nonconvex optimization problem is very interesting

nonconvex optimization problem is very interesting

block compressed sensing of natural images

block compressed sensing of natural images by lu gan

There are two interesting things in this paper.

1 The measurement scheme.

2 The reconstruction method.

1 The measurement scheme.

The paper is in the context of the natural image capture (e.g. capture a image using digital camera).

The target image is divided into several blocks. For each block, a i.i.d Gaussian matrix is used as the measure matrix. Thus, the method is memory efficient and easy to be paralyzed.

2 The reconstruction method.

It is in fact hard thresholding method. They use hard thresholding and wiener filtering several times. But very interestingly, they point out that redundant frame expansions are superior than orthonormal base. e.g. wavelet + curvelet > wavelet.

plus, the POCS(projection onto convex set) is in a similar form of standard ART or sART.

[Practical signal recovery from random projections by E. Candes, etc.]

4D computed tomography reconstruction from few-projection data via temporal nonlocal regularization.

an interesting paper. 4D computed tomography reconstruction from few-projection data via temporal nonlocal regularization. The main idea is that, for cardiac imaging, the motion of the heart is smooth. So They use the images whose corresponding heart phase is near the reconstructed one as constraint.

where J() is the non local mean operator.

Images of all the phases are reconstructed simultaneously.

Under-determined non-cartesian MR reconstruction with non-convex sparsity promoting analysis prior

1 notation:

1.1 Synthesis prior formulation

1.2 analysis prior formulation

2 main results :

2.1 In this paper, the authors present an optimization algorithms for non-convex problem. They use Lp norm to promote sparsity, of cause. The optimization algorithms are based on majorization minimization.

2.2 They claimed that the analysis prior formulation with redundant wavelet transform gives better reconstruction quality than the systhesis prior formulation with orthogonal wavelet transform.

redundant systhesis worse than orthogonal synthesis worse than redundant analysis.

my question : what about analysis synthesis and compared to l1 norm ? (in the paper they use l_0.8 norm)

The paper reminds me of the paper Image reconstruction from few-views by non-convex optimization. In this paper, authors use a TpV norm, which is the lp norm of total variation.

The optimization method is not proved to be convergence. but also the results show the lp norm leads to better reconstruction quality.