Thomas Lewiner

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.

Interactive topology-aware surface reconstruction

The reconstruction of a complete watertight model from scan data is still a difficult process. In particular, since scanned data is often incomplete, the reconstruction of the expected shape is an ill-posed problem. Techniques that reconstruct poorly-sampled areas without any user intervention fail in many cases to faithfully reconstruct the topology of the model. The method that we introduce in this paper is topology-aware: it uses minimal user input to make correct decisions at regions where the topology of the model cannot be automatically induced with a reasonable degree of confidence. We first construct a continuous function over a three-dimensional domain. This function is constructed by minimizing a penalty function combining the data points, user constraints, and a regularization term. The optimization problem is formulated in a mesh-independent manner, and mapped onto a specific mesh using the finite-element method. The zero level-set of this function is a first approximation of the reconstructed surface. At complex under-sampled regions, the constraints might be insufficient. Hence, we analyze the local topological stability of the zero level-set to detect weak regions of the surface. These regions are suggested to the user for adding local inside/outside constraints by merely scribbling over a 2D tablet. Each new user constraint modifies the minimization problem, which is solved incrementally. The process is repeated, converging to a topology-stable reconstruction. Reconstructions of models acquired by a structured-light scanner with a small number of scribbles demonstrate the effectiveness of the method.

Real-Time Gesture Recognition from Depth Data through Key Poses Learning and Decision Forests

Human gesture recognition is a challenging task with many applications. The popularization of real time depth sensors even diversifies potential applications to end-user natural user interface (NUI). The quality of such NUI highly depends on the robustness and execution speed of the gesture recognition. This work introduces a method for real-time gesture recognition from a noisy skeleton stream, such as the ones extracted from Kinect depth sensors. Each pose is described using a tailored angular representation of the skeleton joints. Those descriptors serve to identify key poses through a multi-class classifier derived from Support Vector learning machines. The gesture is labeled on-the-fly from the key pose sequence through a decision forest, that naturally performs the gesture time warping and avoids the requirement for an initial or neutral pose. The proposed method runs in real time and shows robustness in several experiments.

Space-time surface reconstruction using incompressible flow

We introduce a volumetric space-time technique for the reconstruction of moving and deforming objects from point data. The output of our method is a four-dimensional space-time solid, made up of spatial slices, each of which is a three-dimensional solid bounded by a watertight manifold. The motion of the object is described as an incompressible flow of material through time. We optimize the flow so that the distance material moves from one time frame to the next is bounded, the density of material remains constant, and the object remains compact. This formulation overcomes deficiencies in the acquired data, such as persistent occlusions, errors, and missing frames. We demonstrate the performance of our flow-based technique by reconstructing coherent sequences of watertight models from incomplete scanner data.

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.

Competing Fronts for Coarse–to–Fine Surface Reconstruction

We present a deformable model to reconstruct a surface from a point cloud. The model is based on an explicit mesh representation composed of multiple competing evolving fronts. These fronts adapt to the local feature size of the target shape in a coarse–to–fine manner. Hence, they approach towards the finer (local) features of the target shape only after the reconstruction of the coarse (global) features has been completed. This conservative approach leads to a better control and interpretation of the reconstructed topology. The use of an explicit representation for the deformable model guarantees water‐tightness and simple tracking of topological events. Furthermore, the coarse–to–fine nature of reconstruction enables adaptive handling of non‐homogenous sample density, including robustness to missing data in defected areas.

Particle-based viscoplastic fluid/solid simulation

Simulations of viscoplastic materials are traditionally governed by continuum mechanics. The viscous behavior is typically modeled as an internal force, defined by diverse quantities. This work introduces a fluid model to simulate the viscoplastic effect of solid materials, such as plastic, wax, clay and polymer. Our method consists in modeling a solid object through a non-Newtonian fluid with high viscosity. This fluid simulation uses the Smoothed Particle Hydrodynamics method and the viscosity is formulated by using the General Newtonian Fluid model. This model concentrates the viscoplasticity in a single parameter. Our results show clear effects of creep, melting, hardening and flowing.

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.