Juan Cao

A note on Class A Bézier curves

The Class A Bezier curves presented in Farin (2006) were constructed by so-called Class A matrix, which are special matrices satisfying two appropriate conditions. The speciality of the Class A matrix causes the Class A Bezier to possess two properties, which are sufficient conditions for the proof of the curvature and torsion monotonicity. In this paper, we discover that, in Farin (2006), the conditions Class A matrix satisfied cannot guarantee one of the two properties of the Class A Bezier curves, then the proof of the curvature and torsion monotonicity becomes incomplete. Furthermore, we modified the conditions for the Class A matrices to complete the proof.

Isotropic Surface Remeshing Using Constrained Centroidal Delaunay Mesh

We develop a novel isotropic remeshing method based on constrained centroidal Delaunay mesh (CCDM), a generalization of centroidal patch triangulation from 2D to mesh surface. Our method starts with resampling an input mesh with a vertex distribution according to a user‐defined density function. The initial remeshing result is then progressively optimized by alternatively recovering the Delaunay mesh and moving each vertex to the centroid of its 1‐ring neighborhood. The key to making such simple iterations work is an efficient optimization framework that combines both local and global optimization methods. Our method is parameterization‐free, thus avoiding the metric distortion introduced by parameterization, and generating more well‐shaped triangles. Our method guarantees that the topology of surface is preserved without requiring geodesic information. We conduct various experiments to demonstrate the simplicity, efficacy, and robustness of the presented method.

Technical Section: Surface reconstruction using bivariate simplex splines on Delaunay configurations

Recently, a new bivariate simplex spline scheme based on Delaunay configuration has been introduced into the geometric computing community, and it defines a complete spline space that retains many attractive theoretic and computational properties. In this paper, we develop a novel shape modeling framework to reconstruct a closed surface of arbitrary topology based on this new spline scheme. Our framework takes a triangulated set of points, and by solving a linear least-square problem and iteratively refining parameter domains with newly added knots, we can finally obtain a continuous spline surface satisfying the requirement of a user-specified error tolerance. Unlike existing surface reconstruction methods based on triangular B-splines (or DMS splines), in which auxiliary knots must be explicitly added in advance to form a knot sequence for construction of each basis function, our new algorithm completely avoids this less-intuitive and labor-intensive knot generating procedure. We demonstrate the efficacy and effectiveness of our algorithm on real-world, scattered datasets for shape representation and computing.

Spherical DCB-Spline Surfaces with Hierarchical and Adaptive Knot Insertion

This paper develops a novel surface fitting scheme for automatically reconstructing a genus-0 object into a continuous parametric spline surface. A key contribution for making such a fitting method both practical and accurate is our spherical generalization of the Delaunay configuration B-spline (DCB-spline), a new non-tensor-product spline. In this framework, we efficiently compute Delaunay configurations on sphere by the union of two planar Delaunay configurations. Also, we develop a hierarchical and adaptive method that progressively improves the fitting quality by new knot-insertion strategies guided by surface geometry and fitting error. Within our framework, a genus-0 model can be converted to a single spherical spline representation whose root mean square error is tightly bounded within a user-specified tolerance. The reconstructed continuous representation has many attractive properties such as global smoothness and no auxiliary knots. We conduct several experiments to demonstrate the efficacy of our new approach for reverse engineering and shape modeling.

B-spline surface fitting with knot position optimization

In linear least squares fitting of B-spline surfaces, the choice of knot vector is essentially important to the quality of the approximating surface. In this paper, a heuristic criterion for optimal knot positions in the fitting problem is formulated as an optimization problem according to the geometric feature distribution of the input data. Then, the coordinate descent algorithm is used for the optimal knot computation. Based on knot position optimization, an iterative surface fitting framework is developed, which adaptively introduces more knot isolines passing through the regions with more complex geometry or large fitting errors. Hence, the approximation quality of the reconstructed surface is progressively improved up to a pre-specified threshold. We test several models to demonstrate the efficacy of our method in fitting surface with distinct geometric features. Different from the knot placement technique (NKTP method) proposed in Piegl and Tiller 1 and the dominant-column-based fitting method (DOM-based method) (Park 2) which require input data in semi-grid or grid form, our algorithm takes more general data points as input, i.e., any scattered data sets with parameterization. Comparing to NKTP method and DOM-based method, our method efficiently produces more accurate results by using the same number of knots. Graphical abstractDisplay Omitted HighlightsA heuristic criterion is proposed for optimizing knots in the B-spline surface fitting problem.The iterative surface fitting framework can well preserve geometric features.Our method is more efficient and yields more accurate results than DOM-based method.

Approximation by piecewise polynomials on Voronoi tessellation

We propose a novel method to approximate a function on 2D domain by piecewise polynomials. The Voronoi tessellation is used as a partition of the domain, on which the best fitting polynomials in L^2 metric are constructed. Our method optimizes the domain partition and the fitting polynomials simultaneously by minimizing an objective function indicating the approximation quality. We also provide the explicit formula of the gradient of the objective function, which makes an efficient gradient-based algorithm workable for the function minimization. We conduct several experiments to demonstrate the efficacy of our new approach for generating piecewise polynomial approximations of analytic functions and color images.

Line drawing for 3D printing

Abstract This paper focuses on the problem of generating a line drawing from a given image for fused deposition modeling. The abstracted line drawing, comprising of lines with a single color and thickness, would preserve both tone and edges of the input image. We first partition the image into sub-regions manually. The boundaries of the sub-regions are extracted as the feature lines of the image. Next, a proper number of points are randomly placed on the image plane with a density proportional to the darkness of the image. We use Lloyd’s method to push the sampling points away from each other and the feature lines. The points within each sub-region are then connected by solving a travelling salesman problem (TSP). Finally, we further optimize the fairness and the spacing of the lines by minimizing a tailored objective function. A variety of experimental results are presented to show the effectiveness of our method for generating line drawings for 3D printing.

An improvement on the upper bounds of the magnitudes of derivatives of rational triangular Bézier surfaces

National Natural Science Foundation of China [61100105, 61100107, 61170324, 61272300]; Natural Science Foundation of Fujian Province of China [2011J05007, 2012J01291]

Non-uniform B-spline curveswith multiple shape parameters

We introduce a kind of shape-adjustable spline curves defined over a non-uniform knot sequence. These curves not only have the many valued properties of the usual non-uniform B-spline curves, but also are shape adjustable under fixed control polygons. Our method is based on the degree elevation of B-spline curves, where maximum degrees of freedom are added to a curve parameterized in terms of a non-uniform B-spline. We also discuss the geometric effect of the adjustment of shape parameters and propose practical shape modification algorithms, which are indispensable from the user’s perspective.

Point cloud resampling using centroidal Voronoi tessellation methods

Abstract This paper presents a novel technique for resampling point clouds of a smooth surface. The key contribution of this paper is the generalization of centroidal Voronoi tessellation (CVT) to point cloud datasets to make point resampling practical and efficient. In particular, the CVT on a point cloud is efficiently computed by restricting the Voronoi cells to the underlying surface, which is locally approximated by a set of best-fitting planes. We also develop an efficient method to progressively improve the resampling quality by interleaving optimization of resampling points and update of the fitting planes. Our versatile framework is capable of generating high-quality resampling results with isotropic or anisotropic distributions from a given point cloud. We conduct extensive experiments to demonstrate the efficacy and robustness of our resampling method.