Yilda matematika , psevdo-Zernike polinomlari tanilgan va tahlil qilishda keng foydalanilgan optik tizimlar. Ular, shuningdek, keng qo'llaniladi tasvirni tahlil qilish kabi shakl tavsiflovchilari .
Ta'rif
Ular ortogonal to'plami murakkab - baholangan polinomlar sifatida belgilangan
V n m ( x , y ) = R n m ( x , y ) e j m Arktan ( y x ) , {displaystyle V_ {nm} (x, y) = R_ {nm} (x, y) e ^ {jmarctan ({frac {y} {x}})},} qayerda x 2 + y 2 ≤ 1 , n ≥ 0 , | m | ≤ n {displaystyle x ^ {2} + y ^ {2} leq 1, ngeq 0, | m | leq n} birlik disk sifatida berilgan
∫ 0 2 π ∫ 0 1 r [ V n l ( r cos θ , r gunoh θ ) ] ∗ × V m k ( r cos θ , r gunoh θ ) d r d θ = π n + 1 δ m n δ k l , {displaystyle int _ {0} ^ {2pi} int _ {0} ^ {1} r [V_ {nl} (rcos heta, rsin heta)] ^ {*} imes V_ {mk} (rcos heta, rsin heta) , dr, d heta = {frac {pi} {n + 1}} delta _ {mn} delta _ {kl},} bu erda yulduz murakkab konjugatsiyani anglatadi va r 2 = x 2 + y 2 {displaystyle r ^ {2} = x ^ {2} + y ^ {2}} x = r cos θ {displaystyle x = rcos heta} y = r gunoh θ {displaystyle y = rsin heta}
Radial polinomlar R n m {displaystyle R_ {nm}} [1]
R n m ( x , y ) = ∑ s = 0 n − | m | D. n , | m | , s ( x 2 + y 2 ) ( n − s ) / 2 {displaystyle R_ {nm} (x, y) = sum _ {s = 0} ^ {n- | m |} D_ {n, | m |, s} (x ^ {2} + y ^ {2}) ^ {(ns) / 2}}
butun son koeffitsientlari bilan
D. n , m , s = ( − 1 ) s ( 2 n + 1 − s ) ! s ! ( n − m − s ) ! ( n + m − s + 1 ) ! . {displaystyle D_ {n, m, s} = (- 1) ^ {s} {frac {(2n + 1-s)!} {s! (nms)! (n + m-s + 1)!}} .} Misollar
Bunga misollar:
R 0 , 0 = 1 {displaystyle R_ {0,0} = 1}
R 1 , 0 = − 2 + 3 r {displaystyle R_ {1,0} = - 2 + 3r}
R 1 , 1 = r {displaystyle R_ {1,1} = r}
R 2 , 0 = 3 + 10 r 2 − 12 r {displaystyle R_ {2,0} = 3 + 10r ^ {2} -12r}
R 2 , 1 = 5 r 2 − 4 r {displaystyle R_ {2,1} = 5r ^ {2} -4r}
R 2 , 2 = r 2 {displaystyle R_ {2,2} = r ^ {2}}
R 3 , 0 = − 4 + 35 r 3 − 60 r 2 + 30 r {displaystyle R_ {3,0} = - 4 + 35r ^ {3} -60r ^ {2} + 30r}
R 3 , 1 = 21 r 3 − 30 r 2 + 10 r {displaystyle R_ {3,1} = 21r ^ {3} -30r ^ {2} + 10r}
R 3 , 2 = 7 r 3 − 6 r 2 {displaystyle R_ {3,2} = 7r ^ {3} -6r ^ {2}}
R 3 , 3 = r 3 {displaystyle R_ {3,3} = r ^ {3}}
R 4 , 0 = 5 + 126 r 4 − 280 r 3 + 210 r 2 − 60 r {displaystyle R_ {4,0} = 5 + 126r ^ {4} -280r ^ {3} + 210r ^ {2} -60r}
R 4 , 1 = 84 r 4 − 168 r 3 + 105 r 2 − 20 r {displaystyle R_ {4,1} = 84r ^ {4} -168r ^ {3} + 105r ^ {2} -20r}
R 4 , 2 = 36 r 4 − 56 r 3 + 21 r 2 {displaystyle R_ {4,2} = 36r ^ {4} -56r ^ {3} + 21r ^ {2}}
R 4 , 3 = 9 r 4 − 8 r 3 {displaystyle R_ {4,3} = 9r ^ {4} -8r ^ {3}}
R 4 , 4 = r 4 {displaystyle R_ {4,4} = r ^ {4}}
R 5 , 0 = − 6 + 462 r 5 − 1260 r 4 + 1260 r 3 − 560 r 2 + 105 r {displaystyle R_ {5,0} = - 6 + 462r ^ {5} -1260r ^ {4} + 1260r ^ {3} -560r ^ {2} + 105r}
R 5 , 1 = 330 r 5 − 840 r 4 + 756 r 3 − 280 r 2 + 35 r {displaystyle R_ {5,1} = 330r ^ {5} -840r ^ {4} + 756r ^ {3} -280r ^ {2} + 35r}
R 5 , 2 = 165 r 5 − 360 r 4 + 252 r 3 − 56 r 2 {displaystyle R_ {5,2} = 165r ^ {5} -360r ^ {4} + 252r ^ {3} -56r ^ {2}}
R 5 , 3 = 55 r 5 − 90 r 4 + 36 r 3 {displaystyle R_ {5,3} = 55r ^ {5} -90r ^ {4} + 36r ^ {3}}
R 5 , 4 = 11 r 5 − 10 r 4 {displaystyle R_ {5,4} = 11r ^ {5} -10r ^ {4}}
R 5 , 5 = r 5 {displaystyle R_ {5,5} = r ^ {5}}
Lahzalar
Buyurtmaning psevdo-Zernike Moments (PZM) n {displaystyle n} l {displaystyle l}
A n l = n + 1 π ∫ 0 2 π ∫ 0 1 [ V n l ( r cos θ , r gunoh θ ) ] ∗ f ( r cos θ , r gunoh θ ) r d r d θ , {displaystyle A_ {nl} = {frac {n + 1} {pi}} int _ {0} ^ {2pi} int _ {0} ^ {1} [V_ {nl} (rcos heta, rsin heta)] ^ {*} f (rcos heta, rsin heta) r, dr, d heta,} qayerda n = 0 , … ∞ {displaystyle n = 0, ldots infty} l {displaystyle l} tamsayı bo'ysunadigan qadriyatlar | l | ≤ n {displaystyle | l | leq n}
Tasvir funktsiyasini birlik diskidagi psevdo-Zernike koeffitsientlarini kengaytirish orqali tiklash mumkin.
f ( x , y ) = ∑ n = 0 ∞ ∑ l = − n + n A n l V n l ( x , y ) . {displaystyle f (x, y) = sum _ {n = 0} ^ {infty} sum _ {l = -n} ^ {+ n} A_ {nl} V_ {nl} (x, y).} Pseudo-Zernike lahzalari an'anaviylikdan kelib chiqadi Zernike lahzalari va tasvirga nisbatan kuchliroq va kam sezgir bo'lishi ko'rsatilgan shovqin Zernike lahzalaridan ko'ra.[1]
Shuningdek qarang
Adabiyotlar
^ a b Teh, C.-H.; Chin, R. (1988). "Lahzalar usullari bo'yicha tasvirni tahlil qilish to'g'risida". Naqshli tahlil va mashina intellekti bo'yicha IEEE operatsiyalari . 10 (4): 496–513. doi :10.1109/34.3913 . Belkasim, S .; Ahmadi, M .; Shridhar, M. (1996). "Zernike momentlarini tez hisoblashning samarali algoritmi". Franklin instituti jurnali . 333 (4): 577–581. doi :10.1016/0016-0032(96)00017-8 . Xaddadniya, J .; Ahmadi, M .; Faez, K. (2003). "RBF neyron tarmog'iga asoslangan inson yuzini aniqlash tizimida psevdo-zernike momenti bilan xususiyatlarni ajratib olishning samarali usuli" . Amaliy signallarni qayta ishlash bo'yicha EURASIP jurnali . 2003 (9): 890–901. Bibcode :2003EJASP2003..146H . doi :10.1155 / S1110865703305128 T.-W. Lin; Y.-F. Chou (2003). Zernike momentlarini qiyosiy o'rganish . IEEE / WIC veb-razvedka bo'yicha xalqaro konferentsiyasi materiallari. 516-519 betlar. doi :10.1109 / WI.2003.1241255 . ISBN 0-7695-1932-6 Chong, C.-V.; Raveendran, P .; Mukundan, R. (2003). "Psevdo-Zernike lahzalarining o'lchovli invariantlari" (PDF) . Pattern anal. Ariza . 6 (3): 176–184. doi :10.1007 / s10044-002-0183-5 . Chong, Chee-Way; Mukundan, R .; Raveendran, P. (2003). "Psevdo-Zernike momentlarini tez hisoblash uchun samarali algoritm" (PDF) . Int. J. Pattern Recogn. Artif. Int . 17 (6): 1011–1023. doi :10.1142 / S0218001403002769 . hdl :10092/448 . Shutler, Jeymi (1992). "Murakkab Zernike Moments" . 
![{displaystyle int _ {0} ^ {2pi} int _ {0} ^ {1} r [V_ {nl} (rcos heta, rsin heta)] ^ {*} imes V_ {mk} (rcos heta, rsin heta) , dr, d heta = {frac {pi} {n + 1}} delta _ {mn} delta _ {kl},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/214213ac74ccc5cdcb393dd0c96e3614b13da296)
![{displaystyle A_ {nl} = {frac {n + 1} {pi}} int _ {0} ^ {2pi} int _ {0} ^ {1} [V_ {nl} (rcos heta, rsin heta)] ^ {*} f (rcos heta, rsin heta) r, dr, d heta,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69ea395ffd749f711bc22287c5d0feacf6c64c7c)
