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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， Ti>}To}B5  
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， BJ{?S{"6%G  
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？ j_S3<wEJ  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ 3A\Z ]L  
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E CPSE {  
 ZHCrKp  
function z = zernfun(n,m,r,theta,nflag) 7?\r9bD  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. <}F(G-kV6  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N gl!ht@;>ak  
%   and angular frequency M, evaluated at positions (R,THETA) on the \jAI~|3  
%   unit circle.  N is a vector of positive integers (including 0), and ;Hb"SB  
%   M is a vector with the same number of elements as N.  Each element T#HF! GH]  
%   k of M must be a positive integer, with possible values M(k) = -N(k) \\35} 9  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, & d* bQv$  
%   and THETA is a vector of angles.  R and THETA must have the same ?<^^.Si  
%   length.  The output Z is a matrix with one column for every (N,M) *p|->p6,u  
%   pair, and one row for every (R,THETA) pair. m<!CF3g  
% EF;B)y=  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Wj, {lJ,  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), |*L/ m0'L  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral ,2P /[ :  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, "@rHGxK  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized  JY_!G  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. MPLeqk$;  
% 2~h Q   
%   The Zernike functions are an orthogonal basis on the unit circle. 1 /SB[[g  
%   They are used in disciplines such as astronomy, optics, and "WH &BhQYD  
%   optometry to describe functions on a circular domain. `0-i>>  
% V=c?V/pl  
%   The following table lists the first 15 Zernike functions. epcvwM/A  
% |V^f}5gd  
%       n    m    Zernike function           Normalization p$<){,R  
%       -------------------------------------------------- tR_DN  
%       0    0    1                                 1 &09G9GsnQ  
%       1    1    r * cos(theta)                    2 |@hyGu-H+  
%       1   -1    r * sin(theta)                    2 4 &0MB>m  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) E&Sr+D aPD  
%       2    0    (2*r^2 - 1)                    sqrt(3) B0^:nYko  
%       2    2    r^2 * sin(2*theta)             sqrt(6) ~O 4@b/!4  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) HN<e)E38  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Kc+9n%sp  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 8an_s%,AW  
%       3    3    r^3 * sin(3*theta)             sqrt(8) ZQmg;L&7  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) fLV@~T|  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) y/+y |.Xg  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) {wD "|K  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) NSQ)lSW,;  
%       4    4    r^4 * sin(4*theta)             sqrt(10) z0T6a15f!P  
%       -------------------------------------------------- s*vtCdrE.  
% ryTtGx%a  
%   Example 1: {3 >`k.w  
% ,Uy~O(F t  
%       % Display the Zernike function Z(n=5,m=1) =HMuAUa.  
%       x = -1:0.01:1; 6>h"Lsww  
%       [X,Y] = meshgrid(x,x); ^;@!\Rc  
%       [theta,r] = cart2pol(X,Y); O\w-hk  
%       idx = r<=1; d/E0opv  
%       z = nan(size(X)); _;-b ZH  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); VGOdJ|2]Wr  
%       figure &CfzhIi*!  
%       pcolor(x,x,z), shading interp &pAmFe  
%       axis square, colorbar 'JAe =K H  
%       title('Zernike function Z_5^1(r,\theta)') `U{#;  
% >9[wjB2?}  
%   Example 2: ^{-Z3Yxd  
% YwJ<0;:+hS  
%       % Display the first 10 Zernike functions 5=;'LWXCJ  
%       x = -1:0.01:1; 5 gwEr170  
%       [X,Y] = meshgrid(x,x); RR>G}u9 np  
%       [theta,r] = cart2pol(X,Y); Sbj{)  
%       idx = r<=1; D \boF+^  
%       z = nan(size(X)); !1P<A1K  
%       n = [0  1  1  2  2  2  3  3  3  3]; Qq\hD@Z|  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; EtDzmpJR>  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; ?#L5V'ZZ*  
%       y = zernfun(n,m,r(idx),theta(idx)); [8[`V)b  
%       figure('Units','normalized') &y~GTEP  
%       for k = 1:10 &tw{d DD6  
%           z(idx) = y(:,k); ['I5(M@  
%           subplot(4,7,Nplot(k)) ak(P<OC-  
%           pcolor(x,x,z), shading interp t s=+k/Z  
%           set(gca,'XTick',[],'YTick',[]) NaeG)u#+  
%           axis square >F/5`=/'h  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) )lVplAhZD  
%       end !3o]mBH8  
% ~uJO6C6A  
%   See also ZERNPOL, ZERNFUN2. m_UzmWF  
5I5#LQv0  
 ))uki*UNK  
%   Paul Fricker 11/13/2006 1i 6>~  
Mey=%Fv  
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 {-s7_\|p(  
% Check and prepare the inputs: E\~!E20^  
% ----------------------------- 5Veybchy "  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }vQ Y+O  
    error('zernfun:NMvectors','N and M must be vectors.') 3 yM!BTlX  
end  !:|D[1m  
mQ:5(]v  
 y?V#LW[^E  
if length(n)~=length(m) m# I  
    error('zernfun:NMlength','N and M must be the same length.') QBTjiaYGa'  
end C-VkXk  
`wLMJ,@f.  
 5~xv"S(E}  
n = n(:); +p &$`(  
m = m(:); t%30B^Ii%K  
if any(mod(n-m,2)) }I#,o!)Vd  
    error('zernfun:NMmultiplesof2', ... vce1'aW  
          'All N and M must differ by multiples of 2 (including 0).') y3mJO[U0 a  
end 1.q a//'RW  
J ?H| "  
 U( "m}^  
if any(m>n) YDiru  
    error('zernfun:MlessthanN', ... %2rUJaOgy$  
          'Each M must be less than or equal to its corresponding N.') _6MNEoy?  
end I$1~;!<  
<,(Ww   
 WJw %[_W  
if any( r>1 | r<0 ) 98t|G5  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') qvN 5[rb  
end A*+KlhT  
SR&'38UCe  
 m*H6\on:  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ;NrU|g/ksX  
    error('zernfun:RTHvector','R and THETA must be vectors.') e{`DvfY21  
end W8s/"  
7Dwf0Re`  
 |%7OI#t^  
r = r(:); 9y5nG  
theta = theta(:); ^[-3qi  
length_r = length(r); J l9w/T  
if length_r~=length(theta) /? HLEX  
    error('zernfun:RTHlength', ... 1N\-Ku  
          'The number of R- and THETA-values must be equal.') <LOas$  
end `1)n2<B  
)l*6zn`z  
 zrCQEQq  
% Check normalization: +#0,2 wR#  
% -------------------- 'P<T,:z?  
if nargin==5 && ischar(nflag) WG.J-2#3  
    isnorm = strcmpi(nflag,'norm'); Zk75GC  
    if ~isnorm :ODG]-QF  
        error('zernfun:normalization','Unrecognized normalization flag.') F5;x>;r  
    end $sR-J'EE!  
else 9x9~u8j  
    isnorm = false; !Typ_Cs  
end XveG#oyiU  
%y}l^P5z  
 k=qb YGK  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Y'%k G5nF  
% Compute the Zernike Polynomials zWEt< `1M  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b py576GwA  
&nEQ `3~F  
 yu] nK-Y7S  
% Determine the required powers of r: Um\_G@  
% ----------------------------------- ImVHX~ qHJ  
m_abs = abs(m); e4?p(F-x(  
rpowers = []; G%RhNwm  
for j = 1:length(n) 7ZRLSq'S  
    rpowers = [rpowers m_abs(j):2:n(j)]; t|y`Bl2  
end l\{{iAC]I  
rpowers = unique(rpowers); p6e9mSs  
gXI8$W>  
 BSib/)p   
% Pre-compute the values of r raised to the required powers, >,. x'{  
% and compile them in a matrix: |=9=a@l]P  
% ----------------------------- R-2V C  
if rpowers(1)==0 {]CO;5:  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); b];? tP  
    rpowern = cat(2,rpowern{:}); t)/:VImY  
    rpowern = [ones(length_r,1) rpowern]; lGAKHCs  
else N3}jLl/  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); geB]~/-p  
    rpowern = cat(2,rpowern{:}); 9F^rXY.  
end C0@[4a$8f  
t&99ZdE  
 !Cv:,q  
% Compute the values of the polynomials: ]aF!0Fln~  
% -------------------------------------- m=uW:~  
y = zeros(length_r,length(n)); /}=Bi-  
for j = 1:length(n) 9:tn! <^=I  
    s = 0:(n(j)-m_abs(j))/2; }yW*vy6`  
    pows = n(j):-2:m_abs(j); YZH &KGY  
    for k = length(s):-1:1 8hx 3pvmk  
        p = (1-2*mod(s(k),2))* ... rWo&I _{  
                   prod(2:(n(j)-s(k)))/              ... Y +9OP  
                   prod(2:s(k))/                     ... \P` mV9P  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... +`)4jx)r/  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); AT%@T|  
        idx = (pows(k)==rpowers);  H*]B7?S  
        y(:,j) = y(:,j) + p*rpowern(:,idx); NlnmeTLO5  
    end oTqv$IzqP  
     ?g21U97Q  
    if isnorm <(U :v  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); t$W~X~//  
    end ^cy.iolt  
end 0=^A{V!m  
% END: Compute the Zernike Polynomials 9M&uQccY  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Px<*n '~}  
T+y3Ph--^  
 ~3u'=u9l  
% Compute the Zernike functions: IfdgMELk  
% ------------------------------ ,:e##g~k  
idx_pos = m>0; >(aGk{e1  
idx_neg = m<0; b\l +S2  
i!jR>+  
 u(!@6%?-  
z = y; (3>Z NTm  
if any(idx_pos) 5#SD$^  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); &b#O=LF  
end n 7Bua  
if any(idx_neg) `zJTVi4  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); UmRI! WQl  
end )j[rm   
V!Q1o!J  
 rfdT0xfcU  
% EOF zernfun </OZ,3J=  
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)  X0,?~i6Q  
7n-;++a5]  
DDE还是手动输入的呢？ J~6-}z   
c({V[eGY  
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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