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, `4V"s-T'
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, 0I5&a
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? 1{Jb"
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? DK6?E\<
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function z = zernfun(n,m,r,theta,nflag) &cu!Hx
%ZERNFUN Zernike functions of order N and frequency M on the unit circle.
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% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Km!nM$=k
% and angular frequency M, evaluated at positions (R,THETA) on the M4KWN'
% unit circle. N is a vector of positive integers (including 0), and /syVGmS'M
% M is a vector with the same number of elements as N. Each element ka/XK[/'
% k of M must be a positive integer, with possible values M(k) = -N(k) 'e@=^FC
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, }mZVL~|V
% and THETA is a vector of angles. R and THETA must have the same }HRK?.Vj:
% length. The output Z is a matrix with one column for every (N,M) J#Z5^)$
% pair, and one row for every (R,THETA) pair. dlD ki.
% JYm7@gx
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ]6&$|2H?Ni
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^aF8wbuZ
% with delta(m,0) the Kronecker delta, is chosen so that the integral c#lPc>0xb
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, PB9/m-\H
% and theta=0 to theta=2*pi) is unity. For the non-normalized c0ez/q1S
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. M T6/2d
% X}cZxlqc
% The Zernike functions are an orthogonal basis on the unit circle. s+Q;pRZW{
% They are used in disciplines such as astronomy, optics, and (K :]7
% optometry to describe functions on a circular domain. g5S?nHS}
% HjA_g0u
% The following table lists the first 15 Zernike functions. |0.Xl+7
% XIAeCU
% n m Zernike function Normalization LA%bq_>f
% -------------------------------------------------- iiG f'@/
% 0 0 1 1 ,=BLnsg
% 1 1 r * cos(theta) 2 y(a!YicA?
% 1 -1 r * sin(theta) 2 >&S0#>wmyG
% 2 -2 r^2 * cos(2*theta) sqrt(6) qAY%nA>jO
% 2 0 (2*r^2 - 1) sqrt(3) ?LaUed'
% 2 2 r^2 * sin(2*theta) sqrt(6) -*a?<ES`
% 3 -3 r^3 * cos(3*theta) sqrt(8) zt=0o|k
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) k?6z_vu
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) EJ84rSp
% 3 3 r^3 * sin(3*theta) sqrt(8) bAwl:l\`
% 4 -4 r^4 * cos(4*theta) sqrt(10) DmqSQA
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) g{:<