查看“︁泽尔尼克多项式”︁的源代码

{{NoteTA
|G1=Math
}}
[[Image:ZernikePolynome6.svg|360px|thumb|头15个泽尔尼克多项式]]
[[File:Zernike Polynomial J.gif|thumb|360px|20个泽尔尼克多项式 以Noll序列表示]]
'''泽尔尼克多项式'''是一个以1953年获[[诺贝尔物理学奖]]荷兰[[物理学家]][[弗里茨·泽尔尼克]]命名的正交多项式，分为奇、偶两类

奇多项式：

:<math>Z^{m}_n(\rho,\varphi) = R^m_n(\rho)\,\cos(m\,\varphi) \!</math>

偶多项式

:<math>Z^{-m}_n(\rho,\varphi) = R^m_n(\rho)\,\sin(m\,\varphi), \!</math>


其中<math> n \ge m</math> 为非负整数，

<math>\phi</math>为[[方位角]]

 <math>0\le\rho\le 1</math> 为径向距离

如果 ''n''-''m''为偶数则


:<math>R^m_n(\rho) = \sum_{k=0}^{\tfrac{n-m}{2}} \frac{(-1)^k\,(n-k)!}{k!\left (\tfrac{n+m}{2}-k \right )! \left (\tfrac{n-m}{2}-k \right)!} \;\rho^{n-2\,k}</math>

如果''n''-''m''为奇数，则

:<math>R^m_n(\rho) =0</math>

==泽尔尼克多项式的超几何函数表示==
泽尔尼克多项式也可以表示为超几何函数

:<math>\begin{align}
R_n^m(\rho) &= \binom{n}{\tfrac{n+m}{2}}\rho^n \ {}_2F_{1}\left(-\tfrac{n+m}{2},-\tfrac{n-m}{2};-n;\rho^{-2}\right) \\
&= (-1)^{\tfrac{n+m}{2}}\binom{\tfrac{n+m}{2}}{\tfrac{n-m}{2}}\rho^m \ {}_2F_{1}\left(1+n,1-\tfrac{n-m}{2};1+\tfrac{n+m}{2};\rho^2\right)
\end{align}</math>



==Noll 序列==
Noll 用一个J数字表示 [n,m]:如下表

{|class="wikitable"
!n,m
{{!!}} 0,0{{!!}}1,1{{!!}} 1,−1 {{!!}} 2,0{{!!}} 2,−2 {{!!}} 2,2{{!!}}3,−1{{!!}} 3,1 {{!!}} 3,−3 {{!!}} 3,3
|-------
! j
{{!}} 1{{!!}}2{{!!}} 3 {{!!}} 4 {{!!}} 5 {{!!}} 6 {{!!}} 7 {{!!}}8 {{!!}} 9{{!!}} 10
|-----
!n,m
{{!!}}4,0 {{!!}}4,2 {{!!}}4,−2{{!!}}4,4{{!!}}4,−4{{!!}}5,1{{!!}}5,−1{{!!}}5,3 {{!!}}5,−3{{!!}}5,5
|-----
! j
{{!!}}11 {{!!}}12 {{!!}}13 {{!!}}14{{!!}}15{{!!}}16{{!!}} 17 {{!!}} 18 {{!!}}19 {{!!}}20
|}

==泽尔尼克多项式==
由于
:<math>I_j=\int_0^{2\pi} \int_0^1 Z_j^2\,\rho\,d\rho\,d\theta =k_j* \pi.</math>

其中<math>k_j</math>因j而异，

:<math>k_1   =1</math>

:<math>k_2   =\frac{ 1 }{4  }</math>

:<math>k_3   =\frac{  1}{ 4 }</math>

:<math>k_4   =\frac{  1}{3  }</math>

:<math>k_5   =\frac{  1}{ 6 }</math>

必须先归一化

令<math>Z_j=Z_j/\sqrt(k_j)</math>

使得

:<math>I_j=\int_0^{2\pi} \int_0^1 Z_j^2\,\rho\,d\rho\,d\theta = \pi.</math>


归一化泽尔尼克多项式以Noll序列排列如下：
{| class="wikitable"
|-
! Noll index (<math>j</math>) !! Radial degree (<math>n</math>) !! Azimuthal degree (<math>m</math>) !! <math>Z_j</math> !! Classical name
|-
| 1 || 0 || 0 || <math>1</math> || [[Piston (optics)|Piston]]
|-
| 2 || 1 || 1 || <math>2 \rho \cos \theta</math> || [[Tilt (optics)|Tip]] (lateral position) (X-Tilt)
|-
| 3 || 1 || −1 || <math>2 \rho \sin \theta</math> || [[Tilt (optics)|Tilt]] (lateral position) (Y-Tilt)
|-
| 4 || 2 || 0 || <math>\sqrt{3} (2 \rho^2 - 1)</math> || [[Defocus aberration|Defocus]] (longitudinal position)
|-
| 5 || 2 || −2 || <math>\sqrt{6} \rho^2 \sin 2 \theta</math> || [[Astigmatism]]
|-
| 6 || 2 || 2 || <math>\sqrt{6} \rho^2 \cos 2 \theta</math> || Astigmatism
|-
| 7 || 3 || −1 || <math>\sqrt{8} (3 \rho^3 - 2\rho) \sin \theta</math> || [[Coma (optics)|Coma]]
|-
| 8 || 3 || 1 || <math>\sqrt{8} (3 \rho^3 - 2\rho) \cos \theta</math> || Coma
|-
| 9 || 3 || −3 || <math>\sqrt{8} \rho^3 \sin 3 \theta</math> || Trefoil
|-
| 10 || 3 || 3 || <math>\sqrt{8} \rho^3 \cos 3 \theta</math> || Trefoil
|-
| 11 || 4 || 0 || <math>\sqrt{5} (6 \rho^4 - 6 \rho^2 +1)</math> || Third-order [[Spherical aberration|spherical]]
|-
| 12 || 4 || 2 || <math>\sqrt{10} (4 \rho^4 - 3\rho^2) \cos 2 \theta</math> || —
|-
| 13 || 4 || −2 || <math>\sqrt{10} (4 \rho^4 - 3\rho^2) \sin 2 \theta</math> || —
|-
| 14 || 4 || 4 || <math>\sqrt{10} \rho^4 \cos 4 \theta</math> || —
|-
| 15 || 4 || −4 || <math>\sqrt{10} \rho^4 \sin 4 \theta</math> || —

|}

==正交性==
;径向正交性
:<math>\int_0^1 \rho \sqrt{2n+2}R_n^m(\rho)\,\sqrt{2n'+2}R_{n'}^{m}(\rho)\,d\rho = \delta_{n,n'}.</math>
;角度正交性
:<math>\int_0^{2\pi} \cos(m\varphi)\cos(m'\varphi)\,d\varphi=\epsilon_m\pi\delta_{|m|,|m'|},</math>
:<math>\int_0^{2\pi} \sin(m\varphi)\sin(m'\varphi)\,d\varphi=(-1)^{m+m'}\pi\delta_{|m|,|m'|};\quad m\neq 0,</math>
:<math>\int_0^{2\pi} \cos(m\varphi)\sin(m'\varphi)\,d\varphi=0,</math>

其中 <math>\epsilon_m</math> 称为Neumann因子，其数值为 ''2'' 如果满足 <math>m=0</math> ，数值为 ''1''，如果 <math>m\neq 0</math>. 
;径向与角度正交性
:<math>\int Z_n^m(\rho,\varphi)Z_{n'}^{m'}(\rho,\varphi) \, d^2r =\frac{\epsilon_m\pi}{2n+2}\delta_{n,n'}\delta_{m,m'},</math>

其中 <math>d^2r=\rho\,d\rho\,d\varphi</math> 为 雅可比矩阵

<math>n-m</math> 与 <math>n'-m'</math> 都是偶数.


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|url=https://www.researchgate.net/profile/Sajad_Farokhi/publication/261286249_Near_Infrared_Face_Recognition_A_Comparison_of_Moment-Based_Approaches/file/50463533c3e9066a13.pdf?ev=pub_int_doc_dl&origin=publication_list&inViewer=true
|title=Near Infrared Face Recognition: A Comparison of Moment-Based Approaches
|booktitle=The 8th International Conference on Robotic, Vision, Signal Processing & Power Applications
|volume=291
|publisher=Springer
|pages=129–135
|year=2014
|issue=1
|doi=10.1007/978-981-4585-42-2_15
}}
* {{cite journal
|first1=Sajad
|last1=Farokhi
|first2=Siti Mariyam
|last2=Shamsuddin
|first3=Jan
|last3=Flusser
|first4=U.U
|last4=Sheikh
|first5=Mohammad
|last5=Khansari
|first6=Kourosh
|last6=Jafari-Khouzani
|url=http://www.sciencedirect.com/science/article/pii/S1051200414001304
|title=Near infrared face recognition by combining Zernike moments and undecimated discrete wavelet transform
|journal=Digital Signal Processing
|volume=31
|year=2014
|issue=1
|doi=10.1016/j.dsp.2014.04.008
|access-date=2015-01-29
|archive-date=2019-06-02
|archive-url=https://web.archive.org/web/20190602182442/https://www.sciencedirect.com/science/article/pii/S1051200414001304
|dead-url=no
}}

[[Category:正交多项式]]
[[Category:物理光学]]
[[Category:荷兰发明]]