Georgia Institute of Technology College of Sciences

For many problems in science and engineering, one needs to quantitatively compare shapes of objects in images, e.g., anatomical structures in medical images, detected objects in images of natural scenes. One might have large databases of such shapes, and may want to cluster, classify or compare such elements. To be able to perform such analyses, one needs the notion of shape distance quantifying dissimilarity of such entities. In this work, we focus on the elastic shape distance of Srivastava et al. [PAMI, 2011] for closed planar curves. This provides a flexible and intuitive geodesic distance measure between curve shapes in an appropriate shape space, invariant to translation, scaling, rotation and reparametrization. Computing this distance, however, is computationally expensive. The original algorithm proposed by Srivastava et al. using dynamic programming runs in cubic time with respect to the number of nodes per curve. In this work, we propose a new fast hybrid iterative algorithm to compute the elastic shape distance between shapes of closed planar curves. The asymptotic time complexity of our iterative algorithm is O(N log(N)) per iteration. However, in our experiments, we have observed almost a linear trend in the total running times depending on the type of curve data.

Many real-life systems require simulation techniques to evaluate the system performance and facilitate decision making. Stochastic simulation is driven by input model — a collection of probability distributions that model the system stochasticity. The choice of the input model is crucial for successful modeling and analysis via simulation. When there are past observed data of the system stochasticity, we can utilize these data to construct an input model. However, there is only a finite amount of data in practice, so the input model based on data is always subject to uncertainty, which is the so-called input (model) uncertainty. Therefore, a typical stochastic simulation faces two types of uncertainties: one is the input (model) uncertainty, and the other is the intrinsic stochastic uncertainty. In this talk, I will discuss our recent work on how to assess the risk brought by the two types of uncertainties and how to make decisions under these uncertainties.

Many real-world problems reduce to optimization problems that are solved by iterative methods. In this talk, I focus on recently developed efficient algorithms for solving large-scale optimization problems that arises in medical imaging and image processing. In the first part of my talk, I will introduce the Bregman Operator Splitting with Variable Stepsize (BOSVS) algorithm for solving nonsmooth inverse problems. The proposed algorithm is designed to handle applications where the matrix in the fidelity term is large, dense, and ill-conditioned. Numerical results are provided using test problems from parallel magnetic resonance imaging. In the second part, I will focus on the Euler's Elastica-based model which is non-smooth and non-convex, and involves high-order derivatives. I introduce two efficient alternating minimization methods based on operator splitting and alternating direction method of multipliers, where subproblems can be solved efficiently by Fourier transforms and shrinkage operators. I present the analytical properties of each algorithm, as well as several numerical experiments on image inpainting problems, including comparison with some existing state-of-art methods to show the efficiency and the effectiveness of the proposed methods.

An adaptive hybrid level set moment-of-fluid method is developed to study the material solidification of static and dynamic multiphase systems. The main focus is on the solidification of water droplets, which may undergo normal or supercooled freezing. We model the different regimes of freezing such as supercooling, nucleation, recalescence, isothermal freezing and solid cooling accordingly to capture physical dynamics during impact and solidification of water droplets onto solid surfaces. The numerical simulations are validated by comparison to analytical results and experimental observations. The present simulations demonstrate the ability of the method to capture sharp solidification front, handle contact line dynamics, and the simultaneous impact, merging and freezing of a drop. Parameter studies have been conducted, which show the influence of the Stefan number on the regularity of the shape of frozen droplets. Also, it is shown that impacting droplets with different sizes create ice shapes which are uniform near the impact point and become dissimilar away from it. In addition, surface wettability determines whether droplets freeze upon impact or bounce away.