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, v%:deaF
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, #NFB=oJI
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? jC'h54,Mr
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? un 5r9
fQ5v?(
_bCAZa&&
v*!N}1+J
o-@01_j
function z = zernfun(n,m,r,theta,nflag) IB;yL/T
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ;O}%SCF7
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N -yf8
% and angular frequency M, evaluated at positions (R,THETA) on the Q'n+K5&p
% unit circle. N is a vector of positive integers (including 0), and a<&K^M&
% M is a vector with the same number of elements as N. Each element A;L
]=J
% k of M must be a positive integer, with possible values M(k) = -N(k) Tow=B
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Pdf-2
Tx
% and THETA is a vector of angles. R and THETA must have the same ui>jJ(
% length. The output Z is a matrix with one column for every (N,M) }? _KZ)
% pair, and one row for every (R,THETA) pair. )7
Mss/2T
% !MKecRG_
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike @*eY~
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 8H4NNj Oy
% with delta(m,0) the Kronecker delta, is chosen so that the integral :Dty([
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, &za
}THm
% and theta=0 to theta=2*pi) is unity. For the non-normalized )7 & -DI1
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 9I/l+IS"X
% *,z/q6
% The Zernike functions are an orthogonal basis on the unit circle. 4z(~)#'^
% They are used in disciplines such as astronomy, optics, and
b WNa6x
% optometry to describe functions on a circular domain. K[icVT2v~
% G*4I;'6
% The following table lists the first 15 Zernike functions. Q2 !GWz$
% ;d}>8w&tfy
% n m Zernike function Normalization FygNWI '
% -------------------------------------------------- +#eol~j9N
% 0 0 1 1 \1Y|$:T/
% 1 1 r * cos(theta) 2 2OJlE)
.
% 1 -1 r * sin(theta) 2 s;I
@En
% 2 -2 r^2 * cos(2*theta) sqrt(6) svmb~n &x6
% 2 0 (2*r^2 - 1) sqrt(3) R>0[w$
% 2 2 r^2 * sin(2*theta) sqrt(6) /ugWl99.W
% 3 -3 r^3 * cos(3*theta) sqrt(8) ~-k,$J?7
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 5a/A?9?,
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 7K.75%}
% 3 3 r^3 * sin(3*theta) sqrt(8) JH\:9B+:L
% 4 -4 r^4 * cos(4*theta) sqrt(10) )xy>:2!#Y
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) rci,&>L"
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Ga5s9wC
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) @ ;!IPiU
% 4 4 r^4 * sin(4*theta) sqrt(10) c[ZrQJ
% -------------------------------------------------- fx|d"VF[
% yzK<yvN
% Example 1: }B'-*)^|e{
% W+a/>U
% % Display the Zernike function Z(n=5,m=1) .6`r`|=
% x = -1:0.01:1; )l`Ks
% [X,Y] = meshgrid(x,x); =Q<VU/
% [theta,r] = cart2pol(X,Y);
x\Q}fk?{t
% idx = r<=1; k( Sda>-
% z = nan(size(X)); gbzBweWF
% z(idx) = zernfun(5,1,r(idx),theta(idx)); LY0f`RX*&
% figure *1EmK.-'u
% pcolor(x,x,z), shading interp PV#h_X<l%
% axis square, colorbar 7nT|yL?
% title('Zernike function Z_5^1(r,\theta)') Jpduk&u
%
`vH|P
% Example 2: / ]8e[t>!f
% , mz;$z6i
% % Display the first 10 Zernike functions -7&ywgxl
% x = -1:0.01:1; Cdz?+hb
% [X,Y] = meshgrid(x,x); n,FyK`x
% [theta,r] = cart2pol(X,Y); k{mBG9[z
% idx = r<=1; ML>M:Ik+
% z = nan(size(X)); ;J|t-$Z
% n = [0 1 1 2 2 2 3 3 3 3]; 48wt
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; h)Fc<,vwBE
% Nplot = [4 10 12 16 18 20 22 24 26 28]; tn$TyCzckW
% y = zernfun(n,m,r(idx),theta(idx)); rY(7IX
% figure('Units','normalized') `n>|rd
% for k = 1:10 ^>an4UJt
% z(idx) = y(:,k); `F/R:!v
% subplot(4,7,Nplot(k)) KS8@A/f
% pcolor(x,x,z), shading interp kKlNhP(
% set(gca,'XTick',[],'YTick',[]) ufk2zL8y
% axis square
nT> v
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) AM=,:k$
% end P-B5-Nz
% L'Cd`.yVO
% See also ZERNPOL, ZERNFUN2. F?'
{xg=Ym)
X`_tm3HC
% Paul Fricker 11/13/2006 /4(HVua
bhpaC8|
/x@aAJ|
f#&z