A New Image Interpolation Technique using Espline R

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

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

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

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

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

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

• • E 30 splineft: =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])-(-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]^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]^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]^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]^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]^ 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) B 2 (x) = c 2 (x) ¡ 3 c 2 (x ¡ 1) + 3 c 2 (x ¡ 2) ¡ c 2 (x ¡ 3) µ ¶ ( n X+ 1 + n 1 1 x n ; x ¸ 0; cn (x ¡ i ) B n (x) = (¡ 1) i cn (x) = n ! i 0; other wi se i= 0 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® ¢¢¢ ½® )(t) wher e ¡® = (® 1 ; : : : ; ®n ¡ 1 ; ®n ) 1 2 n Pn Pn tn ¡ 1 ® (m ) c + d ½!¡ (t) = e ( m ) (t) m ; n ® m= 1 n= 1 ( n ¡ 1) ! P !¡ = n ( 1 ) n ( 2 ) n(n ) ® (® ; : : : ; ® d ) wher e ( 1) ( 2) (n d ) nd m= 1 n( m ) = n WSCG, January 31, 2007, A New Image Interpolation Technique using E- spline, lbg@dongseo. ac. kr

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

Truncated Exponential (n=2) ® = ¡ 2; : : : ; 2 !¡ = ® (®; ®) ½!¡® (t) !¡ = ® (0; ®) = = !¡ = ® (¡ ®; ®) e® 2 t ¡ e® 1 t 1+ (t) ® 2 ¡ ® 1 e® 1 t e® 2 t + 1+ (t)f g ® 2 ¡ ® 1 ¡ ® 2 if ® 2 = ® 1 then 1+ (t)te® 1 t 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; ¡ ®; ®) ½!¡® (t) = = 1 e® 3 t ¡ e® 2 t e® 3 t ¡ e® 1 t ¡ 1+ (t) f g ® 2 ¡ ® 1 ® 3 ¡ ® 2 ® 3 ¡ ® 1 e® 1 t e® 2 t e® 3 t + + 1+ (t)f g (® 2 ¡ ® 1 )(® 3 ¡ ® 1 ) (® 1 ¡ ® 2 )(® 3 ¡ ® 2 ) (® 1 ¡ ® 3 )(® 2 ¡ ® 3 ) if ® 3 = ® 2 = ® 1 then 1+ (t) 1 t 2 e® 1 t 2 WSCG, January 31, 2007, A New Image Interpolation Technique using E- spline, lbg@dongseo. ac. kr

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

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

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

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

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

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

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

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

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

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

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

Image Interpolation Methods For image resampling, t he int erpolat ion st ep must reconst ruct a two dimensional (2 D) cont inuous signal s(x; y) from it s discret e samples s(k; l) wit h s; x; y 2 R and k; l 2 N. X X = s(x; y) s(k; l) ¢h(x ¡ k; y ¡ l) k l Usually, symmet rical and separable int erpolat ion kernels are used t o reduce t he comput at ional complexity h(x; y) = h(x) ¢h(y) WSCG, January 31, 2007, A New Image Interpolation Technique using E- spline, lbg@dongseo. ac. kr

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

h(x) Kernel functions 1. Nearest Neighbor Int erpolat ion h 1 (x) = 1; 0 · jxj < 1=2 2. Linear Int erpolat ion h 2 (x) = 1 ¡ jxj; 0 · jxj < 1 3. Quadrat ic Approximat ion/ Int erpolat ion ( ¡ 2 ajxj 2 + 1=2(a + 1); Quadh 3 (x) = ajxj 2 ¡ (2 a + 1=2)jxj + 3=4(a + 1); 0 · jxj < 1=2; 1=2 · jxj < 3=2 WSCG, January 31, 2007, A New Image Interpolation Technique using E- spline, lbg@dongseo. ac. kr

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

a= 0 Quadh 3 a= 1 a= 2 WSCG, January 31, 2007, A New Image Interpolation Technique using E- spline, lbg@dongseo. ac. kr

Cubic Interpolation 1. Cubic Int erpolat ion Cubich 2; 4 Cubi ch 2 (x) = 2 jxj 3 ¡ 3 jxj 2 + 1; 0 · jxj < 1 ( Cubi ch 4 (x) = (a + 2)jxj 3 ¡ (a + 3)jxj 2 + 1; ajxj 3 ¡ 5 ajxj 2 + 8 ajxj ¡ 4 a; h(k ¡ ) = h(k + ); C 0 ¡ conti nui ty h`(k ¡ ) = h`(k + ); C 1 ¡ conti nui ty P 1 h(d + k) = 1; 0 · d < 1 k= ¡ 1 0 · jxj < 1; 1 · jxj < 2 Cubich 4 ; ¡ 1; ¡ 3=4; : : : ; 1 WSCG, January 31, 2007, A New Image Interpolation Technique using E- spline, lbg@dongseo. ac. kr

a= 1/ 2 Cubi ch 4 a= 0 WSCG, January 31, 2007, A New Image Interpolation Technique using E- spline, lbg@dongseo. ac. kr

Cubi ch 4 WSCG, January 31, 2007, A New Image Interpolation Technique using E- spline, lbg@dongseo. ac. kr

B-spline Approximation 1. B spline Approximat ion ( (1=2)jxj 3 ¡ jxj 2 + 2=3; h 4 (x) = ¡ (1=6)jxj 3 + jxj 2 ¡ 2 jxj + 4=3; s(k) = k. X+ 2 m = k¡ 2 = 0 · jxj < 1; 1 · jxj < 2 2 t(m) ¢h 4 (k ¡ m) 1 f t(k ¡ 1) + 4 t(k) + t(k + 1)g 6 4 1 6 1 4 1 16 6 6. . 66. 6 4 1 1 4 WSCG, January 31, 2007, A New Image Interpolation Technique using E- spline, lbg@dongseo. ac. kr 3 7 7 7 7 5

B-spline Interpolation Kernel 1. B spline Int erpolat ion Spl i neh (x) = h 4 (x) X 1 p p 3( 3 ¡ 2) j m j ±(x + m) m= ¡ 1 WSCG, January 31, 2007, A New Image Interpolation Technique using E- spline, lbg@dongseo. ac. kr

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

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

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

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

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

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

PSNR calculation • Step Interpolation for different 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=0. 2 PSNR=73. 6695 alpha=1 PSNR=74. 0227 alpha=1. 4 PSNR=74. 2507 alpha=2 PSNR=72. 8553 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) with values of α=0. 2. size(512*512) WSCG, January 31, 2007, A New Image Interpolation Technique using E- spline, lbg@dongseo. ac. kr

Lena interpolated image for ( α, α, -α) with values of α=1. 2, 1. 4. size(512*512) 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 linear B-Spline Lena 31. 3777 31. 4159 31. 5445 31. 6429 31. 5320 31. 4626 31. 3305 29. 3154 30. 4990 31. 3485 Gold-hill 29. 8642 29. 8920 29. 9859 30. 0039 29. 9925 29. 9555 29. 8795 28. 5487 29. 6019 29. 8458 Pepper 30. 9041 30. 9257 30. 9889 30. 9792 30. 9478 30. 8852 30. 7769 29. 2016 29. 8541 30. 8861 Barbara 24. 6432 24. 6508 24. 6731 24. 6694 24. 6581 24. 6355 24. 5961 23. 9500 24. 2930 24. 6364 Boat 28. 2744 28. 2997 28. 3866 28. 4023 28. 3906 28. 3544 28. 2809 26. 9884 27. 7046 28. 2562 WSCG, January 31, 2007, A New Image Interpolation Technique using E- spline, lbg@dongseo. ac. kr