11. Recursive Reconstruction of Sparse Signal Sequences

In this chapter we describe our recent work on the design and analysis of recursive algorithms for causally reconstructing a time sequence of (approximately) sparse signals from a greatly reduced number of linear projection measurements. The signals are sparse in some transform domain referred to as the sparsity basis and their sparsity patterns (support set of the sparsity basis coefficients) can change with time. By “recursive", we mean using only the previous signal’s estimate and the current measurements to get the current signal’s estimate. We also briefly summarize our exact reconstruction results for the noise-free case and our error bounds and error stability results (conditions under which a time-invariant and small bound on the reconstruction error holds at all times) for the noisy case. Connections with related work are also discussed. A key example application where the above problem occurs is dynamic magnetic resonance imaging (MRI) for real-time medical applications such as interventional radiology and MRI-guided surgery, or in functional MRI to track brain activation changes. Cross-sectional images of the brain, heart, larynx or other human organ images are piecewise smooth, and thus approximately sparse in the wavelet domain. In a time sequence, their sparsity pattern changes with time, but quite slowly. The same is also often true for the nonzero signal values. This simple fact, which was first observed in our work, is the key reason that our proposed recursive algorithms can achieve provably exact or accurate reconstruction from very few measurements.