下面这个函数大家都不会陌生,计算zernike函数值的,并根据此可以还原出图像来, A_J!VXq
我输入10阶的n、m,r,theta为38025*1向量,最后得到的z是29525*10阶的矩阵, `n e9&+
这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? %IUTi6P
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那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? = o1&.v2j
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function z = zernfun(n,m,r,theta,nflag) 5@i(pVWZ
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 3J^'x
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N FJsg3D*@J
% and angular frequency M, evaluated at positions (R,THETA) on the k]A$?C0Q<%
% unit circle. N is a vector of positive integers (including 0), and U,~Z 2L
% M is a vector with the same number of elements as N. Each element emS7q|^
% k of M must be a positive integer, with possible values M(k) = -N(k) 95tHire
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, F
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% and THETA is a vector of angles. R and THETA must have the same G\r>3Ys
% length. The output Z is a matrix with one column for every (N,M) l9NET
% pair, and one row for every (R,THETA) pair. <gY.2#6C\%
% 1tCe#*|95
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike U {sT %G
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), x.U:v20`
% with delta(m,0) the Kronecker delta, is chosen so that the integral hOcVxSc.
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 0"c(n0L
% and theta=0 to theta=2*pi) is unity. For the non-normalized mH4Jl1S&
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. thQ)J |1
% j"P}Wn
% The Zernike functions are an orthogonal basis on the unit circle. p=f8A71
% They are used in disciplines such as astronomy, optics, and "nn>I}jK
% optometry to describe functions on a circular domain. 7{u1ynt
% |%Ssb;M
% The following table lists the first 15 Zernike functions. D{,
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% /2]=.bLwz
% n m Zernike function Normalization X&|y|
% -------------------------------------------------- V#d8fRm
% 0 0 1 1 { Em fw9L
% 1 1 r * cos(theta) 2 2?9gf,U
% 1 -1 r * sin(theta) 2 @-jI<g
% 2 -2 r^2 * cos(2*theta) sqrt(6) 8$6^S{M3
% 2 0 (2*r^2 - 1) sqrt(3) 1n+JHXR\
% 2 2 r^2 * sin(2*theta) sqrt(6) ,*{9g6
% 3 -3 r^3 * cos(3*theta) sqrt(8) @ u2P&|:{
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) )Hlc\Mgy
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) rY(h }z
% 3 3 r^3 * sin(3*theta) sqrt(8) &