Compressed sensing with a jackknife and a bootstrap
Keywords:error estimation, visualization, medical imaging, magnetic resonance imaging, compressive sensing, compressive sampling
Compressed sensing proposes to reconstruct more degrees of freedom in a signal than the number of values actually measured (based on a potentially unjustified regularizer or prior distribution). Compressed sensing therefore risks introducing errors -- inserting spurious artifacts or masking the abnormalities that medical imaging seeks to discover. Estimating errors using the standard statistical tools of a jackknife and a bootstrap yields "error bars" in the form of full images that are remarkably qualitatively representative of the actual errors (at least when evaluated and validated on data sets for which the ground truth and hence the actual error is available). These images show the structure of possible errors -- without recourse to measuring the entire ground truth directly -- and build confidence in regions of the images where the estimated errors are small. Further visualizations and summary statistics can aid in the interpretation of such error estimates. Visualizations include suitable colorizations of the reconstruction, as well as the obvious "correction" of the reconstruction by subtracting off the error estimates. The canonical summary statistic would be the root-mean-square of the error estimates. Unfortunately, colorizations appear likely to be too distracting for actual clinical practice in medical imaging, and the root-mean-square gets swamped by background noise in the error estimates. Fortunately, straightforward displays of the error estimates and of the "corrected" reconstruction are illuminating, and the root-mean-square improves greatly after mild blurring of the error estimates; the blurring is barely perceptible to the human eye yet smooths away background noise that would otherwise overwhelm the root-mean-square.