Path Orthogonal Matching Pursuit for k-Sparse Image Reconstruction

We introduce a path-augmentation step to the standard orthogonal matching pursuit algorithm. Our augmentation may be applied to any algorithm that relies on the selection and sorting of high-correlation atoms during an analysis or identification phase by generating a “path” between the two highest-correlation atoms. Here we investigate two types of path: a linear combination (Euclidean geodesic) and a construction relying on an optimal transport map (2-Wasserstein geodesic). We test our extension by generating k-sparse reconstructions of faces using an eigen-face dictionary learned from a subset of the data. We show that our method achieves lower reconstruction error for fixed sparsity levels than either orthogonal matching pursuit or generalized orthogonal matching pursuit.