2008.11.24 dept. of visual contents, dongseo university jung hye kwon kowon.dongseo.ac.kr/~d0076022...
TRANSCRIPT
2008.11.24Dept. of Visual Contents, Dongseo University
Jung Hye Kwon
kowon.dongseo.ac.kr/~d0076022
Image Deformation using Radial Basis Function Interpolation
2 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
Contents
• Image Deformation• Deformation Techniques• Radial Basis Function
– Radial Basis Functions Theory– Image Deformation using RBF
• Experimental Result• Reference
3 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
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
4 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
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 seg
uentially.
『 T. Igarashi, T. Moscovich, and J. F. Hughes, “As-rigid-as-possible shape manipulation.”, ACM Trans. Graph 2005, 24, 3, pp 1134-1141 (2005). 』
5 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
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). 』
6 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
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). 』
7 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
Radial Basis Function
• Radial Basis Functions Theory– Radial basis function originated in the 1970 in Hardy’s cartography appli
cation
– 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). 』
8 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
Radial Basis Function
• A function is known only at a set of discrete points
and desired function values , we can define
with the constraints
wherepj (u) 2 ¦ dr ;
the spaceof polynomial of total degree r in d spatial dimensions
f : Rd1 ! Rd2
U := fu1;u2;¢¢¢;ung V := fv1;v2;¢¢¢;vng
Sf ;U (u) :=nX
i=1
®iÁ(ku ¡ uik) +mX
j =1
¯ j pj (u)
nX
i=1
®ipj (ui ) = 0 j = 1;¢¢¢;m
9 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
Radial Basis Function
• 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
Sf ;U (u)
(n+m) £ (n+m)
11 1 2 1
21 2 2 2
( )( ) ( )
( )( ) ( )
( ) ( ) ( )
m
m
m n m n m n
p up u p u
p up u p uP
p u p u p u
0 0T
A P f
P
11 1 1 2
22 1 2 2
1 2
(|| ||)(|| ||) (|| ||)
(|| ||)(|| ||) (|| ||),
(|| ||) (|| ||) (|| ||)
n
n
n n n n
where
u uu u u u
u uu u u uA
u u u u u u
1
0 0T
A P f
P
10 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
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
1 2: { , , , }nU u u u
1 1 2 2: { , , , }.n nV v u v u v u
, ( ) , 1,2, .f U i i iS u v u i n
, ( )f US u
v
, ( )f Uv u S u
1u
2u
3u
3v
2v3v
Sf ;U (u) :=nX
i=1
®iÁ(ku ¡ uik) +mX
j =1
¯ j pj (u)
11 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
Experimental Result
• Test using Gaussian function and Wendland’s function2 2/( ) r cr e 4
3,1( ) (1 ) (4 1)r rc cr 10c
(a) Original image (b) Gaussian function (c) Wendland’s function
12 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
Experimental Result
• Comparison between our algorithm and [Schaefer et al.].
(a) Original image (b) Our algorithm (c) [Schaefer et al]
13 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
Experimental Result
• An illustration of RBF interpolation and RBF with polynomials
(a) Original image (b) RBF (c) RBF with quadratic
14 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
Experimental Result
• Deformation of a fish image
RBF with quadratic
15 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
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.
16 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
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).
17 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
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).
18 2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
Thank you