Barycentric parameterizations for isotropic BRDFs

A bidirectional reflectance distribution function (BRDF) is often expressed as a function of four real variables: two spherical coordinates in each of the "incoming" and "outgoing" directions. However, many BRDFs reduce to functions of fewer variables. For example, isotropic reflection can be represented by a function of three variables. Some BRDF models can be reduced further. In This work, we introduce new sets of coordinates which we use to reduce the dimensionality of several well-known analytic BRDFs as well as empirically measured BRDF data. The proposed coordinate systems are barycentric with respect to a triangular support with a direct physical interpretation. One coordinate set is based on the BRDF mode) proposed by Lafortune. Another set, based on a model of Ward, is associated with the "halfway" vector common in analytical BRDF formulas. Through these coordinate sets we establish lower bounds on the approximation error inherent in the models on which they are based. We present a third set of coordinates, not based on any analytical model, that performs well in approximating measured data. Finally, our proposed variables suggest novel ways of constructing and visualizing BRDFs.     

矩阵重心坐标

在二维重心坐标——复数重心坐标的基础上引入二维矩阵重心坐标的概念,并利用球面坐标将二维矩阵重心坐标推广到三维.三维矩阵重心坐标适用于三角控制网格、四边形控制网格甚至一般的混合控制网格.对所提出的重心坐标性质进行了研究,发现其满足大部分好的重心坐标所应具有的性质.最后对矩阵重心坐标在三维网格模型中的应用进行了细致的实验,分析了它的优缺点.     

Higher Order Barycentric Coordinates

In recent years, a wide range of generalized barycentric coordinates has been suggested. However, all of them lack control over derivatives. We show how the notion of barycentric coordinates can be extended to specify derivatives at control points. This is also known as Hermite interpolation. We introduce a method to modify existing barycentric coordinates to higher order barycentric coordinates and demonstrate, using higher order mean value coordinates, that our method, although conceptually simple and easy to implement, can be used to give easy and intuitive control at interactive frame rates over local space deformations such as rotations.     

Barycentric Coordinates on Surfaces

This paper introduces a method for defining and efficiently computing barycentric coordinates with respect to polygons on general surfaces. Our construction is geared towards injective polygons (polygons that can be enclosed in a metric ball of an appropriate size) and is based on replacing the linear precision property of planar coordinates by a requirement in terms of center of mass, and generalizing this requirement to the surface setting. We show that the resulting surface barycentric coordinates can be computed using planar barycentric coordinates with respect to a polygon in the tangent plane. We prove theoretically that the surface coordinates properly generalize the planar coordinates and carry some of their useful properties such as unique reconstruction of a point given its coordinates, uniqueness for triangles, edge linearity, similarity invariance, and smoothness; in addition, these coordinates are insensitive to isometric deformations and can be used to reconstruct isometries. We show empirically that surface coordinates are shape‐aware with consistent gross behavior across different surfaces, are well‐behaved for different polygon types/locations on variety of surface forms, and that they are fast to compute. Finally, we demonstrate effectiveness of surface coordinates for interpolation, decal mapping, and correspondence refinement.     

Interior Distance Using Barycentric Coordinates

This paper introduces a framework for defining a shape-aware distance measure between any two points in the interior of a surface mesh. Our framework is based on embedding the surface mesh into a high-dimensional space in a way that best preserves boundary distances between vertices of the mesh, performing a mapping of the mesh volume into this high-dimensional space using barycentric coordinates, and defining the interior distance between any two points simply as their Euclidean distance in the embedding space. We investigate the theoretical properties of the interior distance in relation to properties of the chosen boundary distances and barycentric coordinates, and we investigate empirical properties of the interior distance using diffusion distance as the prescribed boundary distance and mean value coordinates. We prove theoretically that the interior distance is a metric, smooth, interpolating the boundary distances, and reproducing Euclidean distances, and we show empirically that it is insensitive to boundary noise and deformation and quick to compute. In case the barycentric coordinates are non-negative we also show a maximum principle exists. Finally, we use it to define a new geometric property, barycentroid of shape, and show that it captures the notion of semantic center of the shape.     

重心坐标下曲线的像素级生成算法

对重心坐标下的曲线方程以及重心坐标下曲线的生成进行了研究,提出一和像素级的生成算法,由于只涉及到加减法,效率极高。     

A Complex View of Barycentric Mappings

Barycentric coordinates are very popular for interpolating data values on polyhedral domains. It has been recently shown that expressing them as complex functions has various advantages when interpolating two‐dimensional data in the plane, and in particular for holomorphic maps. We extend and generalize these results by investigating the complex representation of real‐valued barycentric coordinates, when applied to planar domains. We show how the construction for generating real‐valued barycentric coordinates from a given weight function can be applied to generating complex‐valued coordinates, thus deriving complex expressions for the classical barycentric coordinates: Wachspress, mean value, and discrete harmonic. Furthermore, we show that a complex barycentric map admits the intuitive interpretation as a complex‐weighted combination of edge‐to‐edge similarity transformations, allowing the design of “home‐made” barycentric maps with desirable properties. Thus, using the tools of complex analysis, we provide a methodology for analyzing existing barycentric mappings, as well as designing new ones.     

Barycentric formulae for some optimal rational approximants involving blaschke products

We want to approximate the valueLf of some bounded linear functionalL (e.g., an integral or a function evaluation) forf∈H ² by a linear combination Σ _j=0 ^j=0 a _j f _j , wheref _j:=f(z _j) for some pointsz _j in the unit disk and the numbersa _j are to be chosen independent off _j. Using ideas of Sard, Larkin has shown that, for the errorLf?Σ _j=0 ^j=0 a _j f _j to be minimal,a _j must be chosen such that Σ _j=0 ^j=0 a _j f _j =Lf ^⊥ for the rational function \(f^ \bot (z) = \sum\nolimits_{j = 0}^n {\{ \prod\nolimits_{k = 0}^n {(1 - \bar z_k z_j )/\prod\nolimits_{k = 0}^n {(1 - \bar z_k z)} } \} l_j } (z)f_j \) , in whichl _j (z) are the Lagrange polynomials. Evaluatingf ^⊥ as given above requriesO(n ²) operations for everyz. We give here formulae, patterned after the barycentric formulae for polynomial, trigonometric and rational interpolation, which permit the evaluation off ^⊥ inO(n) operations for everyz, once some weights (that are independent ofz) have been computed. Moreover, we show that certain rational approximants introduced by F. Stenger (Math. Comp., 1986) can be interpreted as special cases of Larkin's interpolants, and are therefore optimal in the sense of Sard for the corresponding points.     

Barycentric interpolation collocation method for solving the coupled viscous Burgers' equations

The coupled viscous Burgers' equations have been an interesting and hot topic in mathematics and physics for a long time, and they have been solved by many methods. In order to make the numerical solutions more accurate, this paper introduces a new method to solve the equations. Compared to other methods, the present method can obtain higher accuracy with fewer nodes. Several numerical examples show the high accuracy of this method.     

Complex Transfinite Barycentric Mappings with Similarity Kernels

Transfinite barycentric kernels are the continuous version of traditional barycentric coordinates and are used to define interpolants of values given on a smooth planar contour. When the data is two‐dimensional, i.e. the boundary of a planar map, these kernels may be conveniently expressed using complex number algebra, simplifying much of the notation and results. In this paper we develop some of the basic complex‐valued algebra needed to describe these planar maps, and use it to define similarity kernels, a natural alternative to the usual barycentric kernels. We develop the theory behind similarity kernels, explore their properties, and show that the transfinite versions of the popular three‐point barycentric coordinates (Laplace, mean value and Wachspress) have surprisingly simple similarity kernels. We furthermore show how similarity kernels may be used to invert injective transfinite barycentric mappings using an iterative algorithm which converges quite rapidly. This is useful for rendering images deformed by planar barycentric mappings.     

Strong stabilization over polytopes

The problem considered consists of finding a stable controller assigning a closed-loop characteristic polynomial over a stability domain specified as a polytopic region of the space of characteristic polynomial coefficients. The controller is required to be stable with respect to the same polytopic stability region specified for the closed loop system     

Hurwitz testing sets for parallel polytopes of polynomials

In considering robustness of linear systems with uncertain paramenters, one is lead to consider simultaneous stability of families of polynomials. Efficient Hurwitz stability tests for polytopes of polynomials have earlier been developed using evaluations on the imaginary axis. This paper gives a stability criterion for parallel polytopes in terms of Hurwitz stability of a number of corners and edges. The ‘testing set’ of edges and corners depends entirely on the edge directions of the polytope, hence the results are particularly applicable in simultaneous analysis of several polytopes with equal edge directions.It follows as a consequence, that Kharitonov's four polynomial test for independent coefficient uncertainties is replaced by a test of 2q polynomials, when the stability region is a sector Ω = { e^iv | > 0, rπ/q < | v | ≤ π } and r/q is a rational number.     

Homogeneous coordinates for computer graphics

Some mathematical aspects of homogeneous coordinates are presented. It is shown that the usual methods applied by workers in computer graphics are theoretically sound provided care is exercised in defining the range of the coordinate chart. The mechanics of the linear representation of transformations are explained in terms of commutative diagrams. Finally some familiar examples are discussed.     

机器人手眼关系、基坐标系和世界坐标系关系的同时标定

提出了同时确定手眼关系以及机器人坐标系到世界坐标系关系的两种算法．算法１基于矩阵直积理论,可一次算出标定方程的旋转部分和平移部分,与现有的４元数法相比,不存在误差的传递问题;算法２基于粒子群优化理论,同样能一次计算出标定方程的旋转部分和平移部分,与采用ＬｅｖｅｎｂｅｒｇＭａｒｑｕａｒｄｔ的算法相比,该方法只需指定参数的范围．仿真结果表明了算法的有效性,并对于类似问题具有一定参考价值．     

Recursive algorithms for inner ellipsoidal approximation of convex polytopes

In this paper, fast recursive algorithms for the approximation of an n-dimensional convex polytope by means of an inscribed ellipsoid are presented. These algorithms consider at each step a single inequality describing the polytope and, under mild assumptions, they are guaranteed to converge in a finite number of steps. For their recursive nature, the proposed algorithms are better suited to treat a quite large number of constraints than standard off-line solutions, and have their natural application to problems where the set of constraints is iteratively updated, as on-line estimation problems, nonlinear convex optimization procedures and set membership identification.     

Assigning invariant polynomials over polytopes

This paper provides a parameterization of all output feedback controllers yielding closed-loop invariant polynomials varying over some assigned polytopes     

Running Error Analysis of Evaluation Algorithms for Bivariate Polynomials in Barycentric Bernstein Form

Running error analysis for the bivariate de Casteljau algorithm and the VS algorithm is performed. Theoretical results joint with numerical experiments show the better stability properties of the de Casteljau algorithm for the evaluation of bivariate polynomials defined on a triangle in spite of the lower complexity of the VS algorithm. The sharpness of our running error bounds is shown.