Stochastic Gradient Descent (SGD) for Image Processing

Dear all,

I am trying to apply SGD to solve a classical image processing problem as in this link . I am not sure what should I change. Here is the Gradient Descent Code:

niter = 500;    % number of iterations
x = u;          % initial value for x, u is the input noisy image
for i=1:niter
    % smoothed total variation of the image
    gdx = grad(x).^2;
    sgdx=gdx(:,:,1)+gdx(:,:,2);
    NormEps   = sqrt( epsilon^2 + sgdx );
    J         = sum(NormEps(:)) ; % this is a scalar value
    % functional to minimize, lambda is weight of J
    nm=sum((x(:)-u(:)).^2);
    f         = 1/2 * nm^2 + lambda * J;
    % normalized gradient of J
    GradJ     =-div( grad(x)./repmat(NormEps, [1 1 2]) );
    % Gradient Descent update equation
    % the gradient of the functional function f is:
    %  x - y + lambda *    GradJ
    x = x - tau * ( x - u + lambda * GradJ);
end
clf;
imageplot(clamp(x)); % this is the result denoised image

I understand that in SGD we took only random part of the image at each iteration then we compute the minimum, but if I apply this on the input noisy image, I will denoise (badly) small part of the image at each iteration, right? an explanation based on the code above would be excellent!

Best regards,

Ibraheem

[Dan]

Hi Ibraheem, I have updated an existing algorithm to apply SGD : https://www.google.co.il/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&cad=rja&uact=8&ved=0ahUKEwjI9I_R1tTUAhUsCMAKHc0HB0MQFgg5MAM&url=https%3A%2F%2Fwww.mathworks.com%2Fmatlabcentral%2Ffileexchange%2F62921-ecc-registration-100x-faster&usg=AFQjCNGFD-abP0KTXE9Lvsi7dfId74JIUw

Enjoy, Dan

Answer 1

[Xilin Li]

Update a random part of the image at each iteration is not SGD. In SGD, the parameter, say x, you want to optimize for all iterations is the same x, but the gradient used to update x is noisy due to replacing expectation with sample average. I checked your image denoising problem. It is a standard convex optimization, and there are many efficient solvers. You left a comment on my psgd post, and I showed how to use psgd to converge to better solutions faster.

[Ibraheem Al-Dhamari]

many thanks for your answer. I will check your code and try to understand the difference.

[update]

I checked the code and still do not see where is the stochastic part. I saw these lines:

   dX = sqrt(eps)*randn(size(x)); 
   dG = Gradf(y,x+dX,epsilon) - Gradf(y,x,epsilon);

I think the first line add random noise to x (the filtered image) but still, you need to compute the gradient of the whole image not part of it.

From wiki:

   "In stochastic (or "on-line") gradient descent, the true gradient of Q(w) is approximated by a gradient at a single example".

I understood that we need to compute the gradient of part of the image instead of the whole image, right?