2000 entries, Undersampled MRI reconstruction and Signal Recovery on Incoherent Manifolds

The last blog entry was the 2000th blog entry on Nuit Blanche, wow! Does that make me an "Incoherence and Randomness Specialist" yet ? Although, I must admit only 1306 entries are on compressive sensing and 65 on Matrix Factorization. Another item of difficult to picture number (at least to me) is that there have been more than 1 million page views on this blog since about three years ago. Additionally, there are 1370 people on the compressive sensing group on LinkedIn tell your colleagues to join, it has the right mix of academics and industrial folks.

Right now, I am doing some catch-up on some papers I haven't covered yet for the past two weeks:

Undersampled MRI reconstruction with patch-based directional wavelets, Magnetic Resonance Imaging by  Xiaobo Qu, Di Guo, Bende Ning, Yingkun Hou, Yulan Lin, Shuhui Cai, Zhong Chen,  The abstract reads:

Compressed sensing has shown great potential in reducing data acquisition time in MRI. In traditional compressed sensing MRI methods, an image is reconstructed by enforcing its sparse representation with respect to a pre-constructed basis or dictionary. In this paper, patch-based directional wavelets are proposed to reconstruct images from undersampled k-space data. A parameter of patch-based directional wavelets, indicatingthe geometric direction of each patch, is trained from the reconstructed image using conventional compressed sensing MRI methods, and incorporated into the sparsifying transform to provide the sparse representation for the image to be reconstructed. A reconstruction formulation is proposed and solved via an efficient algorithm. Simulation results on phantom and in vivo data indicate that the proposed method outperforms conventional compressed sensing MRI methods on preserving the edges and suppressing the noise. Besides, the proposed method is not sensitive to the initial image when training directions.

You can download high resolution images of the paper at http://www.quxiaobo.org/project/CS_MRI_PBDW/Figures.zip

and

Signal Recovery on Incoherent Manifolds by  Chinmay Hegde and Richard Baraniuk. The abstract reads:

Suppose that we observe noisy linear measurements of an unknown signal that can be modeled as the sum of two component signals, each of which arises from a nonlinear sub-manifold of a high-dimensional ambient space. We introduce Successive Projections onto INcoherent manifolds (SPIN), a first-order projected gradient method to recover the signal components. Despite the nonconvex nature of the recovery problem and the possibility of underdetermined measurements, SPIN provably recovers the signal components, provided that the signal manifolds are incoherent and that the measurement operator satisfies a certain restricted isometry property. SPIN significantly extends the scope of current recovery models and algorithms for low-dimensional linear inverse problems and matches (or exceeds) the current state of the art in terms of performance.

Somebody is going to have to explain to me one day the following: since RIP is providing a not so optimal bound for sparse recovery, then why should it be a concern if it is not optimal in other areas such as when dealing with different manifolds as in this paper on SPIN.  Taking for instance the last statement of the paper:

"...The problem of reconstructing, from a ne measurements, matrices that are a sum of low-rank and sparse matrices has attracted signi cant attention in the recent literature [7, 8, 23]. The key stumbling block is that the manifold of low-rank matrices is not incoherent with the manifold of sparse matrices; indeed, the two manifolds share a nontrivial intersection (i.e., there exist low rank matrices that are also sparse, and vice versa). Phenomena such as these make the analysis of SPIN (or similar algorithms) quite challenging, and it may be that higher-order techniques for signal reconstruction will be needed...."

ok, it will more difficult to prove theoretically that SPIN works for a larger set of issues, but the practionner of the algorithm will likely see those areas much before they are proven. I am just saying...

Anyway, Rich tells me that Chinmay will make a release of SPIN available within a few weeks. Thanks Rich!

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