Horn-Schunck Optical Flow (Averaging Velocity) - matlab

Question

I'm currently trying to implement the HS-method for optical flow but my u and v always seem to have only zeros in them. I can't seem to figure out my error in here:
vid=VideoReader('outback.AVI');
vid.CurrentTime = 1.5;
alpha=1;
iterations=10;
frame_one = readFrame(vid);
vid.CurrentTime = 1.6;
frame_two = readFrame(vid);
% convert to grayscale
fone_gr = rgb2gray(frame_one);
ftwo_gr = rgb2gray(frame_two);
% construct for each image
sobelx=[-1 -2 -1; 0 0 0; 1 2 1];
sobely=sobelx';
time=[-1 1];
fx_fone=imfilter(fone_gr, sobelx);
fy_fone=imfilter(fone_gr, sobely);
ft_fone=imfilter(fone_gr, time);
fx_ftwo=imfilter(ftwo_gr, sobelx);
fy_ftwo=imfilter(ftwo_gr, sobely);
ft_ftwo=imfilter(ftwo_gr, time);
Ix=double(fx_fone+fx_ftwo);
Iy=double(fy_fone+fy_ftwo);
It=double(ft_fone+ft_ftwo);
% flow-variables (velocity = 0 assumption)
velocity_kernel=[0 1 0; 1 0 1; 0 1 0];
u = double(0);
v = double(0);
% iteratively solve for u and v
for i=1:iterations
neighborhood_average_u=conv2(u, velocity_kernel, 'same');
neighborhood_average_v=conv2(v, velocity_kernel, 'same');
data_term = (Ix .* neighborhood_average_u + Iy .* neighborhood_average_v + It);
smoothness_term = alpha^2 + (Ix).^2 + (Iy).^2;
numerator_u = Ix .* data_term;
numerator_v = Iy .* data_term;
u = neighborhood_average_u - ( numerator_u ./ smoothness_term );
v = neighborhood_average_v - ( numerator_v ./ smoothness_term );
end
u(isnan(u))=0;
v(isnan(v))=0;
figure
imshow(frame_one); hold on;
quiver(u, v, 5, 'color', 'b', 'linewidth', 2);
set(gca, 'YDir', 'reverse');
The only thing I'm not really confident about is the computation of the neighborhood average:
velocity_kernel=[0 1 0; 1 0 1; 0 1 0];
u = double(0);
v = double(0);
[..]
neighborhood_average_u=conv2(u, velocity_kernel, 'same');
neighborhood_average_v=conv2(v, velocity_kernel, 'same');
Wouldn't that always result in a convolution matrix with only zeros?
I thought about changing it to the following, since I need to compute the average velocity using the velocity kernel on each pixel of my images:
velocity_kernel=[0 1 0; 1 0 1; 0 1 0];
u = double(0);
v = double(0);
[..]
neighborhood_average_u=conv2(Ix, velocity_kernel, 'same');
neighborhood_average_v=conv2(Iy, velocity_kernel, 'same');
But I still don't know if that would be the correct way. I followed the instructions on the bottom of this MATLAB page:
http://de.mathworks.com/help/vision/ref/opticalflowhs-class.html

Answer 1

I found this paper with some further explanations and also some matlab code.
They compute the average of u and v as follows:
% initial values
u = 0; v = 0;
% weighted average kernel
kernel = [1/12 1/6 1/12; 1/6 0 1/6; 1/12 1/6 1/12];
for i = 1:iterations
uAvg = conv2( u, kernel 'same' );
vAvg = conv2( v, kernel 'same' );
...
end