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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, pd?3_yU  
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, EC<g7_0F  
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? m}54yo  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Fo|6 PoSo  
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function z = zernfun(n,m,r,theta,nflag) s5F,*<  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. sOhQu>gN  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N s"7wG!yf  
%   and angular frequency M, evaluated at positions (R,THETA) on the :G=N|3  
%   unit circle.  N is a vector of positive integers (including 0), and u 0(H!  
%   M is a vector with the same number of elements as N.  Each element oKLL~X>!U  
%   k of M must be a positive integer, with possible values M(k) = -N(k) ]vo&NE  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 52 ? TLID  
%   and THETA is a vector of angles.  R and THETA must have the same ~Gx"gK0  
%   length.  The output Z is a matrix with one column for every (N,M) ..`J-k  
%   pair, and one row for every (R,THETA) pair. !B5 }`*1D  
% _Q_"_*e  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike P~HzN C  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ,_ }  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral F)Oe;z6  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, d:cs8f4>  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized "#anL8  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. }F=scbpXj  
% dtD)VNkBZ  
%   The Zernike functions are an orthogonal basis on the unit circle. 9|R]Lz3PA  
%   They are used in disciplines such as astronomy, optics, and 9. :r;HG  
%   optometry to describe functions on a circular domain. q$6fb)2I]e  
% ^Tgu]t   
%   The following table lists the first 15 Zernike functions. s ~c_9,JK  
% 3 (Bd`=9  
%       n    m    Zernike function           Normalization Q}zAC2@L  
%       -------------------------------------------------- MHar9)$}  
%       0    0    1                                 1 BV_rk^}Ur  
%       1    1    r * cos(theta)                    2 I-<U u 2  
%       1   -1    r * sin(theta)                    2 <mN.6@*{  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) |+K3\b  
%       2    0    (2*r^2 - 1)                    sqrt(3) /R\]tl#2j  
%       2    2    r^2 * sin(2*theta)             sqrt(6) =8:m:Y&|`G  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) EX zA(igS  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) mhU ?N  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) *Y'nDv6_P  
%       3    3    r^3 * sin(3*theta)             sqrt(8) "O@L IR7  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) TN!8J=sx.  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .;nU" a3'  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) qYjR  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) X <QSi   
%       4    4    r^4 * sin(4*theta)             sqrt(10) BSU%.tmI  
%       -------------------------------------------------- .H;[s  
% $3.hZx>  
%   Example 1: [HNWM/ff7+  
% R{={7.As+  
%       % Display the Zernike function Z(n=5,m=1) 0* 7N=  
%       x = -1:0.01:1; `P/7Mf  
%       [X,Y] = meshgrid(x,x); P| c[EUT  
%       [theta,r] = cart2pol(X,Y); B q/<kEgM  
%       idx = r<=1; ,c }R*\  
%       z = nan(size(X)); =SMI,p&  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); $hv o^$  
%       figure 7tbM~+<0  
%       pcolor(x,x,z), shading interp v',%   
%       axis square, colorbar 'VVEd[  
%       title('Zernike function Z_5^1(r,\theta)') "`WcE/(  
% -36pkC 6 \  
%   Example 2: _OR@S%$  
% pHO,][VZ  
%       % Display the first 10 Zernike functions &USKudXmb  
%       x = -1:0.01:1; 8['8ctX  
%       [X,Y] = meshgrid(x,x); W:5,zFW  
%       [theta,r] = cart2pol(X,Y); r5#8V zr  
%       idx = r<=1; ^xz*%2@  
%       z = nan(size(X)); S: b-+w|*  
%       n = [0  1  1  2  2  2  3  3  3  3]; u Uy~$>V  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Ky:y1\K1^K  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; ~%.<rc0  
%       y = zernfun(n,m,r(idx),theta(idx)); @ ={Hx$zL  
%       figure('Units','normalized') xcf`i:\  
%       for k = 1:10 _o,Mji|  
%           z(idx) = y(:,k); kF,_o/Jc  
%           subplot(4,7,Nplot(k)) LHJ}I5zv  
%           pcolor(x,x,z), shading interp ^)wTCkH&y  
%           set(gca,'XTick',[],'YTick',[]) s1"dd7&g'  
%           axis square =H8 LBM  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) >s*ZT%TF  
%       end yv),>4_6  
% 2^3N[pM;  
%   See also ZERNPOL, ZERNFUN2. )7`2FLG  
0 rge]w.X  
"~:AsZ"7  
%   Paul Fricker 11/13/2006 ],.1=iY  
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% Check and prepare the inputs: = ^NTHc^*  
% ----------------------------- 8l<4OgoK  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) * |dz.Tr  
    error('zernfun:NMvectors','N and M must be vectors.') >5%;NI5 G  
end PjxZ3O  
4ju=5D];   
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if length(n)~=length(m) V DN@=/  
    error('zernfun:NMlength','N and M must be the same length.') )fGIe rS  
end {D>@ZC  
I=;.o>  
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n = n(:); z yh #ygH  
m = m(:); _`Lv@T.  
if any(mod(n-m,2)) 'Edm /+  
    error('zernfun:NMmultiplesof2', ... &N+i3l6`  
          'All N and M must differ by multiples of 2 (including 0).') iCZuE:I1K,  
end $F#eD 0|  
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if any(m>n) Vs"1:gi&  
    error('zernfun:MlessthanN', ... `<oNEr+#  
          'Each M must be less than or equal to its corresponding N.') awC:{5R8v  
end c04;2gR  
|qAU\m"Pc  
le*'GgU#  
if any( r>1 | r<0 ) *mVQN1  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') 2d60o~ E  
end D\@)*"  
S A\_U::T  
[ 11D7L%1t  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Y<WA-dYoF  
    error('zernfun:RTHvector','R and THETA must be vectors.') +9fQ YJBA  
end j=U^+jAn  
s` $YY_  
3ss0/\3P  
r = r(:); l'Kx#y$  
theta = theta(:); +qiI;C_P\  
length_r = length(r); Rk$  
if length_r~=length(theta) \r5L7y$9 h  
    error('zernfun:RTHlength', ... *u)#yEJ)  
          'The number of R- and THETA-values must be equal.') . l>.  
end  &e%eIz  
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C&/_mm5  
% Check normalization: ]ME2V  
% -------------------- .Y*jL&!  
if nargin==5 && ischar(nflag) ^U.t5jj  
    isnorm = strcmpi(nflag,'norm'); pl.x_E,HP  
    if ~isnorm C9VtRq  
        error('zernfun:normalization','Unrecognized normalization flag.') tVr^1Y  
    end Evy_I+l  
else )/FEjo  
    isnorm = false; mJYG k_ua  
end }9*NEU) o  
}J0HEpn4  
!M(3[(Ni  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d cG)ql4d  
% Compute the Zernike Polynomials 6=3;(2u[C"  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Bgf'Hm% r  
r=4vN=:  
c*DBa]u2  
% Determine the required powers of r: *Me&> "N"  
% ----------------------------------- @;b @O _  
m_abs = abs(m); s*9lYk0  
rpowers = []; =}OcMM`f  
for j = 1:length(n) E06)&tF  
    rpowers = [rpowers m_abs(j):2:n(j)]; !S}d?8I6  
end 8}.V[,]6  
rpowers = unique(rpowers); Y$<p_X,  
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@@=e-d  
% Pre-compute the values of r raised to the required powers, iu.$P-s  
% and compile them in a matrix: $X ]t}=  
% ----------------------------- z>A;|iL  
if rpowers(1)==0 D0*+7n3  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Y0;66bfh}  
    rpowern = cat(2,rpowern{:}); z:)z]6  
    rpowern = [ones(length_r,1) rpowern]; DPBWw[  
else ?atHZLF  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); w%uM=YmuT  
    rpowern = cat(2,rpowern{:}); Sh;Z\nj  
end <gLq?~e|A  
D&|HS!  
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% Compute the values of the polynomials: %Y-KjSs+l  
% -------------------------------------- Rxl/)H[Lc"  
y = zeros(length_r,length(n)); =j^wa')  
for j = 1:length(n) a?]~Sw"@  
    s = 0:(n(j)-m_abs(j))/2; ()ZP =\L  
    pows = n(j):-2:m_abs(j); eQ80Kf~  
    for k = length(s):-1:1 \o<&s{ 6L  
        p = (1-2*mod(s(k),2))* ... z"$huE>P6  
                   prod(2:(n(j)-s(k)))/              ... n,*E s/\  
                   prod(2:s(k))/                     ... wCw_aXqq  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... }:1*@7eR  
                   prod(2:((n(j)+m_abs(j))/2-s(k)));  +OeoA{-W  
        idx = (pows(k)==rpowers); +Cs.v.GA5  
        y(:,j) = y(:,j) + p*rpowern(:,idx); }S=m: VKH  
    end #]}]ZE  
     SS7C|*-Zd  
    if isnorm m* rw?nLZ  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 5$%CRm  
    end /ULO#CN?;  
end Ox^VU2K;&.  
% END: Compute the Zernike Polynomials n|Gw?@CU7  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~M9&SDT/lB  
-5u. Ix3  
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% Compute the Zernike functions: ~=/.ZUQNX  
% ------------------------------ ~4=XYYcka  
idx_pos = m>0; ]~kgsI[E  
idx_neg = m<0; I7?s+vyds  
4R28S]Gb  
QB6. o6  
z = y; 2U rE>_  
if any(idx_pos) 6'\VPjt  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); x>[f+Tc  
end 6bd{3@   
if any(idx_neg) YE[{Y(5;q  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); .Dt.7G  
end (Z#j^}G_l  
@i>o+>V  
jFG Y`9Zw0  
% EOF zernfun m?]= =9  
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)  &d i=alvv1  
`!7QegJa"  
DDE还是手动输入的呢? @+(a{%~7y  
<:I]0|[  
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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