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boundedOpticalFlow3D.m
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boundedOpticalFlow3D.m
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function [du,dv,dw] = boundedOpticalFlow3D( data1, data2, varargin )
% [du,dv,dw] = boundedOpticalFlow3D( data1, data2, [, 'bound', bound, ...
% 'eta', eta, 'rho', rho ] )
%
% Computes optical flow vectors
%
% Inputs:
% data1/data2 - 3D arrays; optical flow vectors will point from voxels
% in data1 to voxels in data2
%
% Optional Inputs:
% bound is a real positive number specifying the largest possible
% optical flow vector
%
% Written by Nicholas Dwork - Copyright 2015
defaultBound = 1.2;
defaultEta = 8d-10;
defaultRho = 1d-3;
p = inputParser;
p.addRequired( 'data1' );
p.addRequired( 'data2' );
p.addParameter( 'bound', defaultBound, @isnumeric );
p.addParameter( 'eta', defaultEta, @isnumeric );
p.addParameter( 'rho', defaultRho, @isnumeric );
p.parse( data1, data2, varargin{:} );
bound = p.Results.bound;
eta = p.Results.eta;
rho = p.Results.rho;
pyramid1 = makeDataPyramid3( data1, 3 );
pyramid2 = makeDataPyramid3( data2, 3 );
sLevel1 = size( pyramid1{end} );
du = zeros( sLevel1(1), sLevel1(2), sLevel1(3) );
dv = zeros( sLevel1(1), sLevel1(2), sLevel1(3) );
dw = zeros( sLevel1(1), sLevel1(2), sLevel1(3) );
for level=numel(pyramid1):-1:1
disp(['Working on pyramid level ', num2str(level)]);
tmp1 = pyramid1{level};
[nRows, nCols, nPages] = size( pyramid1{level} );
du = medfilt3( du, [5 5 5], 'symmetric' );
dv = medfilt3( dv, [5 5 5], 'symmetric' );
dw = medfilt3( dw, [5 5 5], 'symmetric' );
scale = nRows / size(du,1);
du = resize(du, [nRows nCols nPages], 'bilinear') * scale;
dv = resize(dv, [nRows nCols nPages], 'bilinear') * scale;
dw = resize(dw, [nRows nCols nPages], 'bilinear') * scale;
tmp2 = pyramid2{level};
interp2 = ofInterp3D( tmp2, du, dv, dw );
scaledEta = eta * numel(tmp1(:));
[newDu, newDv, newDw] = boundedOfADMM3D( tmp1, interp2, ...
scaledEta, rho, bound );
du = du + newDu;
dv = dv + newDv;
dw = dw + newDw;
showDiagnostics = 0;
if showDiagnostics==1
close all;
[nRows,nCols,nPages] = size(tmp1);
midRow = ceil(nRows/2);
midCol = ceil(nCols/2);
midPage = ceil(nPages/2);
img1 = squeeze(tmp1(:,:,midPage));
minImg1 = min(img1(:)); maxImg1 = max(img1(:));
figure;
imshow( imresize( img1, ceil(512/nRows), 'nearest' ), ...
[minImg1 maxImg1] );
title('img1', 'FontSize', 20);
img2 = squeeze(tmp2(:,:,midPage));
figure;
imshow( imresize( img2, ceil(512/nRows), 'nearest' ), ...
[minImg1 maxImg1] );
title('img2', 'FontSize', 20);
interped2 = ofInterp3D( tmp2, du, dv, dw );
interpedImg2 = squeeze(interped2(:,:,midPage));
figure;
imshow( imresize( interpedImg2, ceil(512/nRows), 'nearest' ), ...
[minImg1 maxImg1] );
title('interp2', 'FontSize', 20);
duImg = squeeze(du(:,:,midPage));
figure;
imshow( imresize( duImg, ceil(512/nRows), 'nearest' ), [] );
title('du', 'FontSize', 20);
dvImg = squeeze(dv(:,:,midPage));
figure;
imshow( imresize( dvImg, ceil(512/nRows), 'nearest' ), [] );
title('dv', 'FontSize', 20);
dwImg = squeeze(dw(:,:,midPage));
figure;
imshow( imresize( dwImg, ceil(512/nRows), 'nearest' ), [] );
title('dw', 'FontSize', 20);
drawnow;
end
end
end
function [du,dv,dw] = boundedOfADMM3D( data1, data2, eta, rho, bound )
% ADMM parameters
nIter = 500;
% data1 and data2 must be of type double
data1 = double( data1 );
data2 = double( data2 );
% Check to make sure values of data1 and data2 are in a reasonable range
if( max(data1(:)) > 1.5 || max(data2(:)) > 1.5 )
error('Input values should be less than ~1');
end
[nRows, nCols, nPages] = size( data1 );
applyD1 = @(u) cat(2, u(:,2:end,:) - u(:,1:end-1,:),zeros(nRows,1,nPages));
applyD2 = @(u) cat(1, u(2:end,:,:) - u(1:end-1,:,:),zeros(1,nCols,nPages));
applyD3 = @(u) cat(3, u(:,:,2:end) - u(:,:,1:end-1),zeros(nRows,nCols,1));
applyD1Trans = @(u) cat(2,-u(:,1,:),u(:,1:end-2,:) - u(:,2:end-1,:),u(:,end-1,:));
applyD2Trans = @(u) cat(1,-u(1,:,:),u(1:end-2,:,:) - u(2:end-1,:,:),u(end-1,:,:));
applyD3Trans = @(u) cat(3,-u(:,:,1),u(:,:,1:end-2) - u(:,:,2:end-1),u(:,:,end-1));
applyM = @(u) 2*u + applyD1Trans(applyD1(u)) + ...
applyD2Trans(applyD2(u)) + applyD3Trans(applyD3(u));
allOnes = ones(nRows,nCols,nPages);
eigValsM = mirt_dctn(applyM(mirt_idctn(allOnes)));
eigValsMInv = eigValsM.^(-1);
[Iu1, Iv1, Iw1] = imgDeriv3D( data1 );
[Iu2, Iv2, Iw2] = imgDeriv3D( data2 );
Iu = 0.5 * ( Iu1 + Iu2 );
Iv = 0.5 * ( Iv1 + Iv2 );
Iw = 0.5 * ( Iw1 + Iw2 );
It = data2 - data1;
% Initializations
x1 = zeros( nRows, nCols, nPages );
x2 = zeros( nRows, nCols, nPages );
x3 = zeros( nRows, nCols, nPages );
r1 = zeros( nRows, nCols, nPages );
r2 = zeros( nRows, nCols, nPages );
r3 = zeros( nRows, nCols, nPages );
y1 = zeros( nRows, nCols, nPages );
y2 = zeros( nRows, nCols, nPages );
y3 = zeros( nRows, nCols, nPages );
z1_1 = zeros( nRows, nCols, nPages );
z1_2 = zeros( nRows, nCols, nPages );
z1_3 = zeros( nRows, nCols, nPages );
z2_1 = zeros( nRows, nCols, nPages );
z2_2 = zeros( nRows, nCols, nPages );
z2_3 = zeros( nRows, nCols, nPages );
z3_1 = zeros( nRows, nCols, nPages );
z3_2 = zeros( nRows, nCols, nPages );
z3_3 = zeros( nRows, nCols, nPages );
lambda1u = zeros( nRows, nCols, nPages );
lambda1v = zeros( nRows, nCols, nPages );
lambda1w = zeros( nRows, nCols, nPages );
lambda2u = zeros( nRows, nCols, nPages );
lambda2v = zeros( nRows, nCols, nPages );
lambda2w = zeros( nRows, nCols, nPages );
lambda3u_1 = zeros( nRows, nCols, nPages );
lambda3u_2 = zeros( nRows, nCols, nPages );
lambda3u_3 = zeros( nRows, nCols, nPages );
lambda3v_1 = zeros( nRows, nCols, nPages );
lambda3v_2 = zeros( nRows, nCols, nPages );
lambda3v_3 = zeros( nRows, nCols, nPages );
lambda3w_1 = zeros( nRows, nCols, nPages );
lambda3w_2 = zeros( nRows, nCols, nPages );
lambda3w_3 = zeros( nRows, nCols, nPages );
b = -It;
Iub = Iu.*b;
Ivb = Iv.*b;
Iwb = Iw.*b;
M11 = Iu.*Iu + rho; M12 = Iu.*Iv; M13 = Iu.*Iw;
M21 = M12; M22 = Iv.*Iv + rho; M23 = Iv.*Iw;
M31 = M13; M32 = M23; M33 = Iw.*Iw + rho;
KInv = ( M22 - M21./M11.*M12 );
K = 1. / KInv;
CInv = -M31./M11.*M12.*K.*M21./M11.*M13 + M31./M11.*M12.*K.*M23 - ...
M31./M11.*M13 + M32.*K.*M21./M11.*M13 - M32.*K.*M23 + M33;
C = 1. / CInv;
objectives = zeros(nIter,1);
for i=1:nIter
if mod(i,5)==0
disp([ 'Working on iteration ', num2str(i), ' of ', num2str(nIter) ]);
end
thisObjective = ofObjective( Iu, Iv, Iw, b, x1, x2, x3, eta );
objectives(i) = thisObjective;
% Update x
arg1 = r1 + y1 + ...
applyD1Trans(z1_1) + applyD2Trans(z1_2) + applyD3Trans(z1_3) ...
- ( lambda1u + lambda2u ) / rho ...
- ( applyD1Trans(lambda3u_1) + applyD2Trans(lambda3u_2) ...
+ applyD3Trans(lambda3u_3) ) / rho;
x1 = mirt_idctn(eigValsMInv.*mirt_dctn(arg1));
arg2 = r2 + y2 + ...
applyD1Trans(z2_1) + applyD2Trans(z2_2) + applyD3Trans(z2_3) ...
- ( lambda1v + lambda2v ) / rho ...
- ( applyD1Trans(lambda3v_1) + applyD2Trans(lambda3v_2) ...
+ applyD3Trans(lambda3v_3) ) / rho;
x2 = mirt_idctn(eigValsMInv.*mirt_dctn(arg2));
arg3 = r3 + y3 + ...
applyD1Trans(z3_1) + applyD2Trans(z3_2) + applyD3Trans(z3_3) ...
- ( lambda1w + lambda2w ) / rho ...
- ( applyD1Trans(lambda3w_1) + applyD2Trans(lambda3w_2) ...
+ applyD3Trans(lambda3w_3) ) / rho;
x3 = mirt_idctn(eigValsMInv.*mirt_dctn(arg3));
% Update r
rArg1 = x1 + lambda1u / rho;
rArg2 = x2 + lambda1v / rho;
rArg3 = x3 + lambda1w / rho;
rMag = sqrt( rArg1.*rArg1 + rArg2.*rArg2 + rArg3.*rArg3 );
factor = min( bound ./ rMag, 1 );
r1 = rArg1 .* factor;
r2 = rArg2 .* factor;
r3 = rArg3 .* factor;
% Update y
nu1 = Iub + lambda2u + rho*x1;
nu2 = Ivb + lambda2v + rho*x2;
nu3 = Iwb + lambda2w + rho*x3;
y3 = C.*( nu3 - M31./M11.*nu1 + M31./M11.*M12.*K.*nu2 - ...
M31./M11.*M12.*K.*M21./M11.*nu1 - M32.*K.*nu2 + ...
M32.*K.*M21./M11.*nu1 );
y2 = K .* (nu2 - M21./M11.*nu1 + M21./M11.*M13.*y3 - M23.*y3 );
y1 = 1./M11 .* ( nu1 - M12.*y2 - M13.*y3 );
% Update z
z1_1 = softThresh( applyD1(x1) + lambda3u_1/rho, eta/rho );
z1_2 = softThresh( applyD2(x1) + lambda3u_2/rho, eta/rho );
z1_3 = softThresh( applyD3(x1) + lambda3u_3/rho, eta/rho );
z2_1 = softThresh( applyD1(x2) + lambda3v_1/rho, eta/rho );
z2_2 = softThresh( applyD2(x2) + lambda3v_2/rho, eta/rho );
z2_3 = softThresh( applyD3(x2) + lambda3v_3/rho, eta/rho );
z3_1 = softThresh( applyD1(x3) + lambda3w_1/rho, eta/rho );
z3_2 = softThresh( applyD2(x3) + lambda3w_2/rho, eta/rho );
z3_3 = softThresh( applyD3(x3) + lambda3w_3/rho, eta/rho );
% Update lambdas
lambda1u = lambda1u + rho * ( x1 - r1 );
lambda1v = lambda1u + rho * ( x2 - r2 );
lambda1w = lambda1u + rho * ( x3 - r3 );
lambda2u = lambda2u + rho * ( x1 - y1 );
lambda2v = lambda2v + rho * ( x2 - y2 );
lambda2w = lambda2w + rho * ( x3 - y3 );
lambda3u_1 = lambda3u_1 + rho * ( applyD1(x1) - z1_1 );
lambda3u_2 = lambda3u_2 + rho * ( applyD2(x1) - z1_2 );
lambda3u_3 = lambda3u_3 + rho * ( applyD3(x1) - z1_3 );
lambda3v_1 = lambda3v_1 + rho * ( applyD1(x2) - z2_1 );
lambda3v_2 = lambda3v_2 + rho * ( applyD2(x2) - z2_2 );
lambda3v_3 = lambda3v_3 + rho * ( applyD3(x2) - z2_3 );
lambda3w_1 = lambda3w_1 + rho * ( applyD1(x3) - z3_1 );
lambda3w_2 = lambda3w_2 + rho * ( applyD2(x3) - z3_2 );
lambda3w_3 = lambda3w_3 + rho * ( applyD3(x3) - z3_3 );
end
du = x1; dv = x2; dw = x3;
% for diagnostics
showDiagnostics = 0;
if showDiagnostics==1
close all;
admmOptVal = ofObjective( Iu, Iv, Iw, b, x1, x2, x3, eta );
disp(['ADMM Optimal Value: ', num2str(admmOptVal) ] );
figure; plot( objectives );
ofRes = ofResidual3D( Iu, Iv, Iw, It, du, dv, dw );
%figure, imshow( imresize( ofRes, ceil(512/nRows), 'nearest' ), [] );
%title('OF Residual', 'FontSize', 20);
%load( 'star3D.mat' );
%objStar = ofObjective( Iu, Iv, Iw, b, x1Star, x2Star, x3Star, ...
% eta, D1, D2, D3 );
%figure, plot( ( objectives - objStar ) / objStar );
%title('relative error vs iteration');
drawnow;
disp(['Rho: ', num2str(rho)]);
disp(['Max objective: ', num2str(max( objectives ))]);
disp(['Min objective: ', num2str(min( objectives ))]);
close all; figure; plot( objectives ); drawnow;
end
end
function out = augLagrange( Iu, Iv, Iw, It, x1, x2, x3, r1, r2, r3, ...
y1, y2, y3, z1_1, z1_2, z1_3, z2_1, z2_2, z2_3, z3_1, z3_2, z3_3, ...
eta, rho, bound, ...
lambda1u, lambda1v, lambda1w, lambda2u, lambda2v, lambda2w, ...
lambda3u_1, lambda3u_2, lambda3u_3, lambda3v_1, lambda3v_2, ...
lambda3v_3, lambda3w_1, lambda3w_2, lambda3w_3 )
rMag = sqrt( r1.*r1 + r2.*r2 + r3.*r3 );
maxRMag = max( rMag(:) );
if maxRMag > bound
out = Inf;
return
end
[nRows,nCols,nPages] = size( Iu );
applyD1 = @(u) cat(2, u(:,2:end,:) - u(:,1:end-1,:), zeros(nRows,1,nPages));
applyD2 = @(u) cat(1, u(2:end,:,:) - u(1:end-1,:,:), zeros(1,nCols,nPages));
applyD3 = @(u) cat(3, u(:,:,2:end) - u(:,:,1:end-1), zeros(nRows,nCols,1));
D1x1 = applyD1(x1); D1x2 = applyD1(x2); D1x3 = applyD1(x3);
D2x1 = applyD2(x1); D2x2 = applyD2(x2); D2x3 = applyD2(x3);
D3x1 = applyD3(x1); D3x2 = applyD3(x2); D3x3 = applyD3(x3);
b = -It(:);
Ay = Iu .* y1 + Iv .* y2 + Iw .* y3;
gOfW = 0.5*norm(Ay(:) - b, 2)^2 + ...
eta*norm(z1_1(:),1) + eta*norm(z1_2(:),1) + eta*norm(z1_3(:),1) + ...
eta*norm(z2_1(:),1) + eta*norm(z2_2(:),1) + eta*norm(z2_3(:),1) + ...
eta*norm(z3_1(:),1) + eta*norm(z3_2(:),1) + eta*norm(z3_3(:),1);
tmpLam1u = lambda1u .* (x1 - r1);
tmpLam1v = lambda1v .* (x2 - r2);
tmpLam1w = lambda1w .* (x3 - r3);
costLam1 = sum( tmpLam1u(:) + tmpLam1v(:) + tmpLam1w(:) );
tmpLam2u = lambda2u .* (x1 - y1);
tmpLam2v = lambda2v .* (x2 - y2);
tmpLam2w = lambda2w .* (x3 - y3);
costLam2 = sum( tmpLam2u(:) + tmpLam2v(:) + tmpLam2w(:) );
tmpLam3u = lambda3u_1 .* ( D1x1 - z1_1 ) + ...
lambda3u_2 .* ( D2x1 - z1_2 ) + ...
lambda3u_3 .* ( D3x1 - z1_3 );
costLam3u = sum( tmpLam3u(:) );
tmpLam3v = lambda3v_1 .* ( D1x2 - z2_1 ) + ...
lambda3v_2 .* ( D2x2 - z2_2 ) + ...
lambda3v_3 .* ( D3x2 - z2_3 );
costLam3v = sum( tmpLam3v(:) );
tmpLam3w = lambda3w_1 .* ( D1x3 - z3_1 ) + ...
lambda3w_2 .* ( D2x3 - z3_2 ) + ...
lambda3w_3 .* ( D3x3 - z3_3 );
costLam3w = sum( tmpLam3w(:) );
augXR1 = norm( x1(:) - r1(:), 2 )^2;
augXR2 = norm( x2(:) - r2(:), 2 )^2;
augXR3 = norm( x3(:) - r3(:), 2 )^2;
augXY1 = norm( x1(:) - y1(:), 2 )^2;
augXY2 = norm( x2(:) - y2(:), 2 )^2;
augXY3 = norm( x3(:) - y3(:), 2 )^2;
augXZ1_1 = norm( D1x1(:) - z1_1(:), 2 )^2;
augXZ1_2 = norm( D2x1(:) - z1_2(:), 2 )^2;
augXZ1_3 = norm( D3x1(:) - z1_3(:), 2 )^2;
augXZ2_1 = norm( D1x2(:) - z2_1(:), 2 )^2;
augXZ2_2 = norm( D2x2(:) - z2_2(:), 2 )^2;
augXZ2_3 = norm( D3x2(:) - z2_3(:), 2 )^2;
augXZ3_1 = norm( D1x3(:) - z3_1(:), 2 )^2;
augXZ3_2 = norm( D2x3(:) - z3_2(:), 2 )^2;
augXZ3_3 = norm( D3x3(:) - z3_3(:), 2 )^2;
out = gOfW + costLam1 + costLam2 + ...
costLam3u + costLam3v + costLam3w + ...
rho/2 * ( augXR1 + augXR2 + augXR3 + augXY1 + augXY2 + augXY3 + ...
augXZ1_1 + augXZ1_2 + augXZ1_3 + augXZ2_1 + augXZ2_2 + augXZ2_3 + ...
augXZ3_1 + augXZ3_2 + augXZ3_3 );
disp(['Aug Lag: ', num2str(out)]);
end
function [dx, dy, dz] = imgDeriv3D( data )
[M, N, K] = size( data );
dx = zeros(M,N,K);
dy = zeros(M,N,K);
dz = zeros(M,N,K);
dx(:,2:N-1,:) = ( data(:,3:N,:) - data(:,1:N-2,:) ) / 2;
dx(:,1,:) = data(:,2,:) - data(:,1,:);
dx(:,N,:) = data(:,N,:) - data(:,N-1,:);
dy(2:M-1,:,:) = ( data(3:M,:,:) - data(1:M-2,:,:) ) / 2;
dy(1,:,:) = data(2,:,:) - data(1,:,:);
dy(M,:,:) = data(M,:,:) - data(M-1,:,:);
dz(:,:,2:K-1) = ( data(:,:,3:K) - data(:,:,1:K-2) ) / 2;
dz(:,:,1) = data(:,:,2) - data(:,:,1);
dz(:,:,K) = data(:,:,K) - data(:,:,K-1);
end
function ofRes = ofResidual3D( Iu, Iv, Iw, It, du, dv, dw )
ofRes = Iu.*du + Iv.*dv + It + Iw.*dw;
end
function out = softThresh( in, thresh )
out = sign(in) .* max( ( abs(in) - thresh ), 0 );
end
function out = ofObjective( Iu, Iv, Iw, b, du, dv, dw, eta )
[nRows,nCols,nPages] = size(Iu);
applyD1 = @(u) cat(2, u(:,2:end,:) - u(:,1:end-1,:),zeros(nRows,1,nPages));
applyD2 = @(u) cat(1, u(2:end,:,:) - u(1:end-1,:,:),zeros(1,nCols,nPages));
applyD3 = @(u) cat(3, u(:,:,2:end) - u(:,:,1:end-1),zeros(nRows,nCols,1));
D1du = applyD1(du); D1dv = applyD1(dv); D1dw = applyD1(dw);
D2du = applyD2(du); D2dv = applyD2(dv); D2dw = applyD2(dw);
D3du = applyD3(du); D3dv = applyD3(dv); D3dw = applyD3(dw);
Agam = Iu.*du + Iv.*dv + Iw.*dw;
Agamb = Agam - b;
out = 0.5*sum(Agamb(:).*Agamb(:)) + ...
eta*norm(D1du(:),1) + eta*norm(D2du(:),1) + eta*norm(D3du(:),1) + ...
eta*norm(D1dv(:),1) + eta*norm(D2dv(:),1) + eta*norm(D3dv(:),1) + ...
eta*norm(D1dw(:),1) + eta*norm(D2dw(:),1) + eta*norm(D3dw(:),1);
end
function pyramid = makeDataPyramid3( data, nLevels, spacing )
if nargin < 2
nLevels = 4;
end
if nargin < 3
spacing = 2;
end
smooth_sigma = spacing/2;
f = gaussFilt3( smooth_sigma, 2*round(1.5*smooth_sigma)+1 );
ratio = 1. / spacing;
pyramid = cell(nLevels,1);
tmp = data;
for m = 1:nLevels
pyramid{m} = tmp;
sTmp = size( tmp );
newSize = floor( sTmp .* ratio );
tmp = imfilter(tmp, f, 'corr', 'symmetric', 'same'); % Gauss filter
tmp = resize(tmp, newSize, 'bilinear'); % Downsampling
end
end
function h = gaussFilt3( sig, hsize )
siz = (hsize-1)/2;
if ndims(siz) ~= 3 siz = ones(1,3)*siz; end;
if ndims(sig) ~= 3 sig = ones(1,3)*sig; end;
[x,y,z] = ndgrid(-siz(1):siz(1),-siz(2):siz(2),-siz(3):siz(3));
h = exp(-(x.*x/2/sig(1)^2 + y.*y/2/sig(2)^2 + z.*z/2/sig(3)^2));
h = h/sum(h(:));
end