Abstract

Methods of numerical integration of sampled data are compared in terms of their frequency responses and resolving power. Compared, theoretically and by numerical experiments, are trapezoidal, Simpson, Simpson-3/8 methods, method based on cubic spline data interpolation and Discrete Fourier Transform (DFT) based method. Boundary effects associated with DFT- based and spline-based methods are investigated and an improved Discrete Cosine Transform based method is suggested and shown to be superior to all other methods both in terms of approximation to the ideal continuous integrator and of the level of the boundary effects.

© 2005 Optical Society of America

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Appl. Opt. (3)

J. Opt. Soc. Am. (4)

Opt. Lett. (1)

Optical Engineering (1)

A. Moreno, J. Campos, L.P. Yaroslavsky, �??Frequency response of five integration methods to obtain the profile from its slope,�?? Optical Engineering, (2005) (to be published)
[CrossRef]

Proc. SPIE (2)

C. Elster, I. Weingärtner, �??High-accuracy reconstruction of a function f(x) when only df(x)/dx is known at discrete measurements points,�?? Proc. SPIE 4782, (2002)

S. Krey, W.D. van Amstel, J. Campos, A. Moreno, E.J. Lous, �??A fast optical scanning deflectometer for measuring the topography of large silicon wafers,�?? Proc. SPIE, 5523, (SPIE�??s 49th Annual Meeting, Denver), (2004)
[CrossRef]

Other (2)

L. Yaroslavsky, Digital Holography and Digital Image Processing (Kluwer Academic Publishers, 2004)

J. H. Mathew and K. D. Fink, Numerical Methods using MATLAB (Prentice-Hall, Englewood Cliffs, N. J., 1999)

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Figures (5)

Fig 1.
Fig 1.

Comparison of frequency responses of trapezoidal, Simpson, 3/8 Simpson, Cubic Splines and Fourier methods of integrations

Fig. 2.
Fig. 2.

Integration error of periodic sinusoidal signals as a function of the normalized frequency: (a) for all methods; (b) only for DFT-based, trapezoidal and cubic spline methods

Fig. 3.
Fig. 3.

Experimentally obtained integration error versus sample k for DFT-based (FI) method (black) CSI method (red) and DCT-based (Extended) method (blue). Normalized initial frequency: (a) ν=0.273. (b) ν=0.547 and (c) ν=0.820

Fig. 4.
Fig. 4.

Average error evaluated in the 10 first samples of the domain for the same initial normalized frequency (p=0.547) but different N: (a) N=256, (b) N=512, (c) N=1024. The black curve corresponds to Fourier integration method, the blue one to the DCT- based method and the red one to the cubic spline based method.

Fig. 5.
Fig. 5.

Theoretical Profile(a) and Integrated Profiles with Trapezoidal Rule (b), Cubic Spline method (c), and DFT- Method (d). All are shown subsampled.

Equations (63)

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a ¯ ( x ) = a ( x ) d x = i 2 π f α ( f ) exp ( i 2 π f x ) d x = H int ( f ) α ( f ) exp ( i 2 π f x ) d f .
H int ( f ) = i 2 π f
{ a ¯ k } = IDET { η r ( int ) DET { a k } } ,
η r ( int ) = { 0 , r = 0 N h i 2 π r , r = 1 , 2 , N 2 1 h 2 π , r = N 2 η N r * , r = N 2 + 1 , , N 1 ,
η r ( int ) = { 0 , r = 0 N h i 2 π r , r = 1 , 2 , , ( N 1 ) 2 , η N r * , r = N 2 + 1 , , N 1
a ¯ 1 ( T ) = 0 , a ¯ k ( T ) = a ¯ k 1 ( T ) + h 2 ( a k 1 + a k )
a ¯ 1 ( S ) = 0 , a ¯ k ( S ) = a ¯ k 2 ( S ) + h 3 ( a k 2 + 4 a k 1 + a k )
a ¯ 0 ( 3 8 S ) = 0 , a ¯ k ( 3 8 S ) = a k 3 ( 3 8 S ) + 3 h 8 ( a k 3 + 3 a k 2 + 3 a k 1 + a k )
a ¯ k ( T r ) a ¯ k 1 ( T r ) = h 2 ( a k 1 + a k )
a ¯ k ( S ) a ¯ k 2 ( S ) = h 3 ( a k 2 + 4 a k 1 + a k )
a ¯ k ( 3 8 S ) a ¯ k 3 ( 3 8 S ) = 3 h 8 ( a k 3 + 3 k k 2 + 3 a k 1 + a k )
a ¯ k + 1 ( C S ) a ¯ k ( C S ) = h [ 1 2 ( a k + a k + 1 ) h 2 24 ( m k + m k + 1 ) ]
h ( m k 1 + 4 m k + m k + 1 ) = 6 h ( a k + 1 2 a k + a k 1 )
α ¯ r ( T ) ( 1 exp ( i 2 π r N ) ) = h 2 α r [ 1 + exp ( i 2 π r N ) ] ;
α ¯ r ( S ) ( 1 exp ( i 4 π r N ) ) = h 3 α r [ 1 + 4 exp ( i 2 π r N ) + exp ( i 4 π r N ) ] ;
α ¯ r ( 3 8 S ) [ 1 exp ( i 6 π r N ) ] = 3 h 8 α r [ 1 + 3 exp ( i 2 π r N ) + 3 exp ( i 4 π r N ) + exp ( i 6 π r N ) ] ,
α ¯ r ( C S ) = h 4 cos ( π r N ) i sin ( π r N ) [ 1 + 3 cos ( 2 π r N ) + 2 ] α r
η r ( T ) = α ¯ r ( T r ) α r = { 0 , r = 0 , h cos ( π r N ) 2 i sin ( π r N ) , r = 1 , , N 1 ,
η r ( S ) = α ¯ r ( S ) α r = { 0 , r = 0 , h cos ( 2 π r N ) + 2 3 i sin ( 2 π r N ) , r = 1 , , N 1 ,
η r ( 3 S ) = α ¯ r ( 3 S ) α r = { 0 , r = 0 , h cos ( 3 π r N ) + 3 cos ( π r N ) i sin ( 3 π r N ) , r = 1 , , N 1 .
η r ( C S ) = α ¯ r ( C S ) α r = { 0 , r = 0 h 4 i cos ( π r N ) sin ( π r N ) [ 1 + 3 cos ( 2 π r N ) + 2 ] , r = 1 , , N 1
a ˜ k = { a k , k = 0 , 1 , , N 1 a 2 N 1 k , k = N , , 2 N 1
f 0 ( x ) = cos ( 2 π p N x + randomphase )
f 0 ( x ) = 2 π p N cos ( 2 π p N x + randomphase )
error = 1 N k = 0 N [ f i ( k ) f 0 ( k ) ] 2
error p int ( k ) = 1 n s = 0 n [ f i p int + s n ( k ) f 0 p int + s n ( k ) ] 2
b ( x ) = a ( ξ ) h ( x ξ ) d ξ .
b k = n = 0 N h 1 h n a k n
b ( x ) = k = 0 N b 1 b k φ ( r ) ( x k ˜ Δ x ) ,
b ( x ) = k = 0 N b 1 ( n = 0 N h 1 h n a k n ) φ ( r ) ( x k ˜ Δ x ) ,
a m = a ( ξ ) φ ( r ) [ ξ m Δ x ] d ξ
b ( x ) = k = 0 N b 1 [ n = 0 N h 1 h n a ( ξ ) φ ( d ) [ ξ ( k ˜ n ) Δ x ] d ξ ] φ ( r ) ( x k ˜ Δ x ) =
a ( ξ ) [ k = 0 N b q n = 0 N h 1 h n φ ( r ) ( x k ˜ Δ x ) φ ( d ) [ ξ ( k ˜ n ) Δ x ] ] d ξ .
h eq ( x , ξ ) = k = 0 N b 1 n = 0 N h 1 h n φ ( r ) ( x k ˜ Δ x ) φ ( d ) [ ξ ( k ˜ n ) Δ x ]
H e q ( f , p ) = h e q ( x , ξ ) exp [ i 2 π ( f x p ξ ) ] d x d ξ =
[ n = 0 N h 1 h n exp ( i 2 π p n Δ x ) ] [ k = 0 N b 1 exp [ i 2 π ( f p ) k ˜ Δ x ] ] ×
φ ( r ) ( x ) exp ( i 2 π f x ) d x φ ( d ) ( ξ ) exp ( i 2 π p ξ ) d ξ .
H eq ( f , p ) = DFR ( p ) · S V ( f , p ) · Φ ( r ) ( f ) · Φ ( r ) ( p ) .
DFR ( p ) = n = 0 N h 1 h n exp ( i 2 π p n ˜ Δ x ) = exp ( i 2 π p u ( d ) Δ x ) n = 0 N h 1 h n exp ( i 2 π p n Δ x )
SV ( f , p ) = k = 0 N b 1 exp [ i 2 π ( f p ) k ˜ Δ x ] = sin [ π ( f p ) N b Δ x ] sin [ π ( f p ) Δ x ] exp [ i π ( f p ) ( N b + u ( r ) 1 ) Δ x ]
lim N b N b sin [ π ( f p ) N b Δ x ] N b sin [ π ( f p ) Δ x ] = δ ( f p )
Φ ( r ) ( f ) = φ ( r ) ( x ) exp ( i 2 π f x ) d x
Φ ( d ) ( p ) = φ ( d ) ( x ) exp ( i 2 π p x ) d x
η r = 1 N h n = 0 N h 1 h n exp ( i 2 π n + u N h r )
h n = r = 0 N h 1 η n exp ( i 2 π n + u N h r ) = 1 N h r = 0 N h 1 [ η r exp ( i 2 π u r N h ) ] exp [ i 2 π n r N h ]
DFR ( f ) n = 0 N h 1 r = 0 N h 1 η r exp { i 2 π [ ( f Δ x r N h ) n + ( f Δ x u ( d ) u r N h ) ] } =
r = 0 N h 1 η n exp [ i 2 π ( f Δ x u ( d ) u r N h ) ] n = 0 N h 1 exp [ i 2 π ( f Δ x r N h ) n ] =
r = 0 N h 1 η r sin [ π ( f N h Δ x r ) ] sin [ π ( f N h Δ x r ) N h ] exp [ i π ( N h 1 2 u ( d ) ) f Δ x ] exp [ i 2 π ( u + N h 1 2 ) r N h ]
DFR ( f ) r = 0 L N 1 η r sin cd [ N h ; π ( f N h Δ x r ) ] ,
sin cd ( N ; x ) = sin x N sin ( x N ) .
a ˜ k = { a k , k = 0,1 , , N 1 a 2 N 1 k , k = N , , 2 N 1 .
α ˜ r = 1 2 N k = 0 2 N 1 a ˜ k exp ( i 2 π k r 2 N ) =
{ 2 2 N k = 0 N 1 a k cos [ π ( k + 1 2 ) r N ] } exp ( i π r 2 N ) = α r ( DCT ) exp ( i π r 2 N ) ,
α r ( DCT ) = DCT { a k } = 2 2 N k = 0 N 1 a k cos ( π k + 1 2 N r )
α N ( DCT ) = 0 ; α u ( DCT ) = α 2 N u ( DCT ) .
b k = 1 2 N r = 0 2 N 1 α r ( DCT ) exp ( i π r 2 N ) η r exp ( i 2 π k r 2 N ) ,
{ η r = η 2 N r * } ,
b k = 1 2 N { α 0 ( D C T ) η 0 + r = 1 N 1 α r ( DCT ) η r exp ( i 2 π k + 1 2 2 N r ) +
+ r = 1 N 1 α 2 N r ( DCT ) η 2 N r exp [ i 2 π k + 1 2 2 N ( 2 N r ) ] =
1 2 N { α 0 ( DCT ) η 0 + r = 1 N 1 α r ( DCT ) η r [ exp ( i 2 π k + 1 2 2 N r ) + η r * exp ( i 2 π k + 1 2 2 N r ) ]
η r exp ( i 2 π k + 1 2 2 N r ) + η r * exp ( i 2 π k + 1 2 2 N r ) =
2 Re [ η r exp ( i 2 π k + 1 2 2 N r ) ] = η r re cos ( π k + 1 2 N r ) η r im sin ( π k + 1 2 N r )
b k = 1 2 N { α 0 ( DCT ) η 0 + r = 1 N 1 α r ( DCT ) η r re cos ( π k + 1 2 N r ) r = 1 N 1 α r ( DCT ) η r im sin ( π k + 1 2 N r ) .

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