首页 -> 登录 -> 注册 -> 回复主题 -> 发表主题
光行天下 -> ZEMAX,OpticStudio -> 如何从zernike矩中提取出zernike系数啊 [点此返回论坛查看本帖完整版本] [打印本页]

jssylttc 2012-04-23 19:23

如何从zernike矩中提取出zernike系数啊

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, IADHe\.  
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, Qz[^J  
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? hEB5=~A_  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? xZ`z+)  
$$m0mK  
|)!f".`  
BF W b0;+  
kAEq +{h  
function z = zernfun(n,m,r,theta,nflag) R#hy2kA  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. kC.!cPd  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N |qMG@  
%   and angular frequency M, evaluated at positions (R,THETA) on the 5c]:/9&  
%   unit circle.  N is a vector of positive integers (including 0), and *Mhirz% iD  
%   M is a vector with the same number of elements as N.  Each element ]Kq<U%x$  
%   k of M must be a positive integer, with possible values M(k) = -N(k) LXo$\~M8G8  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, o^~ZXF}  
%   and THETA is a vector of angles.  R and THETA must have the same [cnu K  
%   length.  The output Z is a matrix with one column for every (N,M) T$lV+[7  
%   pair, and one row for every (R,THETA) pair. ?\8aT"o  
% gF p3=s0~  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike wL8j i>"  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !-OZ/^l|O`  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral .JLJ(WM  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, id:,\iJ  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized 1k6asz^T  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. AT1cN1:4?  
% eP]y\S*P  
%   The Zernike functions are an orthogonal basis on the unit circle. D@La-K*5  
%   They are used in disciplines such as astronomy, optics, and o5s6$\"  
%   optometry to describe functions on a circular domain. ;=,-C ;`  
% yDqwz[v b  
%   The following table lists the first 15 Zernike functions. 72Bc0Wg  
% u"qu!EY2  
%       n    m    Zernike function           Normalization VF2,(f-*  
%       -------------------------------------------------- vSnVq>-q&  
%       0    0    1                                 1 FXBmatBck  
%       1    1    r * cos(theta)                    2 <CVX[R]U  
%       1   -1    r * sin(theta)                    2 zU!{_Ao9  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) *:H,-@  
%       2    0    (2*r^2 - 1)                    sqrt(3) Z(6.e8fK  
%       2    2    r^2 * sin(2*theta)             sqrt(6) s8,YQ5-  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) 9$$  Ijf  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) /^xv1F{  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) i$] :Y`3h  
%       3    3    r^3 * sin(3*theta)             sqrt(8) nZB ~l=  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) Trs~KcsD  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) W~mo*EJ'^  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) )(G<(eiD  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) @LI;q  
%       4    4    r^4 * sin(4*theta)             sqrt(10) V5lUh#@TN&  
%       -------------------------------------------------- h/tCve3Z  
% ,Sgo_bC/|  
%   Example 1: [UXVL}t k  
% #|E#Rkw!  
%       % Display the Zernike function Z(n=5,m=1) Sk53Lc  
%       x = -1:0.01:1; %q|* }l  
%       [X,Y] = meshgrid(x,x); 5b"=m9{g  
%       [theta,r] = cart2pol(X,Y); =Lkn   
%       idx = r<=1; UUfM 7gq  
%       z = nan(size(X)); N-2#-poDe  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); <2]h$53y!  
%       figure E}4{{{r  
%       pcolor(x,x,z), shading interp Mk#r_:[BS  
%       axis square, colorbar }K 'A/]'  
%       title('Zernike function Z_5^1(r,\theta)') ,5zY1C==Ut  
% Kc[^Pu  
%   Example 2: (DvPdOT+3  
% ^*l dsc  
%       % Display the first 10 Zernike functions o+hp#e  
%       x = -1:0.01:1; dE8f?L'  
%       [X,Y] = meshgrid(x,x); "*#f^/LS  
%       [theta,r] = cart2pol(X,Y); (KC08  
%       idx = r<=1; g"sb0d9  
%       z = nan(size(X)); uH$hMg  
%       n = [0  1  1  2  2  2  3  3  3  3]; Z]Xa:[  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; (QIU3EN  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; ~Zsj@d  
%       y = zernfun(n,m,r(idx),theta(idx)); XwEMF5[  
%       figure('Units','normalized') sRT5i9TQ  
%       for k = 1:10 Po=:-Of:  
%           z(idx) = y(:,k); Ydsnu  
%           subplot(4,7,Nplot(k)) 4'D^>z!c  
%           pcolor(x,x,z), shading interp 'KmM %tN  
%           set(gca,'XTick',[],'YTick',[]) @{qcu\sZ  
%           axis square x=>dmi3  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) =?Ry,^=b  
%       end w@2NXcmw  
% NUnwf h  
%   See also ZERNPOL, ZERNFUN2. vrmMEWPV  
1 sza\pR<  
/|U;_F Pmc  
%   Paul Fricker 11/13/2006 @8J*vY =e  
A?HDY_u  
~uY5~Qs9G  
@}UOm- M  
O~8jz  
% Check and prepare the inputs: ZWyf.VJ  
% ----------------------------- rYPuo  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) |7|'J Ty  
    error('zernfun:NMvectors','N and M must be vectors.') "=]'"'B:  
end (~\HizSl  
=Cf@!wZ^  
w`boQ_Ir  
if length(n)~=length(m) y46sL~HRv  
    error('zernfun:NMlength','N and M must be the same length.') I@N/Y{y#  
end U{EcV%C2  
umPN=0u6  
HHyN\  
n = n(:); a$uD oi  
m = m(:); 1| WDbk  
if any(mod(n-m,2)) T: '<:*pD  
    error('zernfun:NMmultiplesof2', ... }:?_/$};  
          'All N and M must differ by multiples of 2 (including 0).') O:V.;q2]U  
end H RahBTd(z  
: [?7,/w  
_JpTHpqu  
if any(m>n) ve/.q^JeJ  
    error('zernfun:MlessthanN', ... meB9 :w[m  
          'Each M must be less than or equal to its corresponding N.') }rVLWt  
end toG- Dz&  
\o,et9zDJ3  
J'#o6Ud  
if any( r>1 | r<0 ) vG}\Amx+  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') @(/$;I,  
end NSRY(#3  
,!AYeVq  
6a]Qg99\  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) j*VYUM@y1\  
    error('zernfun:RTHvector','R and THETA must be vectors.') !k 'E  
end 2sBYy 8.r  
>0N$R|B&  
vO zUAi  
r = r(:); =;8q`  
theta = theta(:); hsws7sH  
length_r = length(r); JDpW7OrDc  
if length_r~=length(theta) v~^*L iP+  
    error('zernfun:RTHlength', ... .Pe^u%J6F  
          'The number of R- and THETA-values must be equal.') 'Um\m  
end ;cv\v(0  
!M6Km(>  
Fvv/#V^R  
% Check normalization: EffU-=?%!  
% -------------------- ;M#D*<ucI:  
if nargin==5 && ischar(nflag) (=53WbOh/t  
    isnorm = strcmpi(nflag,'norm'); O,&p"K&Z  
    if ~isnorm z[+pN:47  
        error('zernfun:normalization','Unrecognized normalization flag.') 7bW ''J*6  
    end SsL>K*t5  
else ui*CA^ Y  
    isnorm = false; AdF[>Wv  
end yle~hL  
|msQ  
6:bvq?5a5  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5)K?:7  
% Compute the Zernike Polynomials iaaD1 <m  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V+y:!t`  
V,3$>4x  
4y9n,~Qgw  
% Determine the required powers of r: N;av  
% ----------------------------------- "OKsl2e  
m_abs = abs(m); y-7$HWn  
rpowers = []; f,0oCBLPO  
for j = 1:length(n) t7 $2/C  
    rpowers = [rpowers m_abs(j):2:n(j)]; /#4BUfY f  
end aQfrDM<*XS  
rpowers = unique(rpowers); ~u80v h'  
pdR&2fp  
~ @s$  
% Pre-compute the values of r raised to the required powers, ?37Kc,o  
% and compile them in a matrix: 8!dA1]2;  
% ----------------------------- A|GheH!t  
if rpowers(1)==0 cM+s)4TPL  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ki_Py5  
    rpowern = cat(2,rpowern{:}); Zh.9j7 >p  
    rpowern = [ones(length_r,1) rpowern]; |(1z ?Spbe  
else Kd,7x'h`E  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %xI,A'#  
    rpowern = cat(2,rpowern{:}); g~=#8nJ  
end XS"lR |  
!~aDmY 2  
zFV?,"\r  
% Compute the values of the polynomials: 5eSmyj-W  
% -------------------------------------- >&N8Du*[  
y = zeros(length_r,length(n)); X5D}<J2"  
for j = 1:length(n) v.I>B3bEg  
    s = 0:(n(j)-m_abs(j))/2; VFwp .1oa!  
    pows = n(j):-2:m_abs(j); IE9A _u*  
    for k = length(s):-1:1 {p(.ck ze+  
        p = (1-2*mod(s(k),2))* ... iY1JU -S  
                   prod(2:(n(j)-s(k)))/              ... H23-%+*J  
                   prod(2:s(k))/                     ... sHulaX{  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... {e8.E<f-  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); k GzosUt  
        idx = (pows(k)==rpowers); w;Na9tR  
        y(:,j) = y(:,j) + p*rpowern(:,idx); B?J #NFUb  
    end x5}Ru0Z  
     VDq?,4Kb  
    if isnorm !j?2HlIK+  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); QR($KW(  
    end  OL|UOG  
end M7;P)da  
% END: Compute the Zernike Polynomials !'^gqaF+  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n +z5;'my  
k:0HsN!F9  
`PR)7}/<  
% Compute the Zernike functions:  Ju#t^P  
% ------------------------------ mmG+"g$|  
idx_pos = m>0; rOu7r4  
idx_neg = m<0; hqVFb.6[  
lclSzC9  
s`G3SE  
z = y; Zg/ra1n  
if any(idx_pos) 0x\bDWZ_  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); kN*,3)T;}  
end q8{) 27f,  
if any(idx_neg) +Q3i&"QB.  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 2\M^ _x$N  
end 0;j)rmt  
0a??8?Q1G  
W7lR 54%|  
% EOF zernfun K uz /  
phoenixzqy 2012-04-23 20:38
慢慢研究,这个专业性很强的。用的人又少。
sansummer 2012-04-27 10:22
这个太牛了,我目前只能把zygo中的zernike的36项参数带入到zemax中,但是我目前对其结果的可信性表示质疑,以后多交流啊
jssylttc 2012-05-14 11:28
sansummer:这个太牛了,我目前只能把zygo中的zernike的36项参数带入到zemax中,但是我目前对其结果的可信性表示质疑,以后多交流啊 (2012-04-27 10:22)  88]V6Rm9[*  
J3OxM--8"  
DDE还是手动输入的呢? 8LI,'XZ  
xx`xDD  
zygo和zemax的zernike系数,类型对应好就没问题了吧
jssylttc 2012-05-14 11:37
顶顶·········
18257342135 2016-12-13 10:03
支持一下,慢慢研究
查看本帖完整版本: [-- 如何从zernike矩中提取出zernike系数啊 --] [-- top --]

Copyright © 2005-2025 光行天下 蜀ICP备06003254号-1 网站统计