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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， ?:9"X$XR  
我输入10阶的n、m，r,theta为38025*1向量，最后得到的z是29525*10阶的矩阵， hOK8(U0  
这个，跟我们用zygo干涉仪直接拟合出的36项zernike系数，有何关系呢？ lH~[f  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ G=bCNn<  
?+8\.a!  
 3=V &K-  
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function z = zernfun(n,m,r,theta,nflag) cso8xq|b7  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 3o/[t  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N +LJ73 !  
%   and angular frequency M, evaluated at positions (R,THETA) on the @>7%qS  
%   unit circle.  N is a vector of positive integers (including 0), and Y}KNKO;  
%   M is a vector with the same number of elements as N.  Each element <%mRSv  
%   k of M must be a positive integer, with possible values M(k) = -N(k) ;qV>L=a  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, *qpSXmOz  
%   and THETA is a vector of angles.  R and THETA must have the same RPbZ(.  
%   length.  The output Z is a matrix with one column for every (N,M) AQ^u   
%   pair, and one row for every (R,THETA) pair. 0<*<$U  
% :Llb< MY2  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /dIzY0<aO  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), j5h-dK  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral B`sAk %  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 62NsJ<#>  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized N6TH}~62}  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. :rP=t ,  
% 9A#i_#[R  
%   The Zernike functions are an orthogonal basis on the unit circle. y|jq?M<A  
%   They are used in disciplines such as astronomy, optics, and tQ601H>o  
%   optometry to describe functions on a circular domain. Wa~=bH  
% IAyp2  
%   The following table lists the first 15 Zernike functions. ]I6  J7A[  
% l Nv|M)I  
%       n    m    Zernike function           Normalization 9j Gu}V o  
%       -------------------------------------------------- o#N+Y?O  
%       0    0    1                                 1 lmhLM. 2  
%       1    1    r * cos(theta)                    2 dgP3@`YS  
%       1   -1    r * sin(theta)                    2 @E8+C8'  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) _(zG?]y0P  
%       2    0    (2*r^2 - 1)                    sqrt(3) ?r!o~|9|  
%       2    2    r^2 * sin(2*theta)             sqrt(6) 2>59q$ |  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) FC4wwzb  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) CU~PT.  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) [PbOfxxgA  
%       3    3    r^3 * sin(3*theta)             sqrt(8) Fs^Mw g o  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) O.JN ENZf  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) fd9k?,zM  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) o,wUc"CE  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) T6kdS]4-  
%       4    4    r^4 * sin(4*theta)             sqrt(10) ,iwp,=h=  
%       -------------------------------------------------- ABYcH]m  
% >C~6\L`c  
%   Example 1: 8JUwf  
% .o}v#W+st  
%       % Display the Zernike function Z(n=5,m=1) @[v~y"tE}  
%       x = -1:0.01:1; %xt^698&X  
%       [X,Y] = meshgrid(x,x); #Rr%:\*  
%       [theta,r] = cart2pol(X,Y); k|f4Cf,  
%       idx = r<=1; CTA 3*Gn  
%       z = nan(size(X)); x$(f7?s] 1  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); E<*xx#p  
%       figure J?$,c4;W2  
%       pcolor(x,x,z), shading interp EB|}fz  
%       axis square, colorbar [1H^3g '  
%       title('Zernike function Z_5^1(r,\theta)') |vzl. ^"-  
% {(?4!rh  
%   Example 2: -H-~;EzU  
% 6-ils3&  
%       % Display the first 10 Zernike functions PTV:IzoW  
%       x = -1:0.01:1; Ef{Vp;]  
%       [X,Y] = meshgrid(x,x); < FAheE+  
%       [theta,r] = cart2pol(X,Y); YZJyk:H\  
%       idx = r<=1; 2I{"XB  
%       z = nan(size(X)); W=4FFl[  
%       n = [0  1  1  2  2  2  3  3  3  3]; 0Wp|1)ljA  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Srd4))2/0  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; kg\ >k2h  
%       y = zernfun(n,m,r(idx),theta(idx)); |(^PS8wG  
%       figure('Units','normalized') I1&aM}y{G  
%       for k = 1:10 IO:G1;[/2L  
%           z(idx) = y(:,k); f(7GX3?  
%           subplot(4,7,Nplot(k)) *}W_+qo"  
%           pcolor(x,x,z), shading interp bi;1s'Y<D  
%           set(gca,'XTick',[],'YTick',[]) "tpSg  
%           axis square Ny)X+2Ae  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Z;)%%V%o  
%       end &PtJ$0%q  
% ZqO^f*F>h  
%   See also ZERNPOL, ZERNFUN2. R0-j5&^jju  
y1L,0 ]  
 ENY+^7  
%   Paul Fricker 11/13/2006 -d:Jta!}{  
@pU)_d!pJ  
 \Y}8S/]  
8, >P  
 0#^v{DC  
% Check and prepare the inputs: ^_mj  
% ----------------------------- q'MZ R'<@  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) \1Em`nvOX  
    error('zernfun:NMvectors','N and M must be vectors.') +^T@sa`[I  
end 7. ;3e@s  
D. XvG_  
 |LKXOU c  
if length(n)~=length(m) R*r#E{!V;  
    error('zernfun:NMlength','N and M must be the same length.') P7/X|M z  
end ,s;Uf F  
k"w"hg&e  
 iOO)Q\  
n = n(:); Owk|@6!  
m = m(:); UFuX@Lu0  
if any(mod(n-m,2)) uu687|Pm  
    error('zernfun:NMmultiplesof2', ... 3? +Hd  
          'All N and M must differ by multiples of 2 (including 0).') !%0 * z  
end ,zY$8y]  
tIgN$BHR>  
 @F*%9LPv  
if any(m>n) f& '  
    error('zernfun:MlessthanN', ... VP]%Hni]  
          'Each M must be less than or equal to its corresponding N.') HyWCMK6b  
end *;*r 8[U}q  
y'*K|a TG  
 Tnm.A?  
if any( r>1 | r<0 ) 83q6Sv  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') #;nYg?d=  
end yz8jw:d^-  
u.Dz~$T  
 JaGtsi9%.  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) G'A R`"F  
    error('zernfun:RTHvector','R and THETA must be vectors.') wAW5 Z0D  
end @MCg%Afw  
`W*U4?M  
 9q[oa5INd  
r = r(:); z' >_Mc6  
theta = theta(:);  kPLxEwl  
length_r = length(r); <e</m)j  
if length_r~=length(theta) Y0 -n\|  
    error('zernfun:RTHlength', ... e^D]EA ]%  
          'The number of R- and THETA-values must be equal.') b1?'gn~  
end ?.;c$'  
)P|),S,;Z  
 oM`0y@QCf  
% Check normalization: 0KOgw*>_  
% -------------------- p= } Nn(  
if nargin==5 && ischar(nflag) @J`"[%U  
    isnorm = strcmpi(nflag,'norm'); ,nDaqQ-C!!  
    if ~isnorm #4 pB@_  
        error('zernfun:normalization','Unrecognized normalization flag.') B_m8{44zM  
    end OpYY{f  
else s,&Z=zt0R  
    isnorm = false; pcWPH.  
end `RL"AH:+  
WEi2=3dV  
 z~ /` 1  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *CI#+P  
% Compute the Zernike Polynomials | h#u^v3  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]3.;PWa:  
'$%l7  
 Z}Ft:7   
% Determine the required powers of r: iqQD{SRt{  
% ----------------------------------- %M|hA#04vZ  
m_abs = abs(m); -C?ZB}`   
rpowers = []; ?+}_1x`  
for j = 1:length(n) Y glmX"fLf  
    rpowers = [rpowers m_abs(j):2:n(j)]; + @s"zp;F  
end U|Ta4W`k\  
rpowers = unique(rpowers); ;@Y;g(bw:  
%/#NK1&M  
 _^%,x  
% Pre-compute the values of r raised to the required powers, ExL0?FemWV  
% and compile them in a matrix: Cd}<a?m,  
% ----------------------------- |H+UOEiv,p  
if rpowers(1)==0 &Ys<@M7E:  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); IKilr'  
    rpowern = cat(2,rpowern{:}); *mvlb (' &  
    rpowern = [ones(length_r,1) rpowern]; ={@6{-tl  
else [j/9neaye  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); hy"\RW  
    rpowern = cat(2,rpowern{:}); Od,qbU4O  
end pYmk1!]/  
57  
 ,S]7 'UP  
% Compute the values of the polynomials: =R$u[~Xl2X  
% -------------------------------------- )W _v:?A9  
y = zeros(length_r,length(n)); Iom'Y@x  
for j = 1:length(n) +E(L\  
    s = 0:(n(j)-m_abs(j))/2; <&g,Nc'5C  
    pows = n(j):-2:m_abs(j); EaY?aAuS:  
    for k = length(s):-1:1 >$/>#e~  
        p = (1-2*mod(s(k),2))* ... N]=q|D  
                   prod(2:(n(j)-s(k)))/              ... y(yHt= r  
                   prod(2:s(k))/                     ... v\%HPMlh  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... .~~T\rmI  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); a fW@T2  
        idx = (pows(k)==rpowers); 2B&3TLO  
        y(:,j) = y(:,j) + p*rpowern(:,idx); e)? .r9pA;  
    end 6H WE~`ok6  
     lE(HFal0-(  
    if isnorm wc^tgE  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); '/p/8V.O.  
    end DgQp HF  
end tGE$z]1c@  
% END: Compute the Zernike Polynomials a P@N)"  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ww+IWW@  
!")tU+:  
 k|PN0&J  
% Compute the Zernike functions: ]dmrkZz:  
% ------------------------------ K)|G0n*qS  
idx_pos = m>0; 8COGsWK  
idx_neg = m<0;  a a/(N7  
A>;bHf@  
 u$Jz~:=,  
z = y; ,pQZ@I\z  
if any(idx_pos) )e=D(qd  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); u5b|#&-mX  
end Zbt.t] N  
if any(idx_neg) S3*`jF>q  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); XZ]uUP  
end bP$dU,@p~  
.K2qXw"S#  
 1POmP&fI(  
% EOF zernfun b;W3j   
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)  XSe=sHEI  
7$vYo _  
DDE还是手动输入的呢？ Pw7]r<Q  
EyD=q! ZVZ  
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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