Constrained Visualization Using the Shepard Interpolation Family
This paper discusses the problem of visualizing data where there are underlying constraints that must be preserved. For example, we may know that the data are inherently positive. We show how the Modified Quadratic Shepard method, which interpolates scattered data of any dimensionality, can be const...
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| Vydáno v: | Computer graphics forum Ročník 24; číslo 4; s. 809 - 820 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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9600 Garsington Road , Oxford , OX4 2DQ , UK
Blackwell Publishing Ltd
01.12.2005
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| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | This paper discusses the problem of visualizing data where there are underlying constraints that must be preserved. For example, we may know that the data are inherently positive. We show how the Modified Quadratic Shepard method, which interpolates scattered data of any dimensionality, can be constrained to preserve positivity. We do this by forcing the quadratic basis functions to be positive. The method can be extended to handle other types of constraints, including lower bound of 0 and upper bound of 1—as occurs with fractional data. A further extension allows general range restrictions, creating an interpolant that lies between any two specified functions as the lower and upper bounds. |
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| AbstractList | This paper discusses the problem of visualizing data where there are underlying constraints that must be preserved. For example, we may know that the data are inherently positive. We show how the Modified Quadratic Shepard method, which interpolates scattered data of any dimensionality, can be constrained to preserve positivity. We do this by forcing the quadratic basis functions to be positive. The method can be extended to handle other types of constraints, including lower bound of 0 and upper bound of 1 - as occurs with fractional data. A further extension allows general range restrictions, creating an interpolant that lies between any two specified functions as the lower and upper bounds. [PUBLICATION ABSTRACT] This paper discusses the problem of visualizing data where there are underlying constraints that must be preserved. For example, we may know that the data are inherently positive. We show how the Modified Quadratic Shepard method, which interpolates scattered data of any dimensionality, can be constrained to preserve positivity. We do this by forcing the quadratic basis functions to be positive. The method can be extended to handle other types of constraints, including lower bound of 0 and upper bound of 1-as occurs with fractional data. A further extension allows general range restrictions, creating an interpolant that lies between any two specified functions as the lower and upper bounds. |
| Author | Unsworth, K. Asim, M. R. Brodlie, K. W. |
| Author_xml | – sequence: 1 givenname: K. W. surname: Brodlie fullname: Brodlie, K. W. organization: School of Computing, University of Leeds, Leeds LS2 9JT, UK – sequence: 2 givenname: M. R. surname: Asim fullname: Asim, M. R. organization: COMSATS Institute of Information Technology, Off Raiwind Road, Lahore, Pakistan – sequence: 3 givenname: K. surname: Unsworth fullname: Unsworth, K. organization: Applied Computing, Mathematics and Statistics Group, Division of Applied Management and Computing, P.O.Box 84, Lincoln University, Canterbury, New Zealand |
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| Cites_doi | 10.1145/800186.810616 10.1109/38.180119 10.1016/0021-9045(79)90020-0 10.1016/S0377-0427(00)00580-X 10.1016/S0097-8493(03)00084-0 10.1145/45054.45055 10.1007/BF02252986 10.1002/nme.1620151110 10.1016/B978-0-12-587260-7.50008-X 10.2307/2006273 10.1016/0021-9045(73)90020-8 10.1145/45054.356231 |
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| References_xml | – reference: R. Franke and G. Nielson. Smooth interpolation of large sets of scattered data. International Journal of Numerical Methods in Engineering, 15(1980), 1691-1704. – reference: M. Asim and K. Brodlie. Curve drawing subject to positivity and more general constraints. Computers and Graphics, 27(2003), 469-485. – reference: F. I. Utreras. Positive thin plate splines. J. Approximation Theory and its Applications, 1, 3 (1985), 77-108. – reference: R. J. Renka. Algorithm 660: QSHEP2D: Quadratic Shepard method for bivariate interpolation of scattered data. ACM Transactions on Mathematical Software 14, 2 (1988), 149-150. – reference: R. Barnhill G. Birkhoff and W. Gordon. Smooth interpolation in triangles. Journal of Approximation Theory, 8(1973), 114-128. – reference: B. Mulansky and J. Schmidt. Powell-Sabin splines in range restricted interpolation of scattered data. Computing, 53(1994), 137-154. – reference: Y. Xiao and C. Woodbury. Constraining global interpolation methods for sparse data volume visualization. International Journal of Computers and Applications, 21, 2 (1999), 59-64. – reference: R. J. Renka. Multivariate interpolation of large sets of scattered data. ACM Transactions on Mathematical Software, 14, 2 (1988), 139-148. – reference: E. Chan and B. Ong. Range restricted scattered data interpolation using convex combination of cubic Bezier triangles. Journal of Computational and Applied Mathematics, 136(2001), 135-147. – reference: R. Fletcher. Practical Methods of Optimization. Wiley, Chichester and New York , 1987. – reference: W. J. Gordon and J. A. Wixom. Shepard's method of metric interpolation to bivariate and multivariate interpolation. Mathematics of Computation, 32, 141 (1978), 253-264. – reference: P. Lancaster and K. Salkauskas. Curve and Surface Fitting: An Introduction. Academic Press, London , 1986. – reference: G. Nielson. The side-vertex method for interpolation in triangles. 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Approximation Theory and its Applications – volume: 14 start-page: 149 issue: 2 year: 1988 end-page: 150 article-title: Algorithm 660: QSHEP2D: Quadratic Shepard method for bivariate interpolation of scattered data publication-title: ACM Transactions on Mathematical Software – volume: 21 start-page: 59 issue: 2 year: 1999 end-page: 64 article-title: Constraining global interpolation methods for sparse data volume visualization publication-title: International Journal of Computers and Applications – year: 2005 – volume: 27 start-page: 469 year: 2003 end-page: 485 article-title: Curve drawing subject to positivity and more general constraints publication-title: Computers and Graphics – start-page: 259 year: 1996 end-page: 274 – start-page: 60 year: 1993 end-page: 70 article-title: Scattered data modelling publication-title: IEEE Computer Graphics and its Applications – volume: 25 start-page: 318 year: 1979 end-page: 336 article-title: The side‐vertex method for interpolation in triangles publication-title: Journal of Approximation Theory – year: 1987 – start-page: 69 year: 1977 end-page: 120 – volume: 14 start-page: 139 issue: 2 year: 1988 end-page: 148 article-title: Multivariate interpolation of large sets of scattered data publication-title: ACM Transactions on Mathematical Software – volume: 15 start-page: 1691 year: 1980 end-page: 1704 article-title: Smooth interpolation of large sets of scattered data publication-title: International Journal of Numerical Methods in Engineering – year: 2000 – start-page: 189 year: 2000 end-page: 230 – volume: 32 start-page: 253 issue: 141 year: 1978 end-page: 264 article-title: Shepard's method of metric interpolation to bivariate and multivariate interpolation publication-title: Mathematics of Computation – volume: 8 start-page: 114 year: 1973 end-page: 128 article-title: Smooth interpolation in triangles publication-title: Journal of Approximation Theory – volume: 53 start-page: 137 year: 1994 end-page: 154 article-title: Powell‐Sabin splines in range restricted interpolation of scattered data publication-title: Computing – volume: 136 start-page: 135 year: 2001 end-page: 147 article-title: Range restricted scattered data interpolation using convex combination of cubic Bezier triangles publication-title: Journal of Computational and Applied Mathematics – start-page: 517 year: 1968 end-page: 523 – volume-title: Visualization of data subject to positivity constraint year: 2000 ident: e_1_2_8_3_2 – ident: e_1_2_8_2_2 – ident: e_1_2_8_21_2 – ident: e_1_2_8_12_2 doi: 10.1145/800186.810616 – ident: e_1_2_8_15_2 doi: 10.1109/38.180119 – ident: e_1_2_8_7_2 doi: 10.1016/0021-9045(79)90020-0 – volume-title: Practical Methods of Optimization year: 1987 ident: e_1_2_8_20_2 – ident: e_1_2_8_9_2 doi: 10.1016/S0377-0427(00)00580-X – volume: 21 start-page: 59 issue: 2 year: 1999 ident: e_1_2_8_11_2 article-title: Constraining global interpolation methods for sparse data volume visualization publication-title: International Journal of Computers and Applications – ident: e_1_2_8_5_2 doi: 10.1016/S0097-8493(03)00084-0 – ident: e_1_2_8_17_2 doi: 10.1145/45054.45055 – volume: 1 start-page: 77 issue: 3 year: 1985 ident: e_1_2_8_10_2 article-title: Positive thin plate splines publication-title: J. Approximation Theory and its Applications – ident: e_1_2_8_8_2 doi: 10.1007/BF02252986 – ident: e_1_2_8_13_2 doi: 10.1002/nme.1620151110 – ident: e_1_2_8_14_2 doi: 10.1016/B978-0-12-587260-7.50008-X – ident: e_1_2_8_19_2 doi: 10.2307/2006273 – ident: e_1_2_8_4_2 doi: 10.1016/0021-9045(73)90020-8 – start-page: 259 volume-title: Advanced Topics in Multivariate Approximation year: 1996 ident: e_1_2_8_6_2 – ident: e_1_2_8_16_2 doi: 10.1145/45054.356231 – volume-title: Curve and Surface Fitting: An Introduction year: 1986 ident: e_1_2_8_22_2 – ident: e_1_2_8_18_2 |
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| SubjectTerms | Computer graphics constraints G.1.1 Numerical Analysis: Interpolation G.1.6 Numerical Analysis: Optimization I.3.5 Computer Graphics-Computational Geometry and Object Modelling Interpolation positivity shape preservation Shepard's method Studies visualisation Visualization |
| Title | Constrained Visualization Using the Shepard Interpolation Family |
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