How to recover an image from another given image - image

Question

If I have a grey-scale, square image (1) and I rotate a copy of it by 90 degrees. I create a new image (2) where the pixels are the sum of the original and rotated images. My question is if I only have image 2 how can I recover the original image 1?

Answer 1

The short answer is: you can't recover the original image.
Proof:
Assume 2x2 image:
I = [a b]
[c d]
J = I + rot90(I) = [ a + b, b + d] = [E F
[ a + c, c + d] G H]
Now lets try to solve the linear equation system:
E = a + b + 0 + 0
F = 0 + b + 0 + d
G = a + 0 + c + 0
H = 0 + 0 + c + d
A = [a, b, 0, 0 u = [a v = [E
0, b, 0, d b F
a, 0, c, 0 c G
0, 0, c, d] d] H]
v = A*u
In order to extract u, matrix A must be invertibale.
but det(A) = 0, so there are infinite possible solutions.
I tried an iterative approach.
I implemented it in MATLAB.
I played with it a little, and found out that applying bilateral filter, and moderated sharpening, improves the reconstructed result.
There are probably better heuristics, that I can't think off.
Here is the MATLAB implementation:
I = im2double(imread('cameraman.tif'))/2; %Read input sample image and convert to double
J = I + rot90(I); %Sum of I and rotated I.
%Initial guess.
I = ones(size(J))*0.5;
h_fig = figure;
ax = axes(h_fig);
h = imshow(I/2);
alpha = 0.1;
beta = 0.01;
%100000 iterations.
for i = 1:100000
K = I + rot90(I);
E = J - K; %E is the error matrix.
I = I + alpha*E;
if mod(i, 100) == 0
if (i < 100000*0.9)
I = imsharpen(imbilatfilt(I), 'Amount', 0.1);
end
h.CData = I*2;
ax.Title.String = num2str(i);
pause(0.01);
beta = beta * 0.99;
end
end
Sum of I and rot90(I):
Original image:
Reconstructed image: