An international seminar on Geometric Modeling, Geometry Processing, and Computational Geometry

Discrete Willmore Surfaces

This talk will be about the problem of discrete constrained Willmore surfaces: discrete surfaces that have minimal total squared mean curvature while also being discretely conformally equivalent to a given input surface. The Willmore energy is a bending energy, used to model elastic behavior and measure surface smoothness. Adding the conformality constraint turns the problem into a natural extension, in 2D, of classical elastic spline modeling in 1D. This not only makes the use of Willmore functional more practical for a geometric modeling setting, but also leads to more interesting, visually appealing surfaces with rich geometric features. In this talk, I will discuss both theoretical contributions, as well as a practical and efficient algorithm to solve this numerically challenging problem.

Neural 3D Reconstruction

Neural networks have made exciting progress on unstructured 3D geometric data; which is changing the way we fundamentally approach problems in geometry processing. In this talk, I will discuss several works which facilitate 3D reconstruction from several different directions, including consolidating point clouds, estimating a globally consistent point normal orientation, and reconstructing a surface mesh. Finally, I will conclude with ongoing and future work in this direction, as well as other related areas.

Geometrical Challenges in Treating Irregular Heart Beat

This talk presents some of the geometrical aspects involved in treating irregular heart beat rhythm (Arrythmia) using Carto 3 System. Carto 3 is a product of Biosense-Webster, a global leader in the science of diagnosing and treating heart rhythm disorders. CARTO 3 System enables accurate visualization of multiple catheters in a patient’s heart and pinpoints exact location/orientation of a catheter. During arrythmia procedure, a 3D electro-anatomical reconstruction of the heart is built and color coded with the electrical activity in the heart. In this talk, we’ll introduce mesh processing algorithms and discuss industrial challenges encountered in the process of building and coloring geometrical reconstructions of the heart.

Topological and geometric analysis of graphs

In recent years, topological and geometric data analysis (TGDA) has emerged as a new and promising field for processing, analyzing and understanding complex data. Indeed, geometry and topology form natural platforms for data analysis, with geometry describing the ”shape” behind data; and topology characterizing / summarizing both the domain where data are sampled from, as well as functions and maps associated to them.
In this talk, I will show how topological (and geometric ideas) can be used to analyze graph data, which occurs ubiquitously across science and engineering. Graphs could be geometric in nature, such as road networks in GIS, or relational and abstract. I will particularly focus on the reconstruction of hidden geometric graphs from noisy data, as well as graph matching and classification. I will discuss the motivating applications, algorithm development, and theoretical guarantees for these methods. Through these topics, I aim to illustrate the important role that topological and geometric ideas can play in data analysis.

Bio: Yusu Wang is currently Professor in the Halicioglu Data Science Institute at University of California, San Diego. Prior to joining UCSD, she was Professor in the Computer Science and Engineering Department at the Ohio State University. She obtained her PhD degree from Duke University in 2004, where she received the Best PhD Dissertation Award at the Computer Science Department. From 2004-2005, she was a post-doctoral fellow at Stanford University. Yusu Wang primarily works in the fields of Computational geometry, and Computational and applied topology. She is particularly interested in developing effective and theoretically justified algorithms for data analysis using geometric and topological ideas and methods, as well as in applying them to practical domains. She received DOE Early Career Principal Investigator Award in 2006, and NSF Career Award in 2008. Her work received several best paper awards. She is on the editorial boards for SIAM Journal on Computing (SICOMP) and Journal of Computational Geometry (JoCG). She is currently a member of the Computational Geometry Steering Committee. She also serves in SIGACT CATCS (Committee for the Advancement of Theoretical Computer Science) and AWM Meetings Committee.

Deep 3D Generative Modeling

Deep learning has taken the Computer Graphics world by storm. While remarkable progress has been reported in the context of supervised learning, the state of unsupervised learning, in contrast, remains quite primitive. In this talk, we will discuss recent advances where we have combined knowledge from traditional computer graphics and image formation models to enable deep generative modeling workflows. We will describe how we have combined modeling and rendering, in the unsupervised setting, to enable controllable and realistic image and animation production. The work is done in collaboration with various students and research colleagues.

Geometric construction of auxetic metamaterials

Recent advances in digital manufacturing, where computational design, materials science and engineering meet, offer whole new perspectives for tailoring mechanical properties and fabrication of material with applications as diverse as product design, architecture, engineering and art. Auxetic materials are characterized by a negative Poisson’s ratio. This means that they do not behave as usual materials. When stretched in one direction, they do not shrink in the other directions, in contrary they expand. In comparison to standard materials, auxetics are therefore characterized by enhanced mechanical properties such as energy absorption, indentation resistance and acoustic absorption.
This presentation is devoted to our recent work on a category of metamaterials called auxetic structures, or auxetic networks. Whereas regular auxetic networks are well studied, our focus is on irregular, also called disordered auxetic networks. In particular, we are exploring geometrical strategies to generate 2-dimensional disordered auxetic structures.
Starting from an irregular dense network, we seek to reduce the Poisson’s ratio, by pruning bonds (edges) based solely on geometric criteria. To this end, we first deduce some prominent geometric features from regular auxetic networks and then introduce a strategy combining a pure geometric pruning algorithm followed by a physics-based testing phase to determine the resulting Poisson’s ratio of our networks. We provide statistical validation of our approach on large sets of irregular networks, and we additionally show real auxetic networks laser-cut using sheets of rubber. The findings reported here show that it is possible to reduce the Poisson’s ratio by geometric pruning, and that we can generate disordered auxetic networks at lower processing times than a physics-based approach.

Errors in Judgement in Engineering: What Can They Teach Us about the Design Process?

Engineering design is a task that comes with high responsibility: A failed design may easily cause not only monetary damage but, even more importantly, injuries of users. Based on a collection of design flaws [Petroski1994], this presentation will give an overview over modern design approaches that can help to prevent these mistakes in the future. It will touch upon both the topic of conceptual errors and numerical errors.

Accelerating Geometric Algorithms for Freeform Surfaces using Toroidal Patch Approximation

We present a new approach to the acceleration of geometric algorithms for freeform surfaces using a hierarchy of bounding volumes, including those based on the osculating toroidal patches to the surfaces. Using this approach, we revisit some non-trivial conventional geometric algorithms, including those for computing the minimum and Hausdorff distances, the intersection and self-intersection curves, and the integral properties of freeform geometric models. We demonstrate the effectiveness of torus-based geometric computation, by reporting improvement in the speed, stability, and robustness of these algorithms.

Quad-mesh based mappings between surfaces

We discretize mappings between surfaces as correspondences between checkerboard patterns derived from quad meshes. This method captures the degrees of freedom inherent in smooth maps and provides a very simple and efficient computational approach to important types of maps such as conformal or isometric maps. In particular, it enables a natural definition of discrete developable surfaces which is much more flexible in applications than previous concepts of discrete developable surfaces. We discuss geometric modeling of developable surfaces, including cutting, gluing and folding, and present a construction of watertight CAD models consisting of developable spline surfaces. Moreover, we outline further applications of quad-mesh based maps in architectural geometry and computational fabrication.

Hyper-Realistic Rendering: Leveraging Artistic & Mathematical Approaches for Effective Control of Visual Results

My primary goal in this fringe direction of research is to develop a simple, intuitive formal framework for the automatic representation of simplified shapes and materials that can support Hyper-Realism in a wide variety of rendering applications. I observe that with the emphasis on the physical laws in rendering systems, (1) the focus increasingly shifts away from how users perceive the virtual environment, (2) rendering becomes prohibitively difficult to realize desired global illumination effects in real-time, and (3) the true inclusion of human-in-the-loop to control visual results also becomes significantly hard. I have identified two broad categories of artistic and mathematical approaches that can facilitate effective Hyper-Realistic rendering with clear control of visual results: (1) Geometry Representation with Anamorphic Bas-Reliefs and (2) Material Representation with Barycentric Shaders. A significant advantage of these two approaches is that they simplify the reconstruction processes by allowing some of the real-world parameters to be embedded into the representations. In this talk, I will give a wide variety of examples that demonstrate the effectiveness of this approach.

Bio: Ergun Akleman is a Professor in the Departments of Visualization & Computer Science and Engineering. Akleman has been at Texas A&M University for 25 years. He received his Ph.D. degree in Electrical and Computer Engineering from the Georgia Institute of Technology in 1992. Akleman is teaching, research and creative activities are all transdisciplinary. He had more than 150 publications in a wide variety of disciplines from computer graphics, computer-aided design, and mathematics to art, architecture, and social sciences. His most significant and influential contributions as a researcher have been in shape modeling and computer-aided sculpting. He is also a professional cartoonist who published more than 500 cartoons. He has a bi-monthly corner called Computing through Time in the Flagship magazine of IEEE Computer Society, IEEE Computer.

On Reconstructing 3D Thin Structures

Thin structures are ubiquitous, such as ropes, cables, tree branches, wire arts, and wire-frame furniture. It is well known that 3D thin structures are challenging to reconstruct due to their narrow width and lack of distinct features. However, their reconstruction has received relatively little research attention. Clearly, the detection and reconstruction of these thin objects have many applications in computer graphics, computer vision, and robotics. In this talk I will present our recent works on reconstructing 3D thin structures from images, depth maps, and videos. Then I will analyze the limitations of the existing methods and discuss outstanding issues that need to be tackled in order to achieve real-time reconstruction of thin structures in the wild.

Bio: Wenping Wang’s research interests cover computer graphics, computer visualization, computer vision, robotics, medical image processing, and geometric computing, and he has published over 160 journal papers in these fields. He is or has been journal associate editor of several international journals, including Computer Aided Geometric Design (CAGD), Computer Graphics Forum (CGF), IEEE Transactions on Computers, and IEEE Computer Graphics and Applications, and has chaired a number of international conferences, including Pacific Graphics 2012, ACM Symposium on Physical and Solid Modeling (SPM) 2013, SIGGRAPH Asia 2013, and Geometry Summit in 2019. Prof. Wang received the John Gregory Memorial Award for contributions in geometric modeling. He is an IEEE Fellow.

Turning Planar Materials Into Curved Structures

Recent advances in material science and digital fabrication provide promising opportunities for product design, mechanical and biomedical engineering, robotics, architecture, art, and science. Engineered materials and personalized fabrication are revolutionizing manufacturing culture and having a significant impact on various scientific and industrial works. As new fabrication technologies emerge, effective computational tools are needed to fully exploit the potential of digital fabrication.
In this talk, I will discuss how we use the insights from discrete differential geometry to enable designs not possible before and design new materials with specific properties and performance. We introduce a novel computational method for design and fabrication with auxetic materials. The term auxetic refers to solid materials with a negative Poisson ratio — when the material is stretched in one direction, it also expands in all other directions. In particular, we study 2D auxetic materials in the form of a triangular linkage that exhibits auxetic behavior at the macro scale. This stretching, in turn, allows the flat material to approximate doubly-curved surfaces, making it attractive for fabrication. Furthermore, we develop a computational method for designing novel deployable structures via programmable auxetics, i.e., spatially varying triangular linkage optimized to directly and uniquely encode the target 3D surface in the 2D pattern. In contrast to most previous work, our approach is scale-invariant. It can be applied to realize a broad class of complex curved surfaces, ranging from tiny medical implants to large scale architectural domes.

3D Perception for Creativity Assistance

We are living in an era where the digital world is becoming an inevitable part of our professional and daily lives. Digital creation tools are essential for many professions including design, entertainment, gaming etc. In our daily lives, we all take many pictures or capture many videos each day to record and share our memories. There is a stronger demand to transform such digital workflows into life-like experiences. My research focuses on enabling such a transformation by developing computational 3D perception tools to reason about the physical environment, people, objects, and how they interact with each other from 2D digital content including images, sketches, and videos. In this talk, I will focus on some of my recent work in this domain, specifically in the context of garment design and 3D reconstruction of man-made shapes. I will provide examples of how 3D perception can enable 2D creative tools for Adobe’s products and discuss some of the open research challenges.

Freeform quad-based kirigami

Kirigami, the traditional Japanese art of paper cutting and folding generalizes origami and has initiated new research in material science as well as graphics. Jere we use its capabilities to perform geometric modeling with corrugated surface representations possessing an isometric unfolding into a planar domain after appropriate cuts are made.
We initialize our box-based kirigami structures from orthogonal networks of curves, compute a first approximation of their unfolding via mappings between meshes, and complete the process by global optimization. Besides the modeling capabilities we also study the interesting geometry of special kirigami structures from the theoretical side. (This is joint work with Caigui Jiang, Florian Rist, and Helmut Pottmann).

Shape Analysis: From local to non-local information and learning

In this talk we explore two contributions for shape analysis. In a first case, we consider surfaces and how local analysis of the angular oscillations and polynomial radial behavior around surface points leads to accurate normal estimation and new
integral invariants. A direct application of these integral invariants is geometric detail exaggeration. This however assumes that the shape can be represented as a height function over some parameterization plane in a neighborhood of fixed
radius that is the same for all the surface.
In many cases, however, shapes, as they are acquired by laser scanners, might not fulfill this hypothesis: they can have isotropically sampled areas or curve parts. For example, street cables can be considered as curves, depending on the
acquisition accuracy. We call this case the mixed dimension case. Armed with a well-defined probing operator associating a point of the ambient space to a point on the shape, we define Local Probing Fields, and analyze them in a non-local manner.
Hence, we are able to extract and describe data self-similarities. Exploiting these through learning algorithms allows us to revisit various shape processing tasks such as denoising, compression and shape resampling.

Curvature! The starting point of manufacturing

The potential benefits of taking surface curvature into account in engineering design have not yet been fully utilized. One of the problems is that differential geometry is often not offered in the undergraduate engineering program. In this presentation, I focus on a couple of research topics that demonstrate how curvatures play important roles in engineering design. The first topic is the effect of curvature on the energy absorption characteristics of cylindrical corrugated tubes under compression. The second topic is the computation of the differential geometry properties of lines of curvature of parametric surfaces which lends a hand for the formation of curved plates in shipbuilding. The third topic is the fabrication of aesthetically pleasing smooth freeform surfaces using orthogonal principal strips.