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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, S_!R^^ySG9  
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, >1XL;)IL>  
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? ;2W2MZ!TF  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Hz4uZ*7\|  
$T)d!$  
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#6FaIq92V  
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function z = zernfun(n,m,r,theta,nflag) p[P[#IeL  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. aT/KT,!  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N HU3Vv<lz  
%   and angular frequency M, evaluated at positions (R,THETA) on the |7S:l9;  
%   unit circle.  N is a vector of positive integers (including 0), and c20|Cx2m  
%   M is a vector with the same number of elements as N.  Each element qi[(*bFK7  
%   k of M must be a positive integer, with possible values M(k) = -N(k) ]EX--d<_`  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, LI$L9eNv;Y  
%   and THETA is a vector of angles.  R and THETA must have the same 2vXGO|W  
%   length.  The output Z is a matrix with one column for every (N,M) i0&) N,5_  
%   pair, and one row for every (R,THETA) pair. Y=WR6!{  
% 0XQ-   
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike w\v&3T   
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), tYI]=:  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral { ;' :h  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 1(F'~i|5  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized 0eaUorm)  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Oylp:_<aT  
% pgfu+K7?w  
%   The Zernike functions are an orthogonal basis on the unit circle. <VgE39 [  
%   They are used in disciplines such as astronomy, optics, and $@4e(Zrmo  
%   optometry to describe functions on a circular domain. `w(sXkeaI  
% :6sGX p  
%   The following table lists the first 15 Zernike functions. _PdAN= C3  
% gNi}EP5>  
%       n    m    Zernike function           Normalization 0 wYiu  
%       -------------------------------------------------- 1o)=GV1  
%       0    0    1                                 1 F_~6n]Sr  
%       1    1    r * cos(theta)                    2 oYGUjI  
%       1   -1    r * sin(theta)                    2 8ST~$!z$  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) 8s&2gn1  
%       2    0    (2*r^2 - 1)                    sqrt(3) 7vdHR\#;$  
%       2    2    r^2 * sin(2*theta)             sqrt(6) n+S&!PB  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) EXH!glR[$  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) j!"iYtgV  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) > fhSaeN  
%       3    3    r^3 * sin(3*theta)             sqrt(8) w-8)YJ Y  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) KXDz'9_  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) pIrv$^  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 7a27^b  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) N)Qlkz$X  
%       4    4    r^4 * sin(4*theta)             sqrt(10) L)=8mF.  
%       -------------------------------------------------- !c v6 #:  
% MgSp.<!  
%   Example 1: /G[+E&vj  
% N_*u5mfQX  
%       % Display the Zernike function Z(n=5,m=1) vJzxP y|  
%       x = -1:0.01:1; ^S:cNRSW"  
%       [X,Y] = meshgrid(x,x); R\i]O  
%       [theta,r] = cart2pol(X,Y); HK=CP0H  
%       idx = r<=1; /:Rn"0   
%       z = nan(size(X)); c@)pKi#W  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); =k_XKxd  
%       figure G<Th<JF)Q  
%       pcolor(x,x,z), shading interp _g^E%@'W  
%       axis square, colorbar 6qY\7R2+  
%       title('Zernike function Z_5^1(r,\theta)') `mQP{od?"?  
% `8qT['`#R  
%   Example 2: s n=zh1 A  
% #.RG1-L  
%       % Display the first 10 Zernike functions @5JLjCN  
%       x = -1:0.01:1; $&c<T4$d  
%       [X,Y] = meshgrid(x,x); $a)J CErN  
%       [theta,r] = cart2pol(X,Y); Qj{$dqmDN  
%       idx = r<=1; h,Y{t?Of  
%       z = nan(size(X)); `,hW;p>-  
%       n = [0  1  1  2  2  2  3  3  3  3]; 7Q<Kha  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; #%9oQ6nO  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; :4Id7Ce  
%       y = zernfun(n,m,r(idx),theta(idx)); )<m=YI ;<  
%       figure('Units','normalized') ^/ULh,w!fP  
%       for k = 1:10 YY1{v?[  
%           z(idx) = y(:,k); zWP.1 aA&  
%           subplot(4,7,Nplot(k)) f/$-Nl.  
%           pcolor(x,x,z), shading interp ;Hz`0V  
%           set(gca,'XTick',[],'YTick',[]) L5i#Kh_  
%           axis square qBf wN1  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 1 P(&GYc  
%       end KY;uO 8Te  
% Po2_ 0uX  
%   See also ZERNPOL, ZERNFUN2. HMl!?%%  
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oP 6.t-<dU  
%   Paul Fricker 11/13/2006 U4 go8  
^!-E`<jW8  
VPq5xSc?  
Rh05W_?Js  
%Q>~7P  
% Check and prepare the inputs: "^e}C@  
% ----------------------------- {7j6$.7J$&  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) \.XT:B_  
    error('zernfun:NMvectors','N and M must be vectors.') o`JlXuG?o  
end (mOqv9pn  
rH [+/&w5  
+aXMHT"U  
if length(n)~=length(m) $\JQGic`  
    error('zernfun:NMlength','N and M must be the same length.') DKaG?Y,*p  
end )- Wn'C'Z  
dvrvpDoE.  
Lv`8jSt\  
n = n(:); 5 O{Ip-  
m = m(:); 5Tcl<Y6l  
if any(mod(n-m,2)) f0N)N}y  
    error('zernfun:NMmultiplesof2', ... Dn{19V. L  
          'All N and M must differ by multiples of 2 (including 0).') f6dE\  
end 0& SrKn  
tXb7~aO  
Rd;~'gbG  
if any(m>n) ;c \zgs~"T  
    error('zernfun:MlessthanN', ... Occ8Hk/l.  
          'Each M must be less than or equal to its corresponding N.') 7#~m:K@  
end KNUMz4  
af`f*{Co3  
b> >=d)R  
if any( r>1 | r<0 ) NXV~[  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') sEgeS9a{  
end "\R@l Ux.Y  
T[8"u<O96  
1Q2k>q8  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Jte:l:yjtA  
    error('zernfun:RTHvector','R and THETA must be vectors.') [/#k$-  
end ki][qvXJ  
nw]e_sm  
!m/Dd0  
r = r(:); k:HSB</}  
theta = theta(:); }GU6Q|s[u[  
length_r = length(r); s].'@_~s  
if length_r~=length(theta)  c+G:@%  
    error('zernfun:RTHlength', ... c?3F9 w#  
          'The number of R- and THETA-values must be equal.') \I o?ul}za  
end 41+E UMc  
D+vl%(g  
vY+_tpuEH  
% Check normalization: +oKpA\mz  
% -------------------- .AmM%I4K  
if nargin==5 && ischar(nflag) U}C#:Xi>$  
    isnorm = strcmpi(nflag,'norm'); `'WY'\|C  
    if ~isnorm l7r N  
        error('zernfun:normalization','Unrecognized normalization flag.') g`f6gxc  
    end 'zD;:wT  
else J1v0 \  
    isnorm = false; ~%!U,)-  
end =LeVJGF  
9rvxp;  
,h)T(  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _-yF9g"I  
% Compute the Zernike Polynomials D*2p  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LZAj4|~,m  
1wNY}3  
!kk %;XSZ  
% Determine the required powers of r: V2sB[Mw  
% ----------------------------------- Le$u$ulS  
m_abs = abs(m); & b^*N5<Z  
rpowers = []; '>lPq tdZ  
for j = 1:length(n) R.WsC bU  
    rpowers = [rpowers m_abs(j):2:n(j)]; @W5hrei  
end +\(ay"+ d  
rpowers = unique(rpowers); }W>[OY0^A  
lO[jf6gB  
,I:m*.q  
% Pre-compute the values of r raised to the required powers, @Y<ZT;J  
% and compile them in a matrix: CFrHNU  
% ----------------------------- Vh[o[ U  
if rpowers(1)==0 Rb>RjHo S  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); uJ5%JB("E  
    rpowern = cat(2,rpowern{:}); r+.4|u  
    rpowern = [ones(length_r,1) rpowern]; ]TZWFL-  
else +AC-f2  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); p'c<v)ia  
    rpowern = cat(2,rpowern{:}); D"XQ!1B%  
end XTXo xZ#w  
2P> za\  
z&J ow/  
% Compute the values of the polynomials: Mh/>qyS *2  
% -------------------------------------- ^3@a0J=F  
y = zeros(length_r,length(n)); $j2)_(<A%Q  
for j = 1:length(n) 6!D  
    s = 0:(n(j)-m_abs(j))/2; /Rcd}rO  
    pows = n(j):-2:m_abs(j); 3V!&y/c<  
    for k = length(s):-1:1 I)/7M}t`  
        p = (1-2*mod(s(k),2))* ... %oKc?'L0  
                   prod(2:(n(j)-s(k)))/              ... V +<AG*[  
                   prod(2:s(k))/                     ... *SG2k .$  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... b2-|e_x  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); L<>NL$CrN  
        idx = (pows(k)==rpowers); zc~xWy+  
        y(:,j) = y(:,j) + p*rpowern(:,idx); 8q[WfD  
    end nZ+5@( *  
     yl+)I  
    if isnorm iwx0V  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Dj&bHC5%  
    end [?6D1b[  
end Z/UVKJm>:  
% END: Compute the Zernike Polynomials MxA'T(Ay  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6e-h;ylS  
GYmBxX87  
9nAK6$/  
% Compute the Zernike functions: T@.m^|~  
% ------------------------------ V~"d`j  
idx_pos = m>0; R6o<p<fTh  
idx_neg = m<0; )KQv4\0y<  
L*oL KigT  
3Ty{8oUs^  
z = y; tpzdYokh >  
if any(idx_pos) ;4#8#;  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); fv'P!+)t  
end XFAt\g  
if any(idx_neg) h#;K9#x6  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); }ucg!i3C  
end Jm,X~Si  
84\o7@$#  
6]49kHgMhe  
% EOF zernfun CP#MNNvgrw  
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)  :AGQkJb  
ir?9{t/()  
DDE还是手动输入的呢? :UciFIa  
EHjhe z  
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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