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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， Uc/%4Gx   
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， __a9}m4i7x  
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？ n4ISHxM  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ g3y44G CV  
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function z = zernfun(n,m,r,theta,nflag) 'RPe5 vB  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ej0q*TH.  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N !$O +M#  
%   and angular frequency M, evaluated at positions (R,THETA) on the Wg#>2)>  
%   unit circle.  N is a vector of positive integers (including 0), and Z*k}I{0,-  
%   M is a vector with the same number of elements as N.  Each element RSp=If+4  
%   k of M must be a positive integer, with possible values M(k) = -N(k) 5nk]{ G> V  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ETmfy}V8  
%   and THETA is a vector of angles.  R and THETA must have the same RxAZ<8T_  
%   length.  The output Z is a matrix with one column for every (N,M) z 6p.{M  
%   pair, and one row for every (R,THETA) pair. (u~@@d"  
% s8R.?mhH=  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike &n['#7 <(!  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 1%>/%eyn5  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral Ygg+*z  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, WzF/wzR  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized >1W)J3  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. f- _~rQ  
% pJV<#<#Z  
%   The Zernike functions are an orthogonal basis on the unit circle. ;XANIT V  
%   They are used in disciplines such as astronomy, optics, and gx3arVa  
%   optometry to describe functions on a circular domain. 6L8wsz CW  
% Z[ys>\_To  
%   The following table lists the first 15 Zernike functions. p2\@E} z  
% y<kW2<?  
%       n    m    Zernike function           Normalization Fvg>>HVu  
%       -------------------------------------------------- oB+drDp8U  
%       0    0    1                                 1 HG{OkDx]fl  
%       1    1    r * cos(theta)                    2 oPl^tzO  
%       1   -1    r * sin(theta)                    2 G/b $cO}  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) wD4Kil=v  
%       2    0    (2*r^2 - 1)                    sqrt(3) >Ka}v:E  
%       2    2    r^2 * sin(2*theta)             sqrt(6) I_\j05  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) VTS8IXz  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ]e!9{\X,*  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) rQCj^=cf;~  
%       3    3    r^3 * sin(3*theta)             sqrt(8) \Gg6&:Ua  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) ).5$c0`U&  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) I vO#tI  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) \s<L2uRj  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5*C#~gd& F  
%       4    4    r^4 * sin(4*theta)             sqrt(10) 4'[/gMUkw  
%       -------------------------------------------------- O(#DaFJv  
% ^i:%0"[*^i  
%   Example 1: 4YMX;W  
% kocgPO5  
%       % Display the Zernike function Z(n=5,m=1) o-c.D=~  
%       x = -1:0.01:1; e-Mei7{%  
%       [X,Y] = meshgrid(x,x); D5^wT>3>  
%       [theta,r] = cart2pol(X,Y); "&W80,O3  
%       idx = r<=1; WWZ`RY  
%       z = nan(size(X)); v;1<K@UT  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); ~qT5F)$B-  
%       figure  Jpm=V*P  
%       pcolor(x,x,z), shading interp DxoW,G W  
%       axis square, colorbar 54r/s#|-3  
%       title('Zernike function Z_5^1(r,\theta)') ir !/{IQx  
% 6fY-D qF!  
%   Example 2: (l}W\iB' d  
% F!ZE4S_  
%       % Display the first 10 Zernike functions ~Z-o2+xA  
%       x = -1:0.01:1; /-s-W<S[  
%       [X,Y] = meshgrid(x,x); F=EG#<@u  
%       [theta,r] = cart2pol(X,Y); Ce_k&[AJF  
%       idx = r<=1; ip8%9fG\>  
%       z = nan(size(X)); wwaw|$  
%       n = [0  1  1  2  2  2  3  3  3  3]; cao=O \Y7  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; UB^OMB-W.m  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; Y].,}}9k  
%       y = zernfun(n,m,r(idx),theta(idx)); iC2``[m"  
%       figure('Units','normalized') 4 ))ZBq?  
%       for k = 1:10 9lqH  
%           z(idx) = y(:,k); OGcq]ue  
%           subplot(4,7,Nplot(k)) bY&!d.  
%           pcolor(x,x,z), shading interp LT[g +zGB  
%           set(gca,'XTick',[],'YTick',[]) \w#)uYK{i_  
%           axis square < #FxI  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) z;iNfs0i$  
%       end ]sLdz^E3D  
% y90wL U9f  
%   See also ZERNPOL, ZERNFUN2. C2 4"H|D  
#;D@`.#\  
 [|&#A;{F#  
%   Paul Fricker 11/13/2006 z_f^L %J0  
O#Ma Z.=  
 "o)jB~ :L  
i+90##4<?  
 7D&O5Z=%+  
% Check and prepare the inputs: Ua%;hI)j$  
% ----------------------------- \Vb|bw'e(  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) QZ& 4W  
    error('zernfun:NMvectors','N and M must be vectors.') tJ$gH;  
end %\^VxM  
{ k>T*/  
 ,J(shc_F  
if length(n)~=length(m) r(j:C%?}C  
    error('zernfun:NMlength','N and M must be the same length.') P](/5KrK  
end :lW8f~!  
otA'+4\  
 ^u(-v/D9  
n = n(:); [>D5(O  
m = m(:); R0~w F>  
if any(mod(n-m,2)) +wp!hk&C5  
    error('zernfun:NMmultiplesof2', ... ?U+nR/H:6  
          'All N and M must differ by multiples of 2 (including 0).') (<2!^v0.M  
end A01PEVd@A  
>N~orSw%  
 pz=/A  
if any(m>n) RiG]-K:  
    error('zernfun:MlessthanN', ... NY?pvb  
          'Each M must be less than or equal to its corresponding N.') f cnv[B..{  
end /Z~5bb(  
068DC_  
 wEZieHw  
if any( r>1 | r<0 ) '""s%C+  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') L"RE[" m  
end ?f= ~Pn+  
;b}cn!U]  
 .On3ZN  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) !Qq~lAJO;  
    error('zernfun:RTHvector','R and THETA must be vectors.') .=N?;i  
end :}-VLp4b  
Th X6e  
 SohNk9u[8  
r = r(:); r:M0# 2   
theta = theta(:); 4C[kj  
length_r = length(r); Dtox/ ,"  
if length_r~=length(theta) eus@;l*  
    error('zernfun:RTHlength', ... xb2j |KY7  
          'The number of R- and THETA-values must be equal.') WMS~Bk+!  
end 5 qMP u|A  
.qLX jU  
 7+Er}y>  
% Check normalization: 1 F&}e&}c  
% -------------------- 7^;-[? l  
if nargin==5 && ischar(nflag) XMzQ8|]  
    isnorm = strcmpi(nflag,'norm'); ?rY+,nQP  
    if ~isnorm Wcbm,O4u  
        error('zernfun:normalization','Unrecognized normalization flag.') 'U,\5jj'Y  
    end 7)RRCsn  
else /IlO   
    isnorm = false; qOAP_\@T  
end XqGa]/;}  
OepQ Z|2  
 n\< uT1n  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% cT@H49#uB  
% Compute the Zernike Polynomials V/xjI<,  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )Fw#]~Z  
r)B55;*Fh  
 W6RjQ1  
% Determine the required powers of r: "EVf1iQ  
% ----------------------------------- ve\@u@K^  
m_abs = abs(m); ( G~ME>  
rpowers = []; qT01@Bku  
for j = 1:length(n) a^Z=xlJ/uZ  
    rpowers = [rpowers m_abs(j):2:n(j)]; ZENblh8fs  
end 3sgo5D-rMI  
rpowers = unique(rpowers); TrNh,5+b  
{L \TO,  
 qeC^e}h  
% Pre-compute the values of r raised to the required powers, qoZe<jW (  
% and compile them in a matrix: Zq 'FOzs  
% ----------------------------- |"[;0)dw^  
if rpowers(1)==0 _bRgr  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ~Lq`a@]A  
    rpowern = cat(2,rpowern{:});  N!Xn)J  
    rpowern = [ones(length_r,1) rpowern]; F$'po#  
else q,OCA\  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); wc#k@"2AZb  
    rpowern = cat(2,rpowern{:}); k@pEs# a  
end t. HwX9  
iVo-z#  
 X5(oL  
% Compute the values of the polynomials: b>9?gmR{  
% -------------------------------------- v?}rA%so  
y = zeros(length_r,length(n)); 8w#4T:hsuN  
for j = 1:length(n) '*EKi  
    s = 0:(n(j)-m_abs(j))/2; -X3CrW  
    pows = n(j):-2:m_abs(j); a@AIv"q  
    for k = length(s):-1:1 HG)$ W  
        p = (1-2*mod(s(k),2))* ... ,Y16m{<eC  
                   prod(2:(n(j)-s(k)))/              ... 3HNm`b8G4m  
                   prod(2:s(k))/                     ... 6B Hd c  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... hu7o J H  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); N8,g~?r^  
        idx = (pows(k)==rpowers); ]AZCf`7/?  
        y(:,j) = y(:,j) + p*rpowern(:,idx); LoOw]@>  
    end (4IP&^j:\  
     fF2] 7:  
    if isnorm `-Tb=o}.  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); n]nJ$u1u  
    end ,|]k4F  
end s2F[v:|Wq  
% END: Compute the Zernike Polynomials r oBb o  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?T,a(m<i {  
D( y c  
 LoSrXK~0~J  
% Compute the Zernike functions: 3Vk\iJ  
% ------------------------------ Pa=xc>m^  
idx_pos = m>0; aCi)icn$  
idx_neg = m<0; sQ&<cBs2  
I|2dV9y  
 `Gx 5=Bm;  
z = y; ) r"7"i  
if any(idx_pos) n?S)H=  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); |h&okR+_,  
end *|cs_,3  
if any(idx_neg) `=kiqF2P}  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); +i}uRO  
end xa 967Ki9"  
vg1E@rH|}  
 ? :A%$T  
% EOF zernfun ;y)3/46S  
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)  ,8DjQz0ZPo  
J6nH|s8  
DDE还是手动输入的呢？ "rrE_  
N0YJ'.=8,  
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 --]

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