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, 'e6W$?z
我输入10阶的n、m,r,theta为38025*1向量,最后得到的z是29525*10阶的矩阵, Rl 4r 9
这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? ixJUq o
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? +n(H"I7cU
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function z = zernfun(n,m,r,theta,nflag) `w Sg/
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. {d$S~
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N da@y*TO#i
% and angular frequency M, evaluated at positions (R,THETA) on the 1A23G$D
% unit circle. N is a vector of positive integers (including 0), and (.,E6H|zI
% M is a vector with the same number of elements as N. Each element ^_<>o[qE
% k of M must be a positive integer, with possible values M(k) = -N(k) v)JQb-<
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, a@@!Eg
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% and THETA is a vector of angles. R and THETA must have the same y? [*qnPj
% length. The output Z is a matrix with one column for every (N,M) }\u~He%
% pair, and one row for every (R,THETA) pair. C!w@Naj
% gb:Cc,F,%
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ,IUMH]D
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 3w )S=4lB
% with delta(m,0) the Kronecker delta, is chosen so that the integral cFLu+4.jsG
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, hE:P'O1
% and theta=0 to theta=2*pi) is unity. For the non-normalized o*n""m
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. whNRUOK:
% ;J\{r$q
% The Zernike functions are an orthogonal basis on the unit circle. 8O{]ML
% They are used in disciplines such as astronomy, optics, and 'D(Hqdr;:
% optometry to describe functions on a circular domain. 7kn=j6I
% \Y9=dE}
% The following table lists the first 15 Zernike functions. 9[N'HpQ3
% SU#
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% n m Zernike function Normalization p)ZlQ.d#Y
% -------------------------------------------------- G%YD2<V
% 0 0 1 1 |
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% 1 1 r * cos(theta) 2 af{;4Cr
% 1 -1 r * sin(theta) 2 xSb/98;
% 2 -2 r^2 * cos(2*theta) sqrt(6) uMsKF %m
% 2 0 (2*r^2 - 1) sqrt(3) ?vRz}hiy
% 2 2 r^2 * sin(2*theta) sqrt(6) %8o(x 0
% 3 -3 r^3 * cos(3*theta) sqrt(8) NtTLvO6
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) q1dYiG.-Z
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) |ry;'[*
% 3 3 r^3 * sin(3*theta) sqrt(8) Cw{#(xX
% 4 -4 r^4 * cos(4*theta) sqrt(10) jo<