homography matrix problem

Question

i have mini project to do about homography , but i am really a beginner in matlab . i have calculated the homography matrix easily by taking a set pixels from both images but the problem is how to apply this matrix to all pixels of the original image (i think it's difficult to treat each pixel individually ) . here is the original image :

i want to apply the homography to this image ( i have already extracted the homography matrix ) to get a frontal view of the cover.

so with my weak knowledge of matlab , i applied the folowing operations:

i get 4 points from the corners of the original image and the corresponding points in the output image

i have estimated the homography :

0.9638 -0.0960 52.5754; 
0.2449 1.3808 -17.0081; 
-0.0001 0.0013 1.0000

then i have applied it on the image using this poor matlab code:

function [Ired] = affect(I,P)
I1=im2double(I);
for i=1:308
i
for j=1:288
v1 = [j;i;1];
v2 = P*v1;
v3 =v2/v2(3,1);
if (v3(1,1) <0 || v3(2,1)<0 || v3(3,1)<0)
   continue;
else     
Ired(round(1+v3(2,1)),round(1+v3(1,1)))=I1(i,j);
end
end 

end

and the result is the following:

- but the problem with this code is it takes a lot of time (sometimes 10 minutes) to perform the operation on the image especially if the size of the image is high so is there a way to implement this code without loops to make it faster? .

- i get these black dots all over the image , i can do a dilation but the image will be distorted , do you have an idea how to eliminate this dots without distorting the image ?.

Answer 1

Matrix/vector operations are really strongly optimized in Matlab/Octave. Use them whenever you can.

In your case, instead of multiplying the 3x3 homography matrix with a 3x1 vector for NxM times you can easily modify your calculation to do a multiplication with a 3x3 matrix and a 3xM matrix for N times.

Check out this Octave code:

warning ("off", "Octave:broadcast");
w = 1280;
h = 720;
H = rand(3);

tic();
for i=1:h 
    for j=1:w 
        H*[i;j;1]/(H(3,:)*[i;j;1]); 
    end
end; 
toc();

tic();
for i=1:h 
    v=[ones(1,w)*i;1:w;ones(1,w)];
    (H*v)./(H(3,:)*v); 
end; 
toc();

It gives me the following result:

Elapsed time is 25.5651 seconds.
Elapsed time is 0.273893 seconds.

Maybe you can go further by generating a 3x(N*M) matrix with all the image coordinates.

There is a simple solution to the 'black dots' problem. Right now you have the homography $H$ between the original image and the cover image and you generate the points by looping through the pixels of the original image $(x,y)$ to color the points $(x',y')$ in the destination image.

Let's have the homography $H^{-1}$, loop through all the pixels in the destination image and calculate the pixel address of the original image you want to sample the color from.