{ } ( ) ( ) ( 1 ( ) π 2. Trapezoidal rule: a ( ) ( ) Simpson rule: 3/8 Simpson rule: Boole's rule a ( ) ( ) ( ) ( + ) 45 Weddle's Rule Hardy's rule:

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1 L. Yaoslavsy. Selected Topics in Image Pocessing Pat. Fast tansfom methods fo image esampling Lect. 5. Pecise numeical integation and eentiation Conventional numeical integation methods: T T T = S S S a =, a = a + a + 4a + a ( 3/ 8S ( 3/ 8S ( 3/ 8S a, = a = a 3 + 3( a 3 + 3a + 3a + a 8 Bl Bl Bl =, a = a 4 + 7a 4 + 3a 3 + a + 3a 7a Wdl Wdl Wdl a =, a = a a 6 + 5a 5 + a 4 + 6a 3 + a + 5 a + a Hd Hd Hd a =, a = a 6 + a 6 + 6a 5 + a a + 8a Tapezoidal ule: a (, a ( = a ( + ( a + a Simpson ule: ( ( ( ( 3 3/8 Simpson ule: Boole's ule a ( ( ( ( + 45 ( ( ( ( Weddle's Rule Hady's ule: ( ( ( ( m m + m + = 6 a 4 Discete fequency esponses of 4 numeical integation methods ( ( T = : Tapezoidal ule: ( T = i cos( π / N sin( π / N, =,..., N ; Simpson ule: ( S = i [ cos( π / N + ] 3sin( π / N, =,..., N ; 3S 3/8 Simpson ule: = i cos 3π N + 3cos π N sin 3π N, =,..., N a {( ( a a } ( CS ( CS Cubic spline: a = ( a + a ( m m ( [ ( ( ] ( ; ( CS π 3 π Cubic spline integation: = i cos + 4sin, =,.., ; cos( N N π N + N The ideal continuous integato fequency esponse: H int ( f = i πf FFT based numeical integatos: Even N Odd N, = ( int = ( int, = N = i N π, =,..., N / ; = N = ( π, = N / i N π, =,,..., N / { } DFT-based integato: { a } = IDFT ( int DFT{ a } -based integato: a ~ a N = π N = Numeical eentiation methods: a&, = a, =,,..., N = ( int i N π, =,..., N = an, = N,...,N π, = N N, = N +,...,N ( N ( α + / N α N + / sin π = ( cos π N π = N N = N h n= h n Conventional numeical eentiation methods: ( D: = [, ] DD: a n h n ; ( sin( π / N ; ( ( h n = ([,,] + [,, ] sin( π / N h = [ /, 8 /,, 8 /, / ] ; ( 8sin( π / N sin( 4π / N D: ( n Fequency esponse of the ideal continuous eentiato: ( f = iπf H FFT based eentiation method (amp-filteing: { a& } = IDFT( { } DFT( { a } Odd N: Even iπ / N, =,,..., N / iπ / N, =,,..., ( N / = = π /, = N / iπ ( N / N, = ( N + /,..., N iπ ( N / N, = N / +,..., N -based eentiation: N N π + / + π + / a = α sin π = N α N cos π N N = N N N = N SNR Integation and eentiation as an invese poblem: ( ( ( f Hopt f = Hinvese f SNR( f + Filteed bac pojection algoithm fo image econstuction fom pojections ( ( ( ( (

2 Numeical integation in optical metology Lase deflectomety is a technique of measuing pofile of sufaces. It is based on measuing deviation of the incident light caused by its eflection fom the suface. This deviation contains the slope data infomation of the pofile of the test suface. The suface pofile can then be obtained by integation of the slope data. α α Incident Beam Tested Suface Scanning diection x ( N Signal α N ( + Integated Signal -based integation algoithm

3 Integation Eo 5.7 Fouie Fouie Tapezoidal a b.6 Tapezoidal Simpson.5 Cubic Spline Cubic Spline 5 Simpson 3/ Nomalized Fequency Nomalized Fequency Integation Eo Integation eo of peiodic sinusoidal signals as a function of the nomalized fequency: (a fo all methods; (b only fo DFT-based, tapezoidal and cubic spline methods D F T - i n t e g a t o ; F e q u e n c y = S i m p s o n 3 / 8 - i n t e g a t o D F T - i n t e g a t o ; F e q u e n c y = S i m p s o n 3 / 8 - i n t e g a t o Phase invesion phenomenon fo 3/8-Simpson method

4 Resolving powe of the integatos Deivative Deivative Exact integal Exact integal Tapezoidal Simpson Tapezoidal Cubic spline /8-Simpson Cubicspline DFT-based method DFT-based method

5 Numeical eentiation in optical metology and video pocessing: Optical flow computation The pinciple of optical flow computation. Let I ( x y, t spatial ( x, y and time t coodinates and duing time inteval Δ t pixel ( y due to the object motion, to point ( x Δx y + Δy, be image intensity defined in x, moves, +,. Let also assume that the object motion causes no changes in pixel intensity, and the changes may occu solely due to andom factos such as additive signal independent white Gaussian noise that can be attibuted to image senso. Then, given image intensity measuements I ( x, y, t and I( x + Δx, y + Δy, t + Δt in two time moments t and t + Δt, statistically optimal maximum lielihood estimation of movement vecto ( Δ x, Δy is found as a solution of the equation: ( [ ( ( ] Δx, Δy, t = ag min I ξ,, t I ξ + Δx, + Δy, t + Δt dξd ( Δx, Δy ( ξ, ARM whee ARM ( x, y is the object Aea of Rigid Motion centeed at the point ( y x,. Within the accuacy of the Taylo seies expansion of the image intensity function I x, y, t : ( ( Δx, Δy, t = ag min ( Δx, Δy ( ξ, ARM ( ξ,, t I( ξ,, t I( ξ,, t I ξ Δx + ξ, ARM Δy + t ag min I ξ,,t ΔXYT dξd, ( Δx, Δy ( whee ΔXYT is a vecto of space-time shifts ( Δ x; Δy; Δt and (. Δt dξd =. is a scala (inne vecto poduct, I ξ,, t is a vecto of image intensity function space-time deivatives: I ξ,, t = I ( ξ,, t ; I( ξ,, t ; I( ξ,, t. ξ t

6 ( N Signal α N ( + Diffeentiated Signal -based eentiation algoithm

7 Compaison of the accuacy of methods of numeical eentiation [ERR,ERR,ERR_d,ERR_d]=eentiation_test(N,M,Ntest;

8 Ramp -filteing in filteed bac pojection algoithm fo image econstuction fom pojections Radon tansfom: otation and diectional summation Tomogaphic econstuction: amp-filteing pojections, bac pojecting, otation and summation Ramp-filteing of pojections { ( ({ } } { } N, l θ l s= θ ( θs { aˆ } = ROT IDFT H DFT p whee ( s { p } θ ae sampled image pojections obtained fo angle θ { } s, l is a vecto-column of ones, symbolizes matix Konece poduct and ROT è is image otation opeato though angle θ adon_invadon_demo;