Analysis of error propagation in profile measurement by using stitching
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1 Ay o error propgto proe eureet y ug ttchg Ttuy KUME, Kzuhro ENAMI, Yuo HIGASHI, Kej UENO - Oho, Tuu, Ir, 35-8, JAPAN Atrct Sttchg techque whch ee oger eureet rge o proe ro eer eure proe hg prty oerppe eureet rge. Here, ccurcy o the proe ote y ttchg w yze ette coerg tht error ech eureet propgte to error o the ttche proe. A reut, error the ttche proe w expree ucto o eer eureet preter. t c optze the preter.. Itroucto Sttchg prog or hghy prece prooetry wth og eureet egth uch u-eter get o cceertor or ew te oeter o egth Iterto er Coer project [], [] t o ee oeter orer o prooetry X-ry rror hg eter orer o egth. Here, ttchg prooetry w oee ug eer eureet preter ccurcy o the proe ote through ttchg w yze coerg tht error cue ech prt eureet propgte to the ttche proe through ttchg.. Error y ttchg. Moeg o the ttchg Fgure how oe o -te o ttchg prooetry etgte th pper, where (x expre tot proe to e eure t egth, (x expre prty eure proe ro the tot proe ther egth ce ut eureet egth, * expre oerp o the eureet rge, where ee oerppg rto gt the ut eureet egth. Fgure how coecto chee o the two eure proe octe ext to ech other. Here, - (x (x rw y o e expre two eure proe otte e rw oth e o the expre tot proe to e eure. The eure proe re coecte ther et qure pproxto e, y - x - y x tch t oerp. Aug tht there o chge the eure proe except ther ope or oet urg eureet, the eure proe c rtuy e extee how Fg. 3 the retohp etwee two proe c e expree Eq. or etre eureet poto (x to. I Eq., t or ther t (ope th (oet orer coecet o the et qure pproxto e, expre ther erece etwee the two proe.
2 Fg.. Moe o -te o ttchg prooetry. Fg.. Proe coecto ug et qure pproxto e. Fg.3. Vrtu expo o the eure proe to the tot eureet rge (x to.
3 ( x ( x ( x ( Sce retohp expree y Eq. c e ppe or ( to, Eq. ere ( x ( x ( x ( x ( x ( x M ( x ( x ( x M ( x ( x ( x ( x ( x ( x. ( By g oth e o Eq., proe through -te o ttchg (x c e expree ( x ( x ( x, (3 where (x t or proe wthout ttchg. The eco ter o rght h e o Eq. 3 t or uto o the erece etwee two et qure pproxto e o eure proe t o oerp. Thu, proe ote through ttchg c e expree y proe wthout ttchg erece etwee the pproxto e t ech oerp.. Ay o error propgto towr the pproxto e Fg. 4 how eure proe t pproxto e t the oerp. Here, proe (x eure t o eureet pot p ( to wth pg ter et qure pproxto e y x ere ro the eure ue. There retohp og the 4 preter,,. (4 O the other h, ug tht error cue y oy or y-recto error t ech eureet pot,.e. there o error or x-recto, error o coecet or the et qure pproxto e ote ro eure ue t the o eureet pot p, p j,,p -, -p re expree
4 D, (5 D x, (6 repectey. Here, x e x-coorte o ech eureet pot D expree ( ( x x D. (7 Fg.4. Meure proe t pproxto e t the oerp. p ( to expre eureet pot. Sce x-coorte or ech eureet pot utpe o pg ter, Σx Σx re expree ( ( ( ( ( 6 x, (8 ( ( x, (9
5 repectey. Thereore, D expree D ( ( ( Thereore,, re expree, ( ( ( ( ( (. ( Here, ug retohp how Eq. 4, Eq. re trore (, (3 ~ ( ( ( ~ ( ( (, (4 repectey, ce the retohp expree Eq. re truth or, (~ re expree,..3 Ay o error propgto or the ttche proe Error propgto or ttche proe w ette y ug the ue t the e o eureet pot, x, where the ccuute error y ttchg expecte to e xu. Eq. 5 ote y uttutg or x Eq. 3. ( ( ( (5 Fro Eq. 5, error t the e o eureet pot x expree y to o, error (, error the eco ter o the rght h e, tht cue y ttchg. Sce the two error re epeet ech other, the tot error through ttchg expree y qure root o ther qure u how Eq. 6.
6 e (6 Error propgto,, or the two coecet Eq. 5, - -, - - re expree ( ~, (7 ( ~, (8 repectey. Here, -, - re repectey epeet, they re ere ro eeret eureet ue. O the other h, ce coecet or the et qure pproxto e re epeet o, error propgto or ther erece,, re o epeet o. u, error propgto or, whch expree erece etwee the two pproxto e t the oerppg re expree. (9 u ( u Here, re epeet, ce they re ote ro the etc eureet. O the other h, u epeet o. A reut,, error propgto through ttchg expree u (. ( ce ech u epeet. Eq. trore to 6 ( (, ( ug retohp expree Eq Here, uer o ttchg expree
7 ( ( Eq. trore to ( ( ( ( 6. (3 Thereore, the tot error propgto through ttchg e expree ( ( ( ( e 6 4 (4 ug 4 o eureet preter,,, error the ut eureet. 3. Etto or eect o the eureet preter Meureet preter cocerg wth eureet egth Eq. 4,,, re e to eoe y coecet u u, (5, (6 repectey. u e pg coecet t or uer o pg pot wth the ut eureet egth. e eureet egth expo coecet t or expo rto or eureet egth y ttchg. Eq. 4 trore ug u ( ( ( ( e e K u u u u 6 4,(7 where K e expree gcto o error cue y ttchg e error propgto
8 coecet. u,,, Meureet egth expo coecet: Error propgto coecet: Ke Fg. 5. Error propgto coecet K e ucto o eureet egth expo coecet ug pg coecet u preter ce.5 Fg. 5 how K e ucto o ce.5, or u,,,, preter, ug the reto how Eq. 7. Here or expe, error the ttche proe hg tot eureet egth o -te o the ut eureet ce ug u.5 expree to e pproxtey -te o the error the ut eureet. Fgure 6 how K e ucto o ce, or u,,,, preter, ug the reto how Eq. 7. Here or expe, error the ttche proe hg tot eureet egth o -te o the ut eureet ug u expree to e u ce.7 to.8 the error ecoe pproxtey -te o the error the ut eureet.
9 u,,, Oerppg rto: Error propgto coecet: Ke Fg. 6. Error propgto coecet K e ucto o oerppg rto ug pg coecet u preter ce. 4. Cocuo Here, orer to optze eureet preter ce ug ttchg or ppyg u-eter get o cceertor hg ew te oeter o egth Iterto er Coer project. Error ttche proe w yze ug tht error cue y ech ut eureet propgte oeyg error propgto rue. A reut error propgte towr ttche proe w expree y 4 eureet preter (,,,. The reto w geerze y 3 eoe preter (u,,, howg tht eureet coto c e optze y eectg pproprte coto o the preter. It how tht th tuy c o e ppe or prooetry or hghy prece X-ry rror ( ccurcy or oger th egth ot oy or get o er coer. Reerece [] [] 6 prg JSPE u eetg N5 ( Jpee.
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