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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,  J(  
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, U '#Xwax  
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? B3lP#ckh  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? nYcj6?  
7( &\)qf=n  
9"=1 O  
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function z = zernfun(n,m,r,theta,nflag) Wy[Ua#Dd  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. (W~')A"hC'  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N @nuMl5C-`  
%   and angular frequency M, evaluated at positions (R,THETA) on the =q%Q^  
%   unit circle.  N is a vector of positive integers (including 0), and }'y=JV>l  
%   M is a vector with the same number of elements as N.  Each element JFyw,p&xB  
%   k of M must be a positive integer, with possible values M(k) = -N(k) Eu4-=2!4  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Lad8C  
%   and THETA is a vector of angles.  R and THETA must have the same &.zG?e.  
%   length.  The output Z is a matrix with one column for every (N,M) J\kGD  
%   pair, and one row for every (R,THETA) pair. (^:0g.~c  
% Y-Zw'  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 6z6\-45  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 5@YrtZI  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral `Dck$  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1,  =V- ^  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized 6%hr]>L  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. m0I)_R#X[  
% X7cqAi  
%   The Zernike functions are an orthogonal basis on the unit circle. m zh8<w?ns  
%   They are used in disciplines such as astronomy, optics, and )O xsasn)M  
%   optometry to describe functions on a circular domain. K/Q%tr1W0  
% VO (KQx  
%   The following table lists the first 15 Zernike functions. y_>l'{w3^  
% V__|NVoOm  
%       n    m    Zernike function           Normalization Y ,Iv<Hg  
%       -------------------------------------------------- m#\I&(l+  
%       0    0    1                                 1 (;9-8Y&_d  
%       1    1    r * cos(theta)                    2 ; i)NP X  
%       1   -1    r * sin(theta)                    2 x6i7x"  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) #%~PNki  
%       2    0    (2*r^2 - 1)                    sqrt(3) DLigpid  
%       2    2    r^2 * sin(2*theta)             sqrt(6) @O!BQ^'hk#  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) |XDbf3^6  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) +b =X~>vZ  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) pk%%}tP<  
%       3    3    r^3 * sin(3*theta)             sqrt(8) u<VR;p:y  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) 4"om;+\  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) | ODi[~y  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) V0rS^SAF  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ftr8~*]O  
%       4    4    r^4 * sin(4*theta)             sqrt(10) eWN[EJI<  
%       -------------------------------------------------- u8w4e!rKo6  
% AYVkJq?  
%   Example 1: mMNT.a  
% o*fNY  
%       % Display the Zernike function Z(n=5,m=1) %nRz~3X|+v  
%       x = -1:0.01:1; IL]Js W  
%       [X,Y] = meshgrid(x,x); BF="gZoU<  
%       [theta,r] = cart2pol(X,Y); .T>^bLuFy  
%       idx = r<=1; U#qs^f7R  
%       z = nan(size(X)); '%W`:K'  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); m@Q%)sc)  
%       figure !OCb^y  
%       pcolor(x,x,z), shading interp !08\w@  
%       axis square, colorbar G1=/G  
%       title('Zernike function Z_5^1(r,\theta)') T{USzMj  
% kBRy(?Mft&  
%   Example 2: wNfWHaH" m  
% %Kk MWl&:  
%       % Display the first 10 Zernike functions {:63% j  
%       x = -1:0.01:1;  )S8fFV  
%       [X,Y] = meshgrid(x,x); `y^tCJ2u*  
%       [theta,r] = cart2pol(X,Y); ~S85+OJ;M  
%       idx = r<=1; !.EDQ1k  
%       z = nan(size(X)); hhM?I$t:  
%       n = [0  1  1  2  2  2  3  3  3  3]; h1(GzL%i_  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; [ T6MaP?  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; l0v]+>1i:  
%       y = zernfun(n,m,r(idx),theta(idx)); o^\L41x3  
%       figure('Units','normalized') j><8V Qx  
%       for k = 1:10 4Odf6v,*@  
%           z(idx) = y(:,k); Z ? F*Z0y  
%           subplot(4,7,Nplot(k)) ?BEO(;'  
%           pcolor(x,x,z), shading interp -~|E(ys  
%           set(gca,'XTick',[],'YTick',[]) OgOs9=cE{  
%           axis square zm&?G  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ev LZ<|  
%       end PCcI(b>?l  
% 0ECQ>Ux:  
%   See also ZERNPOL, ZERNFUN2. b~u53   
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%   Paul Fricker 11/13/2006 8rNRQOXOa  
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% Check and prepare the inputs: Q2 S!}A  
% ----------------------------- ^h5h kIx0  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) b-"kclK  
    error('zernfun:NMvectors','N and M must be vectors.') OngUZMgdb  
end q qFN4AO  
H7}@56  
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if length(n)~=length(m) # VAL\Z  
    error('zernfun:NMlength','N and M must be the same length.') DZ*m"Bi  
end "/~KB~bB  
t91z<Y|  
tDQo1,(oY  
n = n(:); 6$ \69   
m = m(:); dSkx*#FEE  
if any(mod(n-m,2)) : 6|nXL  
    error('zernfun:NMmultiplesof2', ... O ,Sqh$6U  
          'All N and M must differ by multiples of 2 (including 0).') w dpd`  
end 8IQqDEY^  
q Xj]O3 mm  
'X(Sn3  
if any(m>n) 4^u wZ:  
    error('zernfun:MlessthanN', ... `}o{o  
          'Each M must be less than or equal to its corresponding N.') |j w{7\+  
end +j!$88%Z{  
6EY W:o  
-q")qNt.  
if any( r>1 | r<0 ) OY'6~w9  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') 0\tdxi  
end <%ZlJ_cM  
<)4>"SN&^  
A` )A=L  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ;>jLRx<KC  
    error('zernfun:RTHvector','R and THETA must be vectors.') +h?Rb3=S  
end %&\DCAFk  
N&YQZ^o  
dxk~  
r = r(:); j#D( </T  
theta = theta(:); f)9{D[InM^  
length_r = length(r); c[zaYcbl  
if length_r~=length(theta) qV&ai{G:  
    error('zernfun:RTHlength', ... b[;Zl<  
          'The number of R- and THETA-values must be equal.') ~H\1dCW  
end f J$>VN  
M?I^Od'8  
I>n2# -8  
% Check normalization: &O;' ?/4 S  
% -------------------- cK _:?G  
if nargin==5 && ischar(nflag) _a&Mk  
    isnorm = strcmpi(nflag,'norm'); M?G4k]  
    if ~isnorm &br_opNi  
        error('zernfun:normalization','Unrecognized normalization flag.') geJO#;  
    end GGF;4  
else .~jn N  
    isnorm = false; ?I}0[+)V  
end { cMf_qQ  
#1VejeTi  
y>iote~  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Y1s3 >`  
% Compute the Zernike Polynomials ;UoXj+Z  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GkYD:o=qx  
 Zzea  
jdW#; ]7+y  
% Determine the required powers of r: }v$T1Cw  
% ----------------------------------- J]mq|vE  
m_abs = abs(m); a/s6|ri`0  
rpowers = []; A;e0h)F$-  
for j = 1:length(n) \h UE, ^  
    rpowers = [rpowers m_abs(j):2:n(j)]; $,DX^I%!  
end ~h@<14c{X  
rpowers = unique(rpowers); f3M~2jbv'p  
hJasnY7  
q A?j-H  
% Pre-compute the values of r raised to the required powers, &Rxy]kBA  
% and compile them in a matrix: w?Nx ^)xX  
% ----------------------------- RQt\_x7P  
if rpowers(1)==0 avwhGys#  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); r5(-c]E7  
    rpowern = cat(2,rpowern{:}); <h<4R Rj  
    rpowern = [ones(length_r,1) rpowern]; uU#7SX(uu  
else FEq R7  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); .BqS E   
    rpowern = cat(2,rpowern{:}); GFmVR2z_+  
end `|d&ta[{  
.X(*mmH  
=sa bJsgL  
% Compute the values of the polynomials: 50J"cGs~  
% -------------------------------------- Lf} @v  
y = zeros(length_r,length(n)); m(c5g[6nO  
for j = 1:length(n) WeqQw?-  
    s = 0:(n(j)-m_abs(j))/2; Bvy(vc=UDW  
    pows = n(j):-2:m_abs(j); Kl)PF),  
    for k = length(s):-1:1 6yRxb (  
        p = (1-2*mod(s(k),2))* ... +5C*i@v  
                   prod(2:(n(j)-s(k)))/              ... G: p!PB>=  
                   prod(2:s(k))/                     ... i~04P  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... IsjD-t  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); ]'a9>o  
        idx = (pows(k)==rpowers); *[?DnF+  
        y(:,j) = y(:,j) + p*rpowern(:,idx); J;S@Q/s  
    end 5 qW*/  
     @vcvte  
    if isnorm `fUem,$)1F  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); tzFgPeo$;  
    end p"XQJUuD  
end  _8G  
% END: Compute the Zernike Polynomials I];Hx'/<~  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V3]"ROH  
'5vgpmn  
E*_lT`Hzf  
% Compute the Zernike functions: QA3q9,C"  
% ------------------------------ u  teI[Q  
idx_pos = m>0; QX$i ]y%S  
idx_neg = m<0; s]p3dB#  
L>qLl_.  
;v?!Pml2k  
z = y; 6 NJ5v +  
if any(idx_pos) c>*RQ4vE  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); <Hq|<^_K  
end t6)wR  
if any(idx_neg) ^KsiTVY  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); !Xbr7:UPN1  
end @I '_  
@pKQ}?  
!a[1rQH  
% EOF zernfun h6dVT9  
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)  gsn3]^X  
5'/Ney9N  
DDE还是手动输入的呢? SSKn7`  
BpL,<r,  
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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