[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,T9xm  
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， <;}jf*A  
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？ 7A'd55I4  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ fZ!fwg$  
v3SH+Ej4  
!pY=\vK;  
[!9dA.tF  
<>\s#Jf/  
function z = zernfun(n,m,r,theta,nflag) ip6$Z3[)  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. `|@#~  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N DtkY;Yl  
%   and angular frequency M, evaluated at positions (R,THETA) on the n4
6A  
%   unit circle.  N is a vector of positive integers (including 0), and )QS4Z{)U  
%   M is a vector with the same number of elements as N.  Each element k{_ Op/k}V  
%   k of M must be a positive integer, with possible values M(k) = -N(k) %%
J)@k^vH  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, *opf~B_e  
%   and THETA is a vector of angles.  R and THETA must have the same t}r`~AEa!  
%   length.  The output Z is a matrix with one column for every (N,M) h#a;(F4_7  
%   pair, and one row for every (R,THETA) pair. *{/ ww9fT  
% M =Pn8<h~  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike |Y#KMi ~  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), j/"{tMqQp  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral b=[gK|fu  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, F&?55@b  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized pE.f}  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. -WiOs;2~/  
% +76{S_CZ  
%   The Zernike functions are an orthogonal basis on the unit circle. <s/n8#i=H  
%   They are used in disciplines such as astronomy, optics, and P&PPX#%  
%   optometry to describe functions on a circular domain. zs#s"e:jeR  
% ie4keVlXc  
%   The following table lists the first 15 Zernike functions. O
1TJJ8  
% +oKp>-  
%       n    m    Zernike function           Normalization 1n}q6oa=  
%       -------------------------------------------------- aRFLh  
%       0    0    1                                 1 UUb n7&  
%       1    1    r * cos(theta)                    2 |X&.+RI  
%       1   -1    r * sin(theta)                    2 VA4>!t)  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) 2uonT,W  
%       2    0    (2*r^2 - 1)                    sqrt(3) =@%;6`AVcp  
%       2    2    r^2 * sin(2*theta)             sqrt(6) /7WN,a  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) s|iph~W!L  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) V=yRE  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) JNhHQvi\  
%       3    3    r^3 * sin(3*theta)             sqrt(8) 6{h+(|.(  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) +Kc1a;  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Wn;B~  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) c2M-/ x-:  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {v&c5B~,\  
%       4    4    r^4 * sin(4*theta)             sqrt(10) @\-i3EhR  
%       -------------------------------------------------- zh5'oE&[yC  
% l5sBDiir%  
%   Example 1: =gI;%M\'  
% QmQsNcF~z  
%       % Display the Zernike function Z(n=5,m=1) 3w&fN3 1  
%       x = -1:0.01:1; IT,d(UV_  
%       [X,Y] = meshgrid(x,x); I5RV:e5b  
%       [theta,r] = cart2pol(X,Y); :1%z;  
%       idx = r<=1; .Q'/e>0  
%       z = nan(size(X)); ^X2U A{  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); QuuR_Ao?c'  
%       figure Uh.XL=wY  
%       pcolor(x,x,z), shading interp cG|)z<Z  
%       axis square, colorbar dc#Db~v}k  
%       title('Zernike function Z_5^1(r,\theta)') f1R&Q  
% u<8 f;C_  
%   Example 2: Jvi"K  
% @NBWNgBv  
%       % Display the first 10 Zernike functions $'$#Xn,hU  
%       x = -1:0.01:1; M6n9>aW4  
%       [X,Y] = meshgrid(x,x); Vp3 9`m-W  
%       [theta,r] = cart2pol(X,Y); f"XFf@!  
%       idx = r<=1; }7k!>+eQ  
%       z = nan(size(X));  &tb  
%       n = [0  1  1  2  2  2  3  3  3  3]; _
ED,DM  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; -9BKa~ DVQ  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; V>#iR>w_4,  
%       y = zernfun(n,m,r(idx),theta(idx)); ZLA&<]Ad"$  
%       figure('Units','normalized') ciKkazx
.  
%       for k = 1:10 EZvB#cuL-  
%           z(idx) = y(:,k); urGk_.f  
%           subplot(4,7,Nplot(k)) CbK&.a  
%           pcolor(x,x,z), shading interp $V"NB`T  
%           set(gca,'XTick',[],'YTick',[]) StUiL>9T#  
%           axis square gv=mz,z  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _Q<wb8+/  
%       end by*>w/@9)k  
% 7?6?`no~JJ  
%   See also ZERNPOL, ZERNFUN2. ]h (TZu  
^+Ez[S{8  
/'|'3J]HP  
%   Paul Fricker 11/13/2006 w, 0tY=h6  
]+\@_1<ZI  
MFHPh8P  
YxMOr\B  
Peha{]U  
% Check and prepare the inputs: jE)&`yZ5  
% ----------------------------- D .3Q0a6  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) B`Q.<Lqu  
    error('zernfun:NMvectors','N and M must be vectors.') k*b
fq?E a  
end 4XL*e+UfJ  
$)| l#'r  
VQHJO I  
if length(n)~=length(m) DM6oMT  
    error('zernfun:NMlength','N and M must be the same length.') 5qco4@8  
end NLDmZr
a  
4!lbwqo  
-&Fxg>FrYb  
n = n(:); @+",f]  
m = m(:); )>LQ{X.  
if any(mod(n-m,2)) ?WWnt^  
    error('zernfun:NMmultiplesof2', ... ?{#P.2  
          'All N and M must differ by multiples of 2 (including 0).') sg12C  
end i|>
K  
W38My j!  
~p~8T  
if any(m>n) u(JC 4w'  
    error('zernfun:MlessthanN', ... qs6yEuh#  
          'Each M must be less than or equal to its corresponding N.') jIMaPT  
end *BVkviqxz  
x8p#WB  
ssW+'GD  
if any( r>1 | r<0 ) uF>I0J#z?  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') ;VS;),h/  
end R!xs;|]  
b:7;zOtF  
JJ56d)37.  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 8?W!U*0aS  
    error('zernfun:RTHvector','R and THETA must be vectors.') 9\*xK%T+  
end wgSA6mQZ  
pTZPOv#?Q  

,[+  
r = r(:); VL"ZC:n)-  
theta = theta(:); !mpRLBH  
length_r = length(r); IoNZ'g?d  
if length_r~=length(theta) P*/px4;6  
    error('zernfun:RTHlength', ... !-r@_tn|  
          'The number of R- and THETA-values must be equal.') >H@ dgb  
end e =& abu  
Z~
g~,q  
VS^%PM#:/  
% Check normalization: uc%75TJ@  
% -------------------- W<;i~W
 
if nargin==5 && ischar(nflag) EA75 D&>I  
    isnorm = strcmpi(nflag,'norm'); ;^:~xJFx|  
    if ~isnorm 'q1)W'  
        error('zernfun:normalization','Unrecognized normalization flag.') J),7ukLu^  
    end .CI]8O"3y  
else uW4G!Kw28  
    isnorm = false; %-]j;'6}cX  
end <(d^2-0  
2Iz@lrO6  
t=S94
^g  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ".v9#|  
% Compute the Zernike Polynomials iUA2/ A  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X=(8t2  
^/R@bp#<  
$ sEe0  
% Determine the required powers of r: ZERUvk  
% ----------------------------------- 9`.b   
m_abs = abs(m); -O~WHi5}  
rpowers = []; `(=)8>|e  
for j = 1:length(n) Du$kDCU  
    rpowers = [rpowers m_abs(j):2:n(j)]; H` Q_gy5Z(  
end ]G&?e9OA  
rpowers = unique(rpowers); 4_PMl6qo  
N&S:=x:$S  
/lttJJDU  
% Pre-compute the values of r raised to the required powers, D.qbzJz  
% and compile them in a matrix: S~YrXQ{_>-  
% ----------------------------- xQ1&j,R]  
if rpowers(1)==0 R
NoS7[&  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); -sO EL{  
    rpowern = cat(2,rpowern{:}); :@_CQc*yB  
    rpowern = [ones(length_r,1) rpowern]; H|F>BjXn5  
else
|\?-k  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); S_c#{4n  
    rpowern = cat(2,rpowern{:}); +ls *04  
end ReKnvF~  
}5OlX  
S?hM  
% Compute the values of the polynomials: }'kk}2ej`  
% -------------------------------------- Zi7(lG  
y = zeros(length_r,length(n)); Fxv~;o#  
for j = 1:length(n) k6[t$|lMy  
    s = 0:(n(j)-m_abs(j))/2; :+]6SC0ql  
    pows = n(j):-2:m_abs(j); rVQ:7\=Z  
    for k = length(s):-1:1 {+ [rJ_  
        p = (1-2*mod(s(k),2))* ... `{F8#    
                   prod(2:(n(j)-s(k)))/              ... Gpe h#Q4x  
                   prod(2:s(k))/                     ... X@x: F|/P  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... X/5tZ@  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); 3zWY%(8t4?  
        idx = (pows(k)==rpowers); ?Dd2k%o  
        y(:,j) = y(:,j) + p*rpowern(:,idx); zCO5`%14  
    end w'M0Rd]  
     c)@M7UK[  
    if isnorm jE2ziK  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); b^Rg_,s  
    end }qV4]*+{  
end .vQ2w  
% END: Compute the Zernike Polynomials ]3 0 7.  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L$@RSKYp  
$Ae/NwIlc  
6EX:qp^`  
% Compute the Zernike functions: N@Slc 0  
% ------------------------------ )4GfT  
idx_pos = m>0; 1Lj\"+.  
idx_neg = m<0; #J2856bzS  
Ks7s2vK^  
v%zI~g.L  
z = y; ~&B_ Bswf  
if any(idx_pos) uT;Qo{G^  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); L>@0Nne7  
end dUjdQ  
if any(idx_neg) H7qda'%>  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Mv4JF(,S  
end J=4S\0Z*  
+#&2*nY  
d9Rj-e1x  
% EOF zernfun HLk}E*.mC  

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)  k`TEA?RfQ  

t0?BU~f  
DDE还是手动输入的呢？ ;J?!D x  
0BVMLRB  
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。