Compressive Sensing Take 2 Yubo Paul Yang, Algorithm Interest Group, - - PowerPoint PPT Presentation

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Compressive Sensing Take 2 Yubo Paul Yang, Algorithm Interest Group, - - PowerPoint PPT Presentation

Nov 17, 2022 196 likes β€’360 views

Compressive Sensing Take 2 Yubo Paul Yang, Algorithm Interest Group, Nov. 1 2019 See take 1 by Brian Busemeyer BB cat What is compressive (compressed) sensing? Compressive sensing is a signal processing technique to reconstruct sparse

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Compressive Sensing Take 2

Yubo β€œPaul” Yang, Algorithm Interest Group, Nov. 1 2019 See take 1 by Brian Busemeyer BB cat

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What is compressive (compressed) sensing?

Compressive sensing is a signal processing technique to reconstruct sparse signal from few samples. It solves a system of underdetermined linear equations by imposing sparsity as a constraint.

𝒛 = 𝐡 π’š

when len(y) β‰ͺ len(x) by minimizing the number of non-zero entries in x. solve

= 𝒛

𝐡

π’š Trick: x has to be sparse.

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Simplest example: random transform of a very sparse sample = 𝒛

𝐡 = π‘ π‘π‘œπ‘’π‘π‘› 𝑛𝑏𝑒𝑠𝑗𝑦

… .1 … 1 . . 5

Goal: use y with a small length to recover x Strategy: minimize the L1-norm of x

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Practical application I: digital to analog conversion below Nyquist-Shannon

In practice, constructing the A matrix can be tricky. Signal in time domain, use Fourier transform as A matrix.

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How many samples does it take?

𝑧 𝑒 = ෍

π‘œ=1 π‘œπ‘‘π‘—π‘œ

sin(2𝜌 π‘œ 𝑒) Toy problem: reconstruct a sum of sine waves Number of samples needed for perfect reconstruction is determined by signal sparsity in β€œgood” basis. perfect reconstruction

  • F. Krzakala, M. Mezard, F. Sausset, Y.F. Sun, and L. Zdeborova, Phys. Rev. X 2, 021005 (2012).

sample density signal density perfect reconstruction large error in reconstruction

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How robust is CS to noise?

converged reconstruction but error converges roughly at the same transition sample density as before! Reconstruction is robust up to 5% white noise. Reconstruction noise does increase with more noise.

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Why is compressive sensing useful?

Signal reconstruction while under-sampling (lower average freq. than Nyquist-Shannon) Image reconstruction single-pixel camera fast MRI Digital to analog conversion Map Born-Oppenheimer potential energy surface using phonon directions!

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Practical application II: image compression

In the spirit of Halloween, let us attempt a reconstruction of the Shepp-Logan phantom. 2D images, use wavelet transform as A matrix. pywt package provides forward and inverse transforms

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Practical application II: image compression

My attempt: spooky?

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Application to Physics I: MD vibrational spectrum

  • X. Andrade, J. N. Sanders, and A. Aspuru-Guzik, Proc. Natl. Acad. Sci. U. S. A. 109, 13928-13933 (2012).

Accurate frequency after a few MD time steps Same problem as our practical application I velocity-velocity correlation is sparse in Fourier space

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Application to Physics II: Lattice dynamics

  • F. Zhou, W. Nielson, Y. Xia, V. Ozolins, Phys. Rev. Lett. 113, 18501 (2014).
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Application to Physics II: Lattice dynamics

  • F. Zhou, W. Nielson, Y. Xia, V. Ozolins, Phys. Rev. Lett. 113, 18501 (2014).
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Application to medical imaging: fast MRI

  • E. Candes, β€œCompressive Sensing – A 25 Minute Tour,” Frontiers of Engineering Symposium, Cambridge, UK (2010).
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Conclusions

Compressive sensing is a powerful method for signal reconstruction. Works whenever your problem is connected to a sparse representation by a linear transform. It has already found many applications in many fields!