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Appl Comput Harmon Anal 2020 ;49(3)

Program in Applied Mathematics, Yale University, New Haven, CT 06511, USA.

We present a fast method for evaluating expressions of the form where are real numbers, and are points in a compact interval of . This expression can be viewed as representing the electrostatic potential generated by charges on a line in . While fast algorithms for computing the electrostatic potential of general distributions of charges in exist, in a number of situations in computational physics it is useful to have a simple and extremely fast method for evaluating the potential of charges on a line; we present such a method in this paper, and report numerical results for several examples. Read More

Appl Comput Harmon Anal 2018 May 26;44(3):665-699. Epub 2016 Jul 26.

The Fourier-domain Douglas-Rachford (FDR) algorithm is analyzed for phase retrieval with a single random mask. Since the uniqueness of phase retrieval solution requires more than a single oversampled coded diffraction pattern, the extra information is imposed in either of the following forms: 1) the sector condition on the object; 2) another oversampled diffraction pattern, coded or uncoded. For both settings, the uniqueness of projected fixed point is proved and for setting 2) the local, geometric convergence is derived with a rate given by a spectral gap condition. Read More

Appl Comput Harmon Anal 2016 Sep 2;41(2):470-490. Epub 2016 Mar 2.

Department of Mathematics, University of Iowa, Iowa City, IA 52242, United States.

This paper explores robust recovery of a superposition of distinct complex exponential functions with or without damping factors from a few random Gaussian projections. We assume that the signal of interest is of 2 - 1 dimensions and < 2 - 1. This framework covers a large class of signals arising from real applications in biology, automation, imaging science, etc. Read More

Appl Comput Harmon Anal 2015 Mar;38(2):196-221

University of Vienna, Faculty of Mathematics, NuHAG, Austria.

Gabor frames can advantageously be redefined using the Heisenberg-Weyl operators familiar from harmonic analysis and quantum mechanics. Not only does this redefinition allow us to recover in a very simple way known results of symplectic covariance, but it immediately leads to the consideration of a general deformation scheme by Hamiltonian isotopies ( arbitrary paths of non-linear symplectic mappings passing through the identity). We will study in some detail an associated weak notion of Hamiltonian deformation of Gabor frames, using ideas from semiclassical physics involving coherent states and Gaussian approximations. Read More

Appl Comput Harmon Anal 2013 May 24;34(3):469-487. Epub 2012 Jul 24.

Department of Mathematics and Statistics, Air Force Institute of Technology, Wright-Patterson Air Force Base, OH 45433, USA.

The local histogram transform of an image is a data cube that consists of the histograms of the pixel values that lie within a fixed neighborhood of any given pixel location. Such transforms are useful in image processing applications such as classification and segmentation, especially when dealing with textures that can be distinguished by the distributions of their pixel intensities and colors. We, in particular, use them to identify and delineate biological tissues found in histology images obtained via digital microscopy. Read More

Appl Comput Harmon Anal 2012 Sep;33(2):300-308

School of Interactive Computing, Georgia Institute of Technology, Atlanta, USA.

In this paper we provide rigorous proof for the convergence of an iterative voting-based image segmentation algorithm called Active Masks. Active Masks (AM) was proposed to solve the challenging task of delineating punctate patterns of cells from fluorescence microscope images. Each iteration of AM consists of a linear convolution composed with a nonlinear thresholding; what makes this process special in our case is the presence of additive terms whose role is to "skew" the voting when prior information is available. Read More

Appl Comput Harmon Anal 2011 Jul;31(1):44-58

Department of Mathematics and PACM, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000 USA.

One of the main objectives in the analysis of a high dimensional large data set is to learn its geometric and topological structure. Even though the data itself is parameterized as a point cloud in a high dimensional ambient space ℝ(p), the correlation between parameters often suggests the "manifold assumption" that the data points are distributed on (or near) a low dimensional Riemannian manifold ℳ(d) embedded in ℝ(p), with d ≪ p. We introduce an algorithm that determines the orientability of the intrinsic manifold given a sufficiently large number of sampled data points. Read More

Appl Comput Harmon Anal 2011 Jan;30(1):20-36

Department of Mathematics and PACM, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000 USA,

The angular synchronization problem is to obtain an accurate estimation (up to a constant additive phase) for a set of unknown angles θ(1), …, θ(n) from m noisy measurements of their offsets θ(i) - θ(j) mod 2π. Of particular interest is angle recovery in the presence of many outlier measurements that are uniformly distributed in [0, 2π) and carry no information on the true offsets. We introduce an efficient recovery algorithm for the unknown angles from the top eigenvector of a specially designed Hermitian matrix. Read More

Appl Comput Harmon Anal 2010 May;28(3):296-312

Department of Mathematics, Program in Applied Mathematics, Yale University, 10 Hillhouse Ave. PO Box 208283, New Haven, CT 06520-8283 USA.

Recovering the three-dimensional structure of molecules is important for understanding their functionality. We describe a spectral graph algorithm for reconstructing the three-dimensional structure of molecules from their cryo-electron microscopy images taken at random unknown orientations.We first identify a one-to-one correspondence between radial lines in three-dimensional Fourier space of the molecule and points on the unit sphere. Read More