A Statistical Framework to Investigate the Optimality of Signal-Reconstruction Methods

Abstract

We present a statistical framework to benchmark the performance of reconstruction algorithms for linear inverse problems, in particular, neural-network-based methods that require large quantities of training data. We generate synthetic signals as realizations of sparse stochastic processes, which makes them ideally matched to variational sparsity-promoting techniques. We derive Gibbs sampling schemes to compute the minimum mean-square error estimators for processes with Laplace, Student's t, and Bernoulli-Laplace innovations. These allow our framework to provide quantitative measures of the degree of optimality (in the mean-square-error sense) for any given reconstruction method. We showcase our framework by benchmarking the performance of some well-known variational methods and convolutional neural network architectures that perform direct nonlinear reconstructions in the context of deconvolution and Fourier sampling. Our experimental results support the understanding that, while these neural networks outperform the variational methods and achieve near-optimal results in many settings, their performance deteriorates severely for signals associated with heavy-tailed distributions.

References in corpus (11)

• MoDL: Model Based Deep Learning Architecture for Inverse Problems
• A Review of Convolutional Neural Networks for Inverse Problems in Imaging
• Deep learning for undersampled MRI reconstruction
• CNN-Based Projected Gradient Descent for Consistent Image Reconstruction
• Learned reconstructions for practical mask-based lensless imaging
• CNN-Based Image Reconstruction Method for Ultrafast Ultrasound Imaging
• SIMBA: Scalable Inversion in Optical Tomography using Deep Denoising Priors
• Efficient Gaussian Sampling for Solving Large-Scale Inverse Problems using MCMC Methods
• Pocket Guide to Solve Inverse Problems with GlobalBioIm
• Biomedical Image Reconstruction: From the Foundations to Deep Neural Networks
• High-dimensional Gaussian sampling: a review and a unifying approach based on a stochastic proximal point algorithm