Georgia Institute of Technology College of Sciences

This thesis presents several contributions to the fields of image and geometry processing. In 2D image processing, we propose a method that not only computes the relative depth of objects in bitmap format but also inpaints occluded regions using a PDE-based model and vector representation. Our approach demonstrates both qualitative and quantitative advantages over the state-of-the-art depth-aware bitmap-to-vector conversion models.

In the area of 3D point cloud processing, we introduce a method for generating a robust normal vector field that preserves first order discontinuity while being resistant to noise, supported by a degree of theoretical guarantee. This technique has potential applications in solving PDEs on point clouds, detecting sharp features, and reconstructing surfaces from incomplete and noisy data.

Additionally, we present a dedicated work on surface reconstruction from point cloud data. While many existing models can reconstruct implicit surfaces and some include denoising capabilities, a common drawback is the loss of sharp features: edges and corners are often smoothed out in the process. To address this limitation, we propose a method that not only denoises but also preserves sharp edges and corners during surface reconstruction from noisy data.

In 1970 Matiyasevich, building on earlier work of Davis--Putnam--Robinson, proved that every enumerable subset of Z is Diophantine, thus showing that Hilbert's 10th problem is undecidable for $\mathbb{Z}$. The problem of extending this result to the ring of integers of number fields (and more generally to finitely generated infinite rings) has attracted significant attention and, thanks to the efforts of many mathematicians, the task has been reduced to the problem of constructing, for certain quadratic extensions of number fields $L/K$, an elliptic curve $E/K$ with $rk(E(L))=rk(E(K))>0$. 

In this talk I will explain joint work with Peter Koymans, where we use Green--Tao to construct the desired elliptic curves, settling Hilbert 10 for every finitely generated infinite ring.

This presentation will expound the challenges involved in the generation of digital twins (DT) as valuable tools for supporting innovation and providing informed decision support for the optimization of properties and/or performance of advanced material systems. This presentation will describe the foundational AI/ML (artificial intelligence/machine learning) concepts and frameworks needed to formulate and continuously update the DT of a selected material system. The central challenge comes from the need to establish reliable models for predicting the effective (macroscale) functional response of the heterogeneous material system, which is expected to exhibit highly complex, stochastic, nonlinear behavior. This task demands a rigorous statistical treatment (i.e., uncertainty reduction, quantification and propagation through a network of human-interpretable models) and fusion of insights extracted from inherently incomplete (i.e., limited available information), uncertain, and disparate (due to diverse sources of data gathered at different times and fidelities, such as physical experiments, numerical simulations, and domain expertise) data used in calibrating the multiscale material model. This presentation will illustrate with examples how a suitably designed Bayesian framework combined with emergent AI/ML toolsets can uniquely address this challenge.