exploring compressed sensing fMRI time series

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we are once again solving systems of equations
Table of Contents
compressed sensing from 30% subsampling using the BSBL-BO algorithm
Signal sensing at 30% undersampling using the BSBL-BO algorithm. `$Y_i$` corresponds to time-domain signals whereas `$x_i$` corresponds to frequency-domain signals.

tl;dr An exploration of compressed sensing fMRI time series with 3 different algorithms. Typically, compressed sensing reconstructs a single volume of MRI but fMRI are composed of many volumes; sensing along the time domain could reduce the number of volumes required. Of the 3 algorithms, BSBL-BO performed the best with the error curve elbowing around 30% subsampling.

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Abstract

Compressed sensing reconstructs signals by solving underdetermined linear systems under the con- ditions that the measurements are sparse in the domain and incoherent [1]. In engineering, if measurements are taken within an appropriate basis satisfying the restricted isometry property, i.e., the Gaussian, Bernoulli, or Fourier bases, then this prior structure makes full signal recovery possible [2].

Compressed sensing is challenging with fMRI because the temporal dynamics of hemodynamic signals are relatively slow compared to other fast-acquisition signals that historically benefit from compressed sensing [3]. Additionally, having fewer samples insinuates a loss of statistical power in subsequent analyses. Thankfully, fMRI signals boast two beneficial characteristics that are promising for compressed sensing: 1. they are linear time-invariant, and 2. they lie within, and can transformed by, a Fourier basis [4].

In medical imaging, it is often assumed that an image is sampled at the Nyquist rate, s.t., enough discrete measurements are taken to reconstruct a continuous whole (M > N) without loss of information. If high-fidelity reconstruction is possible sampling below the Nyquist rate, then MRI modalities would benefit since discerning a signal and quickly turning over an analysis reduces real costs. These potential gains beg the question: if an underdetermined linear system can be solved after sampling below the Nyquist rate, can we collect fewer samples and still recover a high-quality fMRI under a compressed sensing paradigm? The purpose of this project is to explore approaches to compressed sensing that yield meaningful signal recoveries from heavy undersampling.

Results

ECOS RMSE and PSNR results
Summary metrics of RMSE (left) and PSNR (right) voxel time series recovery using L1 minimization in a pure convex optimization formulation solved with the ECOS algorithm.
OWL-QN RMSE and PSNR results
Summary metrics of RMSE (left) and PSNR (right) voxel time series recovery using L1 minimization in a pure convex optimization formulation solved with the OWL-QN algorithm.
BSBL-BO RMSE and PSNR results
Summary metrics of RMSE (left) and PSNR (right) voxel time series recovery using L1 minimization in a pure convex optimization formulation solved with the BSBL-BO algorithm.

References

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