AWM at SIAM 2018 Abstracts

Monday, July 9, 2018, 8:30 AM – 10:30 AM,Room: B116, Oregon Convention Center

AWM Workshop: Shape Analysis and Modeling — Part I of II

Medial Fragments for Segmentation of Articulating Objects in Images

Authors: Ellen Gasparovic, Union College, USA,

Abstract. We propose a method for extracting objects from natural images by combining fragments of the Blum medial axis, generated from the Voronoi diagram of an edge map of a natural image, into a coherent whole. Using techniques from persistent homology and graph theory, we combine image cues with geometric cues from the medial fragments in order to aggregate parts of the same object. We demonstrate our method on images containing articulating objects, with an eye to future work applying articulation invariant measures on the medial axis for shape matching between images. This is joint work with Erin Chambers and Kathryn Leonard.

Skeletal Models and Shape Representation

Authors: Kathryn E. Leonard, Occidental College, USA,

Abstract. This talk will consider skeletal models for shapes in two and three dimensions from the perspective of complexity, compression, parts decomposition and recognition, including similarity and articulation. We will discuss mathematical skeletal constructions such as the Blum medial axis, and functions on those skeletons that provide important information about a shape. We will also present results from a massive user study on shape part perception that provides insight into human cognition about shape parts.

Computational Design and Fabrication

Authors: Xiaoting Zhang, Boston University, U.S.,

Abstract. The recent emergence of additive manufacturing – otherwise known as 3D printing – has expanded the accessibility for rapid prototyping objects of interest. However, the applicability of additive manufacturing is often limited by the quality of available models, and by the ability of the designer to produce good representations of the objects. My work has been to alleviate these limitations by developing geometric modeling and processing tools that are premeditative of the mechanical and aesthetic properties of physical objects. I have advanced the quality assurance of additive manufacturing, especially for desktop 3D printers, which is core to customized manufacturing. For instance, I have built support generation systems that optimize aesthetics, material usage and mechanical properties of 3D printed objects. I have also developed computational geometric technologies for additive manufacturing that bring new insights into product complexity and opportunities over traditional manufacturing methods, such as shell models with varying thickness and medical instruments.

Deformation and Rigidity

Authors: Nina Amenta, University of California, Davis, USA,

Abstract. Deformation of a triangle mesh is an important theme in many applied areas such as computer graphics, computer vision and scientific shape analysis. Rigidity, the absence of deformation, has been studied mainly in mathematics, but it seems natural to consider what light the study of rigidity can shed on deformation.

Here’s a classical rigidity result. Given an embedded triangle mesh homeomorphic to the sphere, we consider the vector formed by its edge lengths. Provocatively, the length of this vector matches the number of degrees of freedom of the embedding up to rotation and translation. And indeed for an edge length vector in general position, the mesh is rigid. In other words, if you constructed the Stanford bunny out of toothpicks, it would not fall down, even if the joints at the vertices were totally flexible.

While this classic idea does not seem to be directly helpful for computing with shape deformations (edge lengths are just too intrinsic, and embeddings are not unique), there are other ways we could assign a number to each edge. Each of these leads to a different model of rigidity, some of which might be more useful. We will discuss two: the dihedral angles, and the discrete mean curvature.

AWM Workshop: Shape Analysis and Modeling — Part II of II

Generative Representations of the World

Authors: Ilke Demir, Facebook, USA,

Abstract. Generative approaches enable creating numerous scenarios coherent with the reality, as long as the representations are good approximations of the real world. In this talk, we will discuss extracting such generative representations from 2D and 3D data for mapping, modeling, and reconstruction of spatial data and urban models. We will introduce shape processing algorithms to exploit similarities, grammar discovery approaches to extract procedural rules, and machine learning methods to understand spatial information. The talk will conclude by proposed applications and experimental results on various types of geometric data.

Finding Communities and Roles in Networks

Authors: Carlotta Domeniconi, George Mason University, USA,

Abstract. Community detection and role detection in networks are broad problems with many applications. Existing methods for community discovery frequently make use of edge density and node attributes; however, the methods ultimately have different definitions of community and build strong assumptions about community features into their models. Furthermore, users in online social networks often have very different structural positions which may be attributed to a latent factor: roles.

In this talk, I will present recent advances in community and role discovery in networks, and discuss how their interplay can lead to a new definition of community viewed as interactions of roles.

Stratifying High-dimensional Data Based on Proximity to the Convex Hull Boundary

Authors: Lori Ziegelmeier, Macalester College, USA,

Abstract. The convex hull of a set of points, C, serves to expose extremal properties of C and can help identify elements in C of high interest. For many problems, particularly in the presence of noise, the true vertex set (and facets) may be difficult to determine. One solution is to expand the list of high interest candidates to points lying near the boundary of the convex hull. We propose a quadratic program for the purpose of stratifying points in a data cloud based on proximity to the boundary of the convex hull. For each data point, a quadratic program is solved to determine an associated weight vector. We show that the weight vector encodes geometric information concerning the point’s relationship to the boundary of the convex hull. The computation of the weight vectors can be carried out in parallel, and for a fixed number of points and fixed neighborhood size, the overall computational complexity of the algorithm grows linearly with dimension. As a consequence, meaningful computations can be completed on reasonably large, high-dimensional data sets.

Consistent Shape Matching via Coupled Optimization

Authors: Anastasia Dubrovina, Stanford University, USA,

Abstract. In this talk, I will present a new method for computing accurate point-to-point mappings between pairs of triangle meshes, given imperfect initial correspondences. Unlike the majority of existing techniques, our approach optimizes for a map while leveraging information from the inverse map, yielding results which are highly consistent with respect to map composition. Furthermore, for each point on the source mesh, our method considers only a linear number of candidate points on the target mesh, allowing us to work directly with high resolution meshes. Key to this dimensionality reduction is a novel iterative candidate selection process, combined with an efficient optimization method for intrinsic distortion minimization. Quantitative and qualitative comparison of our method with state-of-the-art techniques show that it provides a powerful matching tool when accurate and consistent correspondences are required.

AWM Workshop – Poster Presentations

A Numerical Study of Steklov Eigenvalue Problem

Weaam Alhejaili, Claremont Graduate University, USA,
Chiu-Yen Kao, Claremont McKenna College, USA,

Abstract. In this research, we focus on the development of numerical approaches to the forward solver and the shape optimization solver for Steklov eigenvalue problems in two dimensions. This problem has a wide range of applications in engineering and applied mathematics. We proposed numerical approaches via spectral methods or finite element methods. To apply spectral methods, we reformulate the Steklov eigenvalue problem in the complex domain via conformal mappings. For shape optimization problem, we use the gradient ascent approach to find the optimal domain which maximizes k- th Steklov eigenvalue with a given k.

Fast Classification of Big Data: Proximal Methods for Sparse Discriminant Analysis

Authors: Summer Atkins, University of Florida,
Brendan P. Ames, University of Alabama, USA,
Gudmundur Einarsson, Technical University of Denmark, Denmark,
Line Clemmensen, Technical University of Denmark, Denmark,

Abstract. Linear discriminant analysis (LDA) fails when interpreting data where the number of features is larger than the number of observations. To address this issue, Clemmensen et al. (2011) developed a sparse version called sparse discriminant analysis (SDA) which allows feature selection and classification to be performed simultaneously. We propose three techniques to improve the efficiency of SDA. We then demonstrate their effectiveness for classifying times series data and simulated data.

Automatic Extraction of a Stroke-based Font Representation

Authors: Elena Balashova, Princeton University, USA,

Abstract. Designing fonts and typefaces is a difficult process for both beginner and expert typographers. Existing workflows require the designer to create every glyph, while adhering to many loosely defined design suggestions. This process can be simplified by exploiting the similar character glyph structure across fonts. To capture these correlations, we propose learning a stroke-based representation from a collection of existing typefaces. We develop a stroke-based geometric model for glyphs and a fitting procedure to re-parametrize arbitrary fonts to our representation.

Stability of Spiral Waves in Cardiac Dynamics

Authors: Stephanie Dodson, Brown University, U.S.,

Abstract. Ventricular fibrillation in the heart is often caused by the formation and breakup of spiral waves in cardiac tissue. The alternans instability, an oscillation in the action potential duration, has been clinically linked to spiral wave breakup. We seek to understand how and why alternans develop and if stable alternans patterns exist. To investigate these questions, we analyze spectral properties of the spiral and use numerical time evolution.

Towards More Efficient Multigrid and Multilevel Methods

Authors: Aditi Ghai, Stony Brook University, USA,

Abstract. Preconditioned Krylov subspace (KSP) methods are widely used for solving large and sparse linear systems arising from PDE discretizations. In this work, we present a systematic comparison of some KSP methods, with different classes of preconditioners including incomplete LU factorization (ILUT, ILUTP and multilevel ILU), and algebraic multigrid (including classical AMG and smoothed aggregation). We will also address multilevel ILU preconditioner for predominantly symmetric systems. This study helps establish some practical guidelines for choosing preconditioned KSP methods and to increase the efficiency and robustness of multigrid methods.

Robust Residual-based and Residual-free Greedy Algorithms for Reduced Basis Methods

Authors: Jiahua Jang, University of Massachusetts, Dartmouth, USA,

Abstract. While the offline-enhanced RBM has demonstrated the potential of efficiently solving large scale parametrized problem by identifying a subset of the training set, many opportunities for extending the scope of RBM remain. In this work, the offline stage of RBM is improved by mitigating the computational cost of error estimate over the training set. We present two error estimation mechanisms: the classical one with enhanced implementation and the RB-coefficient-only error indicator. The former method is a more efficient and accurate implementation of the classical approach rendering it capable of approximating the truth solution to machine accuracy (as opposed to its square root) for coercive problems. The latter error indicator is efficiently evaluated solely from the RB coefficients, which is motivated by the need to minimize the Lebesgue constant. We achieved enhancement of the efficiency while guaranteeing the accuracy of the proposed algorithm, both demonstrated through numerical experiments.

Regularization in Tomographic Reconstruction

Authors: Ratna Khatri, George Mason University, USA,

Abstract. Tomographic reconstruction is a non-invasive 2D/3D image recovery technique. It is widely used at the Advanced Photon Source at Argonne National Laboratory. One way of solving this problem is via linear least squares optimization formulation assuming the experimental data follows a Gaussian distribution. Due to limited data, the problem is usually ill-conditioned. We study regularization techniques for tomographic reconstruction and provide a performance comparison among different types of regularizers.

Anatomical Biomarker for Alzheimer’s Disease Progression in the Transentorhinal Cortex

Authors: Sue Kulason, Johns Hopkins University, USA,
Daniel Tward, Johns Hopkins University, USA,
Chelsea Sicat, Johns Hopkins University, USA,
Arnold Bakker, Johns Hopkins University, USA,
Michela Gallagher, Johns Hopkins University, USA,
Marilyn Albert, Johns Hopkins University, USA,
Michael I. Miller, Johns Hopkins University, USA,

Abstract. The purpose of this research was to develop a more robust biomarker for early Alzheimer’s disease. We manually segmented for the entorhinal cortex from 3T MRIs and reduced longitudinal variability in boundaries using longitudinal diffeomorphometry. Cortical thickness was estimated using a technique called normal geodesic flow. Permutation tests of resulting model showed significant thickness atrophy localized to the transentorhinal cortex. Linear discriminant analysis confirmed sensitivity/specificity were improved using this thickness of the transentorhinal cortex over global entorhinal cortex thickness and hippocampal volume.

Semiclassical Sine-Gordon Equation, Universality at Gradient Catastrophe

Authors: Bingying Lu, University of Michigan, USA,

Peter D. Miller, University of Michigan, Ann Arbor, USA,

Abstract. We consider the semi-classical sine-Gordon equation with a class of initial data. In a neighbourhood of a gradient catastrophe point, the solution behave like modulated plane waves in one region and localize like “spikes” in other. The “spike” locations correspond to the poles of the Triotronque\’e solution. We give first correction of the solution near the catastrophe, and study the shape of the localized structures. The asymptotic behaviour is nonsensitive to initial data.

Subset Selection with an Extension of DEIM

Authors: Emily Hendryx, Rice University, USA,

Abstract. Index selection via the discrete empirical interpolation method (DEIM) can be used to identify representative subsets of data from a larger data matrix. However, the rank of the data matrix limits the number of representatives that can be in the identified subset–an issue, for instance, when the number of classes present in the data is greater than the number of features observed. This work presents a novel extension of DEIM to allow for the selection of additional representatives along with experimental results that support the use of such an extension.