Hélio Lopes

Efficient implementation of Marching Cubes' cases with topological guarantees

Abstract Marching Cubes methods first offered visual access to experimental and theoretical volumetric data. The implementation of this method usually relies on a small look-up table; many enhancements and optimizations of Marching Cubes still use it. However, this look-up table can lead to cracks and inconsistent topology. This paper introduces a full implementation of Chernyaevs technique to ensure a topologically correct result, i.e., a manifold mesh, for any input data. It completes the original paper for the ambiguity resolution and for the feasibility of the implementation. Moreover, the cube interpolation provided here can be used in a wider range of methods. The source code is available online.

Curvature and torsion estimators based on parametric curve fitting

Many applications of geometry processing and computer vision rely on geometric properties of curves, particularly, their curvature. Several methods have already been proposed to estimate the curvature of a planar curve, most of them for curves in digital spaces. This work proposes a new scheme for estimating curvature and torsion of planar and spatial curves, based on weighted least-squares fitting and local arc-length approximation. The method is simple enough to admit a convergence analysis that takes into account the effect of noise in the samples. The implementation of the method is compared to other curvature estimation methods showing a good performance. Applications to prediction in geometry compression are presented both as a practical application and as a validation of this new scheme.

Applications of Forman's discrete Morse theory to topology visualization and mesh compression

Morse theory is a powerful tool for investigating the topology of smooth manifolds. It has been widely used by the computational topology, computer graphics, and geometric modeling communities to devise topology-based algorithms and data structures. Forman introduced a discrete version of this theory which is purely combinatorial. We aim to build, visualize, and apply the basic elements of Formans discrete Morse theory. We intend to use some of those concepts to visually study the topology of an object. As a basis, an algorithmic construction of optimal Formans discrete gradient vector fields is provided. This construction is then used to topologically analyze mesh compression schemes, such as Edgebreaker and Grow&Fold. In particular, we prove that the complexity class of the strategy optimization of Grow&Fold is MAX-SNP hard.

Optimal discrete Morse functions for 2-manifolds

Morse theory is a powerful tool in its applications to computational topology, computer graphics and geometric modeling. It was originally formulated for smooth manifolds. Recently, Robin Forman formulated a version of this theory for discrete structures such as cell complexes. It opens up several categories of interesting objects (particularly meshes) to applications of Morse theory. Once a Morse function has been defined on a manifold, then information about its topology can be deduced from its critical elements. The main objective of this paper is to introduce a linear algorithm to define optimal discrete Morse functions on discrete 2-manifolds, where optimality entails having the least number of critical elements. The algorithm presented is also extended to general finite cell complexes of dimension at most 2, with no guarantee of optimality.

Structural operators for modeling 3-manifolds

In this work, it is presented a complete set of operators, called Morse operators, to build and unbuild any combinatorial orientable &dimensional manifolds embedded in R”(n 3 3). Also, a suitable data structure, called Handl~Face, is introduced in order to represent 3manifolds with or without boundary. The main emphasis is on the boundary of the manifold as a key ingredient either for the data structure or operators.

Toward Optimality in Discrete Morse Theory

Morse theory is a fundamental tool for investigating the topology of smooth manifolds. This tool has been extended to discrete structures by Forman, which allows combinatorial analysis and direct computation. This theory relies on discrete gradient vector fields, whose critical elements describe the topology of the structure. The purpose of this work is to construct optimal discrete gradient vector fields, where optimality means having the minimum number of critical elements. The problem is equivalently stated in terms of maximal hyperforests of hypergraphs. Deduced from this theoretical result, a algorithm constructing almost optimal discrete gradient fields is provided. The optimal parts of the algorithm are proved, and the part of exponential complexity is replaced by heuristics. Although reaching optimality is MAX-SNP hard, the experiments on odd topological models are almost always optimal.

Edgebreaker: a simple compression for surfaces with handles

The Edgebreaker is an efficient scheme for compressing triangulated surfaces. A surprisingly simple implementation of Edgebreaker has been proposed for surfaces homeomorphic to a sphere. It uses the Corner-Table data structure, which represents the connectivity of a triangulated surface by two tables of integers, and encodes them with less than 2 bits per triangle. We extend this simple formulation to deal with triangulated surfaces with handles and present the detailed pseudocode for the encoding and decoding algorithms (which take one page each). We justify the validity of the proposed approach using the mathematical formulation of the Handlebody theory for surfaces, which explains the topological changes that occur when two boundary edges of a portion of a surface are identified.

Meshless Helmholtz-Hodge Decomposition

Vector fields analysis traditionally distinguishes conservative (curl-free) from mass preserving (divergence-free) components. The Helmholtz-Hodge decomposition allows separating any vector field into the sum of three uniquely defined components: curl free, divergence free and harmonic. This decomposition is usually achieved by using mesh-based methods such as finite differences or finite elements. This work presents a new meshless approach to the Helmholtz-Hodge decomposition for the analysis of 2D discrete vector fields. It embeds into the SPH particle-based framework. The proposed method is efficient and can be applied to extract features from a 2D discrete vector field and to multiphase fluid flow simulation to ensure incompressibility.

Robust adaptive meshes for implicit surfaces

This work introduces a robust algorithm for computing good polygonal approximations of implicit surfaces, where robustness entails recovering the exact topology of the implicit surface. Furthermore, the approximate triangle mesh adapts to the geometry and to the topology of the real implicit surface. This method generates an octree subdivided according to the interval evaluation of the implicit function in order to guarantee the robustness, and to the interval automatic differentiation in order to adapt the octree to the geometry of the implicit surface. The triangle mesh is then generated from that octree through an enhanced dual marching

Learning good views through intelligent galleries

The definition of a good view of a 3D scene is highly subjective and strongly depends on both the scene content and the 3D application. Usually, camera placement is performed directly by the user, and that task may be laborious. Existing automatic virtual cameras guide the user by optimizing a single rule, e.g. maximizing the visible silhouette or the projected area. However, the use of a static pre‐defined rule may fail in respecting the users subjective understanding of the scene. This work introduces intelligent design galleries, a learning approach for subjective problems such as the camera placement. The interaction of the user with a design gallery teaches a statistical learning machine. The trained machine can then imitate the user, either by pre‐selecting good views or by automatically placing the camera. The learning process relies on a Support Vector Machines for classifying views from a collection of descriptors, ranging from 2D image quality to 3D features visibility. Experiments of the automatic camera placement demonstrate that the proposed technique is efficient and handles scenes with occlusion and high depth complexities. This work also includes user validations of the intelligent gallery interface.