Modulated Unit-Norm Tight Frames for Compressed Sensing - implementation -

Lu Gan just sent me the following:

Dear Igor,

.... You have mentioned our work about "scrambled hadamard transform" and "spinning disk for terahertz imaging" on your blog before. I really appreciate that.
With this Email, I would to let you know 2 of our recent research work

1) We proposed a new unified framework for the construction of structured random matrices for compressed sensing. Under this framework, the RIP results of some popular structured sensing matrices (e.g. compressive multiplexing, random demodulation) can be easily analyzed and improved. We also propose several new structured sensing matrices based on the framework. The paper will be published on IEEE Trans. on Signal Processing soon.

You can find the paper from either of the following links:
http://arxiv.org/abs/1411.7630

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7093188&queryText=modulated+unit+norm+tight+frames&newsearch=true&searchField=Search_All

The code related to the paper is available at:
https://github.com/p-zhang/p-zhang.github.io/tree/master/archive/myresearch/udb_matlab_code

2) We have also done some work on dictionary learning for 3D terahertz imaging, which was published on Elsevier Digital Signal processing. The abstract can be found on the following link:

http://www.sciencedirect.com/science/article/pii/S1051200415001426
Subsampled terahertz data reconstruction based on spatio-temporal dictionary learning

Thanks in advance and have a nice weekend!

Best,
Lu

 Thank you Lu !

Modulated Unit-Norm Tight Frames for Compressed Sensing by Peng Zhang, Lu Gan, Sumei Sun, Cong Ling

In this paper, we propose a compressed sensing (CS) framework that consists of three parts: a unit-norm tight frame (UTF), a random diagonal matrix and a column-wise orthonormal matrix. We prove that this structure satisfies the restricted isometry property (RIP) with high probability if the number of measurements m=O(slog2slog2n) for s-sparse signals of length n and if the column-wise orthonormal matrix is bounded. Some existing structured sensing models can be studied under this framework, which then gives tighter bounds on the required number of measurements to satisfy the RIP. More importantly, we propose several structured sensing models by appealing to this unified framework, such as a general sensing model with arbitrary/determinisic subsamplers, a fast and efficient block compressed sensing scheme, and structured sensing matrices with deterministic phase modulations, all of which can lead to improvements on practical applications. In particular, one of the constructions is applied to simplify the transceiver design of CS-based channel estimation for orthogonal frequency division multiplexing (OFDM) systems.

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