep learning-based denoising for diffusion
%       tensor MRI. NeuroImage, 2022; 253: 119033.
%
% (c) Qiyuan Tian, Harvard, 2022

load bvecs from Human Connectome Project

clear, clc, close all

bvals_all = dlmread('hcp_bval.txt'); % all bvals
bvecs_all = dlmread('hcp_bvec.txt'); % all bvecs

bvals_all = round(bvals_all / 100) * 100;
disp(unique(bvals_all));

dirs = bvecs_all(bvals_all > 100 & bvals_all < 1500, :); % only use bvecs for b=1000
dirs_vis = dirs .* sign(dirs(:, 3)); % directions flipped to z > 0 for visualization

figure; % display diffusion encoding directions, all flipped to z > 0
plot3(dirs_vis(:, 1), dirs_vis(:, 2), dirs_vis(:, 3), '.');
grid on, axis equal
zlim([-1, 1])
title('90 diffusion encoding directions');

           0
        1000
        2000
        3000

select all sets of 6 encoding directions optimal for tensor fitting

% 6 optimized directions from the DSM scheme that minimizes the condition
% number of the diffusion tensor transformation matrix
% from S Skare et al., J Magn Reson. 2000;147(2):340-52
% 10, 20, 30 (dsm10.m, dsm20.m, dsm30.m) or more directions should be used for
% very noisy data

dsm6 = [0.91, 0.416, 0; ...
       0, 0.91, 0.416; ...
       0.416, 0, 0.91; ...
       0.91, -0.416, 0; ...
       0, 0.91, -0.416; ...
      -0.416, 0, 0.91];

dsm6_norm = dsm6 ./ sqrt(dsm6(:, 1) .^ 2 + dsm6(:, 2) .^ 2 + dsm6(:, 3) .^ 2); % normalize vectors

% randomly rotate the DSM6 dirs, select their nearest 6 dirs from acquired
% dirs, keep those with low condition number

% selected dirs are not perfectly DSM6 dirs, so condition number might be
% large, which is not optimal for tensor fitting and amplifies noise, which
% is especially a problem for a small number of encoding directions (e.g.,
% lower than 60)

rotang_all = [];
condnum_all = [];
ind_all = [];
for ii = 1 : 100000 % number of iterations can be increased

    rotangs = rand(1, 3) * 2 * pi; % random angles to rotate around x, y, z axis
    R = rot3d(rotangs); % rotation matrix
    dsm6_rot = (R * dsm6_norm')'; % roated directions

    % find 6 nearest dirs in acquired dirs
    angerrors = acosd(abs(dsm6_rot * dirs')); % angles btw rotated DSM6 dirs and acquired dirs
    [minerrors, ind] = min(angerrors, [], 2); % 6 dirs with min angles compared to rotated DSM6 dirs

    condnum = cond(amatrix(dirs(ind, :))); % cond number of tensor tx matrix of selected dirs

    % only use dirs with low cond number
    if condnum < 1.6 % threshold should be determined based on given data, 1.6 is good in our experience
        if isempty(ind_all) || ~any(sum(ind_all == sort(ind'), 2) == 6) % make sure no repetition

            % record params for satisfied sets
            condnum_all = cat(1, condnum_all, condnum);
            ind_all = cat(1, ind_all, sort(ind'));
            rotang_all = cat(1, rotang_all, rotangs);
        end
    end
end
disp(size(ind_all, 1))

for ii = 1 : 2
    figure; % display two selected sets of 6 optimal directions
    plot3(dirs_vis(:, 1), dirs_vis(:, 2), dirs_vis(:, 3), '.'); % all dirs
    hold on

    visdirs_use = dirs_vis(ind_all(ii, :), :);
    plot3(visdirs_use(:, 1), visdirs_use(:, 2), visdirs_use(:, 3), 'o'); % selected dirs

    R = rot3d(rotang_all(ii, :));
    dsm6_rot = (R * dsm6_norm')';
    dsm6_rot_vis = dsm6_rot .* sign(dsm6_rot(:, 3));
    plot3(dsm6_rot_vis(:, 1), dsm6_rot_vis(:, 2), dsm6_rot_vis(:, 3), 'x'); % rotated DSM6 dirs

    grid on, axis equal
    xlim([-1, 1])
    ylim([-1, 1])
    zlim([-1, 1])
    title(['cond num=' num2str(condnum_all(ii))]);
end

    21

select N sets out of all selected sets of 6 dirs

% directions from all sets are uniformly distributed on z>0 hemisphere
% (since v and -v are identical for diffusion encoding) to maximize angular
% coverage as well as on whole sphere

N = 3; % select 3 sets as an example, so in total of 3x6=18 dirs
bvecs_hemi = []; % designed diffusion encoding directions
unif_min = 10^10; % init using a large number
ind_min = [];

for ii = 1 : 100000 % number of iterations can be increased for better results

    r = randperm(size(ind_all, 1), N);
    ind = ind_all(r, :);

    dirs_all = dirs(ind(:), :); % all 30 directions

    % [v; -v] is used for optimization such that v is uniformly distributed
    % on z>0 hemisphere
    dirs_tmp = [dirs_all; -dirs_all];

    % uniformly distributed directions have lowest electrostatic potential energy
    % details see Jones Magn. Reson. Med., 51 (2004), pp. 807-815
    unif = potentialenergy(dirs_tmp); % uniformity of all 30 directions

    if unif < unif_min % keep new directions if uniformity is better
        unif_min = unif;
        bvecs_hemi = dirs_all;
        ind_min = ind;

        disp(unif_min)
    end
end

n = size(bvecs_hemi, 1) / N;

figure
for ii = 1 : N
    idxs = (ii - 1) * n + 1;
    idxe = ii  * n;
    bvecs_vis = bvecs_hemi(idxs:idxe, :);

    tmp = sign(bvecs_vis(:, 3));
    tmp(tmp == 0) = 1;
    bvecs_vis = bvecs_vis .* tmp; % flip dirs with z<0 to z>0

    plot3(bvecs_vis(:, 1), bvecs_vis(:, 2), bvecs_vis(:, 3), '.'); % each color represents a set
    hold on
end

grid on, axis equal
xlim([-1, 1])
ylim([-1, 1])
zlim([-1, 1])
title('bvecs are uniform on both hemi and whole sphere');


figure
for ii = 1 : N
    idxs = (ii - 1) * n + 1;
    idxe = ii  * n;
    bvecs_vis = bvecs_hemi(idxs:idxe, :);

    tmp = sign(bvecs_vis(:, 3));
    tmp(tmp == 0) = 1;
    bvecs_vis = bvecs_vis .* tmp; % flip dirs with z<0 to z>0

    plot3(bvecs_vis(:, 1), bvecs_vis(:, 2), bvecs_vis(:, 3), 'o'); % each color represents a set
    hold on
end
plot3(dirs_vis(:, 1), dirs_vis(:, 2), dirs_vis(:, 3), 'k.');

grid on, axis equal
xlim([-1, 1])
ylim([-1, 1])
zlim([-1, 1])
title('selected directions out of all directions');

   1.0061e+03

  996.6677

  995.7374

  995.0967

  993.7528

  989.3878

DeepDTI pipeline

% subsequent data preparation and CNN training and application follows DeepDTI
% see codes here: https://github.com/qiyuantian/DeepDTI

% diffusion weighted images (DWIs) from each set, which are optimal for
% diffusion tensor fitting, serve as input for CNNs

% DWIs from all sets serve as target of CNNs

Published with MATLAB® R2018b