# How to find the value of regularization parameter/s when having one or multiple priors

I am working on a image reconstruction framework, specifically interferometry. In simple words we have data that is pertubated by noise, say

$$V^o_i = V_i + \sigma_i$$

Where $$V^o$$ is a complex value and usually called the measured/observed visibility which is corrupted by Gaussian noise, $$V$$ is the Fourier Transform of an image $$I$$ and $$\sigma$$ is the noise of each measurement.

We can solve the problem solving a non-linear equation such as:

$$\Phi(I)= \arg \min_{I} ||\frac{V^o - V^m(I)}{\sigma}||^2$$

where we call $$V^m$$ model visibilities. I don't want to extend much more here, but since we have an infinity number of solutions and given that a chi-squared will return a noisy image we want an image that represents the highest possible fidelity to the data.

In that sense we want to add prior to regularize the solution. Here, we can think of Maximum Entropy, L1 Norm, Total Squared Variation, etc. Consequently, the equation turns into:

$$\Phi(I, \lambda)= \arg \min_{I} ||\frac{V^o - V^m(I)}{\sigma}||^2 + \lambda P(I)$$

where $$\lambda$$ is the regularization parameter and P is the prior function.

I've been reading papers about how to find the $$\lambda$$ parameter and most of them use an estimation calculating the variance of the noise. But in that case then $$\lambda = \text{Var}(\sigma)$$? This is also called the discrepancy principle (See Regularization in Image Restoration and Reconstruction by W. Clem Karl) In that case, $$\lambda$$ would be the same for each prior right? But it has been seen doing the L-curve plots for same dataset and different prior that the "best" $$\lambda$$s for each one of the cases are different.

Moreover, since I am constructing a framework, the idea is to plug different priors, and then the equation will turn to:

$$\Phi(I, \lambda)= \arg \min_{I} ||\frac{V^o - V^m(I)}{\sigma}||^2 + \lambda_1 P_1(I) + \lambda_2 P_2(I)+..+\lambda_n P_n(I)$$

On that case the calculation of the different $$\lambda$$s is not that trivial as before, I think....

I would appreciate if you can give me a slight hand with both of these cases.

Cheers

I have just found this paper that calculates the $$\lambda_i$$ parameters on an L-surface: Efficient determination of multiple regularization parameters in a generalized L-curve framework.