How to recover an image from another given image

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?

Solution

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: