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velocity_from_tensors.m
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function [v, cout] = velocity_from_tensors(T, model, kernelsize, sigma, cin)
% [V, COUT] = VELOCITY_FROM_TENSORS(T, MODEL, KERNELSIZE, SIGMA, CIN)
%
% Compute a velocity field from a tensor field, using the algorithm
% described in section 6.5 of Gunnar Farnebäck's thesis "Polynomial
% Expansion for Orientation and Motion Estimation". Normalized averaging is
% performed separably with a Gaussian filter.
%
% T - Tensor field of size height x width x 3 x 3.
% MODEL - Velocity field model. Supported are 'constant', 'affine',
% and 'eightparam', with definitions according to equations
% 6.4, 6.5, and 6.6.
% KERNELSIZE - Spatial size of the Gaussian averaging filter.
% SIGMA - Standard deviation of the Gaussian averaging filter
% CIN - Input certainty for the tensor field, used in the normalized
% averaging step. Typically cin should be reduced close to the
% borders of the tensor field.
% V - Estimated velocity field of size height x width x 2
% COUT - Output confidence field, according to equation 6.23. Notice
% that a small value indicates a high confidence.
%
% Author: Gunnar Farnebäck
% Computer Vision Laboratory
% Linköping University, Sweden
% Extract the spatial size of the tensor field
sides = size(T);
sides = sides(1:2);
% Remove the isotropic part of the tensors.
T = remove_isotropic(T);
switch model
case 'constant'
% Tensor averaging, which in this case is equivalent with computation of Q.
T = T(:,:,[1 5 9 2 3 6]);
app = gaussian_app(kernelsize, 1, sigma);
T = conv3(conv3(T.*repmat(cin, [1 1 6]), app), app') ./ ...
(eps + repmat(conv3(conv3(cin, app), app'), [1 1 6]));
T = reshape(T(:,:,[1 4 5;4 2 6;5 6 3]), [sides 3 3]);
%Solve the equation Qv=-q.
a = T(:,:,1,1);
b = T(:,:,2,2);
c = T(:,:,1,2);
d = T(:,:,1,3);
e = T(:,:,2,3);
% Q = [a c;c b], inv(Q) = [b -c;-c a]/(a*b-c^2), q = [d;e]
v = -cat(3, d.*b-c.*e, a.*e-c.*d) ./ (eps + repmat(a.*b-c.^2, [1 1 2]));
%Compute output certainty
if nargout > 1
cout = T(:,:,3,3) + d.*v(:,:,1) + e.*v(:,:,2);
end
case 'affine'
% Build the quadratic forms for the affine motion
[x,y] = meshgrid(1:sides(2),1:sides(1));
Q = zeros([sides 25]);
Q(:,:,1) = x.*x.*T(:,:,1,1);
Q(:,:,2) = y.*y.*T(:,:,1,1);
Q(:,:,3) = T(:,:,1,1);
Q(:,:,4) = x.*x.*T(:,:,2,2);
Q(:,:,5) = y.*y.*T(:,:,2,2);
Q(:,:,6) = T(:,:,2,2);
Q(:,:,7) = T(:,:,3,3);
Q(:,:,8) = x.*y.*T(:,:,1,1);
Q(:,:,9) = x.*T(:,:,1,1);
Q(:,:,10) = x.*x.*T(:,:,1,2);
Q(:,:,11) = x.*y.*T(:,:,1,2);
Q(:,:,12) = x.*T(:,:,1,2);
Q(:,:,13) = x.*T(:,:,1,3);
Q(:,:,14) = y.*T(:,:,1,1);
Q(:,:,15) = y.*y.*T(:,:,1,2);
Q(:,:,16) = y.*T(:,:,1,2);
Q(:,:,17) = y.*T(:,:,1,3);
Q(:,:,18) = T(:,:,1,2);
Q(:,:,19) = T(:,:,1,3);
Q(:,:,20) = x.*y.*T(:,:,2,2);
Q(:,:,21) = x.*T(:,:,2,2);
Q(:,:,22) = x.*T(:,:,2,3);
Q(:,:,23) = y.*T(:,:,2,2);
Q(:,:,24) = y.*T(:,:,2,3);
Q(:,:,25) = T(:,:,2,3);
app = gaussian_app(kernelsize, 1, sigma);
Q = conv3(conv3(Q.*repmat(cin, [1 1 25]), app), app') ./ ...
(eps + repmat(conv3(conv3(cin, app), app'), [1 1 25]));
Q = reshape(Q(:,:,[ 1 8 9 10 11 12 13;
8 2 14 11 15 16 17;
9 14 3 12 16 18 19;
10 11 12 4 20 21 22;
11 15 16 20 5 23 24;
12 16 18 21 23 6 25;
13 17 19 22 24 25 7]),[sides 7 7]);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The code above is somewhat obfuscated in order to improve speed. What
% we really want to do is to compute Q = S^T T S as in equation 5.17 and
% Q_{tot} according to equation 5.31, with the affine motion model.
% A more direct, but slower, implementation is shown below:
%
% [x,y] = meshgrid(1:sides(2),1:sides(1));
% Q = zeros([sides 7 7]);
% Q(:,:,1,1) = x.*x.*T(:,:,1,1);
% Q(:,:,2,2) = y.*y.*T(:,:,1,1);
% Q(:,:,3,3) = T(:,:,1,1);
% Q(:,:,4,4) = x.*x.*T(:,:,2,2);
% Q(:,:,5,5) = y.*y.*T(:,:,2,2);
% Q(:,:,6,6) = T(:,:,2,2);
% Q(:,:,7,7) = T(:,:,3,3);
%
% Q(:,:,1,2) = x.*y.*T(:,:,1,1);
% Q(:,:,1,3) = x.*T(:,:,1,1);
% Q(:,:,1,4) = x.*x.*T(:,:,1,2);
% Q(:,:,1,5) = x.*y.*T(:,:,1,2);
% Q(:,:,1,6) = x.*T(:,:,1,2);
% Q(:,:,1,7) = x.*T(:,:,1,3);
%
% Q(:,:,2,3) = y.*T(:,:,1,1);
% Q(:,:,2,4) = x.*y.*T(:,:,1,2);
% Q(:,:,2,5) = y.*y.*T(:,:,1,2);
% Q(:,:,2,6) = y.*T(:,:,1,2);
% Q(:,:,2,7) = y.*T(:,:,1,3);
%
% Q(:,:,3,4) = x.*T(:,:,1,2);
% Q(:,:,3,5) = y.*T(:,:,1,2);
% Q(:,:,3,6) = T(:,:,1,2);
% Q(:,:,3,7) = T(:,:,1,3);
%
% Q(:,:,4,5) = x.*y.*T(:,:,2,2);
% Q(:,:,4,6) = x.*T(:,:,2,2);
% Q(:,:,4,7) = x.*T(:,:,2,3);
%
% Q(:,:,5,6) = y.*T(:,:,2,2);
% Q(:,:,5,7) = y.*T(:,:,2,3);
%
% Q(:,:,6,7) = T(:,:,2,3);
%
% Q(:,:,2,1) = Q(:,:,1,2);
% Q(:,:,3,1) = Q(:,:,1,3);
% Q(:,:,4,1) = Q(:,:,1,4);
% Q(:,:,5,1) = Q(:,:,1,5);
% Q(:,:,6,1) = Q(:,:,1,6);
% Q(:,:,7,1) = Q(:,:,1,7);
% Q(:,:,3,2) = Q(:,:,2,3);
% Q(:,:,4,2) = Q(:,:,2,4);
% Q(:,:,5,2) = Q(:,:,2,5);
% Q(:,:,6,2) = Q(:,:,2,6);
% Q(:,:,7,2) = Q(:,:,2,7);
% Q(:,:,4,3) = Q(:,:,3,4);
% Q(:,:,5,3) = Q(:,:,3,5);
% Q(:,:,6,3) = Q(:,:,3,6);
% Q(:,:,7,3) = Q(:,:,3,7);
% Q(:,:,5,4) = Q(:,:,4,5);
% Q(:,:,6,4) = Q(:,:,4,6);
% Q(:,:,7,4) = Q(:,:,4,7);
% Q(:,:,6,5) = Q(:,:,5,6);
% Q(:,:,7,5) = Q(:,:,5,7);
% Q(:,:,7,6) = Q(:,:,6,7);
%
% Q = Q(:,:,:);
% app = gaussian_app(kernelsize,1,sigma);
% Q = conv3(conv3(Q.*repmat(cin,[1 1 49]),app),app') ./ ...
% (eps+repmat(conv3(conv3(cin,app),app'),[1 1 49]));
% Q = reshape(Q,[sides 7 7]);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Solve the equation Qp=-q.
p = min_quadform(Q);
v = zeros([sides 2]);
v(:,:,1) = sum(p(:,:,1:3).*cat(3,x,y,ones(sides)),3);
v(:,:,2) = sum(p(:,:,4:6).*cat(3,x,y,ones(sides)),3);
if nargout > 1
cout = Q(:,:,7,7) + sum(p.*Q(:,:,[1:6],7),3);
end
case 'eightparam'
% Build the quadratic forms for the eight parameter motion
[x,y] = meshgrid(1:sides(2), 1:sides(1));
Q = zeros([sides 9 9]);
Q(:,:,1,1) = T(:,:,1,1);
Q(:,:,2,2) = x.^2.*T(:,:,1,1);
Q(:,:,3,3) = y.^2.*T(:,:,1,1);
Q(:,:,4,4) = T(:,:,2,2);
Q(:,:,5,5) = x.^2.*T(:,:,2,2);
Q(:,:,6,6) = y.^2.*T(:,:,2,2);
Q(:,:,7,7) = x.^4.*T(:,:,1,1) + 2*x.^3.*y.*T(:,:,1,2) + ...
x.^2.*y.^2.*T(:,:,2,2);
Q(:,:,8,8) = x.^2.*y.^2.*T(:,:,1,1) + 2*x.*y.^3.*T(:,:,1,2) + ...
y.^4.*T(:,:,2,2);
Q(:,:,9,9) = T(:,:,3,3);
Q(:,:,1,2) = x.*T(:,:,1,1);
Q(:,:,1,3) = y.*T(:,:,1,1);
Q(:,:,1,4) = T(:,:,1,2);
Q(:,:,1,5) = x.*T(:,:,1,2);
Q(:,:,1,6) = y.*T(:,:,1,2);
Q(:,:,1,7) = x.^2.*T(:,:,1,1) + x.*y.*T(:,:,1,2);
Q(:,:,1,8) = x.*y.*T(:,:,1,1) + y.^2.*T(:,:,1,2);
Q(:,:,1,9) = T(:,:,1,3);
Q(:,:,2,3) = x.*y.*T(:,:,1,1);
Q(:,:,2,4) = x.*T(:,:,1,2);
Q(:,:,2,5) = x.^2.*T(:,:,1,2);
Q(:,:,2,6) = x.*y.*T(:,:,1,2);
Q(:,:,2,7) = x.^3.*T(:,:,1,1) + x.^2.*y.*T(:,:,1,2);
Q(:,:,2,8) = x.^2.*y.*T(:,:,1,1) + x.*y.^2.*T(:,:,1,2);
Q(:,:,2,9) = x.*T(:,:,1,3);
Q(:,:,3,4) = y.*T(:,:,1,2);
Q(:,:,3,5) = x.*y.*T(:,:,1,2);
Q(:,:,3,6) = y.^2.*T(:,:,1,2);
Q(:,:,3,7) = x.^2.*y.*T(:,:,1,1) + x.*y.^2.*T(:,:,1,2);
Q(:,:,3,8) = x.*y.^2.*T(:,:,1,1) + y.^3.*T(:,:,1,2);
Q(:,:,3,9) = y.*T(:,:,1,3);
Q(:,:,4,5) = x.*T(:,:,2,2);
Q(:,:,4,6) = y.*T(:,:,2,2);
Q(:,:,4,7) = x.^2.*T(:,:,1,2) + x.*y.*T(:,:,2,2);
Q(:,:,4,8) = x.*y.*T(:,:,1,2) + y.^2.*T(:,:,2,2);
Q(:,:,4,9) = T(:,:,2,3);
Q(:,:,5,6) = x.*y.*T(:,:,2,2);
Q(:,:,5,7) = x.^3.*T(:,:,1,2) + x.^2.*y.*T(:,:,2,2);
Q(:,:,5,8) = x.^2.*y.*T(:,:,1,2) + x.*y.^2.*T(:,:,2,2);
Q(:,:,5,9) = x.*T(:,:,2,3);
Q(:,:,6,7) = x.^2.*y.*T(:,:,1,2) + x.*y.^2.*T(:,:,2,2);
Q(:,:,6,8) = x.*y.^2.*T(:,:,1,2) + y.^3.*T(:,:,2,2);
Q(:,:,6,9) = y.*T(:,:,2,3);
Q(:,:,7,8) = x.^3.*y.*T(:,:,1,1) + 2*x.^2.*y.^2.*T(:,:,1,2) + ...
x.*y.^3.*T(:,:,2,2);
Q(:,:,7,9) = x.^2.*T(:,:,1,3) + x.*y.*T(:,:,2,3);
Q(:,:,8,9) = x.*y.*T(:,:,1,3) + y.^2.*T(:,:,2,3);
Q(:,:,2,1) = Q(:,:,1,2);
Q(:,:,3,1) = Q(:,:,1,3);
Q(:,:,4,1) = Q(:,:,1,4);
Q(:,:,5,1) = Q(:,:,1,5);
Q(:,:,6,1) = Q(:,:,1,6);
Q(:,:,7,1) = Q(:,:,1,7);
Q(:,:,8,1) = Q(:,:,1,8);
Q(:,:,9,1) = Q(:,:,1,9);
Q(:,:,3,2) = Q(:,:,2,3);
Q(:,:,4,2) = Q(:,:,2,4);
Q(:,:,5,2) = Q(:,:,2,5);
Q(:,:,6,2) = Q(:,:,2,6);
Q(:,:,7,2) = Q(:,:,2,7);
Q(:,:,8,2) = Q(:,:,2,8);
Q(:,:,9,2) = Q(:,:,2,9);
Q(:,:,4,3) = Q(:,:,3,4);
Q(:,:,5,3) = Q(:,:,3,5);
Q(:,:,6,3) = Q(:,:,3,6);
Q(:,:,7,3) = Q(:,:,3,7);
Q(:,:,8,3) = Q(:,:,3,8);
Q(:,:,9,3) = Q(:,:,3,9);
Q(:,:,5,4) = Q(:,:,4,5);
Q(:,:,6,4) = Q(:,:,4,6);
Q(:,:,7,4) = Q(:,:,4,7);
Q(:,:,8,4) = Q(:,:,4,8);
Q(:,:,9,4) = Q(:,:,4,9);
Q(:,:,6,5) = Q(:,:,5,6);
Q(:,:,7,5) = Q(:,:,5,7);
Q(:,:,8,5) = Q(:,:,5,8);
Q(:,:,9,5) = Q(:,:,5,9);
Q(:,:,7,6) = Q(:,:,6,7);
Q(:,:,8,6) = Q(:,:,6,8);
Q(:,:,9,6) = Q(:,:,6,9);
Q(:,:,8,7) = Q(:,:,7,8);
Q(:,:,9,7) = Q(:,:,7,9);
Q(:,:,9,8) = Q(:,:,8,9);
Q = Q(:,:,:);
app = gaussian_app(kernelsize, 1, sigma);
Q = conv3(conv3(Q.*repmat(cin, [1 1 81]), app), app') ./ ...
(eps + repmat(conv3(conv3(cin, app), app'), [1 1 81]));
Q = reshape(Q, [sides 9 9]);
% Solve the equation Qp = -q.
p = min_quadform(Q);
v = zeros([sides 2]);
v(:,:,1) = sum(p(:,:,[1 2 3 7 8]).*cat(3,ones(sides),x,y,x.^2,x.*y),3);
v(:,:,2) = sum(p(:,:,4:8).*cat(3,ones(sides),x,y,x.*y,y.^2),3);
if nargout > 1
cout = Q(:,:,9,9) + sum(p.*Q(:,:,[1:8],9), 3);
end
otherwise
error('invalid motion model')
end