A Quantitative Vainberg Method for Black Box Scattering

We give a quantitative version of Vainberg's method relating pole free regions to propagation of singularities for black box scatterers. In particular, we show that there is a logarithmic resonance free region near the real axis of size $\tau$ with polynomial bounds on the resolvent if and only if the wave propagator gains derivatives at rate $\tau$. Next we show that if there exist singularities in the wave trace at times tending to infinity which smooth at rate $\tau$, then there are resonances in logarithmic strips whose width is given by $\tau$. As our main application of these results, we give sharp bounds on the size of resonance free regions in scattering on geometrically nontrapping manifolds with conic points. Moreover, these bounds are generically optimal on exteriors of nontrapping polygonal domains.Comment: 22 pages, 1 figur

Quantitative Analysis by the Point-Centered Quarter Method

This document is an introduction to the use of the point-centered quarter method. It briefly outlines its history, its methodology, and some of the practical issues (and modifications) that inevitably arise with its use in the field. Additionally this paper shows how data collected using point-centered quarter method sampling may be used to determine importance values of different species of trees and describes and derives several methods of estimating plant density and corresponding confidence intervals. New to this revision is an appendix of R functions to carry out these calculations.Comment: 56 pages, 12 figures, 16 tables. Corrected typos. Expanded Appendix B on Angle-Order Methods. Added Appendix D containing R functions to carry out all calculations. Added references. Original version: 34 pages, 6 figures, 16 table

A variational method for quantitative photoacoustic tomography with piecewise constant coefficients

In this article, we consider the inverse problem of determining spatially heterogeneous absorption and diffusion coefficients from a single measurement of the absorbed energy (in the steady-state diffusion approximation of light transfer). This problem, which is central in quantitative photoacoustic tomography, is in general ill-posed since it admits an infinite number of solution pairs. We show that when the coefficients are known to be piecewise constant functions, a unique solution can be obtained. For the numerical determination of the coefficients, we suggest a variational method based based on an Ambrosio-Tortorelli-approximation of a Mumford-Shah-like functional, which we implemented numerically and tested on simulated two-dimensional data

3D quantitative microwave imaging with a regularized Gauss-Newton method for breast cancer detection

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Measurements of wavelength dependent scattering and backscattering coefficients by low-coherence spectroscopy

Quantitative measurements of scattering properties are invaluable for optical techniques in medicine. However, noninvasive, quantitative measurements of scattering properties over a large wavelength range remain challenging. We introduce low-coherence spectroscopy as a noninvasive method to locally and simultaneously measure scattering μs and backscattering μb coefficients from 480 to 700 nm with 8 nm spectral resolution. The method is tested on media with varying scattering properties (μs = 1 to 34 mm−1 and μb = 2.10−6 to 2.10−3 mm−1), containing different sized polystyrene spheres. The results are in excellent agreement with Mie theor