Multichannel blind deconvolution in the simplest formulation: how to solve?

Question

Recently I began to study deconvolution algorithms and met the following acquisition model:

where f is the original (latent) image, g is the input (observed) image, h is the point spread function (degradation kernel), n is a random additive noise and * is the convolution operator. If we know g and h, then we can recover f using Richardson-Lucy algorithm:

where , (W,H) is the size of rectangular support of h and multiplication and division are pointwise. Simple enough to code in C++, so I did just so. It turned out that approximates to f while i is less then some m and then it starts rapidly decay. So the algorithm just needed to be stopped at this m - the most satisfactory iteration.

If the point spread function g is also unknown then the problem is said to be blind, and the modification of Richardson-Lucy algorithm can be applied:

For initial guess for f we can take g, as before, and for initial guess for h we can take trivial PSF, or any simple form that would look similar to observed image degradation. This algorithm also works quit fine on the simulated data.

Now I consider the multiframe blind deconvolution problem with the following acquisition model:

Is there a way to develop Richardson-Lucy algorithm for solving the problem in this formulation? If no, is there any other iterative procedure for recovering f, that wouldn't be much more complicated than the previous ones?

Answer 1

According to your acquisition model, latent image (f) remains same while the observed images are different due to different psf and noise models. One way to look at it, is a motion-blur problem where a sharp and noise-free image(f) is corrupted by the motion blur kernel. As this is an ill-posed problem, in most of the literature it's solved iteratively by estimating the blur kernel and the latent image. The way you solve this depends entirely on your objective function. For example in some papers IRLS is used to estimate the blur kernel. You can find a lot of literature on this.

• If you want to use Richardson Lucy Blind deconvolution, then use it on just one frame.
• One strategy can be in each iteration while recovering f, assign different weights for contribution from each g(observed images). You can incorporate different weights in the objective function or calculate them according to the estimated blur kernel.

Answer 2

Is there a way to develop Richardson-Lucy algorithm for solving the problem in this formulation?

I'm not a specialist in this area, but I don't think that such way to construct an algorithm exists, at least not straightforwardly. Here is my argument for this. The first problem you described (when the psf is known) is already ill-posed due to the random nature of the noise and loss of information about convolution near image edges. The second problem on your list — single-channel blind deconvolution — is the extention of the previous one. In this case in addition it's underdetermined, so the ill-posedness expands, and so it's natural that the method to solve this problem is developed from the method for solving the first problem. Now when we consider the multichannel blind deconvolution formulation, we add a bunch of additional information to our previous model and so the problem goes from underdetermined to overdetermined. This is the whole other kind of ill-posedness and hence different approaches to solution are required.

is there any other iterative procedure for recovering f, that wouldn't be much more complicated than the previous ones?

I can recommend the algorithm introduced by Šroubek and Milanfar in [1]. I'm not sure whether it's much more complicated on your opinion or not so much, but it's by far one of the most recent and robust. The formulation of the problem is precisely the same as you wrote. The algorithm takes as input K>1 number of images, the upper bound of the psf size L, and four tuning parameters: alpha, beta, gamma, delta. To specify gamma, for example, you will need to estimate the variance of the noise on your input images and take the largest variance var, then gamma = 1/var. The algorithm solves the following optimization problem using alternating minimization:

where F is the data fidelity term and Q and R are regularizers of the image and blurs, respectively.

For detailed analysis of the algorithm see [1], for a collection of different deconvolution formulation and their solutions see [2]. Hope it helps.

Referenses:

1. Filip Šroubek, Peyman Milanfar. —- Robust Multichannel Blind Deconvolution via Fast Alternating Minimization. -— IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 4, APRIL 2012

2. Patrizio Campisi, Karen Egiazarian. —- Blind Image Deconvolution: Theory and Applications