1. using Radial Basis Function Interpolation
Image Deformation
using Radial Basis Function Interpolation
Good afternoon to all of you. I’m Jung Hye Kwon from Dongseo Univ. in pusan, korea.
Here I’m going to present my paper on Image deformation using radial basis functions interpolation
I hope that all of you will pay attention to me and cooperate me during my presentation.
Dept. of Visual Contents, Dongseo University
Jung Hye Kwon
kowon.dongseo.ac.kr/~d

2. Contents Image Deformation Deformation Techniques
Radial Basis Function
Radial Basis Functions Theory
Image Deformation using RBF
Experimental Result
Reference
My presentation will go on through there main steps
First of al I’m going to introduce image deformation
Next I come to the deformation techniques and then radial basis function theory and image deformation using 귤
Finally, I’ll explain our experimental result
I’ll describe each step in detail with help of presentation slides
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ.,

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
Image deformation as one field of computer graphics, is a method which is changing image to be wanted by user.
Image deformation has a number of uses from 2D,3D computer animation, morphing, medical imaging
To perform these deformations the user selects a set of handles to control the deformation.
These handles may take the form of points, lines, or even polygon grids.
We deformed image using the points
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ.,

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.
The deformation techniques divide into two parts.
One is to deform the shape using mesh base method, the other is to deform the shape using approximation method.
Igarshi et al presented an interactive system that allows the user to deform a 2D shape by manipulating a few points.
The shape is represented by a triangle mesh and the user moves several vertices of the mesh as constrained handles.
The system then computes the positions of the remaining free vertices by minimizing the distortion of each triangle.
To make the problem linear, they present a two-step closed-form algorithm. The first step to compute the rotation and the second step to compute the scale.
This divides the problem into two least-squares minimization problems that can be solved sequentially.
『 T. Igarashi, T. Moscovich, and J. F. Hughes,
“As-rigid-as-possible shape manipulation.”, ACM Trans. Graph 2005, 24, 3, pp (2005).』
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ.,

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.
Weng et.al. presents a novel 2D shape deformation algorithm based on nonlinear least squares optimization.
The algorithm have to preserve two local shape properties: Laplacian coordinates of the boundary curve and local area of the shape interior, which are together represented in a non-quadratic energy function.
The result is an interactive shape deformation system that can achieve physically plausible results.
『 Y. Weng, W. Xu, Y. Wu, K. Zhou, B. Guo, “2D shape deformation using nonlinear least squares Optimization .”,
The visual computer , pp (2006).』
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ.,

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
In second parts, Schaefer et al. [Sch06b] provided an image deformation based on Moving Least squares using various classes of linear functions including affine, similarity and rigid transformations.
These deformations are realistic and give the user the impression of manipulation real-world objects.
They also allow the user to specify the deformations using either sets of points of line segments.
Our paper builds on a recent paper by Schaefer et al.
Therefore our paper provides a simple image deformation technology using the Radial Basis Function interpolation in approximation theory
『S. Schaefer, T. McPhail, J. Warren, “Image deformation using moving least squares.”,
Proceedings of ACM SIGGRAPH , pp (2006).』
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ.,

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
Radial Basis Functions originated in the 1970’s in Hardy’s [Har71a] cartography application and since that time has been successfully used in a various applications.
Until now, to approximate multidimensional scattered data the radial basis function method has been developed.
The strong point of this radial basis function is
Easy to use, calculation quickly and various basis function
『 R. L. Hardy, “Multiquadric equations of topography and other irregular surfaces”, J. Geophys. Res., pp (1971). 』
『 R. Franke, “Scattered data interpolation: tests of some methods.”, Math. Comp., pp (1982).』
『 C. A. Micchelli, “ Interpolation of scattered data : distance matrices and conditionally positive definite functions”,
Constr. Approx., pp (1986).』
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ.,

8. Radial Basis Function
A function is known only at a set of discrete points
and desired function values , we can define
with the constraints f : R d 1 ! 2 U : = f u 1 ; 2 n g V : = f v 1 ; 2 n g S f ; U ( u ) : = n X i 1 Á k + m j p n X i = 1 p j ( u ) ; m
There are discrete points U={u1,u2,un} and desired function values V={v1,v2,vn}
We can define Radial basis function interpolation form following equations
In order to solve the polynomial term, we’ll give constraints w h e r p j ( u ) 2 d ; t s a c o f l y n m i g
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ.,

9. 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 S f ; U ( u ) ( n + m )
Calculate the coefficients of interpolation that are acquired to satisfy the (n+m)*(n+m)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.,

10. 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
We construct a deformation function interpolation relates the u_i points in the set of control points to their v_i-u_i to the counter parts in the deformed position where v_i-u_i is difference vector
If we have given two sets data U and V. we solve for the radial basis function interpolation this equation satisfying
Finally, we obtain a deformed position v with this equation S f ; U ( u ) : = n X i 1 Á k + m j p
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ.,

11. Experimental Result
Test using Gaussian function and Wendland’s function
We have tested some of the images using Gaussian and Wendland’s [Wen95a] functions as basis functions, respectively, to evaluate the proposed algorithm.
Both characteristic feature of the function monotonically decreases with distance from the centre. Wendland’s functions are known as basis functions with compact-support basis function.
To help illustrate the behavior of a number of the basis functions consider the cat image shown in this picture .
To expend only the cat’s ears, that is, to deform image locally, the Gaussian function have to use 5 control-points while Wendland’s one does only 2-control points. That’s due to compactly supported property.
(a) Original image
(b) Gaussian function
(c) Wendland’s function
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ.,

12. Experimental Result
Comparison between our algorithm and [Schaefer et al.].
This picture shows the comparison results of our algorithm and Schaefer et al of algorithm (c). Schaefer algorithm have to set many control-points on the image in order to avoid image translation.
As a result, deformation brings the lack of smoothness while our algorithm can get smoother result than Schaefer algorithm only with fewer control points.
(a) Original image
(b) Our algorithm
(c) [Schaefer et al]
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ.,

13. Experimental Result
An illustration of RBF interpolation and RBF with polynomials
Figure 9 shows the deformation of a hexagonal image to make circle from the original image.
(a) Original image
(b) RBF
(c) RBF with quadratic
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ.,

14. Experimental Result Deformation of a fish image RBF with quadratic
The deformation of fish image is shown in Figure 10 and its continuous frames made by moving a tail can take effect as if the fish swim.
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ.,

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.
In this paper, we implemented the deformation based on RBF.
As you saw in experimental results, the various results of deformation were obtained by a term of controlling polynomial of degree. The 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, we can select it according to global or partial changes. In this paper, we used two basis functions. Further research will be extended to uses of other basis functions and line or curve segments instead of points as control attribute.
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ.,

16. Reference
[Fra82a] R. Franke, “Scattered data interpolation: tests of some methods.”, Math. Comp., pp (1982).
[Har71a] R. L. Hardy, “Multiquadric equations of topography and other irregular surfaces”, J. Geophys. Res., pp (1971).
[Iga05a] T. Igarashi, T. Moscovich, and J. F. Hughes, “As-rigid-as-possible shape manipulation.”, ACM Trans. Graph 2005, 24, 3, pp (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 (2006).
[Mic86a] C. A. Micchelli, “ Interpolation of scattered data : distance matrices and conditionally positive definite functions”, Constr. Approx., pp (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 ( 2000).
Here are the research papers
I’ll show the moving picture about our experiment result
~~~
It’s butterfly image using RBF with quadratic term
it seems like 3d image
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ.,

17. Reference
[Sch95a] R. Schaback, “Error estimates and condition numbers for radial basis function interpolation”, Advances in Computational Mathematics, pp (1995).
[Sch06b] S. Schaefer, T. McPhail, J. Warren, “Image deformation using moving least squares.”, Proceedings of ACM SIGGRAPH , pp (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 (1995).
[Wen06a] Y. Weng, W. Xu, Y. Wu, K. Zhou, B. Guo, “2D shape deformation using nonlinear least squares Optimization .”, The visual computer , pp (2006).
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ.,

18. Thank you
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ.,