[讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

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，  C
~T*Wlk  
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， SZ~lCdWad  
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？ eQ<Vky^SJ  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ nxe9^h7m  
':]Hj8t_  
t\f[->f  
Av!xI  
'u6n,yRm  
function z = zernfun(n,m,r,theta,nflag) d2Ta&Md  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ywA7hm  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N HJt '@t=Ak  
%   and angular frequency M, evaluated at positions (R,THETA) on the AYfL}X<Ig  
%   unit circle.  N is a vector of positive integers (including 0), and b"w@am>&  
%   M is a vector with the same number of elements as N.  Each element |qpFR)l  
%   k of M must be a positive integer, with possible values M(k) = -N(k) D/+l$aBz  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, f( <O~D  
%   and THETA is a vector of angles.  R and THETA must have the same K?>sP%m)  
%   length.  The output Z is a matrix with one column for every (N,M) co-1r/ -O  
%   pair, and one row for every (R,THETA) pair. Cng_*\=O  
% 4<Kxo\\S  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike FmgMd)#  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), WAJKP"  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral jtgj h\Nt  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, :"cKxd  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized S2>$S^[U  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. MhIHfW]b  
% dF*M"|
[  
%   The Zernike functions are an orthogonal basis on the unit circle. B_>r|^Vh  
%   They are used in disciplines such as astronomy, optics, and eo^C[# .  
%   optometry to describe functions on a circular domain. l[[^]__  
% QwL*A `@  
%   The following table lists the first 15 Zernike functions. v>_83P`  
% ~RV"_8`V9  
%       n    m    Zernike function           Normalization z>)lp$  
%       -------------------------------------------------- oWEzzMRz  
%       0    0    1                                 1 S3&n?\CO:  
%       1    1    r * cos(theta)                    2 yQf(/Uxk*x  
%       1   -1    r * sin(theta)                    2 .@$A~/ YU  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) )>@%;\qV  
%       2    0    (2*r^2 - 1)                    sqrt(3) #Y'ewu;qJ  
%       2    2    r^2 * sin(2*theta)             sqrt(6) i`=%
X{9  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) LIT`~D  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Z/d {v:)  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Y(gai?  
%       3    3    r^3 * sin(3*theta)             sqrt(8) @WiTh'w0  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) TeFi[1  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) syCT)}T6z  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) WJMmt XO  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;
te( {u+  
%       4    4    r^4 * sin(4*theta)             sqrt(10) Q:Ma3El\  
%       -------------------------------------------------- tlB-s;  
% `26.+>Z7  
%   Example 1: v#e*RI2}  
% uPE Ab2u="  
%       % Display the Zernike function Z(n=5,m=1) <C451+95  
%       x = -1:0.01:1; q*kLi~Oe  
%       [X,Y] = meshgrid(x,x); .o]9 HbIk5  
%       [theta,r] = cart2pol(X,Y); Y*IKPnPot2  
%       idx = r<=1; n3j_=(  
%       z = nan(size(X)); (LJ7xoJ^  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); ?Ezy0>j  
%       figure +O^} t  
%       pcolor(x,x,z), shading interp Gte\=0Wr  
%       axis square, colorbar Ihv@2{*(b  
%       title('Zernike function Z_5^1(r,\theta)') D !{e  
% CeM%?fr5  
%   Example 2: }pGjc_:']  
% "=LeHY=9  
%       % Display the first 10 Zernike functions K(HrwH`a{  
%       x = -1:0.01:1; l dp$jrNLr  
%       [X,Y] = meshgrid(x,x); =woP~+  
%       [theta,r] = cart2pol(X,Y); /F6"uZSt4  
%       idx = r<=1; q_98=fyE6  
%       z = nan(size(X)); Q<KF<K'0hg  
%       n = [0  1  1  2  2  2  3  3  3  3]; f4&;l|R0a  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ?FwHqyFVlQ  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; GVfRy@7n  
%       y = zernfun(n,m,r(idx),theta(idx)); w9n0p0xr<  
%       figure('Units','normalized') Ya(3Z_f+VZ  
%       for k = 1:10 &Pc.[k  
%           z(idx) = y(:,k); m/,80J8L+f  
%           subplot(4,7,Nplot(k)) +ej5C:El_}  
%           pcolor(x,x,z), shading interp h<8c{RuoZC  
%           set(gca,'XTick',[],'YTick',[]) C](djkA$  
%           axis square wQ[!~>A  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 9+/D\|"{  
%       end \HG4i/V:h  
% 1_l)$"  
%   See also ZERNPOL, ZERNFUN2. /a)^)  
N(3Bzd)  
'Gamb+[  
%   Paul Fricker 11/13/2006 PZO.$'L|7  
Cl3L)  
t=|}?lN<  
Qvel#*-4  
L\5:od[EP  
% Check and prepare the inputs: TjI&8#AWBA  
% ----------------------------- '-Oh$hqCx|  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) W39J)~D^@  
    error('zernfun:NMvectors','N and M must be vectors.') 2#
#mVEo.(  
end G9GHBwT  
f6nuh&!-  
hpYv*WH:  
if length(n)~=length(m) 4mtO"'|  
    error('zernfun:NMlength','N and M must be the same length.') TBky+]p@  
end .m
cohfR  
-$_FKny  
aof'shS8  
n = n(:); N9s.nu  
m = m(:); Z'l!/l!  
if any(mod(n-m,2)) :RwURv+kT  
    error('zernfun:NMmultiplesof2', ... PgHmOs  
          'All N and M must differ by multiples of 2 (including 0).') !ZvVj\{  
end HKJ^6|'  
3<N2ehi?  
2oOos%0  
if any(m>n) X.FoX  
    error('zernfun:MlessthanN', ... c5:0`~5Fn  
          'Each M must be less than or equal to its corresponding N.') l!W!Gz0to  
end _MuzD&^qE  
UEt78eN  
H8B2{]HAt  
if any( r>1 | r<0 ) `T{CB) ?9  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') N}<!k#d E  
end Iza;~8dH5  
s&Al4>}.f  
@ &rf?:  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) j]`hy"  
    error('zernfun:RTHvector','R and THETA must be vectors.') Gpcordt/  
end qn{4AWmJ  
Ciz,1IV  
13)6p|6x  
r = r(:); 6@3v+Vf'  
theta = theta(:); b$_qG6)IJO  
length_r = length(r); j9GKz1
 
if length_r~=length(theta) .*xO/pn  
    error('zernfun:RTHlength', ... 7GG`9!l]D  
          'The number of R- and THETA-values must be equal.') 8 nqF i  
end #3eI4KJ4+l  
mG\9Qkom|  
;]=@;? 9  
% Check normalization: W$&*i1<a+  
% -------------------- R
>1oF]w  
if nargin==5 && ischar(nflag) #7]>ozKm  
    isnorm = strcmpi(nflag,'norm'); ="f-I9y  
    if ~isnorm vpOGyvI  
        error('zernfun:normalization','Unrecognized normalization flag.') Pth4_]US  
    end +ZGH  
else mA_EvzXk\  
    isnorm = false; <<Y]P+uU  
end ;=E}PbZt2  
RBg2iG$8|  
~m0=YAlk?  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S4_ZG>\VT  
% Compute the Zernike Polynomials *f{4_ts  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yB=R7E7  
zf5%|7o  
O U9{Y9e  
% Determine the required powers of r: yd'cLZd<}  
% ----------------------------------- 5p:2gsk  
m_abs = abs(m); YcR: _ac  
rpowers = []; rM6S%rS  
for j = 1:length(n) ;05lwP*r]  
    rpowers = [rpowers m_abs(j):2:n(j)]; Z![#Uz.z  
end yp@cn(:~  
rpowers = unique(rpowers); 9$VdYw7D  
-e
m3 #V  
b j<T`M!  
% Pre-compute the values of r raised to the required powers, 7~ZG"^k  
% and compile them in a matrix: kkj@!1q(wO  
% ----------------------------- R$MR|
 
if rpowers(1)==0 UYQ@ub  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); HM"(cB(n`  
    rpowern = cat(2,rpowern{:}); rq1~%S  
    rpowern = [ones(length_r,1) rpowern]; 6& hiW]Adm  
else 8{{^pW?x  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); */TO$ ^s  
    rpowern = cat(2,rpowern{:}); b}u#MU  
end -xJ\/"A  
kI5LG6  
<Nc9F['&#  
% Compute the values of the polynomials: `ZP[-:`  
% -------------------------------------- ]^{5`  
y = zeros(length_r,length(n)); KVViTpZ  
for j = 1:length(n) 4"{g{8  
    s = 0:(n(j)-m_abs(j))/2; 2"P1I
  
    pows = n(j):-2:m_abs(j); ?V_v=X%w  
    for k = length(s):-1:1 >SYOtzg%  
        p = (1-2*mod(s(k),2))* ... I<xcVY9L  
                   prod(2:(n(j)-s(k)))/              ... !VrBoU4<d  
                   prod(2:s(k))/                     ... c\tw#;\9  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ?6I`$ &OA  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); rfZg  
        idx = (pows(k)==rpowers); *]kE3  
        y(:,j) = y(:,j) + p*rpowern(:,idx); Yx ;j  
    end 0&r}
'f?  
     `fVzY"Qv k  
    if isnorm  TL
Vfu4  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); *0'{n*>  
    end Esg:  
end qzo)\,  
% END: Compute the Zernike Polynomials -ucR@P]  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #}Ays#wA>?  
a{?>F&vnU  
6jl{^dI  
% Compute the Zernike functions:
(
m.jC}J  
% ------------------------------ 8@T0]vH&  
idx_pos = m>0; F1
`mq2^@  
idx_neg = m<0; =aehhs>  
PM {L}tEQ  
~ r$I&8  
z = y; MUN:}S  
if any(idx_pos) >4#\ U!  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); otP2qAI  
end )*o) iN 7l  
if any(idx_neg) 5=4-IO6W[]  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Ja@?.gW  
end ZQ[s:  
Ww{-(Ktx  
2Paw*"U  
% EOF zernfun 1fF\k#BE-%  

支持一下，慢慢研究

顶顶·········

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)  ]7{-HuQ8>}  

uw@|Y{(K r  
DDE还是手动输入的呢？ O8ZHI
s  
!b+Kasss9  
zygo和zemax的zernike系数，类型对应好就没问题了吧

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

慢慢研究，这个专业性很强的。用的人又少。