Treffer: Sparse recovery algorithms for 3D imaging using point spread function engineering

Ph.D. ; In this thesis, we are concerned with the high-resolution imaging and localization problem of 3D point source image recovery from 2D data using methods based on point spread function (PSF) design. A new technique patented by S. Prasad for applying rotating point spread functions with a single lobe to obtain depth from defocus is considered. Applications include high-resolution single molecule localization microscopy, as well as localization of space debris using a space-based telescope. We develop and apply several recovery algorithms for this rotating PSF problem. First, we encode the depth position of a single point source with the amount of rotation of the PSF. Principal component analysis and total variation are used to determine the rotation angle, from which we can obtain the axial location of the point source. Based on another PSF fitting technique, we can estimate the transverse location and flux. For multiple point sources, finding the locations and fluxes is a large-scale sparse 3D inverse problem. We have developed solution algorithms based on matching pursuit and non-convex optimization. The distribution of point sources is discretized on a cubical lattice where the indexes of nonzero entries represent the 3D locations of the point sources. The values of these entries are the point source fluxes. Ecient implementation schemes including 3D fast Fourier transforms are proposed for both techniques. In the matching pursuit case, we develop and apply a single best replacement (SBR) algorithm for our 3D localization problem. Acceleration techniques on searching processing and precomputation are proposed. In the nonconvex optimization case, we consider two kinds of noise models. The continuous exact ℓ₀ model (CEL0) with a least squares data-fitting term is applied for the Gaussian noise model, and a new nonconvex regularization method with data-fitting term based on Kullback-Leibler (KL) divergence is proposed for the Poisson noise model. In addition, we propose a new scheme of estimation of the ...