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
– Image Deformation using RBF
• Experimental Result
• Reference
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
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., [email protected]
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., [email protected]
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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., [email protected]
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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., [email protected]
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., [email protected]
7
Radial Basis Function
• A function
f : Rd1 ! Rd2
U := f u1 ; u2 ; ¢¢¢; un g
is known only at a set of discrete points
V := f v1 ; v2 ; ¢¢¢; vn g
and desired function values
,
we can define
Xn
Sf ;U (u) :=
Xm
®i Á(ku ¡ u i k) +
i= 1
¯j pj (u)
j=1
with the constraints
Xn
®i pj (ui ) = 0
j = 1; ¢¢¢; m
i= 1
wher e pj (u) 2 ¦ d ;
r
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., [email protected]
8
Radial Basis Function
Sf ;U (u)
• Calculate
the
of
that are acquired to satisfy
(n + m)
£ coefficients
(n + m)
the
system of linear equations. We may be
written in matrix form as
 A
 PT

P     f 
 


0     0 
• Unique solution is obtained in case of the inverse of matrix
where
   A
     PT
  
  (|| u1  u1 ||)  (|| u1  u2 ||)

 (|| u2  u1 ||)  (|| u2  u2 ||)
A


  (|| un  u1 ||)  (|| un  u2 ||)
1
P  f 
0   0 
 (|| u1  un ||) 
 (|| u2  un ||) 


 (|| un  un ||) 
,
 p1 (u1 )

p (u )
P 1 2


 pm (un )
p2 (u1 )
p2 (u2 )
pm (un )
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
pm (u1 ) 
pm (u2 ) 


pm (un ) 
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Image deformation using RBF
• If we have given two sets data U : {u1, u2 , , un} and
V : {v  u , v  u , , v  u }. we solve for the radial basis function
interpolation S f ,U (u) , satisfying
1
1
2
2
n
n
S f ,U (ui )  vi  ui , i  1,2,
u1
n.
v3
Finally, we obtain a deformed position v
v  u  S f ,U (u )
where
Xn
Sf ;U (u) :=
v3
u3
Xm
®i Á(ku ¡ u i k) +
i= 1
v2
u2
¯j pj (u)
j=1
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
10
Experimental Result
• Test using Gaussian function and Wendland’s function
 (r )  e r
(a) Original image
2
/ c2
3,1 (r )  (1  cr ) 4  (4 cr  1)
(b) Gaussian function
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
c  10
(c) Wendland’s function
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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., [email protected]
(c) [Schaefer et al]
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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., [email protected]
13
Experimental Result
• Deformation of a fish image
RBF with quadratic
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
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., [email protected]
15
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., [email protected]
16
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.653660 (2006).
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
17
Thank you
2008 – Image Deformation using Radial Basis Function Interpolation,
Jung Hye Kwon, Dongseo Univ., [email protected]
18