Seminar in Inverse Problems [Spring 2019]
Inverse problems is a field of mathematics comprised of many areas including analysis, modeling, PDE's and scientific computation. Inverse problems arise in abundance in engineering, biology, physics, geophysics and more. This seminar addresses fundamental topics in inverse problems in a variety of applications.
Regular meeting times & location: Thursdays at 2 pm in Weber 15
Speaker: Alexei Novikov, Penn State University
Time and location: Thursday, Mar. 12, 2 pm, Weber 015
Title: Imaging with incomplete and corrupted data
Abstract: We consider the problem of imaging sparse scenes from a few noisy data using an l1-minimization approach. This problem can be cast as a linear system of the form Ax=b. The dimension of the unknown sparse vector x is much larger than the dimension of the data vector b. The l1-minimization alone, however, is not robust for imaging with noisy data. To improve its performance we propose to solve instead the augmented linear system [A|C]x=b, where the matrix C is a noise collector. It is constructed so as its column vectors provide a frame on which the noise of the data can be well approximated with high probability. This approach gives rise to a new parameter-free imaging method that has a zero false discovery rate for any level of noise. We also obtain exact support recovery if the noise is not too large.
Spring 2020 Schedule
| Mar. 5 | Electrical impedance tomography, enclosure method and machine learning | Samuli Siltanen, University of Helsinki, Finland |
| Mar. 12 | Imaging with incomplete and corrupted data | Alexei Novikov, Penn State |
| Apr. 2 | How to solve Bayesian inverse problems in practice | Wolfgang Bangerth, Dept. of Math., CSU |
| Apr. 9 | Reconstruction of a conductivity inclusion using the Faber polynomials - Rescheduled to the fall semester | Mikyoung Lim, KAIST, Seoul Korea |
| Apr. 16 | A Mathematical Approach to Neuronal Network Reconstruction | Paulina Vosolov, Dept. of Math., RPI |
Fall 2019 Schedule
| Sept. 5 | Sparsity-Based Inpainting and Data Separation | Emily King, Dept. of Math., CSU |
| Sept. 19 | A hybrid approach combining analytical and iterative regularization methods for Electrical Impedance Tomography and Diffuse Optical Tomography problems | Sanwar Ahmad, Dept. of Math., CSU |
| Oct. 17 | Rescheduled to Oct. 31 | Roozbeh Gharakhloo, Dept. of Math., CSU |
| Oct. 31 | A Riemann-Hilbert approach to asymptotic analysis of a bordered Toeplitz determinant and the next-to-diagonal correlations of the anisotropic square lattice Ising model. | Roozbeh Gharakhloo, Dept. of Math., CSU |
| Nov. 7 | No Seminar - Math Day | |
| Nov. 14 | Spectral approaches to d-bar systems | Christian Klein, Mathematics Institute of Bourgogne, France |
| Nov. 21 | Pulmonary Imaging using Electrical Impedance Tomography with a Low-Frequency Ultrasound Prior | Melody Alsaker, Gonzaga University |
Abstracts
Alexei Novikov, Penn State March. 12, 2020
Title: Imaging with incomplete and corrupted data
Abstract: We consider the problem of imaging sparse scenes from a few noisy data using an l1-minimization approach. This problem can be cast as a linear system of the form Ax=b. The dimension of the unknown sparse vector x is much larger than the dimension of the data vector b. The l1-minimization alone, however, is not robust for imaging with noisy data. To improve its performance we propose to solve instead the augmented linear system [A|C]x=b, where the matrix C is a noise collector. It is constructed so as its column vectors provide a frame on which the noise of the data can be well approximated with high probability. This approach gives rise to a new parameter-free imaging method that has a zero false discovery rate for any level of noise. We also obtain exact support recovery if the noise is not too large.
Wolfgang Bangerth, Dept. of Math., CSU Apr. 2, 2020
Title: How to solve Bayesian inverse problems in practice
Abstract: In inverse problems, one wants to infer properties of an object under investigation from observing its response to external stimuli. Biomedical imaging problems are an example, but so is non-destructive testing of materials or seismic imaging of the earth. Traditionally, people have asked what properties "match observations best", but since the late 1980s, we understand that because measurements are noisy, we should instead ask for how likely certain values of these properties are. This is called "Bayesian inversion". I'll give a short overview of how these are derived, and then ask how one actually *computes* such probability distributions. This is relevant because Bayesian formulations are typically exceedingly expensive to solve, primarily because they are high-dimensional. There are many methods to sample such probability distributions, but no generally accepted benchmark that would allow for fair comparisons of these methods. The second half of the talk will therefore be a discussion of a benchmark David Aristoff and I have been working on that models some of the complexities of real inverse problems. I will also show some preliminary data we have obtained by spending ~15 years of CPU time.
Mikyoung Lim, KAIST, Seoul Korea April 10, 2020
Title: Reconstruction of a conductivity inclusion using the Faber polynomials
Abstract: In this talk, I present analytical shape recovery methods for the conductivity inclusion in two dimensions, obtained in collaboration with Doosung Choi and Junbeom Kim. For a simply connected planar domain, there exists a function that conformally maps a region outside a circular disk to the region outside the domain. The conformal mapping then defines the so-called Faber polynomials, which form a basis for analytic functions. An electrical inclusion, inserted in a homogeneous background, induces a perturbation in the background potential. This perturbation admits a far-field expansion, whose coefficients are the so-called generalized polarization tensors (GPTs). GPTs can be obtained from the exterior measurements. As a modification of GPTs, we recently introduced the Faber polynomial polarization tensors (FPTs). We design two analytical approaches for the shape recovery of a conductivity inclusion by employing the concept of FPTs. Numerical experiments demonstrate the validity of the proposed analytical methods.
Emily King, Dept. of Math., CSU Sept. 5
Title: Sparsity-Based Inpainting and Data Separation
Abstract: Scientific and commercial data is often incomplete. Recovery of the missing information is often an important step in data analysis. Data recovery in images is known as inpainting. Furthermore, real-world data can in many cases be represented as a superposition of two or more different types of structures. For example, images may be decomposed into texture and cartoon-like components. Both inpainting and data separation are underdetermined inverse problems that benefit from sparsity-based regularization. In this talk, theoretical guarantees for successful inpainting and data separation via (analysis-side) sparsity regularization will be discussed. By leveraging tools from microlocal analysis, one may apply those guarantees to prove asymptotic results about inpainting and separation using specific dictionaries, like shearlets. Some numerical results comparing different algorithms will also be presented.
Sanwar Ahmad, Dept. of Math., CSU Sept. 19
Title: A hybrid approach combining analytical and iterative regularization methods for Electrical Impedance Tomography and Diffuse Optical Tomography problems
Abstract: Electrical impedance tomography (EIT) and Diffuse Optical Tomography (DOT) are imaging methods that have been gaining more popularity due to their ease of use and non-invasiveness. EIT and DOT can potentially be used as alternatives to traditional imaging techniques, such as computed tomography (CT) scans, to reduce the damaging effects of radiation on the tissue. For EIT, the inner distribution of resistivity, which corresponds to different resistivity properties of different tissues, is estimated from the voltage potentials measured on the boundary of the object being imaged. In DOT, the optical properties of the tissue, mainly scattering and absorption, are estimated by measuring the light on the boundary of the tissue illuminated by a near-infrared source at the tissue's surface. In this presentation, we discuss a direct method for solving the EIT inverse problem using mollifier regularization, which is then modified and extended to solve the inverse problem in DOT. A comprehensive numerical and computational comparison for EIT is presented. Based on the comparative results, a novel hybrid method combining the mollifier and iterative method is proposed.
Roozbeh Gharakhloo, Dept. of Math., CSU Oct. 31
Title: A Riemann-Hilbert approach to asymptotic analysis of a bordered Toeplitz determinant and the next-to-diagonal correlations of the anisotropic square lattice Ising model.
Abstract (pdf)
Christian Klein, Mathematics Institute of Bourgogne, France Nov. 14
Title: Spectral approaches to d-bar systems
Abstract: We discuss numerical approaches to d-bar systems appearing in the context of electrical impedance tomography and the inverse scattering theory of the Davey-Stewartson II equation. The approaches are spectral, i.e., the numerical error decreases exponentially with the resolution for functions being smooth on the considered domains. We present algorithms for the case of potentials in the Schwartz class of rapidly decreasing smooth functions, and for functions with compact support. In the first case we consider Fourier techniques with an analytical regularization of the singular integrands combined with Krylov subspace techniqes. In the second case we use polar cooordinates and Fourier techniques in the angular variable and polynomial interpolation in the radial variable. This is a joint work with Olga Assainova, Johannes Sjoestrand, Nikola Stoilov from IMB Dijon, Ken Mclaughlin from Colorado State University and Peter Miller from Michigan University
Melody Alsaker, Gonzaga University Nov. 21
Title: Pulmonary Imaging using Electrical Impedance Tomography with a Low-Frequency Ultrasound Prior
Abstract: Electrical impedance tomography (EIT) has been widely studied for use in pulmonary imaging applications, and the inclusion of prior-known spatial information from thoracic CT scans has been demonstrated to improve pulmonary EIT images. However, obtaining a CT scan has a number of drawbacks to the patient, including ionizing radiation, expense, and a need to move the patient to the machine. Ultrasound tomography (UST), on the other hand, does not have these drawbacks, but has only been minimally studied for use in pulmonary imaging due to difficulties with wave propagation. Low-frequency UST, however, has the potential to propagate in lung tissue, and therefore may be suitable for use as a prior in EIT. In this talk, we present numerically simulated results demonstrating the potential use of UST as a spatial prior in pulmonary EIT imaging.
Past Seminars
[Spring 2015] [Fall 2014]