arXiv subjects

Michael Tao

Publications and source records attributed to Michael Tao.

6 recordsLinked to original sources

arXiv2026

BijectiveRemesh: Maintaining Bijective Mappings for Data Transfer Across Remeshed Manifolds

We introduce BijectiveRemesh, a robust algorithm for maintaining a continuous, bijective mapping across complex remeshing sequences on both 2D triangle surfaces and 3D tetrahedral meshes. Unlike traditional data transfer methods that rely on interpolation or projection, our approach constructs a mathematically rigorous composite map from the input mesh to the output mesh by chaining local bijective atlases defined for each primitive remeshing operation. Our framework represents the overall mapping as a composition of local bijective atlases, one per remeshing operation. Building upon successive self-parameterization, we introduce a Shared Scaffold structure for 2D triangle meshes that enforces global bijectivity through local orientation preservation. We extend this approach to handle edge splits, edge swaps, and vertex smoothing beyond the original edge collapses. For 3D tetrahedral meshes, we generalize the local atlas construction using Steinitz's Theorem and Maxwell-Cremona lifting to ensure valid embeddings. This enables exact tracking of geometric entities, including points, curves, and surfaces, across remeshing, with applications from texture transfer to volumetric simulations.

cs.GR↗

arXiv2026

Topology-First B-Rep Meshing

Parametric boundary representation models (B-Reps) are the de facto standard in CAD, graphics, and robotics, yet converting them into valid meshes remains fragile. The difficulty originates from the unavoidable approximation of high-order surface and curve intersections to low-order primitives: the resulting geometric realization often fails to respect the exact topology encoded in the B-Rep, producing meshes with incorrect or missing adjacencies. Existing meshing pipelines address these inconsistencies through heuristic feature-merging and repair strategies that offer no topological guarantees and frequently fail on complex models. We propose a fundamentally different approach: the B-Rep topology is treated as an invariant of the meshing process. Our algorithm enforces the exact B-Rep topology while allowing a single user-defined tolerance to control the deviation of the mesh from the underlying parametric surfaces. Consequently, for any admissible tolerance, the output mesh is topologically correct; only its geometric fidelity degrades as the tolerance increases. This decoupling eliminates the need for post-hoc repairs and yields robust meshes even when the underlying geometry is inconsistent or highly approximated. We evaluate our method on thousands of real-world CAD models from the ABC and Fusion 360 repositories, including instances that fail with standard meshing tools. The results demonstrate that topological guarantees at the algorithmic level enable reliable mesh generation suitable for downstream applications.

cs.GR↗

arXiv2025

Codimensional MultiMeshing: Synchronizing the Evolution of Multiple Embedded Geometries

Complex geometric tasks such as geometric modeling, physical simulation, and texture parametrization often involve the embedding of many complex sub-domains with potentially different dimensions. These tasks often require evolving the geometry and topology of the discretizations of these sub-domains, and guaranteeing a \emph{consistent} overall embedding for the multiplicity of sub-domains is required to define boundary conditions. We propose a data structure and algorithmic framework for hierarchically encoding a collection of meshes, enabling topological and geometric changes to be automatically propagated with coherent correspondences between them. We demonstrate the effectiveness of our approach in surface mesh decimation while preserving UV seams, periodic 2D/3D meshing, and extending the TetWild algorithm to ensure topology preservation of the embedded structures.

cs.GR↗

arXiv2021

Computational Design of Active 3D-Printed Multi-State Structures for Shape Morphing

Active structures have the ability to change their shape, properties, and functionality as a response to changing operational conditions, which makes them more versatile than their static counterparts. However, most active structures currently lack the capability to achieve multiple, different target states with a single input actuation or require a tedious material programming step. Furthermore, the systematic design and fabrication of active structures is still a challenge as many structures are designed by hand in a trial and error process and thus are limited by engineers' knowledge and experience. In this work, a computational design and fabrication framework is proposed to generate structures with multiple target states for one input actuation that don't require a separate training step. A material dithering scheme based on multi-material 3D printing is combined with locally applied copper coil heating elements and sequential heating patterns to control the thermo-mechanical properties of the structures and switch between the different deformation modes. A novel topology optimization approach based on power diagrams is used to encode the different target states in the structure while ensuring the fabricability of the structures and the compatibility with the drop-in heating elements. The versatility of the proposed framework is demonstrated for four different example structures from engineering and computer graphics. The numerical and experimental results show that the optimization framework can produce structures that show the desired motion, but experimental accuracy is limited by current fabrication methods. The generality of the proposed method makes it suitable for the development of structures for applications in many different fields from aerospace to robotics to animated fabrication in computer graphics.

cs.CE↗

arXiv2019

Beyond Pixel Norm-Balls: Parametric Adversaries using an Analytically Differentiable Renderer

Many machine learning image classifiers are vulnerable to adversarial attacks, inputs with perturbations designed to intentionally trigger misclassification. Current adversarial methods directly alter pixel colors and evaluate against pixel norm-balls: pixel perturbations smaller than a specified magnitude, according to a measurement norm. This evaluation, however, has limited practical utility since perturbations in the pixel space do not correspond to underlying real-world phenomena of image formation that lead to them and has no security motivation attached. Pixels in natural images are measurements of light that has interacted with the geometry of a physical scene. As such, we propose the direct perturbation of physical parameters that underly image formation: lighting and geometry. As such, we propose a novel evaluation measure, parametric norm-balls, by directly perturbing physical parameters that underly image formation. One enabling contribution we present is a physically-based differentiable renderer that allows us to propagate pixel gradients to the parametric space of lighting and geometry. Our approach enables physically-based adversarial attacks, and our differentiable renderer leverages models from the interactive rendering literature to balance the performance and accuracy trade-offs necessary for a memory-efficient and scalable adversarial data augmentation workflow.

cs.LG↗

arXiv2017

An empirical analysis of the optimization of deep network loss surfaces

The success of deep neural networks hinges on our ability to accurately and efficiently optimize high-dimensional, non-convex functions. In this paper, we empirically investigate the loss functions of state-of-the-art networks, and how commonly-used stochastic gradient descent variants optimize these loss functions. To do this, we visualize the loss function by projecting them down to low-dimensional spaces chosen based on the convergence points of different optimization algorithms. Our observations suggest that optimization algorithms encounter and choose different descent directions at many saddle points to find different final weights. Based on consistency we observe across re-runs of the same stochastic optimization algorithm, we hypothesize that each optimization algorithm makes characteristic choices at these saddle points.

cs.LG↗