MS01: Recent developments in computational inverse problems in imaging


Session A: Oct.1, 10:40am-12:00pm, Classroom Building 102
Session B: Oct.1, 3:10pm-4:30pm, Classroom Building 102

10:40am-11:00am (Oct 1)
CLB 102

Thu Le,
Kansas State University 

Orthogonality sampling method for inverse elastic scattering from anisotropic media


In this talk, we investigate the inverse scattering problem of time-harmonic elastic waves for anisotropic inclusions in an isotropic homogeneous background. The elasticity tensor and mass density are allowed to be heterogeneous inside the inclusion and may be discontinuous across the background-inclusion interface. We derive the far-field pattern of the scattered wave using the Lippmann-Schwinger integral equation of the scattering problem. Using the far-field pattern as the data for the inverse problem we construct indicator functions of orthogonality sampling types for the scattering objects. These functions are very robust to noise, computationally cheap, and do not involve any regularization process. We provide some theoretical analysis as well as numerical simulations for the proposed indicator functions. This is joint work with Dinh-Liem Nguyen and Trung Truong.


11:00am-11:20am (Oct 1)
CLB 102

Heejin Lee,
Purdue University
An inverse scattering obstacle problem with partial thin coatings
  We consider the inverse problem of recovering the shape and boundary coefficients of an obstacle from far-field measurements of the scattered field. More specifically the scatterer is impenetrable with Dirichlet boundary condition on a part of its boundary and anisotropic generalized impedance boundary condition on the complementary boundary. The latter is an approximate model for complicated thin anisotropic, absorbing layer and is given as second-order surface differential operator. A deep analysis of the far-field operator (otherwise known as the relative scattering operator) for this scattering problem leads to unique determination results and reconstruction methods for the shape of the scatterer as well as the boundary coefficients. Our reconstruction method is non-iterative and uses no a priori information on the topology and physics of the unknown object. This inversion approach is mathematically rigorous, it resolves nonlinear information from the range properties of the linear far-field operator, and it is easy to implement.


11:20am-11:40am (Oct 1)
CLB 102

Govanni Granados,
Purdue University
Asymptotic analysis applied to small volume inverse shape problems
  We consider two inverse shape problems coming from diffuse optical tomography and inverse scattering. For both problems, we assume that there are small volume subregions that we wish to recover using the measured Cauchy data. We will derive an asymptotic expansion involving their respective fields. Using the asymptotic expansion, we derive a MUSIC-type algorithm for the Reciprocity Gap Functional, which we prove can recover the subregion(s) with a finite amount of Cauchy data. Numerical examples will be presented for both problems in two dimensions in the unit circle.


11:40am-12:00pm (Oct 1)
CLB 102

Fatemeh Pourahmadian

University of Colorado Boulder

A comparative study of time- vs. frequency- domain inverse elastic scattering using laboratory test data.
  In this talk, we first review the time-domain linear sampling method for elastic-wave imaging of fractures as a complement to the existing LSM framework in the frequency domain. This opens the door for a comparative performance analysis between time- and frequency- domain waveform inversions using the laboratory test data in [1,2]. The experiments reported by [1] (resp. [2]) capture interaction of ultrasonic waves with a stationary (resp. evolving) fracture in a plate whose footprint is measured on the boundary. The resulting ultrasonic measurements are then used to computed TLSM maps which successfully reconstruct the evolution of damage in time and space. It is further shown that the time-domain inversion with sparse or reduced-aperture data remain robust at moderate noise levels. Further, a comparative analysis is conducted between the TLSM reconstructions and the corresponding multifrequency LSM maps of [1] using same data. A remarkable contrast in image quality is observed between the time- and frequency- domain inversions.

[1] F. Pourahmadian and H. Yue. Laboratory application of sampling approaches to inverse scattering. Inverse Problems, 37(5):055012, 2021.

[2] F. Pourahmadian. Experimental validation of differential evolution indicators for ultrasonic imaging in unknown backgrounds. Mechanical Systems and Signal Processing, 161:108029, 2021.


3:10pm-3:30pm (Oct 1)
CLB 102

Trung Truong,
Kansas State University

A sampling-type method combined with deep learning for inverse scattering with one incident wave.


We are interested in the inverse scattering problem that aims to reconstruct the geometry of a bounded object from measured data of the scattered wave. When the scattered wave data was obtained for only one incident wave, existing reconstruction methods cannot provide satisfactory results for relatively complex geometries. This lack of data is common in practice due to technical difficulties. Therefore, we study a sampling-type method that is fast, simple, regularization-free, stable against high levels of noise, and combine it with a deep neural network to solve the inverse scattering problem in which the scattering data is only provided for one incident wave. This combined method can be understood as a network using the image computed by the sampling method for the first layer and followed by the U-net architecture for the remaining layers. The network is trained on simulated data sets of simple scattering objects and is validated by objects with more complex geometries and real data without any additional transfer training. This is joint work with Thu Le, Dinh-Liem Nguyen, and Vu Nguyen.


3:30pm-3:50pm (Oct 1)
CLB 102

Christina Frederick,
New Jersey Institute of Technology

Machine learning techniques for inverse problems in sonar imaging.


In this talk we discuss machine learning for inverse problems in high frequency underwater acoustics, where the goal is to recover detailed characteristics of the seafloor from measured backscatter data generated from SONAR systems. The key to successful inversion is the use of accurate forward modeling that captures of the dependence of the backscatter on seafloor properties, such as sediment type, roughness, and thickness of layers. To enable a rapid, remote, and accurate recovery of the seafloor, we propose an approach that combines high fidelity forward modeling and simulation of the entire physical wave propagation and scattering process and machine learning strategies. The idea is to partition large underwater acoustic environments, on the order of kilometers in spatial width, into much smaller “template” domains, a few meters in spatial width, in which the sediment layer can be described using a limited number of parameters. Hybrid prediction models can be created by embedding localized simulations of Helmholtz equations on each template domain in a large-scale geometric optics model for larger domains. To solve the inverse problem, machine learning strategies applied to a reference library of acoustic templates can be used to estimate the seafloor parameters that describe the full domain.


3:50pm-4:10pm (Oct 1)
CLB 102

Mikhail Zaslavsky,
Southern Methodist University

Lippmann-Schwinger-Lanczos algorithm for imaging problems.


Data-driven reduced order models (ROMs) are combined with the Lippmann-Schwinger integral equation to produce a direct nonlinear inversion method. The ROM is viewed as a Galerkin projection and is sparse due to Lanczos orthogonalization. Embedding into the continuous problem, a data-driven internal solution is produced. This internal solution is then used in the Lippmann-Schwinger equation, thus making further iterative updates unnecessary. We show numerical experiments for spectral domain data for which our inversion is far superior to the Born inversion and works as well as when the true internal solution is known.


4:10pm-4:30pm (Oct 1)
CLB 102

Thi-Phong Nguyen,
New Jersey Institute of Technology

A direct approach for inverse source problems in photoacoustic tomography.


This talk will discuss a direct approach to solve numerically the inverse source problem of recovering the initial condition of a wave equation in time using the boundary measurements. This problem is generally encountered for example in photoacoustic tomography, where the initial condition is generated by the expansion of the part which has been heated by photons. By applying an orthonormal basis in time, we discretize the original equation into a coupled system of elliptical equations. The source is now represented as a Fourier series where the Fourier coefficients are obtained by solving the coupled system above. The representation of the source allows one to recover the source directly instead of applying inverse methods, so the results are robust with respect to noise. This is a joint work with Loc H. Nguyen, Thuy T. Le, and William Powell at the UNC at Charlotte.