Chongyang Deng

Chongyang Deng
Hangzhou Dianzi University · School of Science

Dr.

About

51
Publications
8,227
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468
Citations
Additional affiliations
June 2014 - November 2014
Università della Svizzera italiana
Position
  • Professor
January 2014 - February 2014
City University of Hong Kong
Position
  • Research Associate
June 2013 - September 2013
City University of Hong Kong
Position
  • Research Associate

Publications

Publications (51)
Article
Full-text available
We propose Gauss–Seidel progressive iterative approximation (GS-PIA) for subdivision surface interpolation by combining the Gauss–Seidel iterative method for linear systems and progressive iterative approximation (PIA) for free-form curve and surface interpolation. We address the details of GS-PIA for Loop and Catmull–Clark surface interpolation an...
Article
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We are interested in constructing more generalized barycentric coordinates (GBC) over arbitrary polygon in the 2D setting. We propose a constrained minimization over the class of infinitely differentiable functions subject to the GBC constraints of preserving linear functions and the non-negativity condition. It includes the harmonic GBC, biharmoni...
Article
Full-text available
Barycentric coordinates provide a simple way of expressing the linear interpolant to data given at the vertices of a triangle and have numerous applications in computer graphics and other fields. The generalization of barycentric coordinates to polygons with more than three vertices is not unique and many constructions have been proposed. Among the...
Article
Full-text available
Volumetric modeling is an important topic for material modeling and isogeometric simulation. In this paper, two kinds of interpolatory Catmull-Clark volumetric subdivision approaches over unstructured hexahedral meshes are proposed based on the limit point formula of Catmull-Clark subdivision volume. The basic idea of the first method is to constru...
Article
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We derive a formula for weighted polynomial least squares approximation which expresses the approximant as a convex combination of interpolants. There is a similar formula for L2 approximation and the same principle applies to multivariate approximation.
Article
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Generalized barycentric coordinates (GBC) are widely used in computer graphics and related areas and there are a few kinds of GBC in the literature. In this paper we propose positive and smooth Gordon-Wixom coordinates (PSGWC) for the interior points of planar polygons, which are smooth and inherit all nice properties of positive Gordon-Wixom coord...
Article
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Feasibility condition, which ensures that the solution space does not violate any constraints, and optimality condition, which guarantees that all points of the solution space are optimal, are very significant conditions for the solution space of interval linear programming (ILP) problems. Among the existing methods for ILP problems, the best-worst...
Article
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A planar spiral is a kind of curve with single-signed, monotone increasing or decreasing curvature. A spiral can only interpolate a group of admissible G² Hermite data (G² Hermite data which can be interpolated by spiral). In this paper, we present a biarc-based method for interpolating C-shaped inadmissible G² Hermite data {A,TA,OA;B,TB,OB} by pie...
Article
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The GS-PIA algorithm for non-uniform cubic B-spline curve interpolation has the advantages of simplicity, stability, fast convergence and so on. In this paper, we elaborate the detailed geometric meaning of the GS-PIA algorithm and prove the convergence of the algorithm. We first define comparison matrix of configuration matrix of the algorithm. An...
Article
In this paper we first derive a recursive relation of the generating functions of a family of dual 2n-point subdivision schemes. Based on the recursive relation we design repeated local operations for implementing the 2n-point subdivision schemes. Associated interpolation properties of the limit curve sequence of the dual 2n-point subdivision schem...
Article
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Bilinear and rational bilinear maps defined by planar quadrilaterals play important roles in computer graphics and geometric design. We show that the inverse of a rational bilinear map has a close relationship with a conic, which is named as ”characteristic conic” in this paper. We show that the characteristic conic of a rational bilinear map can b...
Article
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Geometric iterative methods (GIM), including the progressive–iterative approximation (PIA) and the geometric interpolation/approximation method, are a class of iterative methods for fitting curves and surfaces with clear geometric meanings. In this paper, we provide an overview of the interpolatory and approximate geometric iteration methods, prese...
Article
This paper introduces a new formula to evaluate the deviation of a binary interpolatory subdivision curve from its data polygon. We first bound the deviation of the new control points of each subdivision step from its data polygon by accumulating the distances between the new control points and the midpoints of their corresponding edges. Then, by f...
Article
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Univariate pseudo-splines are a generalization of uniform B-splines and interpolatory $2n$-point subdivision schemes. Each pseudo-spline is characterized as the subdivision scheme with least possible support among all schemes with specific degrees of polynomial generation and reproduction. In this paper we consider the problem of constructing the s...
Article
This paper has been recommended for acceptance by Konrad Polthier.
Article
We propose to implement the m-ary 2. N-point Dubuc-Deslauriers subdivision scheme (DDSS) using a series of repeated local operations, which are based on a recursive formula between the newly inserted points of m-ary 2. N-point DDSS and those of m-ary (2N-2)-point and m-ary (2N-4)-point DDSSs. Numerical analysis reveals the robustness of our method.
Article
The Lebesgue constant of interpolation operators plays an important role in approximation theory. Several upper bounds have been achieved on the Lebesgue constant of Berrut’s rational interpolation operator. The aim of this paper is to investigate the Lebesgue constant of Berrut’s rational interpolant at equidistant nodes, and present a tight upper...
Article
Full-text available
Barycentric coordinates are commonly used to represent a point inside a polygon as an affine combination of the polygon's vertices and to interpolate data given at these vertices. While unique for triangles, various generalizations to arbitrary simple polygons exist, each satisfying a different set of properties. Some of these generalized barycentr...
Article
Recently, Floater (2015) proposed a one-parameter family of generalized barycentric coordinates for quadrilaterals. In this note we gave a proof to his observation on the limit of such coordinates: the coordinates approach the piecewise linear coordinates obtained by triangulating the quadrilateral using one or other of the diagonals if the paramet...
Article
In this paper we present an efficient framework for the evaluation of subdivision schemes with polynomial reproduction property. For all interested rational parameters between 0 and 1 with the same denominator, their exact limit positions on the subdivision curve can be obtained by solving a system of linear equations. When the framework is applied...
Article
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Based on the Jacobi iterative method for solving the system of linear equations, we propose a progressive iterative approximation method for interpolating a set of points by non-uniform cubic B-spline curves, (abbr. Jacobi-PIA). In Jacobi-PIA, the control points of the initial cubic B-spline curve are set as the points to be interpolated, then cont...
Article
In this paper, we derive a bound on derivatives of rational conic Bézier curves. Both theoretical analysis and numerical examples show that our bound is sharper than some existed ones. Discussions on optimal bounds for the derivatives of rational conic Bézier curves are also provided.
Article
Pseudo-splines provide a rich family of subdivision schemes with a wide range of choices that meet various demands for balancing the approximation power, the length of the support, and the regularity of the limit functions. Special cases of pseudo-splines include uniform odd-degree B-splines and the interpolatory 2n-point subdivision schemes, and t...
Article
Cutdown polygon, constructed by directly connecting some selected control points of the Bézier piece, is proposed for Bézier piece approximation by Zhang and Ma. However, for Bézier piece with odd degree, there are multiple solutions for selecting control points of cutdown polygon. In this note we propose to approximate odd degree Bézier piece by a...
Article
We present a simple method for C-shaped G2G2 Hermite interpolation by a rational cubic Bézier curve with conic precision. For the interpolating rational cubic Bézier curve, we derive its control points according to two conic Bézier curves, both matching the G1G1 Hermite data and one end curvature of the given G2G2 Hermite data, and the weights are...
Article
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The progressive and iterative approximation (PIA) method is an efficient and intuitive method for data fitting. However, in the classical PIA method, the number of the control points is equal to that of the data points. It is not feasible when the number of data points is very large. In this paper, we develop a new progressive and iterative approxi...
Article
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This paper presents efficient and necessary coincidence conditions for two quartic Bezier curves, i.e., either their two control polygons are coincident or the two curves can be reparameterized or degenerated into the same quadratic Bezier curve.
Article
This paper considers optimal solutions of general interval linear programming problems. Necessary and sufficient conditions of (A,b)-strong and (A,b,c)-strong optimal solutions to the interval linear programming with inequality constraints are proposed. The features of the proposed methods are illustrated by some examples.
Article
Conti et al. (2012, Remark 3.4) conjecture that the norm of the interpolatory 2n-point Dubuc–Deslauriers subdivision scheme is bounded from above by 4 for any n∈Nn∈N. We disprove their conjecture by showing that the norm grows logarithmically in n and therefore diverges as n increases.
Article
Full-text available
In this paper we solve the open problem, finding the solutions for privacy-preserving horizontally partitioned linear programs with inequality constraints, proposed recently by Mangasarian (Optim Lett 2011, doi:10.1007/s11590-010-0268-9).
Article
According to some identities and inequalities about the intermediate weights and control points of the de Casteljau algorithm for rational Bézier curve, we derive a new bound of the derivative of rational Bézier curves. The comparison of the new bounds with some existing ones is also presented. Based on some previous bounds, the bound presented in...
Article
Full-text available
For approximating subdivision schemes, there are several unified frameworks for effectively constructing subdivision surfaces generalizing splines of an arbitrary degree. In this article, we present a similar unified framework for interpolatory subdivision schemes. We first decompose the 2n-point interpolatory curve subdivision scheme into repeated...
Article
Usually we obtain the B-spline control points of a cubic uniform spline interpolant by solving a system of linear equations. In this paper, we derive an explicit formula for the control points of periodic spline interpolants. As an application of this formula, we derive the least possible bound for the deviation of the periodic cubic spline interpo...
Article
Based on the technique of C-shaped G1G1 Hermite interpolation by a cubic Pythagorean-hodograph (PH) curve, we present a simple method for C-shaped G2G2 Hermite interpolation by a rational cubic Bézier curve. The method reproduces a circular arc when the input data come from it. Both the Bézier control points, which have a well-understood geometrica...
Article
Spirals are curves with single-signed, monotone increasing or decreasing curvature. A spiral can only interpolate certain G2G2 Hermite data that is referred to as admissible G2G2 Hermite data. In this paper we propose a biarc-based subdivision scheme that can generate a planar spiral matching an arbitrary set of given admissible G2G2 Hermite data,...
Article
This paper proposes a weighted progressive method for constructing a Loop subdivision surface interpolating a given mesh. The convergent rate of the weighted progressive interpolation can be controlled by adjusting the weight of the iteration. For different weights in the available range, the limit meshes are the same as that of the reverse solutio...
Article
Using the generalized interval arithmetic we give a generalized cholesky decomposition. Generalized intervals (intervals whose bounds are not constrained to be increasingly ordered) extend classical intervals providing better algebraic properties. In particular, the generalized interval arithmetic is a group for addition and for multiplication of z...
Article
In this paper we derive some new derivative bounds of rational Bézier curves according to some existing identities and inequalities. The comparison of the new bounds with some existing ones is also presented.
Conference Paper
Full-text available
This paper presents an interpolatory subdivision scheme derived from the Doo-Sabin subdivision scheme. We first present the relations among three curve subdivision schemes, namely a four point interpolatory subdivision scheme, a cubic B-spline curve subdivision scheme, and the Chaikin’s algorithm that generates uniform quadratic B-spline curves. By...
Article
Machine learning and linear programming with time dependent cost are two popular intelligent optimization tools to handle uncertainty in real world problems. Thus, combining these two technologies is quite attractive. This paper proposed an effective framework to deal with uncertainty in practice, based on combing introducing learning parameter int...
Article
Full-text available
Using the limit point formula of the Loop subdivision scheme, we propose a very simple and efficient method for constructing interpolation surface of triangular meshes by Loop subdivision scheme. The excellent properties of the method are: (1) Locality: the perturbation of a given vertex only influences the surface shape near this vertex. (2) Effic...
Article
Full-text available
Interpolating an arbitrary topology mesh by a smooth surface plays important role in geometric modeling and computer graphics. In this paper we present an efficient new algorithm for constructing Catmull–Clark surface that interpolates a given mesh. The control mesh of the interpolating surface is obtained by one Catmull–Clark subdivision of the gi...
Article
To avoid the complex computation in the process of interpolating triangular mesh by 3 subdivision scheme, we propose a simple and efficient algorithm for interpolating closed triangular meshes by 3 subdivision scheme using the limit point formula of 3 subdivision scheme. Given the interpolated triangular mesh, by subdividing it with a new geometric...
Article
A new geometry driven subdivision scheme for curve interpolation is presented in this paper. Given a sequence of points and associated tangent vectors, we get a smooth curve interpolating the initial points by inserting new points iteratively. The new point corresponding to an edge is the incenter of a triangle, which is formed by the edge and the...
Article
Full-text available
Spirals are curves with one-signed, monotone increasing or decreasing curvature. They are commonly useful in a variety of applications, either for aesthetic or for engineering requirements. In this paper we propose a new iterative subdivision scheme for generating planar spiral segments from two points and their tangent vectors. The subdivision pro...
Conference Paper
Full-text available
Interpolating an arbitrary topology mesh by a smooth surface plays an important role in geometric modeling and computer graphics. In this paper we present an efficient new algorithm for constructing a Doo-Sabin subdivision surface that interpolates a given mesh. By introducing additional degrees of freedom, the control vertices of the Doo-Sabin sub...
Article
Full-text available
In this paper we present a local fitting algorithm for converting smooth planar curves to B-splines. For a smooth planar curve a set of points together with their tangent vectors are first sampled from the curve such that the connected polygon approximates the curve with high accuracy and inflexions are detected by the sampled data efficiently. The...
Article
In this paper we prove that the degree elevation of B-spline curves can be interpreted as corner cutting process in theory. We also discover the geometric meaning of the auxiliary control points during the corner cutting. Our main idea is to gradually elevate the degree of B-spline curves one knot interval by one knot interval. To this end, a new c...
Article
Full-text available
In this paper a class of new inequalities about Bernstein polynomial is established. With these inequalities, the estimation of heights, the derivative bounds of B�zier curves and rational B�zier curves can be improved greatly.

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Projects

Projects (2)
Project
Because of their many excellent properties, generalized barycentric coordinates (GBCs) are widely used in Computer Aided Geometric Design, Computer Graphics, Image Processing, Finite Element Analysis and related areas. This project focuses on key problems of GBCs for complex boundary. This project will help to extend the theoretic foundations of GBCs and then promote its further application in related areas.
Project
Subdivision is a standard modeling tool in computer graphics and geometric design for generating curves and surfaces by iteratively applying local refinement rules to an initial control polygon or mesh. There are two classes of subdivision schemes, namely interpolatory and approximating schemes, depending on whether the limit curve or surface interpolates the initial control vertices or not. For approximating schemes, it is known how to split the refinement rules into sequences of very local operations, resulting in an efficient process for computing new vertices, and yet obtaining limit curves and surfaces with high order of continuity. However, little work has been done on how to factorize interpolatory schemes this way. Up to now, interpolatory schemes with local rules yield only C1 continuous limit curves and surfaces. In this project, we propose to develop new interpolatory subdivision schemes based on simple local refinement operations, similar to those of the Lane-Riesenfeld algorithm for generating uniform B-splines. The continuity of the resulting limit curves and surfaces can be of an arbitrary high order, except at a limited number of extraordinary vertices where C1 continuity is obtained in the surface case. The key aspects of this project include: (1) designing interpolatory subdivision schemes for different types of meshes and topological splitting with higher order continuity at regular vertices; (2) decompose subdivision rules at regular vertices into very local operations; (3) using the Fourier transform and its inverse to extend the construction of local operations to extraordinary vertices; (4) simple methods for analyzing the smoothness of the limit surface at the extraordinary vertices; and (5) incorpotating crease and boundary features on the limit surface with also repeated local operations. Overall, this project proposes to develop simple and efficient algorithms for implementing interpolatory subdivision schemes with highly continuous limit curves and surfaces, and to lay the foundations for applying interpolatory subdivision schemes in both computer graphics and geometric design.