A Compressed Sensing Algorithm for Sparse-View Pinhole Single Photon Emission Computed Tomography

Single Photon Emission Computed Tomography (SPECT) systems are being developed with multiple cameras and without gantry rotation to provide rapid dynamic acquisitions. However, the resulting data is angularly undersampled, due to the limited number of views. We propose a novel reconstruction algorithm for sparse-view SPECT based on Compressed Sensing (CS) theory. The algorithm models Poisson noise by modifying the Iterative Hard Thresholding algorithm to minimize the Kullback-Leibler (KL) distance by gradient descent. Because the underlying objects of SPECT images are expected to be smooth, a discrete wavelet transform (DWT) using an orthogonal spline wavelet kernel is used as the sparsifying transform. Preliminary feasibility of the algorithm was tested on simulated data of a phantom consisting of two Gaussian distributions. Single-pinhole projection data with Poisson noise were simulated at 128, 60, 15, 10, and 5 views over 360 degrees. Image quality was assessed using the coefficient of variation and the relative contrast between the two objects in the phantom. Overall, the results demonstrate preliminary feasibility of the proposed CS algorithm for sparse-view SPECT imaging

A range description for the planar circular Radon transform

The transform considered in the paper integrates a function supported in the unit disk on the plane over all circles centered at the boundary of this disk. Such circular Radon transform arises in several contemporary imaging techniques, as well as in other applications. As it is common for transforms of Radon type, its range has infinite co-dimension in standard function spaces. Range descriptions for such transforms are known to be very important for computed tomography, for instance when dealing with incomplete data, error correction, and other issues. A complete range description for the circular Radon transform is obtained. Range conditions include the recently found set of moment type conditions, which happens to be incomplete, as well as the rest of conditions that have less standard form. In order to explain the procedure better, a similar (non-standard) treatment of the range conditions is described first for the usual Radon transform on the plane.Comment: submitted for publicatio

Reconstruction of Planar Domains from Partial Integral Measurements

We consider the problem of reconstruction of planar domains from their moments. Specifically, we consider domains with boundary which can be represented by a union of a finite number of pieces whose graphs are solutions of a linear differential equation with polynomial coefficients. This includes domains with piecewise-algebraic and, in particular, piecewise-polynomial boundaries. Our approach is based on one-dimensional reconstruction method of [Bat]* and a kind of "separation of variables" which reduces the planar problem to two one-dimensional problems, one of them parametric. Several explicit examples of reconstruction are given. Another main topic of the paper concerns "invisible sets" for various types of incomplete moment measurements. We suggest a certain point of view which stresses remarkable similarity between several apparently unrelated problems. In particular, we discuss zero quadrature domains (invisible for harmonic polynomials), invisibility for powers of a given polynomial, and invisibility for complex moments (Wermer's theorem and further developments). The common property we would like to stress is a "rigidity" and symmetry of the invisible objects. * D.Batenkov, Moment inversion of piecewise D-finite functions, Inverse Problems 25 (2009) 105001Comment: Proceedings of Complex Analysis and Dynamical Systems V, 201

Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations

In a nonlinear oscillatory system, spectral submanifolds (SSMs) are the smoothest invariant manifolds tangent to linear modal subspaces of an equilibrium. Amplitude-frequency plots of the dynamics on SSMs provide the classic backbone curves sought in experimental nonlinear model identification. We develop here a methodology to compute analytically both the shape of SSMs and their corresponding backbone curves from a data-assimilating model fitted to experimental vibration signals. Using examples of both synthetic and real experimental data, we demonstrate that this approach reproduces backbone curves with high accuracy.Comment: 32 pages, 4 figure

Measurement of the spin of the M87 black hole from its observed twisted light

We present the first observational evidence that light propagating near a rotating black hole is twisted in phase and carries orbital angular momentum (OAM). This physical observable allows a direct measurement of the rotation of the black hole. We extracted the OAM spectra from the radio intensity data collected by the Event Horizon Telescope from around the black hole M87* by using wavefront reconstruction and phase recovery techniques and from the visibility amplitude and phase maps. This method is robust and complementary to black-hole shadow circularity analyses. It shows that the M87* rotates clockwise with an estimated rotation parameter $a=0.90\pm0.05$ with $\sim 95\%$ confidence level (c.l.) and inclination $i=17^\circ \pm2^\circ$, equivalent to a magnetic arrested disk with inclination $i=163^\circ\pm2^\circ$. From our analysis we conclude, within a 6 $\sigma$ c.l., that the M87* is rotating.Comment: Small addition on coherence. 5 pages, 2 figures Accepted for publication in MNRAS Letter