3 research outputs found
Compressed sensing with approximate message passing: measurement matrix and algorithm design
Compressed sensing (CS) is an emerging technique that exploits the properties of a sparse or
compressible signal to efficiently and faithfully capture it with a sampling rate far below the
Nyquist rate. The primary goal of compressed sensing is to achieve the best signal recovery
with the least number of samples. To this end, two research directions have been receiving
increasing attention: customizing the measurement matrix to the signal of interest and optimizing
the reconstruction algorithm. In this thesis, contributions in both directions are made
in the Bayesian setting for compressed sensing. The work presented in this thesis focuses on
the approximate message passing (AMP) schemes, a new class of recovery algorithm that takes
advantage of the statistical properties of the CS problem.
First of all, a complete sample distortion (SD) framework is presented to fundamentally quantify
the reconstruction performance for a certain pair of measurement matrix and recovery
scheme. In the SD setting, the non-optimality region of the homogeneous Gaussian matrix
is identified and the novel zeroing matrix is proposed with an improved performance. With the
SD framework, the optimal sample allocation strategy for the block diagonal measurement matrix
are derived for the wavelet representation of natural images. Extensive simulations validate
the optimality of the proposed measurement matrix design.
Motivated by the zeroing matrix, we extend the seeded matrix design in the CS literature to
the novel modulated matrix structure. The major advantage of the modulated matrix over the
seeded matrix lies in the simplicity of its state evolution dynamics. Together with the AMP
based algorithm, the modulated matrix possesses a 1-D performance prediction system, with
which we can optimize the matrix configuration. We then focus on a special modulated matrix
form, designated as the two block matrix, which can also be seen as a generalization of the
zeroing matrix. The effectiveness of the two block matrix is demonstrated through both sparse
and compressible signals. The underlining reason for the improved performance is presented
through the analysis of the state evolution dynamics.
The final contribution of the thesis explores improving the reconstruction algorithm. By taking
the signal prior into account, the Bayesian optimal AMP (BAMP) algorithm is demonstrated
to dramatically improve the reconstruction quality. The key insight for its success is that it
utilizes the minimum mean square error (MMSE) estimator for the CS denoising. However, the
prerequisite of the prior information makes it often impractical. A novel SURE-AMP algorithm
is proposed to address the dilemma. The critical feature of SURE-AMP is that the Stein’s
unbiased risk estimate (SURE) based parametric least square estimator is used to replace the
MMSE estimator. Given the optimization of the SURE estimator only involves the noisy data,
it eliminates the need for the signal prior, thus can accommodate more general sparse models