compressed Sensing Reconstruction via Belief Propagation

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Introduction

One of the key theorem in digital signal processing is Shannon/Nyquist theorem. This theorem specifies the conditions on which a band limited signal can be reconstructed uniquely from its discrete samples. This property of band limited signals made signal processing viable on natural analog signals. However, in some applications even the sampled signal lays in a extremely high dimensional vector space. To make signal processing algorithms computationally tractable, many researchers work on compressiblity of signals. Here, it is assumed that most of the information content of a signal lays in a few samples with large magnitude. This lead to study and investigation on a class of signals, known as compressible signals.

Compressible signals can be stored efficiently by ignoring small value coefficients. Naturally it means during sampling procedure loosing those samples is unimportant. This lead to a natural questions: Can we sample compressible signals in a compressed way? Is there any method to sense only those large value coefficients? In parallel works by Donoho <ref name="R1"> D. Donoho, “Compressed Sensing,” in IEEE Trans. on Info. theory, vol 52, no 4,pp. 1289–1306, Apr. 2006.</ref> and Candes et. al <ref name="R2"> E. Candes, J. Romberg, J.; T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” in IEEE Trans. on Info. theory, vol 52, no 2,pp. 489–509, Feb. 2006.</ref> this question was answered. They showed that procedure can be done by measuring a small number of random linear projection of the source signal. They also provided a decoding scheme for reconstructing the source signal using those measurements.

Compressed Sensing

Compressed Sensing Reconstruction Algorithms

Connecting CS decoding to graph decoding algorithms

CS-LDPC decoding of sparse signals

Source model
Decoding via statistical inference
Exact solution to CS statistical inference
Approximate solution to CS statistical inference via message passing

Numerical results

Conclusion

References

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