Optimal transport registration

function  [u dt cur energy] = match_pdfs_2d_haker( pdf1, pdf2)

% normalize pdfs
cumsum1 = cumIntegrate(pdf1(:));
cumsum2 = cumIntegrate(pdf2(:));
pdf1 = pdf1/cumsum1(end);
pdf2 = pdf2/cumsum2(end);

[M N] = size(pdf1);

% find an initial map, by doing 1D matchings in x and then in y.
proj1 = zeros(1,N);
proj2 = zeros(1,N);
for i=1:N
    cumsum1 = cumIntegrate(pdf1(:,i));
    cumsum2 = cumIntegrate(pdf2(:,i));
    proj1(i) = cumsum1(end);
    proj2(i) = cumsum2(end);
end;

a = find_warp(proj2, proj1);
ga = gradientAccurate(a);

[X,Y] = meshgrid(1:N, 1:M);
interpPDF2 = interp2(X,Y,pdf2, a, 1:M, 'spline',0.);
b=zeros(M,N);
for i=1:N
    b(:,i) = find_warp( interpPDF2(:, i).*ga(i), pdf1(:,i));
end;

%u^0 = (a,b)
u=reshape([repmat(a', [M 1]) b], M, N,2);

%let's do a constant number of iterations
for t=1:3000

    %%%%%%%% compute grad^orthog Laplace^{-1} div(u^orthog) %%%%%%%%%
    %%%%%%%% where orthog stands for a 90 degrees rotations,%%%%%%%%%
    %%%%%%%% and Laplace^-1(g) solves Laplacian f = g       %%%%%%%%%

    [dudx dudy] = gradientAccurate(u);
    divorthog = -dudy(:,:,1)+dudx(:,:,2); %div(u^orthog)
    f=reshape(poicalc(divorthog(:),1,1,M,N), M, N); %Laplace^{-1} div(u^orthog)
    [dfdx dfdy] = gradientAccurate(f); %grad Laplace^{-1} div(u^orthog)

    % 1./pdf1 * grad^orthog Laplace^{-1} div(u^orthog)
    update1 = repmat(1./pdf1, [1 1 2]).*reshape([-dfdy dfdx], size(u));

    %%%%%%%% upwind for Du %%%%%%%%%%%%%%%%%
    [dudx dudy] = gradientAccurate(u);
    dudxm = dudx;
    dudym = dudy;
    dudxp = dudx;
    dudyp = dudy;
    dudxm(:, 3:end-2, :) = (3*u(:, 3:end-2, :) - 4*u(:, 2:end-3, :) + u(:, 1:end-4, :))*0.5;
    dudxp(:, 3:end-2, :) = (-3*u(:, 3:end-2, :) + 4*u(:, 4:end-1, :) - u(:, 5:end, :))*0.5;
    dudym(3:end-2, :, :) = (3*u(3:end-2, :, :) - 4*u(2:end-3, :, :) + u(1:end-4, :, :))*0.5;
    dudyp(3:end-2, :, :) = (-3*u(3:end-2, :, :) + 4*u(4:end-1, :, :) - u(5:end, :, :))*0.5;

    Dupp = abs(dudxp(:,:,1).*dudyp(:,:,2)-dudxp(:,:,2).*dudyp(:,:,1));
    Dupm = abs(dudxp(:,:,1).*dudym(:,:,2)-dudxp(:,:,2).*dudym(:,:,1));
    Dump = abs(dudxm(:,:,1).*dudyp(:,:,2)-dudxm(:,:,2).*dudyp(:,:,1));
    Dumm = abs(dudxm(:,:,1).*dudym(:,:,2)-dudxm(:,:,2).*dudym(:,:,1));

    Du =  (update1(:, :,1)>0).*(update1(:, :,2)>0).*Dupp ...
        + (update1(:, :,1)>0).*(update1(:, :,2)<0).*Dupm ...
        + (update1(:, :,1)<0).*(update1(:, :,2)>0).*Dump ...
        + (update1(:, :,1)<0).*(update1(:, :,2)<0).*Dumm;

    dt(t) = 0.2*min(1./abs(update1(:))); %dt according to stability conditions

    % update : du/dt = 1./pdf1 * det(Jac(u)) * update1
    u = u+dt(t).*repmat(Du, [1 1 2]).*update1;
end;



function y = find_warp(pdf1, pdf2) %1D matching
cdf1 = cumIntegrate(pdf1);
cdf2 = cumIntegrate(pdf2);
cdf1 = cdf1/cdf1(end);
cdf2 = cdf2/cdf2(end);
N = length(cdf1);
[ucdf1, uxx] = unique(cdf1, 'first');
y = transpose(interp1(ucdf1,uxx,cdf2,'spline',N));


function y = cumIntegrate(f)  % generic function to approximate the cumulative integral
 y= cumtrapz(f);  % or y=cumsum, or y=intgrad1 with the external library

function varargout=gradientAccurate(f)
if (length(f)==length(f(:)))
    [varargout{1}] = gradient2(f);  %or my 5th order gradient2(f)  code
else
    [varargout{1} varargout{2}] = gradient2(f);  %or my 5th order gradient2(f) code
end