Least Squares with Total Variation Regularization - How to Set the Lambda ($ \lambda $) Parameter?

Question

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. Anyone has any idea about how to deal with it?

Answer 1

Usually lambda would be determined by some kind of cross-validation or bootstrapping, i.e. fit random subsets of the data.