Bijective Parameterization Research Articles

Globally Injective Flattening via a Reduced Harmonic Subspace

We present a highly efficient-and-robust method for free-boundary flattening of disk-like triangle meshes in a globally injective manner. We show that by restricting the solution to a low-dimensional subspace of harmonic maps, we can dramatically accelerate the process while obtaining a low-distortion result. The algorithm consists of two main steps. A linear subspace construction, and a nonlinear nonconvex optimization for finding a low-distortion globally injective map within that subspace. The complexity of the first step dominates the algorithm's runtime and is merely that of solving a linear system. We combine recent results for computing locally-and-globally injective maps with that of harmonic maps into a conceptually simple algorithm that guarantees global injectivity. We demonstrate the great efficiency of our method over a dataset of 100 large scale models with more than 2M triangles each. Our algorithm is 10 times faster on average compared to the state-of-the-art Efficient Bijective Parameterizations (EBP) method [Su et al. 2020], on these high-resolution meshes, and more than 20 times faster on challenging examples (Figures 1,11). The speedup over [Jiang et al. 2017; Smith and Schaefer 2015] is even more dramatic.

Volumetric parameterization with truncated hierarchical B-splines for isogeometric analysis

Constructing spline parameterizations for computational domains is one of the fundamental problems in isogeometric analysis. In this paper, we present an efficient method for parameterizing volumetric computational domains using truncated hierarchical B-splines (THB). To ensure the injectivity of the parameterization, we utilize a collocation-based framework where the foldovers are eliminated on an increasing collocation set until the injectivity can be verified. A local–global solver is employed to eliminate any foldover effectively by projecting the mapping onto a bounded conformal distortion space. Local refinement of THB splines is also performed to enlarge the injectivity space. Various sufficient conditions are proposed to check the injectivity of the mapping efficiently. The quality of the bijective mapping is further improved by optimizing a geometrical energy, which is also embedded in the collocation framework to ensure injectivity. A coarse-to-fine variant of our method, which is more robust, is also presented. THB splines not only reduce the number of degrees of freedom in the parametric representation, but also increase the possibility of producing a bijective parameterization by local refinement. Some examples are provided to demonstrate the efficiency, effectiveness and robustness of our method.

Efficient Texture Parameterization Driven by Perceptual‐Loss‐on‐Screen

AbstractTexture mapping is a ubiquitous technique to enrich the visual effect of a mesh, which represents the desired signal (e.g. diffuse color) on the mesh to a texture image discretized by pixels through a bijective parameterization. To achieve high visual quality, large number of pixels are generally required, which brings big burden in storage, memory and transmission. We propose to use a perceptual model and a rendering procedure to measure the loss coming from the discretization, then optimize a parameterization to improve the efficiency, i.e. using fewer pixels under a comparable perceptual loss. The general perceptual model and rendering procedure can be very complicated, and non‐isotropic property rooted in the square shape of pixels make the problem more difficult to solve. We adopt a two‐stage strategy and use the Bayesian optimization in the triangle‐wise stage. With our carefully designed weighting scheme, the mesh‐wise optimization can take the triangle‐wise perceptual loss into consideration under a global conforming requirement. Comparing with many parameterizations manually designed, driven by interpolation error, or driven by isotropic energy, ours can use significantly fewer pixels with comparable perception loss or vise vesa.

Constructing planar domain parameterization with HB-splines via quasi-conformal mapping

Constructing a high-quality parameterization of a computational domain is a fundamental research problem in isogeometric analysis, which has been extensively investigated so far. However, most of the current approaches employ non-uniform rational B-splines (NURBS) as the geometric representation of the physical domain. NURBS introduce redundant degrees of freedom due to their tensor-product structure. In this paper, we propose a new parameterization method for planar domains by adopting hierarchical B-splines (HB-splines) as the geometric representation that possess local refinement abilities. Starting from an initial parameterization such as a harmonic map, our method repeats the following two steps until a bijective parameterization with low distortion is achieved. First, a non-linear optimization model is proposed to compute a quasi-conformal map represented by HB-splines, and an efficient algorithm is provided to deal with this model by alternatively solving two quadratic optimization problems. Second, the parameterization is refined locally through HB-splines based on the bijectivity and conformal distortion of the parameterization. Several examples are demonstrated to verity the effectiveness and advantages of the proposed approach.

Memory‐Efficient Bijective Parameterizations of Very‐Large‐Scale Models

AbstractAs high‐precision 3D scanners become more and more widespread, it is easy to obtain very‐large‐scale meshes containing at least millions of vertices. However, processing these very‐large‐scale meshes is still a very challenging task due to memory limitations. This paper focuses on a fundamental geometric processing task, i.e., bijective parameterization construction. To this end, we present a spline‐enhanced method to compute bijective and low distortion parameterizations for very‐large‐scale disk topology meshes. Instead of computing descent directions using the mesh vertices as variables, we estimate descent directions for each vertex by optimizing a proxy energy defined in spline spaces. Since the spline functions contain a small set of control points, it significantly decreases memory requirement. Besides, a divide‐and‐conquer method is proposed to obtain bijective initializations, and a submesh‐based optimization strategy is developed to reduce distortion further. The capability and feasibility of our method are demonstrated over various complex models. Compared to the existing methods for bijective parameterizations of very‐large‐scale meshes, our method exhibits better scalability and requires much less memory.

Efficient bijective parameterizations

We propose a novel method to efficiently compute bijective parameterizations with low distortion on disk topology meshes. Our method relies on a second-order solver. To design an efficient solver, we develop two key techniques. First, we propose a coarse shell to substantially reduce the number of collision constraints that are used to guarantee overlap-free boundaries. During the optimization process, the shell ensures the Hessian matrix with a fixed nonzero structure and a low density, thereby significantly accelerating the optimization. The second is a triangle inequality-based barrier function that effectively ensures non-intersecting boundaries. Our barrier function is C ∞ inside the locally supported region and its convex second-order approximation is able to be analytically obtained. Compared to state-of-the-art methods for optimizing bijective parameterizations, our method exhibits better scalability and is about six times faster. The performance of our bijective parameterization algorithm is comparable to state-of-the-art methods of locally flip-free parameterizations. A large number of experimental results have shown the capability and feasibility of our method.

Robust atlas generation via angle-based segmentation

We present a robust method to generate atlases with low isometric distortion and high packing efficiency. Given a surface that has been cut to disk topology, the algorithm contains four steps: (1) computing a bijective parameterization with low isometric distortion; (2) partitioning the input parameterized charts into a set of rectangle-like patches; (3) mapping rectangle-like patches onto rectangles that are packed into a rectangular domain; and (4) reducing parameterization distortion while maintaining high packing efficiency and bijection. Since there have been elegant and robust solutions for the first, third, and fourth steps, we focus on the second step. Central to the second step is a novel and robust segmentation technique. To improve the robustness of the segmentation, we first deform the parameterized charts to straighten their boundaries while aligning with the axes as much as possible, bounding the deformation distortion, and not introducing intersecting boundaries. Since rectangle-like patches are preferred for the third and fourth steps, we segment the deformed charts by decomposing concave interior angles into several interior angles close to π2 or π. To this end, we develop four types of operations: (i) bridging operation, (ii) extension operation, (iii) vertical line operation, and (iv) bisector operation. The pre-deformation process makes our heuristic segmentation very simple and effective. We demonstrate the efficacy of our method on various complex models. Compared to other state-of-the-art methods, our method achieves higher robustness.

Low-rank Parameterization of Volumetric Domains for Isogeometric Analysis

Constructing a volumetric spline parameterization of a three dimensional domain with given boundary is a fundamental problem in isogeometric analysis. Recently it is shown that the rank of the parameterization has great impact in subsequent numerical simulation, and lower rank leads to higher computational efficiency. In this paper, we propose a method for low-rank spline parameterization of volumetric domains using low-rank tensor approximation technique. We model the problem as a non-linear optimization problem, the objective function of which consists of the MIPS (Most Isometric Parameterizations) term and the low-rank regularization term. An algorithm based on the alternating direction method of multipliers (ADMM) is presented to solve the optimization problem efficiently. To further obtain a low-rank spline representation of the parameterization, a modified CANDECOMP/PARAFAC (CP) decomposition algorithm is proposed. Experimental examples demonstrate that our approach can produce low-rank and bijective parameterizations with high quality.

Hybrid geometry / topology based mesh segmentation for reverse engineering

Mesh segmentation and parameterization are crucial for Reverse Engineering (RE). Bijective parameterizations of the sub-meshes are a sine-qua-non test for segmentation. Current segmentation methods use either (1) topologic or (2) geometric criteria to partition the mesh. Reported topology-based segmentations produce large sub-meshes which reject parameterizations. Geometry-based segmentations are very sensitive to local variations in dihedral angle or curvatures, thus producing an exaggerated large number of small sub-meshes. Although small sub-meshes accept nearly isometric parameterizations, this significant granulation defeats the intent of synthesizing a usable Boundary Representation (compulsory for RE). In response to these limitations, this article presents an implementation of a hybrid geometry / topology segmentation algorithm for mechanical workpieces. This method locates heat transfer constraints (topological criterion) in low frequency neighborhoods of the mesh (geometric criterion) and solves for the resulting temperature distribution on the mesh. The mesh partition dictated by the temperature scalar map results in large, albeit parameterizable, sub-meshes. Our algorithm is tested with both benchmark repository and physical piece scans data. The experiments are successful, except for the well - known cases of topological cylinders, which require a user - introduced boundary along the cylinder generatrices.

Weighted area/angle distortion minimization for Mesh Parameterization

PurposeMesh Parameterization is central to reverse engineering, tool path planning, etc. This work synthesizes parameterizations with un-constrained borders, overall minimum angle plus area distortion. This study aims to present an assessment of the sensitivity of the minimized distortion with respect to weighed area and angle distortions.Design/methodology/approachA Mesh Parameterization which does not constrain borders is implemented by performing: isometry maps for each triangle to the plane Z = 0; an affine transform within the plane Z = 0 to glue the triangles back together; and a Levenberg–Marquardt minimization algorithm of a nonlinear F penalty function that modifies the parameters of the first two transformations to discourage triangle flips, angle or area distortions. F is a convex weighed combination of area distortion (weight: α with 0 ≤ α ≤ 1) and angle distortion (weight: 1 − α).FindingsThe present study parameterization algorithm has linear complexity [\U0001d4aa(n), n = number of mesh vertices]. The sensitivity analysis permits a fine-tuning of the weight parameter which achieves overall bijective parameterizations in the studied cases. No theoretical guarantee is given in this manuscript for the bijectivity. This algorithm has equal or superior performance compared with the ABF, LSCM and ARAP algorithms for the Ball, Cow and Gargoyle data sets. Additional correct results of this algorithm alone are presented for the Foot, Fandisk and Sliced-Glove data sets.Originality/valueThe devised free boundary nonlinear Mesh Parameterization method does not require a valid initial parameterization and produces locally bijective parameterizations in all of our tests. A formal sensitivity analysis shows that the resulting parameterization is more stable, i.e. the UV mapping changes very little when the algorithm tries to preserve angles than when it tries to preserve areas. The algorithm presented in this study belongs to the class that parameterizes meshes with holes. This study presents the results of a complexity analysis comparing the present study algorithm with 12 competing ones.

Orbifold Tutte embeddings

Injective parameterizations of surface meshes are vital for many applications in Computer Graphics, Geometry Processing and related fields. Tutte's embedding, and its generalization to convex combination maps, are among the most popular approaches for computing parameterizations of surface meshes into the plane, as they guarantee injectivity, and their computation only requires solving a sparse linear system. However, they are only applicable to disk-type and toric surface meshes. In this paper we suggest a generalization of Tutte's embedding to other surface topologies, and in particular the common, yet untreated case, of sphere-type surfaces. The basic idea is to enforce certain boundary conditions on the parameterization so as to achieve a Euclidean orbifold structure. The orbifold-Tutte embedding is a seamless, globally bijective parameterization that, similarly to the classic Tutte embedding, only requires solving a sparse linear system for its computation. In case the cotangent weights are used, the orbifold-Tutte embedding globally minimizes the Dirichlet energy and is shown to approximate conformal and four-point quasiconformal mappings. As far as we are aware, this is the first fully-linear method that produces bijective approximations to conformal mappings. Aside from parameterizations, the orbifold-Tutte embedding can be used to generate bijective inter-surface mappings with three or four landmarks and symmetric patterns on sphere-type surfaces.

Bijective parameterization with free boundaries

We present a fully automatic method for generating guaranteed bijective surface parameterizations from triangulated 3D surfaces partitioned into charts. We do so by using a distortion metric that prevents local folds of triangles in the parameterization and a barrier function that prevents intersection of the chart boundaries. In addition, we show how to modify the line search of an interior point method to directly compute the singularities of the distortion metric and barrier functions to maintain a bijective map. By using an isometric metric that is efficient to compute and a spatial hash to accelerate the evaluation and gradient of the barrier function for the boundary, we achieve fast optimization times. Unlike previous methods, we do not require the boundary be constrained by the user to a non-intersecting shape to guarantee a bijection, and the boundary of the parameterization is free to change shape during the optimization to minimize distortion.