Hodge Decomposition of Single-Cell RNA Velocity.

RNA velocity has the ability to capture the cell dynamic information in the biological processes; yet, a comprehensive analysis of the cell state transitions and their associated chemical and biological processes remains a gap. In this work, we provide the Hodge decomposition, coupled with discrete exterior calculus (DEC), to unveil cell dynamics by examining the decomposed curl-free, divergence-free, and harmonic components of the RNA velocity field in a low dimensional representation, such as a UMAP or a t-SNE representation. Decomposition results show that the decomposed components distinctly reveal key cell dynamic features such as cell cycle, bifurcation, and cell lineage differentiation, regardless of the choice of the low-dimensional representations.

ILEE: Algorithms and toolbox for unguided and accurate quantitative analysis of cytoskeletal images.

The eukaryotic cytoskeleton plays essential roles in cell signaling and trafficking, broadly associated with immunity and diseases in humans and plants. To date, most studies describing cytoskeleton dynamics and function rely on qualitative/quantitative analyses of cytoskeletal images. While state-of-the-art, these approaches face general challenges: the diversity among filaments causes considerable inaccuracy, and the widely adopted image projection leads to bias and information loss.

Fast Computation of Neck-Like Features.

Locating neck-like features, or locally narrow parts, of a surface is crucial in various applications such as segmentation, shape analysis, path planning, and robotics. Topological methods are often utilized to find the set of shortest loops around handles and tunnels. However, there are abundant neck-like features on genus-0 shapes without any handles.

Metric-Driven 3D Frame Field Generation.

Controlling the size and shear of elements is crucial in pure hex or hex-dominant meshing. To this end, non-orthonormal frame fields that are almost everywhere integrable (except for the singularities) can play a key role. However, it is often challenging or impossible to generate such a frame field under the tight control of a general Riemannian metric field.

EVOLUTIONARY DE RHAM-HODGE METHOD.

The de Rham-Hodge theory is a landmark of the 20 Century's mathematics and has had a great impact on mathematics, physics, computer science, and engineering. This work introduces an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from a filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to close manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropriate analysis and treatment of boundary conditions on differential forms to maintain proper topological properties.

HERMES: PERSISTENT SPECTRAL GRAPH SOFTWARE.

Persistent homology (PH) is one of the most popular tools in topological data analysis (TDA), while graph theory has had a significant impact on data science. Our earlier work introduced the persistent spectral graph (PSG) theory as a unified multiscale paradigm to encompass TDA and geometric analysis. In PSG theory, families of persistent Laplacian matrices (PLMs) corresponding to various topological dimensions are constructed via a filtration to sample a given dataset at multiple scales.

The de Rham-Hodge Analysis and Modeling of Biomolecules.

Biological macromolecules have intricate structures that underpin their biological functions. Understanding their structure-function relationships remains a challenge due to their structural complexity and functional variability. Although de Rham-Hodge theory, a landmark of twentieth-century mathematics, has had a tremendous impact on mathematics and physics, it has not been devised for macromolecular modeling and analysis.

Protein pocket detection via convex hull surface evolution and associated Reeb graph.

Motivation: Protein pocket information is invaluable for drug target identification, agonist design, virtual screening and receptor-ligand binding analysis. A recent study indicates that about half holoproteins can simultaneously bind multiple interacting ligands in a large pocket containing structured sub-pockets. Although this hierarchical pocket and sub-pocket structure has a significant impact to multi-ligand synergistic interactions in the protein binding site, there is no method available for this analysis.

Stain Formation on Deforming Inelastic Cloth.

We propose a novel approach to simulating the formation and evolution of stains on cloths in motion. We accurately capture the diffusion of a pigmented solution over a complex knitted or woven fabric through homogenization of its inhomogeneous and/or anisotropic properties into bulk anisotropic diffusion tensors. Secondary effects such as absorption, adsorption and evaporation are also accounted for through physically-based modeling.

Divide-and-conquer strategy for large-scale Eulerian solvent excluded surface.

Motivation: Surface generation and visualization are some of the most important tasks in biomolecular modeling and computation. Eulerian solvent excluded surface (ESES) software provides analytical solvent excluded surface (SES) in the Cartesian grid, which is necessary for simulating many biomolecular electrostatic and ion channel models. However, large biomolecules and/or fine grid resolutions give rise to excessively large memory requirements in ESES construction.

Coloring 3D Printed Surfaces by Thermoforming.

Decorating the surfaces of 3D printed objects with color textures is still not readily available in most consumer-level or even high-end 3D printers. Existing techniques such as hydrographics color transfer suffer from the issues of air pockets in concave regions and discoloration in overly stretched regions. We propose a novel thermoforming-based coloring technique to alleviate these problems as well as to simplify the overall procedure.

Adaptive 3D Face Reconstruction from Unconstrained Photo Collections.

Given a photo collection of "unconstrained" face images of one individual captured under a variety of unknown pose, expression, and illumination conditions, this paper presents a method for reconstructing a 3D face surface model of the individual along with albedo information. Unlike prior work on face reconstruction that requires large photo collections, we formulate an approach to adapt to photo collections with a high diversity in both the number of images and the image quality. To achieve this, we incorporate prior knowledge about face shape by fitting a 3D morphable model to form a personalized template, following by using a novel photometric stereo formulation to complete the fine details, under a coarse-to-fine scheme.

ESES: Software for Eulerian solvent excluded surface.

Solvent excluded surface (SES) is one of the most popular surface definitions in biophysics and molecular biology. In addition to its usage in biomolecular visualization, it has been widely used in implicit solvent models, in which SES is usually immersed in a Cartesian mesh. Therefore, it is important to construct SESs in the Eulerian representation for biophysical modeling and computation.

Frame Field Singularity Correction for Automatic Hexahedralization.

We present an automatic hexahedralization tool, based on a systematic treatment that removes some of the singularities that would lead to degenerate volumetric parameterization. Such singularities could be abundant in automatically generated frame fields guiding the interior and boundary layouts of the hexahedra in an all hexahedral mesh. We first give the mathematical definitions of the inadmissible singularities prevalent in frame fields, including newly introduced surface singularity types.

Persistent homology for the quantitative prediction of fullerene stability.

Persistent homology is a relatively new tool often used for qualitative analysis of intrinsic topological features in images and data originated from scientific and engineering applications. In this article, we report novel quantitative predictions of the energy and stability of fullerene molecules, the very first attempt in using persistent homology in this context. The ground-state structures of a series of small fullerene molecules are first investigated with the standard Vietoris-Rips complex.

FaceWarehouse: a 3D facial expression database for visual computing.

We present FaceWarehouse, a database of 3D facial expressions for visual computing applications. We use Kinect, an off-the-shelf RGBD camera, to capture 150 individuals aged 7-80 from various ethnic backgrounds. For each person, we captured the RGBD data of her different expressions, including the neutral expression and 19 other expressions such as mouth-opening, smile, kiss, etc.

Multiscale geometric modeling of macromolecules I: Cartesian representation.

This paper focuses on the geometric modeling and computational algorithm development of biomolecular structures from two data sources: Protein Data Bank (PDB) and Electron Microscopy Data Bank (EMDB) in the Eulerian (or Cartesian) representation. Molecular surface (MS) contains non-smooth geometric singularities, such as cusps, tips and self-intersecting facets, which often lead to computational instabilities in molecular simulations, and violate the physical principle of surface free energy minimization. Variational multiscale surface definitions are proposed based on geometric flows and solvation analysis of biomolecular systems.

Multiscale geometric modeling of macromolecules II: Lagrangian representation.

Geometric modeling of biomolecules plays an essential role in the conceptualization of biolmolecular structure, function, dynamics, and transport. Qualitatively, geometric modeling offers a basis for molecular visualization, which is crucial for the understanding of molecular structure and interactions. Quantitatively, geometric modeling bridges the gap between molecular information, such as that from X-ray, NMR, and cryo-electron microscopy, and theoretical/mathematical models, such as molecular dynamics, the Poisson-Boltzmann equation, and the Nernst-Planck equation.

Choking loops on surfaces.

We present a method for computing "choking" loops--a set of surface loops that describe the narrowing of the volumes inside/outside of the surface and extend the notion of surface homology and homotopy loops. The intuition behind their definition is that a choking loop represents the region where an offset of the original surface would get pinched. Our generalized loops naturally include the usual 2g handles/tunnels computed based on the topology of the genus-g surface, but also include loops that identify chokepoints or bottlenecks, i.

TransCut: interactive rendering of translucent cutouts.

We present TransCut, a technique for interactive rendering of translucent objects undergoing fracturing and cutting operations. As the object is fractured or cut open, the user can directly examine and intuitively understand the complex translucent interior, as well as edit material properties through painting on cross sections and recombining the broken pieces—all with immediate and realistic visual feedback. This new mode of interaction with translucent volumes is made possible with two technical contributions.

Geometric modeling of subcellular structures, organelles, and multiprotein complexes.

Recently, the structure, function, stability, and dynamics of subcellular structures, organelles, and multiprotein complexes have emerged as a leading interest in structural biology. Geometric modeling not only provides visualizations of shapes for large biomolecular complexes but also fills the gap between structural information and theoretical modeling, and enables the understanding of function, stability, and dynamics. This paper introduces a suite of computational tools for volumetric data processing, information extraction, surface mesh rendering, geometric measurement, and curvature estimation of biomolecular complexes.

A deformable surface model for real-time water drop animation.

A water drop behaves differently from a large water body because of its strong viscosity and surface tension under the small scale. Surface tension causes the motion of a water drop to be largely determined by its boundary surface. Meanwhile, viscosity makes the interior of a water drop less relevant to its motion, as the smooth velocity field can be well approximated by an interpolation of the velocity on the boundary.

Interactive Shape Interpolation through Controllable Dynamic Deformation.

In this paper, we introduce an interactive approach to generate physically based shape interpolation between poses. We extend linear modal analysis to offer an efficient and robust numerical technique to generate physically-plausible dynamics even for very large deformation. Our method also provides a rich set of intuitive editing tools with real-time feedback, including control over vibration frequencies, amplitudes, and damping of the resulting interpolation sequence.

Age-invariant face recognition.

One of the challenges in automatic face recognition is to achieve temporal invariance. In other words, the goal is to come up with a representation and matching scheme that is robust to changes due to facial aging. Facial aging is a complex process that affects both the 3D shape of the face and its texture (e.