Image Deformation using Radial Basis Function Interpolation 2008

Image Deformation using Radial Basis Function Interpolation 2008. 11. 24 Dept. of Visual Contents, Dongseo University Jung Hye Kwon kowon. dongseo. ac. kr/~d 0076022

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 2

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 3

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 4

Deformation Techniques • Mesh editing method – 2 D Shape deformation using nonlinear least squares optimization – Weng et al • Based on nonlinear least squares optimization • A 2 D 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, “ 2 D 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 5

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. Mc. Phail, 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 6

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 7

Radial Basis Function • A function f : Rd 1 ! Rd 2 U : = f u 1 ; u 2 ; ¢¢¢; un g is known only at a set of discrete points = and desired function values V : f v 1 ; v 2 ; ¢¢¢; vn g, we can define Sf ; U (u) : = Xn ®i Á(ku ¡ ui k) + i= 1 Xm ¯j pj (u) j=1 with the constraints Xn ®i pj (ui ) = 0 j = 1; ¢¢¢; m i= 1 wher e pj (u) 2 ¦ dr ; the space of polynomi al of total degr ee r i n d spati al di mensi ons 2008 – Image Deformation using Radial Basis Function Interpolation, Jung Hye Kwon, Dongseo Univ. , kjh@dit. dongseo. ac. kr 8

Radial Basis Function Sf ; U (u) • Calculate the coefficients of that are acquired to satisfy (n + m) £ (n + m) 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 9

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

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

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

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

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 14

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 15

Reference • [Fra 82 a] R. Franke, “Scattered data interpolation: tests of some methods. ”, Math. Comp. , pp. 181 -200 (1982). • [Har 71 a] R. L. Hardy, “Multiquadric equations of topography and other irregular surfaces”, J. Geophys. Res. , pp. 1905 -1915 (1971). • [Iga 05 a] T. Igarashi, T. Moscovich, and J. F. Hughes, “As-rigid-as-possible shape manipulation. ”, ACM Trans. Graph 2005, 24, 3, pp 1134 -1141 (2005). • [Lee 06 a] 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). • [Mic 86 a] C. A. Micchelli, “ Interpolation of scattered data : distance matrices and conditionally positive definite functions”, Constr. Approx. , pp. 11 -22 (1986). • [Noh 00 a] 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 16

Reference • [Sch 95 a] R. Schaback, “Error estimates and condition numbers for radial basis function interpolation”, Advances in Computational Mathematics, pp. 251 -264 (1995). • [Sch 06 b] S. Schaefer, T. Mc. Phail, J. Warren, “Image deformation using moving least squares. ”, Proceedings of ACM SIGGRAPH , pp. 533 -540 (2006). • [Toi 08 a] Wilna du Toit, “Radial Basis Function Interpolation. ”, the degree of Master of Science at the University of Stellenbosch (2008). • [Wen 95 a] H. Wendland, “Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. ”, Advances in Computational Mathematics, pp. 389 -396 (1995). • [Wen 06 a] Y. Weng, W. Xu, Y. Wu, K. Zhou, B. Guo, “ 2 D 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 17

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