1. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr A New Image Interpolation Technique using E-spline R.B.Gupta, B.G.Lee, J.J.Lee Graduate School of Design and IT Dongseo Univ. Busan, Korea lbg@dongseo.ac.kr http://kowon.dongseo.ac.kr/~lbg/

2. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr E-spline & Interpolation Kernel

3. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr B-spline basis functions B 1 ( x ) = 8 > < > : x ; 0 · x < 1 ; 2 ¡ x ; 1 · x < 2 ; 0 ; o t h erw i se B 2 ( x ) = 8 > > > < > > > : 1 2 x 2 ; 0 · x < 1 ; ¡ 3 2 + 3 x ¡ x 2 ; 1 · x < 2 ; 1 2 ( ¡ 3 + x ) 2 ; 2 · x < 3 ; 0 ; o t h erw i se B ( x: 0 ; 1 ; 2 ) B ( x: 0 ; 1 ; 2 ; 3 ) B 0 ( x ) = ( 1 ; 0 · x < 1 ; 0 ; o t h erw i se

4. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Recursive Formula for B-spline

5. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Repeated integration for B-spline B 0 ( x ) = ( 1 ; 0 · x < 1 ; 0 ; o t h erw i se p ( x ) ­ q ( x ) = Z 1 ¡ 1 p ( t ) q ( x ¡ t ) d t B n ( x ) = Z 1 0 B n ¡ 1 ( x ¡ t ) d t = B 0 ( x ) ­ B n ¡ 1 ( x )

6. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr E-spline ¯ ¡ ! ® ( t ) = ( ¯ ® 1 ­ ¯ ® 2 ­ ¢¢¢ ­ ¯ ® n )( t ) ¯ ® 1 ( t ) ¯ ( ® 1 ; ® 2 ) ( t ) ¯ ® ( t ) = ( e ® t ; 0 · t < 1 ; 0 ; o t h erw i se

7. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr E30splineft:=proc(a, t)local r; r:=(-a[1]^2*exp(a[3])*a[2]+a[1]*exp(a[2])*a[4]^2+a[1]*exp(a[3])*a[2]^2+a[1]^2*exp(a[2])*a[3]- a[1]^2*exp(a[2])*a[4]+a[1]^2*exp(a[3])*a[4]+exp(a[3])*a[4]^2*a[2]-exp(a[3])*a[2]^2*a[4]-exp(a[1])*a[2]^2*a[3]+a[2]^2*exp(a[1])*a[4]- a[1]*exp(a[3])*a[4]^2-a[1]*exp(a[2])*a[3]^2+exp(a[1])*a[4]^2*a[3]+exp(a[1])*a[2]*a[3]^2-a[2]*exp(a[1])*a[4]^2+exp(a[4])*a[1]^2*a[2]- exp(a[2])*a[4]^2*a[3]+exp(a[2])*a[4]*a[3]^2-exp(a[1])*a[4]*a[3]^2+exp(a[4])*a[2]^2*a[3]-exp(a[4])*a[1]*a[2]^2-exp(a[4])*a[1]^2*a[3]- exp(a[4])*a[2]*a[3]^2+exp(a[4])*a[1]*a[3]^2)/(-a[1]*a[4]*a[3]+a[3]^2*a[1]+a[1]*a[4]*a[2]-a[1]*a[3]*a[2]-a[3]^3+a[4]*a[3]^2- a[3]*a[4]*a[2]+a[3]^2*a[2])/(-a[1]^2+a[2]*a[1]+a[4]*a[1]-a[4]*a[2])/(-a[2]+a[4])/(2*(- a[1]^2*exp(a[3])*a[2]+a[1]*exp(a[2])*a[4]^2+a[1]*exp(a[3])*a[2]^2+a[1]^2*exp(a[2])*a[3]- a[1]^2*exp(a[2])*a[4]+a[1]^2*exp(a[3])*a[4]+exp(a[3])*a[4]^2*a[2]-exp(a[3])*a[2]^2*a[4]-exp(a[1])*a[2]^2*a[3]+a[2]^2*exp(a[1])*a[4]- a[1]*exp(a[3])*a[4]^2-a[1]*exp(a[2])*a[3]^2+exp(a[1])*a[4]^2*a[3]+exp(a[1])*a[2]*a[3]^2-a[2]*exp(a[1])*a[4]^2+exp(a[4])*a[1]^2*a[2]- exp(a[2])*a[4]^2*a[3]+exp(a[2])*a[4]*a[3]^2-exp(a[1])*a[4]*a[3]^2+exp(a[4])*a[2]^2*a[3]-exp(a[4])*a[1]*a[2]^2-exp(a[4])*a[1]^2*a[3]- exp(a[4])*a[2]*a[3]^2+exp(a[4])*a[1]*a[3]^2)/(-a[1]*a[4]*a[3]+a[3]^2*a[1]+a[1]*a[4]*a[2]-a[1]*a[3]*a[2]-a[3]^3+a[4]*a[3]^2- a[3]*a[4]*a[2]+a[3]^2*a[2])/(-a[1]^2+a[2]*a[1]+a[4]*a[1]-a[4]*a[2])/(-a[2]+a[4])/(2*(- a[1]^2*exp(a[3])*a[2]+a[1]*exp(a[2])*a[4]^2+a[1]*exp(a[3])*a[2]^2+a[1]^2*exp(a[2])*a[3]- a[1]^2*exp(a[2])*a[4]+a[1]^2*exp(a[3])*a[4]+exp(a[3])*a[4]^2*a[2]-exp(a[3])*a[2]^2*a[4]-exp(a[1])*a[2]^2*a[3]+a[2]^2*exp(a[1])*a[4]- a[1]*exp(a[3])*a[4]^2-a[1]*exp(a[2])*a[3]^2+exp(a[1])*a[4]^2*a[3]+exp(a[1])*a[2]*a[3]^2-a[2]*exp(a[1])*a[4]^2+exp(a[4])*a[1]^2*a[2]- exp(a[2])*a[4]^2*a[3]+exp(a[2])*a[4]*a[3]^2-exp(a[1])*a[4]*a[3]^2+exp(a[4])*a[2]^2*a[3]-exp(a[4])*a[1]*a[2]^2-exp(a[4])*a[1]^2*a[3]- exp(a[4])*a[2]*a[3]^2+exp(a[4])*a[1]*a[3]^2)/(-a[1]*a[4]*a[3]+a[3]^2*a[1]+a[1]*a[4]*a[2]-a[1]*a[3]*a[2]-a[3]^3+a[4]*a[3]^2- a[3]*a[4]*a[2]+a[3]^2*a[2])/(-a[1]^2+a[2]*a[1]+a[4]*a[1]-a[4]*a[2])/(-a[2]+a[4])-(-exp(a[4])*exp(a[2])*a[3]^2*a[2]- exp(a[4])*a[2]*exp(a[1])*a[4]^2-exp(a[4])*a[2]^2*exp(a[3])*a[4]+exp(a[4])*a[2]*exp(a[3])*a[4]^2+a[1]*exp(a[3])*exp(a[2])*a[2]^2- exp(a[2])*exp(a[3])*a[1]*a[3]^2- a[1]^2*exp(a[2])*exp(a[1])*a[4]+exp(a[4])*exp(a[2])*a[3]^2*a[4]+exp(a[4])*exp(a[1])*a[4]^2*a[3]+exp(a[4])*exp(a[3])*a[2]^2*a[3]- a[2]*exp(a[1])*exp(a[2])*a[4]^2-exp(a[4])*a[1]^2*exp(a[3])*a[3]-exp(a[4])*exp(a[2])*a[4]^2*a[3]-exp(a[4])*exp(a[1])*a[4]*a[3]^2- exp(a[4])*a[1]^2*exp(a[1])*a[3]-exp(a[4])*exp(a[3])*a[3]^2*a[2]-exp(a[2])*exp(a[1])*a[1]*a[3]^2+a[1]*exp(a[3])*exp(a[1])*a[2]^2- exp(a[4])*a[1]*exp(a[2])*a[2]^2-exp(a[4])*a[1]*exp(a[3])*a[4]^2- exp(a[4])*a[1]*exp(a[1])*a[2]^2+exp(a[1])*exp(a[3])*a[3]*a[4]^2+exp(a[3])*exp(a[1])*a[1]^2*a[4]- a[1]^2*exp(a[3])*exp(a[2])*a[2]+a[1]^2*exp(a[2])*exp(a[1])*a[3]+a[1]^2*exp(a[2])*exp(a[3])*a[3]-a[1]^2*exp(a[3])*exp(a[1])*a[2]- exp(a[1])*exp(a[2])*a[2]^2*a[3]+exp(a[4])*a[2]^2*exp(a[1])*a[4]+exp(a[4])*exp(a[1])*a[1]*a[3]^2+exp(a[1])*exp(a[3])*a[3]^2*a[2]+exp(a[2 ])*exp(a[3])*a[4]*a[3]^2-exp(a[1])*exp(a[3])*a[2]^2*a[3]-exp(a[1])*exp(a[3])*a[4]*a[3]^2+a[2]^2*exp(a[1])*exp(a[2])*a[4]- exp(a[2])*exp(a[3])*a[4]^2*a[3]- exp(a[4])*exp(a[2])*a[1]^2*a[4]+exp(a[4])*exp(a[2])*a[4]^2*a[1]+exp(a[4])*exp(a[3])*a[1]*a[3]^2+exp(a[1])*exp(a[2])*a[3]^2*a[2]- a[2]^2*exp(a[3])*exp(a[2])*a[4]+exp(a[4])*a[1]^2*exp(a[2])*a[2]+a[1]*exp(a[2])*exp(a[1])*a[4]^2+exp(a[4])*a[1]^2*exp(a[1])*a[2]- exp(a[3])*exp(a[1])*a[4]^2*a[1]+a[2]*exp(a[3])*exp(a[2])*a[4]^2+exp(a[4])*a[1]^2*exp(a[3])*a[4]+exp(a[4])*exp(a[2])*a[2]^2*a[3])/(- a[1]*a[4]*a[2]-a[1]*a[3]*a[2]+a[3]*a[4]*a[2]+a[2]*a[1]^2+a[4]*a[1]^2-a[1]^3-a[1]*a[4]*a[3]+a[3]*a[1]^2)/(a[4]*a[2]-a[4]*a[3]+a[3]*a[2]- a[2]^2)/(-a[3]+a[4]))-(-exp(a[4])*exp(a[2])*a[3]^2*a[2]-exp(a[4])*a[2]*exp(a[1])*a[4]^2- exp(a[4])*a[2]^2*exp(a[3])*a[4]+exp(a[4])*a[2]*exp(a[3])*a[4]^2+a[1]*exp(a[3])*exp(a[2])*a[2]^2-exp(a[2])*exp(a[3])*a[1]*a[3]^2- a[1]^2*exp(a[2])*exp(a[1])*a[4]+exp(a[4])*exp(a[2])*a[3]^2*a[4]+exp(a[4])*exp(a[1])*a[4]^2*a[3]+exp(a[4])*exp(a[3])*a[2]^2*a[3]- a[2]*exp(a[1])*exp(a[2])*a[4]^2-exp(a[4])*a[1]^2*exp(a[3])*a[3]-exp(a[4])*exp(a[2])*a[4]^2*a[3]-exp(a[4])*exp(a[1])*a[4]*a[3]^2- exp(a[4])*a[1]^2*exp(a[1])*a[3]-exp(a[4])*exp(a[3])*a[3]^2*a[2]-exp(a[2])*exp(a[1])*a[1]*a[3]^2+a[1]*exp(a[3])*exp(a[1])*a[2]^2- exp(a[4])*a[1]*exp(a[2])*a[2]^2-exp(a[4])*a[1]*exp(a[3])*a[4]^2- exp(a[4])*a[1]*exp(a[1])*a[2]^2+exp(a[1])*exp(a[3])*a[3]*a[4]^2+exp(a[3])*exp(a[1])*a[1]^2*a[4]- a[1]^2*exp(a[3])*exp(a[2])*a[2]+a[1]^2*exp(a[2])*exp(a[1])*a[3]+a[1]^2*exp(a[2])*exp(a[3])*a[3]-a[1]^2*exp(a[3])*exp(a[1])*a[2]- exp(a[1])*exp(a[2])*a[2]^2*a[3]+exp(a[4])*a[2]^2*exp(a[1])*a[4]+exp(a[4])*exp(a[1])*a[1]*a[3]^2+exp(a[1])*exp(a[3])*a[3]^2*a[2]+exp(a[2 ])*exp(a[3])*a[4]*a[3]^2-exp(a[1])*exp(a[3])*a[2]^2*a[3]-exp(a[1])*exp(a[3])*a[4]*a[3]^2+a[2]^2*exp(a[1])*exp(a[2])*a[4]- exp(a[2])*exp(a[3])*a[4]^2*a[3]- exp(a[4])*exp(a[2])*a[1]^2*a[4]+exp(a[4])*exp(a[2])*a[4]^2*a[1]+exp(a[4])*exp(a[3])*a[1]*a[3]^2+exp(a[1])*exp(a[2])*a[3]^2*a[2]- a[2]^2*exp(a[3])*exp(a[2])*a[4]+exp(a[4])*a[1]^2*exp(a[2])*a[2]+a[1]*exp(a[2])*exp(a[1])*a[4]^2+exp(a[4])*a[1]^2*exp(a[1])*a[2]- exp(a[3])*exp(a[1])*a[4]^2*a[1]+a[2]*exp(a[3])*exp(a[2])*a[4]^2+exp(a[4])*a[1]^2*exp(a[3])*a[4]+exp(a[4])*exp(a[2])*a[2]^2*a[3])/(- a[1]*a[4]*a[2]-a[1]*a[3]*a[2]+a[3]*a[4]*a[2]+a[2]*a[1]^2+a[4]*a[1]^2-a[1]^3-a[1]*a[4]*a[3]+a[3]*a[1]^2)/(a[4]*a[2]-a[4]*a[3]+a[3]*a[2]- a[2]^2)/(-a[3]+a[4])/(2*(-a[1]^2*exp(a[3])*a[2]+a[1]*exp(a[2])*a[4]^2+a[1]*exp(a[3])*a[2]^2+a[1]^2*exp(a[2])*a[3]- a[1]^2*exp(a[2])*a[4]+a[1]^2*exp(a[3])*a[4]+exp(a[3])*a[4]^2*a[2]-exp(a[3])*a[2]^2*a[4]-exp(a[1])*a[2]^2*a[3]+a[2]^2*exp(a[1])*a[4]- a[1]*exp(a[3])*a[4]^2-a[1]*exp(a[2])*a[3]^2+exp(a[1])*a[4]^2*a[3]+exp(a[1])*a[2]*a[3]^2-a[2]*exp(a[1])*a[4]^2+exp(a[4])*a[1]^2*a[2]- exp(a[2])*a[4]^2*a[3]+exp(a[2])*a[4]*a[3]^2-exp(a[1])*a[4]*a[3]^2+exp(a[4])*a[2]^2*a[3]-exp(a[4])*a[1]*a[2]^2-exp(a[4])*a[1]^2*a[3]- exp(a[4])*a[2]*a[3]^2+exp(a[4])*a[1]*a[3]^2)/(-a[1]*a[4]*a[3]+a[3]^2*a[1]+a[1]*a[4]*a[2]-a[1]*a[3]*a[2]-a[3]^3+a[4]*a[3]^2- a[3]*a[4]*a[2]+a[3]^2*a[2])/(-a[1]^2+a[2]*a[1]+a[4]*a[1]-a[4]*a[2])/(-a[2]+a[4])-(-exp(a[4])*exp(a[2])*a[3]^2*a[2]- exp(a[4])*a[2]*exp(a[1])*a[4]^2-exp(a[4])*a[2]^2*exp(a[3])*a[4]+exp(a[4])*a[2]*exp(a[3])*a[4]^2+a[1]*exp(a[3])*exp(a[2])*a[2]^2- exp(a[2])*exp(a[3])*a[1]*a[3]^2- a[1]^2*exp(a[2])*exp(a[1])*a[4]+exp(a[4])*exp(a[2])*a[3]^2*a[4]+exp(a[4])*exp(a[1])*a[4]^2*a[3]+exp(a[4])*exp(a[3])*a[2]^2*a[3]- a[2]*exp(a[1])*exp(a[2])*a[4]^2-exp(a[4])*a[1]^2*exp(a[3])*a[3]-exp(a[4])*exp(a[2])*a[4]^2*a[3]-exp(a[4])*exp(a[1])*a[4]*a[3]^2- exp(a[4])*a[1]^2*exp(a[1])*a[3]-exp(a[4])*exp(a[3])*a[3]^2*a[2]-exp(a[2])*exp(a[1])*a[1]*a[3]^2+a[1]*exp(a[3])*exp(a[1])*a[2]^

8. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Truncated powers for B-spline B 1 ( x ) = c 1 ( x ) ¡ 2 c 1 ( x ¡ 1 ) + c 1 ( x ¡ 2 ) c n ( x ) = ( 1 n ! x n ; x ¸ 0 ; 0 ; o t h erw i se B 2 ( x ) = c 2 ( x ) ¡ 3 c 2 ( x ¡ 1 ) + 3 c 2 ( x ¡ 2 ) ¡ c 2 ( x ¡ 3 ) B n ( x ) = n + 1 X i = 0 ( ¡ 1 ) i µ n + 1 i ¶ c n ( x ¡ i )

9. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Truncated Exponential ½ ® ( t ) = 1 + ( t ) e ® t ½ ¡ ! ® ( t ) = ( ½ ® 1 ­ ½ ® 2 ­ ¢¢¢ ­ ½ ® n )( t ) w h ere ¡ ! ® = ( ® 1 ;:::; ® n ¡ 1 ; ® n ) ½ ¡ ! ® ( t ) = P n d m = 1 P n ( m ) n = 1 c m ; n t n ¡ 1 + ( n ¡ 1 ) ! e ® ( m ) ( t ) ¡ ! ® = ( ® n ( 1 ) ( 1 ) ; ® n ( 2 ) ( 2 ) ;:::; ® n ( n d ) ( n d ) ) w h ere P n d m = 1 n ( m ) = n

10. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Truncated Exponential (n=1) ¡ ! ® = ( ® ) ½ ¡ ! ® ( t ) = 1 + ( t ) e ® t ® = ¡ 2 ;:::; 2

11. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Truncated Exponential (n=2) ® = ¡ 2 ;:::; 2 ¡ ! ® = ( 0 ; ® ) ¡ ! ® = ( ® ; ® ) ¡ ! ® = ( ¡ ® ; ® ) ½ ¡ ! ® ( t ) = 1 + ( t ) e ® 2 t ¡ e ® 1 t ® 2 ¡ ® 1 = 1 + ( t )f e ® 1 t ® 2 ¡ ® 1 + e ® 2 t ® 1 ¡ ® 2 g i f ® 2 = ® 1 t h en 1 + ( t ) t e ® 1 t

12. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Truncated Exponential (n=3) ® = ¡ 2 ;:::; 2 ¡ ! ® = ( 0 ; 0 ; ® ) ¡ ! ® = ( 0 ; ¡ ® ; ® ) ¡ ! ® = ( 0 ; ® ; ® ) i f ® 3 = ® 2 = ® 1 t h en 1 + ( t ) 1 2 t 2 e ® 1 t

13. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Truncated Exponential (n=4) ¡ ! ® = ( 0 ; 0 ; ® ; ® ) ¡ ! ® = ( 0 ; 0 ; ¡ ® ; ® ) ¡ ! ® = ( ¡ ® ; ¡ ® ; ® ; ® ) ® = ¡ 2 ;:::; 2 ½ ¡ ! ® ( t ) = 1 + ( t ) P n i = 1 e ® i t Q j 6 = i ( ® j ¡ ® i )

14. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr E-spline (n) [ 1, -e ® 1 ] ½ ® ( t ) = 1 + ( t ) e ® t ¯ ® ( t ) = ½ ® ( t ) ¡ e ® ½ ® ( t ¡ 1 ) = ( e ® t ; 0 · t < 1 ; 0 ; o t h erw i se [ 1, - ( e ® 1 + e ® 2 ) ; e ® 1 e ® 2 ] [ 1, - ( e ® 1 + e ® 2 + e ® 3 ) ; e ® 1 e ® 2 + e ® 1 e ® 3 + e ® 2 e ® 3 ; ¡ e ® 1 e ® 2 e ® 3 ] n Y i = 1 ( 1 ¡ e ® i x )

15. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Linear E-spline ® 1 = ¡ 1 8 ; ® 2 = 0 ( ¯ ® 1 ­ ¯ ® 2 )( t )

16. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Linear E-spline ® 1 = 1 8 ; ® 2 = 0 ( ¯ ® 1 ­ ¯ ® 2 )( t )

17. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Linear E-spline ( ¯ ® 1 ­ ¯ ® 2 )( t ) ® 1 = ¡ 1 8 ; ® 2 = ¡ 1 4

18. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Linear E-spline ( ¯ ® 1 ­ ¯ ® 2 )( t ) ; ® 2 = ¡ 1 ::: 1

19. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Normalized Linear E-spline ¯ ( 0 ; ® ) ( t ) = ® e ® ¡ 1 ( e ® t ¡ 1 ® ; 0 · t < 1 ; e ® ¡ e ® ( t ¡ 1 ) ® ; 1 · t < 2 Partition of unity ( 0, - 2 )( 1, - 2 )

20. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Quadratic E-spline ¯ ( ® 1 ; ® 2 ; ® 3 ) ( t ) = ( ¯ ® 1 ­ ¯ ® 2 ­ ¯ ® 3 )( t ) ( 0, -® ; ® ) ¯ ( ® 1 ; ® 2 ; ® 3 ) ( ¡ 1 = 2 ) + ¯ ( ® 1 ; ® 2 ; ® 3 ) ( 1 = 2 )

21. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Cubic E-spline ( 0, 0, -® ; ® )( 0, -® ; ® ; ® ) ( -® ; ¡ ® ; ® ; ® ) ( 0, -® ; ¡ ® ; ® ) ( 0, 0, 0, ® )

22. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Quadratic E-spline

23. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Quadratic E-spline

24. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Quadratic E-spline

25. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Image Interpolation Methods

26. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr h(x) Kernel functions I d ea l h ( x ) = s i nc ( x ) = s i n ( ¼x ) ¼x h ( 0 ) = 1, h ( x ) = 0, j x j = 1 ; 2 ;::: P 1 k = ¡ 1 h ( d + k ) = 1 ; 0 · d < 1

27. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr h(x) Kernel functions

28. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr h 2 h 1 Q ua d h 3

29. Q ua d h 3 a = 0 a = 2 a = 1

30. Cubic Interpolation C u b i c h 4 ; ¡ 1 ; ¡ 3 = 4 ;:::; 1 h ( k ¡ ) = h ( k + ) ; C 0 ¡ con t i nu i t y h` ( k ¡ ) = h` ( k + ) ; C 1 ¡ con t i nu i t y P 1 k = ¡ 1 h ( d + k ) = 1 ; 0 · d < 1

31. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr C u b i c h 4 a = 0 a = - 1 / 2 a = - 1

32. C u b i c h 4

33. B-spline Approximation s ( k ) = k + 2 X m = k ¡ 2 t ( m ) ¢ h 4 ( k ¡ m ) = 1 6 f t ( k ¡ 1 ) + 4 t ( k ) + t ( k + 1 )g 1 6 2 6 6 6 6 6 6 6 4 41 141 141... 141 14 3 7 7 7 7 7 7 7 5 1. B -sp l i ne A pprox i ma t i on h 4 ( x ) = ( ( 1 = 2 )j x j 3 ¡ j x j 2 + 2 = 3 ; 0 · j x j < 1 ; ¡ ( 1 = 6 )j x j 3 + j x j 2 ¡ 2 j x j + 4 = 3 ; 1 · j x j < 2

34. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr B-spline Interpolation Kernel 1. B -sp l i ne I n t erpo l a t i on S p l i ne h ( x ) = h 4 ( x ) 1 X m = ¡ 1 p 3 ( p 3 ¡ 2 ) j m j ± ( x + m )

35. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr

36. Discrete B-spline Interpolation Kernel

37. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Discrete E-spline Interpolation Kernel ( 0, 0, -® ; ® )

38. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Discrete E-spline Interpolation Kernel ( -® ; ¡ ® ; ® ; ® )

39. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Discrete E-spline Interpolation Kernel ( -® ; ¡ ® ; ® ; ® )

40. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Frequency domain response Fourier domain magnitude plot for (α, α, -α, -α) with different values of α. ( -® ; ¡ ® ; ® ; ® )

41. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Partition of unity Sum of sampled interpolation kernels as a function of the displacement for different α.

42. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr 1-D Interpolation

43. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr PSNR calculation Step Interpolation for different alpha alpha=0.2 PSNR=83.9830 alpha=1 PSNR=88.0330 alpha=1.4 PSNR=92.5440 alpha=2 PSNR=77.4848 Ramp interpolation for different alpha alpha=0.2 PSNR=73.6695 alpha=1 PSNR=74.0227 alpha=1.4 PSNR=74.2507 alpha=2 PSNR=72.8553

44. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Original Lena image and Lena interpolated image and for (α, α, -α, -α) with values of α=0.2. size(512*512)

45. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr Lena interpolated image for (α, α, -α, -α) with values of α=1.2,1.4. size(512*512)

46. WSCG, January 31, 2007, A New Image Interpolation Technique using E-spline, lbg@dongseo.ac.kr PSNR Alpha= 0.2 Alpha= 0.5 Alpha= 1.0 Alpha= 1.2 Alpha= 1.3 Alpha= 1.4 Alpha= 1.5 Alpha= 2.0 linearB-Spline Lena31.377731.415931.544531.642931.532031.462631.330529.315430.499031.3485 Gold-hill29.864229.892029.985930.003929.992529.955529.879528.548729.601929.8458 Pepper30.904130.925730.988930.979230.947830.885230.776929.201629.854130.8861 Barbara24.643224.650824.673124.669424.658124.635524.596123.950024.293024.6364 Boat28.274428.299728.386628.402328.390628.354428.280926.988427.704628.2562