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, 3ew8m}A{O
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, hIJ)MZU|
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? utlpY1#q/
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? /cFzotr"9
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function z = zernfun(n,m,r,theta,nflag) kJs^ z
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Ap\AP{S4
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N lo&#(L+2
% and angular frequency M, evaluated at positions (R,THETA) on the =wi*Nd7L
% unit circle. N is a vector of positive integers (including 0), and E{Pgf8
% M is a vector with the same number of elements as N. Each element S06Hs~>Y
% k of M must be a positive integer, with possible values M(k) = -N(k) L3(^{W]|
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Cup@TET35
% and THETA is a vector of angles. R and THETA must have the same $trAC@3O@
% length. The output Z is a matrix with one column for every (N,M) ps:f=6m2
% pair, and one row for every (R,THETA) pair. 9O,,m~B
% tZWrz
e^
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike d6 _C"r
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), >x:EJV
% with delta(m,0) the Kronecker delta, is chosen so that the integral R@T6U:1
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, S(eQ{rSs
% and theta=0 to theta=2*pi) is unity. For the non-normalized IF
k
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. B`#h{ )[
% ZC^C
% The Zernike functions are an orthogonal basis on the unit circle. \[wCp*;1}
% They are used in disciplines such as astronomy, optics, and ?Ce#BwQ>
% optometry to describe functions on a circular domain. ?T:
jk4+
% oholt/gb+0
% The following table lists the first 15 Zernike functions. N1--~e
% iy5R5L2
% n m Zernike function Normalization QBE@(2G}C
% -------------------------------------------------- U!q[e`B
% 0 0 1 1 h=RDO
% 1 1 r * cos(theta) 2 GSVdb/+
% 1 -1 r * sin(theta) 2 FJM;X-UOY
% 2 -2 r^2 * cos(2*theta) sqrt(6) *ftC_v@p5
% 2 0 (2*r^2 - 1) sqrt(3) 73NZ:h%=
% 2 2 r^2 * sin(2*theta) sqrt(6) q{4|Kpx@
% 3 -3 r^3 * cos(3*theta) sqrt(8) %I4zQiJ%
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) f!GHEhQ9
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) dXu {p
% 3 3 r^3 * sin(3*theta) sqrt(8) T.?k>Ak
% 4 -4 r^4 * cos(4*theta) sqrt(10) ]= x
1`j
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) aSnp/g
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 7$T8&Mh
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !)H*r|*[
% 4 4 r^4 * sin(4*theta) sqrt(10) z)L}ECZh9
% -------------------------------------------------- Y\t_&