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2008.11.24 Dept. of Visual Contents, Dongseo University Jung Hye Kwon kowon.dongseo.ac.kr/~d0076022

Image Deformation using Radial Basis Function Interpolation. 2008.11.24 Dept. of Visual Contents, Dongseo University Jung Hye Kwon kowon.dongseo.ac.kr/~d0076022. Contents. Image Deformation Deformation Techniques Radial Basis Function Radial Basis Functions Theory

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2008.11.24 Dept. of Visual Contents, Dongseo University Jung Hye Kwon kowon.dongseo.ac.kr/~d0076022

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  1. Image Deformation using Radial Basis Function Interpolation 2008.11.24 Dept. of Visual Contents, Dongseo University Jung Hye Kwon kowon.dongseo.ac.kr/~d0076022

  2. Contents • Image Deformation • Deformation Techniques • Radial Basis Function • Radial Basis Functions Theory • Image Deformation using RBF • Experimental Result • Reference 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

  3. Image Deformation • As one field of computer graphics • The deformation method of changing image to be wanted by user • Used in the filed of computer animation, morphing and medical image • To perform deformation the user selects some set of handle • Points, lines, or grids 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

  4. Deformation Techniques • Mesh base method • As-Rigid-As-Possible Shape Manipulation – Igarashi et al • Constrained Delaunay triangulation • The user manipulates the shape by indication handles on the shape and then interactively moving the handles • two step close form algorithm (rotation and scaling) • The problem into two least-squares minimization problems that can be solved seguentially. 『 T. Igarashi, T. Moscovich, and J. F. Hughes, “As-rigid-as-possible shape manipulation.”, ACM Trans. Graph 2005, 24, 3, pp 1134-1141 (2005).』 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

  5. Deformation Techniques • Mesh editing method • 2D Shape deformation using nonlinear least squares optimization – Weng et al • Based on nonlinear least squares optimization • A 2D shape with the boundary represented as a simple closed polygon. • All these methods try to minimize a nonlinear energy function representing local properties of the surface. • The results is an interactive shape deformation system that can achieve plausible results more than previous linear least squares methods. 『 Y. Weng, W. Xu, Y. Wu, K. Zhou, B. Guo, “2D shape deformation using nonlinear least squares Optimization .”, The visual computer , pp.653-660 (2006).』 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

  6. Deformation Techniques • Approximation method • Image Deformation using Moving Least Squares – Schaefer et al • Based on Moving Least Squares • Using various linear function : affine, similarity, rigid • These deformations are realistic and give the user the impression of manipulation real-world objects. • The deformations using either sets of points or line segments 『S. Schaefer, T. McPhail, J. Warren, “Image deformation using moving least squares.”, Proceedings of ACM SIGGRAPH , pp. 533-540 (2006).』 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

  7. Radial Basis Function • Radial Basis Functions Theory • Radial basis function originated in the 1970 in Hardy’s cartography application • Franke’s of 1982, as well as the work done by Micchelli in 1986 into providing the non-singularity of the interpolation matrix. • The strong point of this radial basis function • easy to use. • calculating quickly • various basis function 『 R. L. Hardy, “Multiquadric equations of topography and other irregular surfaces”, J. Geophys. Res., pp. 1905-1915 (1971). 』 『 R. Franke, “Scattered data interpolation: tests of some methods.”, Math. Comp., pp.181-200 (1982).』 『 C. A. Micchelli, “ Interpolation of scattered data : distance matrices and conditionally positive definite functions”, Constr. Approx., pp. 11-22 (1986).』 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

  8. Radial Basis Function d ( ) h ¦ 2 d d w e r e p u j f R R f g 1 2 V ; r : ! f g ¢ ¢ ¢ : v v v U = n 1 2 n n ¢ ¢ ¢ m : u u u ; ; ; = 1 2 n ; ; ; X h f l l f l X d d l d X i i i i i t t t t ( ) j e s p a c e o p o y n o m a o o a e g r e e r n s p a a m e n s o n s 0 1 ( ) ( k k ) ( ) Á ¯ S ¢ ¢ ¢ ® p u m ¡ + = = i j i u : ® u u p u = f U i i j j ; ; ; i 1 = i j 1 1 = = • A function is known only at a set of discrete points and desired function values , we can define with the constraints 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

  9. Radial Basis Function ( ) ( ( ) ) S + £ + n m u n m f U ; • Calculate the coefficients of that are acquired to satisfy the system of linear equations. We may be written in matrix form as • Unique solution is obtained in case of the inverse of matrix 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

  10. Image deformation using RBF n m X X ( ) ( k k ) ( ) Á ¯ S ¡ + u : ® u u p u = f U i i j j ; i j 1 1 = = • If we have given two sets data and we solve for the radial basis function interpolation , satisfying Finally, we obtain a deformed position where 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

  11. Experimental Result • Test using Gaussian function and Wendland’s function 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

  12. Experimental Result • Comparison between our algorithm and [Schaefer et al.]. 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

  13. Experimental Result • An illustration of RBF interpolation and RBF with polynomials 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

  14. Experimental Result • Deformation of a fish image RBF with quadratic 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

  15. Conclusion & Future work • Our proposed method is faster by simple calculation and its result is better than the previous methods. • The term of controlling polynomial degree has many possibilities for various application fields. • . Especially, in field of animation. • Further research will be extended to uses of other basis functions. • line or curve segments instead of points as control attribute. 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

  16. Reference • [Fra82a] R. Franke, “Scattered data interpolation: tests of some methods.”, Math. Comp., pp.181-200 (1982). • [Har71a] R. L. Hardy, “Multiquadric equations of topography and other irregular surfaces”, J. Geophys. Res., pp. 1905-1915 (1971). • [Iga05a] T. Igarashi, T. Moscovich, and J. F. Hughes, “As-rigid-as-possible shape manipulation.”, ACM Trans. Graph 2005, 24, 3, pp 1134-1141 (2005). • [Lee06a] Byung-Gook Lee, Yeon Ju Lee, and Jungho Yoon, “Stationary binary subdivision schemes using radial basis function interpolation.”, Advances in Computational Mathematics, pp.57-72 (2006). • [Mic86a] C. A. Micchelli, “ Interpolation of scattered data : distance matrices and conditionally positive definite functions”, Constr. Approx., pp. 11-22 (1986). • [Noh00a] Jun-Yong Noh, D. Fidaleo, U. Neumann ," Animated deformations with radial basis functions", Proceedings of the ACM symposium on Virtual reality software and technology, pp.166-174 ( 2000). 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

  17. Reference • [Sch95a] R. Schaback, “Error estimates and condition numbers for radial basis function interpolation”, Advances in Computational Mathematics, pp. 251-264 (1995). • [Sch06b] S. Schaefer, T. McPhail, J. Warren, “Image deformation using moving least squares.”, Proceedings of ACM SIGGRAPH , pp. 533-540 (2006). • [Toi08a] Wilna du Toit, “Radial Basis Function Interpolation .”, the degree of Master of Science at the University of Stellenbosch (2008). • [Wen95a] H. Wendland, “Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree.”, Advances in Computational Mathematics, pp.389-396 (1995). • [Wen06a] Y. Weng, W. Xu, Y. Wu, K. Zhou, B. Guo, “2D shape deformation using nonlinear least squares Optimization .”, The visual computer , pp.653-660 (2006). 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

  18. Thank you 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ., kjh@dit.dongseo.ac.kr

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