Two-Level Structural Sparsity Regularization for Finding Lattice Locations and Defects in Noisy Image Data

This paper presents a regularized regression model with two-level structural sparsity penalties and applies it for locating individual atoms in a noisy electron microscope image. For crystalline materials, the locations of atoms have spatial symmetries, forming a few regular lattice groups. Therefore, by simply estimating the underlying lattice groups seen in the image, one can locate most atoms in the image accurately. Identifying the few underlying lattice groups is formulated as a sparse group selection problem. On the other hand, some positions on the lattice groups can be vacant due to atomic defects, so simply finding the lattice groups may result in many false detections on the vacant positions. To minimize such false detections, the proposed model includes an individual sparsity regularization in addition to the group sparsity for a within-group selection, which results in a regularization regression model with two-level sparsities. We propose a modification of the group orthogonal matching pursuit (gOMP) algorithm with a thresholding step to solve the problem. The convergence analysis and statistical analysis of the proposed algorithm are presented. The proposed algorithm is also evaluated through numerical experiments with two simulated images and three real images.

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