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

2008.11.24Dept. of Visual Contents, Dongseo University

Jung Hye Kwon

kowon.dongseo.ac.kr/~d0076022

Image Deformation using Radial Basis Function Interpolation

http://kowon.dongseo.ac.kr/~d0076022

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



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



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). 』




• 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). 』




• 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). 』



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). 』




• 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




• 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



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)



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



Experimental Result

• Comparison between our algorithm and [Schaefer et al.].

(a) Original image (b) Our algorithm (c) [Schaefer et al]



Experimental Result

• An illustration of RBF interpolation and RBF with polynomials

(a) Original image (b) RBF (c) RBF with quadratic



Experimental Result

• Deformation of a fish image

RBF with quadratic



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.



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).



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).



Thank you

Documents

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