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Scattered Data Interpolation with Multilevel Splines
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
, 1997
"... This paper describes a fast algorithm for scattered data interpolation and approximation. Multilevel Bsplines are introduced to compute a C²continuous surface through a set of irregularly spaced points. The algorithm makes use of a coarsetofine hierarchy of control lattices to generate a sequen ..."
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Cited by 158 (10 self)
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This paper describes a fast algorithm for scattered data interpolation and approximation. Multilevel Bsplines are introduced to compute a C²continuous surface through a set of irregularly spaced points. The algorithm makes use of a coarsetofine hierarchy of control lattices to generate a sequence of bicubic Bspline functions whose sum approaches the desired interpolation function. Large performance gains are realized by using Bspline refinement to reduce the sum of these functions into one equivalent Bspline function. Experimental results demonstrate that highfidelity reconstruction is possible from a selected set of sparse and irregular samples.
Scattered Data Interpolation Methods for Electronic Imaging Systems: A Survey
, 2002
"... Numerous problems in electronic imaging systems involve the need to interpolate from irregularly spaced data. One example is the calibration of color input/output devices with respect to a common intermediate objective color space, such as XYZ or L*a*b*. In the present report we survey some of the m ..."
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Cited by 65 (0 self)
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Numerous problems in electronic imaging systems involve the need to interpolate from irregularly spaced data. One example is the calibration of color input/output devices with respect to a common intermediate objective color space, such as XYZ or L*a*b*. In the present report we survey some of the most important methods of scattered data interpolation in twodimensional and in threedimensional spaces. We review both singlevalued cases, where the underlying function has the form f:R #R, and multivalued cases, where the underlying function is f:R . The main methods we review include linear triangular (or tetrahedral) interpolation, cubic triangular (CloughTocher) interpolation, triangle based blending interpolation, inverse distance weighted methods, radial basis function methods, and natural neighbor interpolation methods. We also review one method of scattered data fitting, as an illustration to the basic differences between scattered data interpolation and scattered data fitting.
Scattered Data Fitting on the Sphere
 in Mathematical Methods for Curves and Surfaces II
, 1998
"... . We discuss several approaches to the problem of interpolating or approximating data given at scattered points lying on the surface of the sphere. These include methods based on spherical harmonics, tensorproduct spaces on a rectangular map of the sphere, functions defined over spherical triangulat ..."
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Cited by 50 (5 self)
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. We discuss several approaches to the problem of interpolating or approximating data given at scattered points lying on the surface of the sphere. These include methods based on spherical harmonics, tensorproduct spaces on a rectangular map of the sphere, functions defined over spherical triangulations, spherical splines, spherical radial basis functions, and some associated multiresolution methods. In addition, we briefly discuss spherelike surfaces, visualization, and methods for more general surfaces. The paper includes a total of 206 references. x1. Introduction Let S be the unit sphere in IR 3 , and suppose that fv i g n i=1 is a set of scattered points lying on S. In this paper we are interested in the following problem: Problem 1. Given real numbers fr i g n i=1 , find a (smooth) function s defined on S which interpolates the data in the sense that s(v i ) = r i ; i = 1; : : : ; n; (1) or approximates it in the sense that s(v i ) ß r i ; i = 1; : : : ; n: (2) Data f...
Parametric Tilings and Scattered Data Approximation
 International Journal of Shape Modeling
, 1998
"... Abstract: This paper is concerned with methods for mapping meshes in IR 3 to meshes in IR 2 in such a way that the local geometry of the mesh is as far as possible preserved. Two linear methods are analyzed and compared. The first, which is more general and more stable, is based on convex combinatio ..."
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Cited by 21 (5 self)
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Abstract: This paper is concerned with methods for mapping meshes in IR 3 to meshes in IR 2 in such a way that the local geometry of the mesh is as far as possible preserved. Two linear methods are analyzed and compared. The first, which is more general and more stable, is based on convex combinations while the second is based on minimizing weighted squared lengths of edges. These methods can be used to approximate scattered data points in IR 3 with smooth parametric spline surfaces. 1.
A general method for overlap control in image warping
 Computers and Graphics
, 2001
"... Anewgeneral solution for the construction of onetoone image warping functions is presented. The algorithm takes a set of user speci ed translations and constructs a set of onetoone warps by interpolation and scaling. These are concatenated to produce a single onetoone warping function, which ..."
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Cited by 6 (0 self)
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Anewgeneral solution for the construction of onetoone image warping functions is presented. The algorithm takes a set of user speci ed translations and constructs a set of onetoone warps by interpolation and scaling. These are concatenated to produce a single onetoone warping function, which prevents overlap in the resultant warped image.
Augmenting gridbased contours to improve thin plate dem generation. Photogrammetric Engineering
 Remote Sensing
, 2005
"... We present two new preprocessing techniques that improve thin plate Digital Elevation Model (DEM) approximations from gridbased contour data. One method computes gradients from an initial interpolated or approximated surface. The aspects are used to create gradient lines that are interpolated usin ..."
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Cited by 5 (3 self)
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We present two new preprocessing techniques that improve thin plate Digital Elevation Model (DEM) approximations from gridbased contour data. One method computes gradients from an initial interpolated or approximated surface. The aspects are used to create gradient lines that are interpolated using CatmullRom splines. The computed elevations are added to the initial contour data set. Thin plate methods are applied to all of the data. The splines allow information to flow across contours, improving the final surface. The second method successively computes new, intermediate contours in between existing isolines, which provide additional data for subsequent thin plate processing. Both methods alleviate artifacts visible in previous thin plate methods. The surfaces are tested with published methods to show qualitative and quantitative improvements over previous methods. 2 1
Investigation of Fuzzy Rule Interpolation Techniques and the Universal Approximation Property of Fuzzy Controllers
, 1999
"... Fuzzy control is the most successfull application area of fuzzy theory. The advantage of fuzzy controllers against conventionel ones is that, they can by used for modelling systems with complicated (non linearizable) or unknown behaviour by means of linguistic variables anf fuzzy Ifthen rules. Late ..."
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Cited by 4 (1 self)
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Fuzzy control is the most successfull application area of fuzzy theory. The advantage of fuzzy controllers against conventionel ones is that, they can by used for modelling systems with complicated (non linearizable) or unknown behaviour by means of linguistic variables anf fuzzy Ifthen rules. Later on, the first approximating model can be tuned to obtain appropriate result. However, an essential problem of these algorithms is that their time complexity grows exponentially with the number of input variables. Fuzzy rule interpolation methods are one of the technique developed to reduce the complexity of fuzzy reasoning approaches. Purposes of this work are the following:  Modification of the first published KóczyHirota (KH) interpolation method to alleviate the socalled abnormal conclusion while maintaining its advantageous complexity behaviour.  Investigation of the mathematical stability of the KHmethod.  Examination of the universal approximation property of certain fuzzy controllers. A modification of the original KH approach was proposed, whose main idea is the following. The consequent fuzzy sets are transformed by a proper coordinate transformation to such a space where the convexity of these consequents excludes abnormality of the conclusion. After the conclusion is calculated in this space, the inverse of the aforementioned transformation is used to obtain the corresponding conclusion in the original output space. The proposed method is closed for convex and normal fuzzy sets (Theorem 2.1). The new interpolation method was compared with the KHapproach one in several aspects. It was investigated how the proposed method differs form linear between characteristic points, and finally a comparison among the main interpolation techniques is given with respect to the relation of the observation's and conclusion's fuzziness. It was proven that the inputoutput function of the KH interpolation converges uniformly to the arbitrary approximated continuous function if the measurement points are uniformly distributed on the domain. A generalization of this theorem is also given for a wider class of interpolatory operators. It was pointed out that the stability of the wellknown Shepardinterpolation (investigated extensively by approximation theorists) is can be derived from the one of the KH interpolation. The third main statement characterizes a set of certain type fuzzy controllers with bounded number of rules concerning the universal approximation property. As a generalization of Moser's result, it was shown that this property does not hold for the set of Tcontrollers (which includes Sugeno, TakagiSugeno, TakagiSugenoKang inference methods) if the number of rules is prerestricted, although that is a considerable practical limitation. It contradicts to those statements which state that fuzzy controllers are universal approximators, i.e., they lie dense in the space of continuous functions.
Spline Approximation of General Volumetric Data
, 2004
"... We present an efficient algorithm for approximating huge general volumetric data sets, i.e. the data is given over arbitrarily shaped volumes and consists of up to millions of samples. The method is based on cubic trivariate splines, i.e. piecewise polynomials of total degree three defined w.r.t. un ..."
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Cited by 2 (0 self)
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We present an efficient algorithm for approximating huge general volumetric data sets, i.e. the data is given over arbitrarily shaped volumes and consists of up to millions of samples. The method is based on cubic trivariate splines, i.e. piecewise polynomials of total degree three defined w.r.t. uniform type6 tetrahedral partitions of the volumetric domain. Similar as in the recent bivariate approximation approaches (cf. [10, 15]), the splines in three variables are automatically determined from the discrete data as a result of a twostep method (see [40]), where local discrete least squares polynomial approximations of varying degrees are extended by using natural conditions, i.e. the continuity and smoothness properties which determine the underlying spline space. The main advantages of this approach with linear algorithmic complexity are as follows: no tetrahedral partition of the volume data is needed, only small linear systems have to be solved, the local variation and distribution of the data is automatically adapted, BernsteinBézier techniques wellknown in Computer Aided Geometric Design (CAGD) can be fully exploited, noisy data are automatically smoothed. Our numerical examples with huge data sets for synthetic data as well as some realworld data confirm the efficiency of the methods, show the high quality of the spline approximation, and illustrate that the rendered isosurfaces inherit a visual smooth appearance from the volume approximating splines.
Systolic computation of interpolation polynomials
 Computing
, 1990
"... Several timeoptimal and spacetimeoptimal systolic arrays are presented for computing a process dependence graph corresponding to the Aitken algorithm. It is shown that these arrays also can be used to compute the generalized divided differences, i.e., the coefficients of the Hermite interpolating ..."
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Several timeoptimal and spacetimeoptimal systolic arrays are presented for computing a process dependence graph corresponding to the Aitken algorithm. It is shown that these arrays also can be used to compute the generalized divided differences, i.e., the coefficients of the Hermite interpolating polynomial. Multivariate generalized divided differences are shown to be efficiently computed on a 2dimensional systolic array. The techniques also are applied to the Neville algorithm, producing similar results.