Poster Session

Title: Bagged Deep Image Prior for Recovering Images in the Presence of Speckle Noise.

Abstract: We investigate both the theoretical and algorithmic aspects of likelihood-based methods for recovering a complex-valued signal from multiple sets of measurements, referred to as looks, affected by speckle (multiplicative) noise. Our theoretical contributions include establishing the first existing theoretical upper bound on the Mean Squared Error (MSE) of the maximum likelihood estimator under the deep image prior hypothesis. Our theoretical results capture the dependence of MSE upon the number of parameters in the deep image prior, the number of looks, the signal dimension, and the number of measurements per look. On the algorithmic side, we introduce the concept of bagged Deep Image Priors (Bagged-DIP) and integrate  them with projected gradient descent. Furthermore, we show how employing Newton-Schulz algorithm for calculating matrix inverses within the iterations of PGD reduces the computational complexity of the algorithm. We will show that this method achieves state-of-the-art performance.

Title: Reconstruction of Small Volume Regions in EIT with a Robin Transmission Condition

Abstract: We consider a small volume, inverse shape problem coming from electrical impedance tomography with a Robin transmission condition. In general, a boundary condition of Robin type models corrosion. In this presentation, we study a method for recovering an interior corroded region from electrostatic data. We derive an asymptotic expansion of the current gap operator applied to imposed voltage and prove that a MUSIC-type algorithm can be used to recover the small volume region. Numerical examples will be presented in two dimensions in the unit circle.

Title: Efficient numerical methods for inverse scattering problems in complex media

Abstract: The Linear Sampling Method, introduced by Colton and Krisch in 1996, along with other qualitative-type methods, are not just effective numerical methods for shape reconstruction problems in inverse scattering theory, but they are also practical and cost-effective solutions. These methods, which are low cost and require little prior knowledge about the physical parameters of the background, function by building an appropriate indicator function for each sampling point in the research domain using measurements of scattered waves. This talk will present two applications of these methods. The first application is to reconstruct local defects in a bi-periodic domain without needing the physical parameters of the periodic background. The second one is identifying cracks in a finite body in 2D using elastic waves.

Title: Magneto-Acoustic Tomography With Magnetic Induction (MAT-MI)

Abstract: Imaging modalities like magnetic resonance imaging (MRI), computed tomography (CT), and electrical impedance tomography (EIT) are widely used in the scientific and medical fields. Their beauty and utility lie in their ability to elucidate internal information of an object from data on the object’s boundary. However, current modalities like these fail to deliver images with high contrast and high resolution, which results in difficult to interpret images. The MAT-MI process is a novel hybrid imaging modality for reconstructing the internal conductivity map of a conductive body from its internal acoustic properties with improved resolution and contrast. The first step, where a physical ”scan” is taken, calculates the object’s internal acoustic properties from the acoustic properties on the object’s boundary. The second step is itself a twostep quasi-Newton method that takes this acoustic source data and reconstructs the object’s conductivity map, effectively reconstructing the internal image of a solid body. This has obvious applications to the medical field in both research and patient care settings. Established results on the MAT-MI process prove stability of the inverse problem and convergence of the algorithm while providing numerical results to reinforce the mathematical claims. In this talk, I will explain the steps of the MAT-MI process, provide an overview of some stability and convergence results, and present a gallery of numerical experiments. Given time, I will also explore some interesting future directions for this project and their associated limitations.

Title: Theoretical analysis of binary mask in snapshot compressive imaging.

Abstract: This poster presents a theoretical analysis of binary masks in snapshot compressive imaging (SCI) systems, demonstrating that optimal performance is achieved with a non-zero element probability of less than 0.5. It examines independently distributed masks and those generated by the Markov process, offering insights for enhancing SCI system design and optimization.

Title: On the Duality between Scattering Poles and Transmission Eigenvalues.

Abstract: This poster focuses on a characterization method of scattering poles for different acoustic and electromagnetic scattering problems. This characterization is based on the duality between scattering poles and interior eigenvalues, first developed in the paper [1], and consists in relating the scattering poles to the kernel of an interior scattering problem, whereas the interior eigenvalues are related to the kernel of an exterior scattering operator. In particular, we develop here the case of an anisotropic media for the electromagnetic case, where the duality takes place between the scattering poles and the transmission eigenvalues [2]. In a second part we develop the approach for an acoustic scattering problem with an impedance boundary condition, for which the duality is between scattering poles and the interior impedance eigenvalues, and finally we show some numerical results [3].

[1] Cakoni, F., Colton, D., & Haddar, H. (2020). A duality between scattering poles and transmission eigenvalues in scattering theory. Proceedings of the Royal Society A, 476(2244), 20200612.

[2] (article in preparation), Cakoni, F., Haddar, H., Zilberberg, D. (2022). A duality between scattering poles and transmission eigenvalues in electromagnetism.

[3] (article in preparation),Cakoni, F., Haddar, H., Zilberberg, D. (2022). On the duality between scattering poles and impedance eigenvalues, and a numerical application.