[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， #BS]wj2#  
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， :EgdV  
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？ PL~k `L  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ UShn)3F  
R,Zuy(g  
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function z = zernfun(n,m,r,theta,nflag) 3\=8tg p  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. C*Ws6s>+z  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N IX7d[nm39  
%   and angular frequency M, evaluated at positions (R,THETA) on the mMN oR]  
%   unit circle.  N is a vector of positive integers (including 0), and C,2IE
T  
%   M is a vector with the same number of elements as N.  Each element ~$r^Ur!E\  
%   k of M must be a positive integer, with possible values M(k) = -N(k) ^e@c Ozt  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, W}L=JJo},  
%   and THETA is a vector of angles.  R and THETA must have the same lG#&Pv>-  
%   length.  The output Z is a matrix with one column for every (N,M) sbK0OA  
%   pair, and one row for every (R,THETA) pair. q4Ye  
% /oiAAB27  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike %J.Rm0F
D:  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), W\eB   
%   with delta(m,0) the Kronecker delta, is chosen so that the integral !c6lP'U  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3tXtt@Yy  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized z*yN*M6t  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. !FHNKh  
% ](MXP,R  
%   The Zernike functions are an orthogonal basis on the unit circle.
5\|[)~b  
%   They are used in disciplines such as astronomy, optics, and }QJE9;<e  
%   optometry to describe functions on a circular domain. Y2<#%@%4  
% *<9D]  
%   The following table lists the first 15 Zernike functions. J=zZGd%  
% nWXI*%m5  
%       n    m    Zernike function           Normalization K:'pK1zy  
%       -------------------------------------------------- |lJXI:GG  
%       0    0    1                                 1 ?'T>/<(  
%       1    1    r * cos(theta)                    2 00;=6q]TA  
%       1   -1    r * sin(theta)                    2 $6y1';A  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) D<xPx  
%       2    0    (2*r^2 - 1)                    sqrt(3) .\U+`>4av  
%       2    2    r^2 * sin(2*theta)             sqrt(6) ybS7uo  
%       3   -3    r^3 * cos(3*theta)             sqrt(8)  ~-
M7  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) bO2$0!=I  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) EiJSLL  
%       3    3    r^3 * sin(3*theta)             sqrt(8) T$}<So|  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) f[|xp?ef  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) d=>5%$:v  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) :hMuxHr  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |qI_9#M\(  
%       4    4    r^4 * sin(4*theta)             sqrt(10) %J|EDf,M  
%       -------------------------------------------------- &q":o 'q  
% #G*z{BRQ  
%   Example 1: gVG :z_6  
% i}wu+<Mk  
%       % Display the Zernike function Z(n=5,m=1) <EBp X  
%       x = -1:0.01:1; H[>_LYZ8  
%       [X,Y] = meshgrid(x,x); }1_gemlf  
%       [theta,r] = cart2pol(X,Y); i(c2NPbX  
%       idx = r<=1; MH !CzV&  
%       z = nan(size(X)); 5lU`o  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); 6eS#L21*  
%       figure B1Lnu
B%  
%       pcolor(x,x,z), shading interp r`S]`&#}(  
%       axis square, colorbar hlUF9
}  
%       title('Zernike function Z_5^1(r,\theta)')  bM-Y4[  
% k*-+@U"+  
%   Example 2: ?<nz2 piP,  
% }>Os@]*'^(  
%       % Display the first 10 Zernike functions KO5
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%       x = -1:0.01:1; DJ<c  
%       [X,Y] = meshgrid(x,x); 'm2,7]  
%       [theta,r] = cart2pol(X,Y); cA/2,i  
%       idx = r<=1; ^ g4)aaBZ  
%       z = nan(size(X)); s#d# *pgzh  
%       n = [0  1  1  2  2  2  3  3  3  3]; *g=*}
2  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Q]|+Y0y}X  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; VS}Vl  
%       y = zernfun(n,m,r(idx),theta(idx)); !4 hs9b  
%       figure('Units','normalized') Aga7X@fV(  
%       for k = 1:10 _aDx('  
%           z(idx) = y(:,k); ~Yr.0i.W  
%           subplot(4,7,Nplot(k)) N@A#e/8  
%           pcolor(x,x,z), shading interp Jhj]rsGk  
%           set(gca,'XTick',[],'YTick',[]) [{zekF~)@  
%           axis square qlgh$9  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) <v2R6cj5  
%       end {;-$;\D  
% 2XXEg>CU  
%   See also ZERNPOL, ZERNFUN2. >K &b,o,[  
u5,IH2BU  
O!j@8~='  
%   Paul Fricker 11/13/2006 $K,aLcu  
:JN3@NsK  
d@w I: 7  
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Lte\;Se.tu  
% Check and prepare the inputs: F;_;lRAb  
% ----------------------------- u#P7~9ZG-  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) '8Gw{&&  
    error('zernfun:NMvectors','N and M must be vectors.') j2\G1@05  
end t*<c+Ixu  
XG[%oL  
1/fvk  
if length(n)~=length(m) nut7b  
    error('zernfun:NMlength','N and M must be the same length.') ,>g 6OU2~6  
end *Z0}0< D@Z  
5$#<z1M.&  
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n = n(:); Ec[:6}  
m = m(:); A%2!Hr  
if any(mod(n-m,2)) xj~6,;83xR  
    error('zernfun:NMmultiplesof2', ... {Ise (>V  
          'All N and M must differ by multiples of 2 (including 0).') ^{Vm,nAQqs  
end stDn{x.
 
Th

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[cH/Y2[  
if any(m>n) mb/3 #)  
    error('zernfun:MlessthanN', ... .DX#:?@4@Y  
          'Each M must be less than or equal to its corresponding N.') ~kHir]jc
 
end %EpK=;51U  
3gM{lS}h#  
E?zp?t:a  
if any( r>1 | r<0 ) H}$#aXEAn  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') lu{}j4  
end P*LcWrK  
'0=U+Egp  
l-Xx
v  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) NMDNls&)k  
    error('zernfun:RTHvector','R and THETA must be vectors.') 7]^Cg;EtM:  
end eGE%c1H9a  
|'J3"am'  
hh?'tb{  
r = r(:); =?2y <B  
theta = theta(:); lfKknp#B/O  
length_r = length(r); *H$
nydQ:  
if length_r~=length(theta) /qCYNwWH9  
    error('zernfun:RTHlength', ... H{V-C_  
          'The number of R- and THETA-values must be equal.') G]SE A  
end PU>;4l  
m=K XMX  
:vr,@1c  
% Check normalization: 2ReulL8j  
% -------------------- kj8zWG4KH  
if nargin==5 && ischar(nflag) \uYUX~}i"  
    isnorm = strcmpi(nflag,'norm'); =MXF`k^}  
    if ~isnorm <V,?!}V  
        error('zernfun:normalization','Unrecognized normalization flag.') !Q#b4f  
    end 3xe8DD  
else eS"gHldz  
    isnorm = false; OBZ|W**N"  
end a~%ej.)l  
A/
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<|8l;  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m^=, RfUUd  
% Compute the Zernike Polynomials X1-s,[j'  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% oY]VP+b!  
{)[i\=,`{  
-3V~YhG  
% Determine the required powers of r: <,GHy/u\  
% ----------------------------------- q5A+%#  
m_abs = abs(m); -r2cK{Hhp&  
rpowers = []; wx!*fy4hL  
for j = 1:length(n) d#*5U9\z  
    rpowers = [rpowers m_abs(j):2:n(j)]; zm:=d>D..  
end e!8_3BE  
rpowers = unique(rpowers); 6?lg 6a/eO  
Yyo|W;a]  
{aL$vgYT1  
% Pre-compute the values of r raised to the required powers, 98]t"ny [  
% and compile them in a matrix: <Z;7=k  
% ----------------------------- G225Nz;Y*  
if rpowers(1)==0 `SW " RLS3  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); GKSy|z  
    rpowern = cat(2,rpowern{:}); RmQt%a7\{  
    rpowern = [ones(length_r,1) rpowern]; VB#31T#q?  
else vP4Ij  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); cg.e(@(  
    rpowern = cat(2,rpowern{:}); ^ZlV1G;/W@  
end g#
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16y$;kf8  
% Compute the values of the polynomials: kziBHis!  
% -------------------------------------- #R8l"]fxr?  
y = zeros(length_r,length(n)); ]Yu+M3Fq  
for j = 1:length(n) -FR;:  
    s = 0:(n(j)-m_abs(j))/2; vw]nqS~N  
    pows = n(j):-2:m_abs(j); D5>~'N3b  
    for k = length(s):-1:1 <f6PULm  
        p = (1-2*mod(s(k),2))* ... tb{{oxa,k  
                   prod(2:(n(j)-s(k)))/              ... _pGviGR  
                   prod(2:s(k))/                     ... FNM"!z  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... >l1Yhxd_0*  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); h%s  
        idx = (pows(k)==rpowers); T/;hIX:R  
        y(:,j) = y(:,j) + p*rpowern(:,idx); \.a .'l  
    end nc~d*K\!  
     [J`G`s!  
    if isnorm Zsogx}i-  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); B|=maz:_  
    end N]}+F w\5  
end +I n"OR%  
% END: Compute the Zernike Polynomials 2S6EDXc  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ug,|'<G+  
RG3G},Q   
t"p#iia  
% Compute the Zernike functions: wKlCx  
% ------------------------------ yTt (fn:;  
idx_pos = m>0; } XU:DE  
idx_neg = m<0; --YUiNh
h  
S1`0d9ds#  
&U*J{OP
|  
z = y; BDRVT Y(s  
if any(idx_pos) ()#tR^T  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); &i^NStqu  
end ?1:/ 6  
if any(idx_neg) @!/fvP  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); DB%AO:8  
end pf@}4PN}  
(I ds<n"  
VQ<i$ I  
% EOF zernfun zlztF$Bo  

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)  Ld\LKwo  

/lr1hW~Dbk  
DDE还是手动输入的呢？ yE.495  
sb}K%-  
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。