Topology verification for isosurface extraction

Journal ArticleThe broad goals of verifiable visualization rely on correct algorithmic implementations. We extend a framework for verification of isosurfacing implementations to check topological properties. Specifically, we use stratified Morse theory and digital topology to design algorithms which verify topological invariants. Our extended framework reveals unexpected behavior and coding mistakes in popular publicly available isosurface codes

Lattice cleaving: a multimaterial tetrahedral meshing algorithm with guarantees

pre-printWe introduce a new algorithm for generating tetrahedral meshes that conform to physical boundaries in volumetric domains consisting of multiple materials. The proposed method allows for an arbitrary number of materials, produces high-quality tetrahedral meshes with upper and lower bounds on dihedral angles, and guarantees geometric fidelity. Moreover, the method is combinatoric so its implementation enables rapid mesh construction. These meshes are structured in a way that also allows grading, to reduce element counts in regions of homogeneity. Additionally, we provide proofs showing that both element quality and geometric fidelity are bounded using this approach

Doctor of Philosophy

dissertationIn this dissertation, we advance the theory and practice of verifying visualization algorithms. We present techniques to assess visualization correctness through testing of important mathematical properties. Where applicable, these techniques allow us to distinguish whether anomalies in visualization features can be attributed to the underlying physical process or to artifacts from the implementation under verification. Such scientific scrutiny is at the heart of verifiable visualization - subjecting visualization algorithms to the same verification process that is used in other components of the scientific pipeline. The contributions of this dissertation are manifold. We derive the mathematical framework for the expected behavior of several visualization algorithms, and compare them to experimentally observed results in the selected codes. In the Computational Science & Engineering community CS&E, this technique is know as the Method of Manufactured Solution (MMS). We apply MMS to the verification of geometrical and topological properties of isosurface extraction algorithms, and direct volume rendering. We derive the convergence of geometrical properties of isosurface extraction techniques, such as function value and normals. For the verification of topological properties, we use stratified Morse theory and digital topology to design algorithms that verify topological invariants. In the case of volume rendering algorithms, we provide the expected discretization errors for three different error sources. The results of applying the MMS is another important contribution of this dissertation. We report unexpected behavior for almost all implementations tested. In some cases, we were able to find and fix bugs that prevented the correctness of the visualization algorithm. In particular, we address an almost 2 0 -year-old bug with the core disambiguation procedure of Marching Cubes 33, one of the first algorithms intended to preserve the topology of the trilinear interpolant. Finally, an important by-product of this work is a range of responses practitioners can expect to encounter with the visualization technique under verification

Chain-Based Representations for Solid and Physical Modeling

In this paper we show that the (co)chain complex associated with a decomposition of the computational domain, commonly called a mesh in computational science and engineering, can be represented by a block-bidiagonal matrix that we call the Hasse matrix. Moreover, we show that topology-preserving mesh refinements, produced by the action of (the simplest) Euler operators, can be reduced to multilinear transformations of the Hasse matrix representing the complex. Our main result is a new representation of the (co)chain complex underlying field computations, a representation that provides new insights into the transformations induced by local mesh refinements. Our approach is based on first principles and is general in that it applies to most representational domains that can be characterized as cell complexes, without any restrictions on their type, dimension, codimension, orientability, manifoldness, connectedness

Geometric Modeling of Cellular Materials for Additive Manufacturing in Biomedical Field: A Review

Advances in additive manufacturing technologies facilitate the fabrication of cellular materials that have tailored functional characteristics. The application of solid freeform fabrication techniques is especially exploited in designing scaffolds for tissue engineering. In this review, firstly, a classification of cellular materials from a geometric point of view is proposed; then, the main approaches on geometric modeling of cellular materials are discussed. Finally, an investigation on porous scaffolds fabricated by additive manufacturing technologies is pointed out. Perspectives in geometric modeling of scaffolds for tissue engineering are also proposed

What's the Situation with Intelligent Mesh Generation: A Survey and Perspectives

Intelligent Mesh Generation (IMG) represents a novel and promising field of research, utilizing machine learning techniques to generate meshes. Despite its relative infancy, IMG has significantly broadened the adaptability and practicality of mesh generation techniques, delivering numerous breakthroughs and unveiling potential future pathways. However, a noticeable void exists in the contemporary literature concerning comprehensive surveys of IMG methods. This paper endeavors to fill this gap by providing a systematic and thorough survey of the current IMG landscape. With a focus on 113 preliminary IMG methods, we undertake a meticulous analysis from various angles, encompassing core algorithm techniques and their application scope, agent learning objectives, data types, targeted challenges, as well as advantages and limitations. We have curated and categorized the literature, proposing three unique taxonomies based on key techniques, output mesh unit elements, and relevant input data types. This paper also underscores several promising future research directions and challenges in IMG. To augment reader accessibility, a dedicated IMG project page is available at \url{https://github.com/xzb030/IMG_Survey}