I am trying to use total variation minimization for an image reconstruction problem. Essentially, I am trying to penalize different in the intensity of the two pixels in the reconstructed image. For this I minimize |Ax-b|+ \lambda |F(X)| where F(x)= (x_i - x_i+1)^2 is a quadratic function that penalizes the difference intensity of two nearest pixels. I am however unable to decide how to fix the value of \lambda (strength of regularization). In literature I found ways to find \lambda for the case of norm regularization using Lagrange multipliers. However, I am unable to find/formulate a method to find optimal \lambda for this case. Any1 has any idea about how to deal with it?
Usually lambda would be determined by some kind of cross-validation or bootstrapping, i.e. fit random subsets of the data.