arXiv subjects

Marco Livesu

Publications and source records attributed to Marco Livesu.

16 recordsLinked to original sources

arXiv2025

Volumetric Functional Maps

Computing volumetric correspondences between 3D shapes is a prominent tool for medical and industrial applications. In this work, we pave the way for spectral volume mapping, extending for the first time the surface-based functional maps framework. We show that the eigenfunctions of the volumetric Laplace operator define a functional space that is suitable for high-quality signal transfer. We also experiment with various techniques that edit this functional space, porting them to volume domains. We validate our method on novel volumetric datasets and on tetrahedralizations of well established surface datasets, also showcasing practical applications involving both discrete and continuous signal mapping, for segmentation transfer, mesh connectivity transfer and solid texturing. Finally, we show that the volumetric spectrum greatly improves the accuracy for classical shape matching tasks among surfaces, consistently outperforming surface-only spectral methods.

cs.GR↗

arXiv2024

Advancing Front Mapping

We present Advancing Front Mapping (AFM), a provably robust algorithm for the computation of surface mappings to simple base domains. Given an input mesh and a convex or star-shaped target domain, AFM installs a (possibly refined) version of the input connectivity into the target shape, generating a piece-wise linear mapping between them. The algorithm is inspired by the advancing front meshing paradigm, which is revisited to operate on two embeddings at once, thus becoming a tool for compatible mesh generation. AFM extends the capabilities of existing robust approaches, such as Tutte or Progressive Embedding, by providing the same theoretical guarantees of injectivity and at the same time introducing two key advantages: support for a broader set of target domains (star-shaped polygons) and local mesh refinement, which is used to automatically open the space of solutions in case a valid mapping to the target domain does not exist. AFM relies solely on two topological operators (split and flip), and on the computation of segment intersections, thus permitting to compute provably injective mappings without solving any numerical problem. This makes the algorithm predictable, easy to implement, debug and deploy. We validated the capabilities of AFM extensively, executing more than one billion advancing front moves on 36K mapping tasks, proving that our theoretical guarantees nicely transition to a robust and practical implementation.

cs.CG↗

arXiv2022

Interactive and Robust Mesh Booleans

Boolean operations are among the most used paradigms to create and edit digital shapes. Despite being conceptually simple, the computation of mesh Booleans is notoriously challenging. Main issues come from numerical approximations that make the detection and processing of intersection points inconsistent and unreliable, exposing implementations based on floating point arithmetic to many kinds of degeneracy and failure. Numerical methods based on rational numbers or exact geometric predicates have the needed robustness guarantees, that are achieved at the cost of increased computation times that, as of today, has always restricted the use of robust mesh Booleans to offline applications. We introduce the first algorithm for Boolean operations with robustness guarantees that is capable of operating at interactive frame rates on meshes with up to 200K triangles. We evaluate our tool thoroughly, considering not only interactive applications but also batch processing of large collections of meshes, processing of huge meshes containing millions of elements and variadic Booleans of hundreds of shapes altogether. In all these experiments, we consistently outperform prior art by at least one order of magnitude.

cs.CG↗

arXiv2022

Hex-Mesh Generation and Processing: a Survey

In this article, we provide a detailed survey of techniques for hexahedral mesh generation. We cover the whole spectrum of alternative approaches to mesh generation, as well as post processing algorithms for connectivity editing and mesh optimization. For each technique, we highlight capabilities and limitations, also pointing out the associated unsolved challenges. Recent relaxed approaches, aiming to generate not pure-hex but hex-dominant meshes, are also discussed. The required background, pertaining to geometrical as well as combinatorial aspects, is introduced along the way.

cs.GR↗

arXiv2021

Deterministic Linear Time Constrained Triangulation using Simplified Earcut

Triangulation algorithms that conform to a set of non-intersecting input segments typically proceed in an incremental fashion, by inserting points first, and then segments. Inserting a segment amounts to: (1) deleting all the triangles it intersects; (2) filling the so generated hole with two polygons that have the wanted segment as shared edge; (3) triangulate each polygon separately. In this paper we prove that these polygons are such that all their convex vertices but two can be used to form triangles in an earcut fashion, without the need to check whether other polygon points are located within each ear. The fact that any simple polygon contains at least three convex vertices guarantees the existence of a valid ear to cut, ensuring convergence. Not only this translates to an optimal deterministic linear time triangulation algorithm, but such algorithm is also trivial to implement. We formally prove the correctness of our approach, also validating it in practical applications and comparing it with prior art.

cs.CG↗

arXiv2021

Optimal Dual Schemes for Adaptive Grid Based Hexmeshing

Hexahedral meshes are an ubiquitous domain for the numerical resolution of partial differential equations. Computing a pure hexahedral mesh from an adaptively refined grid is a prominent approach to automatic hexmeshing, and requires the ability to restore the all hex property around the hanging nodes that arise at the interface between cells having different size. The most advanced tools to accomplish this task are based on mesh dualization. These approaches use topological schemes to regularize the valence of inner vertices and edges, such that dualizing the grid yields a pure hexahedral mesh. In this paper we study in detail the dual approach, and propose four main contributions to it: (i) we enumerate all the possible transitions that dual methods must be able to handle, showing that prior schemes do not natively cover all of them; (ii) we show that schemes are internally asymmetric, therefore not only their implementation is ambiguous, but different implementation choices lead to hexahedral meshes with different singular structure; (iii) we explore the combinatorial space of dual schemes, selecting the minimum set that covers all the possible configurations and also yields the simplest singular structure in the output hexmesh; (iv) we enlarge the class of adaptive grids that can be transformed into pure hexahedral meshes, relaxing one of the tight requirements imposed by previous approaches, and ultimately permitting to obtain much coarser meshes for same geometric accuracy. Last but not least, for the first time we make grid-based hexmeshing truly reproducible, releasing our code and also revealing a conspicuous amount of technical details that were always overlooked in previous literature, creating an entry barrier that was hard to overcome for practitioners in the field.

cs.GR↗

arXiv2020

Mapping Surfaces with Earcut

Mapping a shape to some parametric domain is a fundamental tool in graphics and scientific computing. In practice, a map between two shapes is commonly represented by two meshes with same connectivity and different embedding. The standard approach is to input a meshing of one of the two domains plus a function that projects its boundary to the other domain, and then solve for the position of the interior vertices. Inspired by basic principles in mesh generation, in this paper we present the reader a novel point of view on mesh parameterization: we consider connectivity as an additional unknown, and assume that our inputs are just two boundaries that enclose the domains we want to connect. We compute the map by simultaneously growing the same mesh inside both shapes.This change in perspective allows us to recast the parameterization problem as a mesh generation problem, granting access to a wide set of mature tools that are typically not used in this setting. Our practical outcome is a provably robust yet trivial to implement algorithm that maps any planar shape with simple topology to a homotopic domain that is weakly visible from an inner convex kernel. Furthermore, we speculate on a possible extension of the proposed ideas to volumetric maps, listing the major challenges that arise. Differently from prior methods, for which we show that a volumetric extension is not possible, our analysis leaves us reasoneable hopes that the robust generation of volumetric maps via compatible mesh generation could be obtained in the future.

cs.CG↗

arXiv2020

A Survey of Algorithms for Geodesic Paths and Distances

Numerical computation of shortest paths or geodesics on curved domains, as well as the associated geodesic distance, arises in a broad range of applications across digital geometry processing, scientific computing, computer graphics, and computer vision. Relative to Euclidean distance computation, these tasks are complicated by the influence of curvature on the behavior of shortest paths, as well as the fact that the representation of the domain may itself be approximate. In spite of the difficulty of this problem, recent literature has developed a wide variety of sophisticated methods that enable rapid queries of geodesic information, even on relatively large models. This survey reviews the major categories of approaches to the computation of geodesic paths and distances, highlighting common themes and opportunities for future improvement.

cs.GR↗

arXiv2020

Obtaining a Canonical Polygonal Schema from a Greedy Homotopy Basis with Minimal Mesh Refinement

Any closed manifold of genus g can be cut open to form a topological disk and then mapped to a regular polygon with 4g sides. This construction is called the canonical polygonal schema of the manifold, and is a key ingredient for many applications in graphics and engineering, where a parameterization between two shapes with same topology is often needed. The sides of the 4g-gon define on the manifold a system of loops, which all intersect at a single point and are disjoint elsewhere. Computing a shortest system of loops of this kind is NP-hard. A computationally tractable alternative consists in computing a set of shortest loops that are not fully disjoint in polynomial time, using the greedy homotopy basis algorithm proposed by Erickson and Whittlesey, and then detach them in post processing via mesh refinement. Despite this operation is conceptually simple, known refinement strategies do not scale well for high genus shapes, triggering a mesh growth that may exceed the amount of memory available in modern computers, leading to failures. In this paper we study various local refinement operators to detach cycles in a system of loops, and show that there are important differences between them, both in terms of mesh complexity and preservation of the original surface. We ultimately propose two novel refinement approaches: the former minimizes the number of new elements in the mesh, possibly at the cost of a deviation from the input geometry. The latter allows to trade mesh complexity for geometric accuracy, bounding deviation from the input surface. Both strategies are trivial to implement, and experiments confirm that they allow to realize canonical polygonal schemas even for extremely high genus shapes where previous methods fail.

cs.CG↗

arXiv2019

Real-time Deformation with Coupled Cages and Skeletons

Skeleton-based and cage-based deformation techniques represent the two most popular approaches to control real-time deformations of digital shapes and are, to a vast extent, complementary to one another. Despite their complementary roles, high-end modeling packages do not allow for seamless integration of such control structures, thus inducing a considerable burden on the user to maintain them synchronized. In this paper, we propose a framework that seamlessly combines rigging skeletons and deformation cages, granting artists with a real-time deformation system that operates using any smooth combination of the two approaches. By coupling the deformation spaces of cages and skeletons, we access a much larger space, containing poses that are impossible to obtain by acting solely on a skeleton or a cage. Our method is oblivious to the specific techniques used to perform skinning and cage-based deformation, securing it compatible with pre-existing tools. We demonstrate the usefulness of our hybrid approach on a variety of examples.

cs.GR↗

arXiv2019

Parametric shape optimization for combined additive-subtractive manufacturing

In the industrial practice, additive manufacturing processes are often followed by post-processing operations such as subtractive machining, milling, etc. to achieve the desired surface quality and dimensional accuracy. Hence, a given part must be 3D printed with extra material to enable such finishing phase. This combined additive/subtractive technique can be optimized to reduce manufacturing costs by saving printing time and reducing material and energy usage. In this work, a numerical methodology based on parametric shape optimization is proposed for optimizing the thickness of the extra material, allowing for minimal machining operations while ensuring the finishing requirements. Moreover, the proposed approach is complemented by a novel algorithm for generating inner structures leading to reduced distortion and improved weight reduction. The computational effort induced by classical constrained optimization methods is alleviated by replacing both the objective and constraint functions by their sparse-grid surrogates. Numerical results showcase the effectiveness of the proposed approach.

cs.CE↗

arXiv2019

Loopy Cuts: Surface-Field Aware Block Decomposition for Hex-Meshing

We present a new fully automatic block-decomposition hexahedral meshing algorithm capable of producing high quality meshes that strictly preserve feature curve networks on the input surface and align with an input surface cross-field. We produce all-hex meshes on the vast majority of inputs, and introduce localized non-hex elements only when the surface feature network necessitates those. The input to our framework is a closed surface with a collection of geometric or user-demarcated feature curves and a feature-aligned surface cross-field. Its output is a compact set of blocks whose edges interpolate these features and are loosely aligned with this cross-field. We obtain this block decomposition by cutting the input model using a collection of simple cutting surfaces bounded by closed surface loops. The set of cutting loops spans the input feature curves, ensuring feature preservation, and is obtained using a field-space sampling process. The computed loops are uniformly distributed across the surface, cross orthogonally, and are loosely aligned with the cross-field directions, inducing the desired block decomposition. We validate our method by applying it to a large range of complex inputs and comparing our results to those produced by state-of-the-art alternatives. Contrary to prior approaches, our framework consistently produces high-quality field aligned meshes while strictly preserving geometric or user-specified surface features.

cs.GR↗

arXiv2019

Surface2Volume: Surface Segmentation Conforming Assemblable Volumetric Partition

Users frequently seek to fabricate objects whose outer surfaces consist of regions with different surface attributes, such as color or material. Manufacturing such objects in a single piece is often challenging or even impossible. The alternative is to partition them into single-attribute volumetric parts that can be fabricated separately and then assembled to form the target object. Facilitating this approach requires partitioning the input model into parts that conform to the surface segmentation and that can be moved apart with no collisions. We propose Surface2Volume, a partition algorithm capable of producing such assemblable parts, each of which is affiliated with a single attribute, the outer surface of whose assembly conforms to the input surface geometry and segmentation. In computing the partition we strictly enforce conformity with surface segmentation and assemblability, and optimize for ease of fabrication by minimizing part count, promoting part simplicity, and simplifying assembly sequencing. We note that computing the desired partition requires solving for three types of variables: per-part assembly trajectories, partition topology, i.e. the connectivity of the interface surfaces separating the different parts, and the geometry, or location, of these interfaces. We efficiently produce the desired partitions by addressing one type of variables at a time: first computing the assembly trajectories, then determining interface topology, and finally computing interface locations that allow parts assemblability. We algorithmically identify inputs that necessitate sequential assembly, and partition these inputs gradually by computing and disassembling a subset of assemblable parts at a time. We demonstrate our method....

cs.GR↗

arXiv2019

HexaLab.net: an online viewer for hexahedral meshes

We introduce HexaLab: a WebGL application for real time visualization, exploration and assessment of hexahedral meshes. HexaLab can be used by simply opening www.hexalab.net. Our visualization tool targets both users and scholars. Practitioners who employ hexmeshes for Finite Element Analysis, can readily check mesh quality and assess its usability for simulation. Researchers involved in mesh generation may use HexaLab to perform a detailed analysis of the mesh structure, isolating weak points and testing new solutions to improve on the state of the art and generate high quality images. To this end, we support a wide variety of visualization and volume inspection tools. Our system offers also immediate access to a repository containing all the publicly available meshes produced with the most recent techniques for hexmesh generation. We believe HexaLab, providing a common tool for visualizing, assessing and distributing results, will push forward the recent strive for replicability in our scientific community.

cs.GR↗

arXiv2018

FETI-DP preconditioners for the Virtual Element Method on general 2D meshes

We analyze the performance of a state-of-the-art domain decomposition approach, the Finite Element Tearing and Interconnecting Dual Primal (FETI-DP) method, for the efficient solution of very large linear systems arising from elliptic problems discretized by the Virtual Element Method. We provide numerical experiments on a model linear elliptic problem with highly heterogeneous diffusion coefficients on arbitrary Voronoi meshes, which we modify by adding nodes and edges deriving from the intersection with an unrelated coarse decomposition. The experiments confirm also in this case that the FETI-DP method is numerically scalable with respect to both the problem size and number of subdomains, and its performance is robust with respect to jumps in the diffusion coefficients and shape of the mesh elements.

math.NA↗

arXiv2017

From 3D Models to 3D Prints: an Overview of the Processing Pipeline

Due to the wide diffusion of 3D printing technologies, geometric algorithms for Additive Manufacturing are being invented at an impressive speed. Each single step, in particular along the Process Planning pipeline, can now count on dozens of methods that prepare the 3D model for fabrication, while analysing and optimizing geometry and machine instructions for various objectives. This report provides a classification of this huge state of the art, and elicits the relation between each single algorithm and a list of desirable objectives during Process Planning. The objectives themselves are listed and discussed, along with possible needs for tradeoffs. Additive Manufacturing technologies are broadly categorized to explicitly relate classes of devices and supported features. Finally, this report offers an analysis of the state of the art while discussing open and challenging problems from both an academic and an industrial perspective.

cs.GR↗