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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, ?.~E:8  
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, \L}aTCvG  
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? E9TWLB5A)(  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? hZf0q 2  
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function z = zernfun(n,m,r,theta,nflag) pYIm43r H  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. #8iRWm0*6  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N  "_t2R &A  
%   and angular frequency M, evaluated at positions (R,THETA) on the u^T)4~(  
%   unit circle.  N is a vector of positive integers (including 0), and @T[}] e  
%   M is a vector with the same number of elements as N.  Each element xU+c?OLi  
%   k of M must be a positive integer, with possible values M(k) = -N(k) DjUif "v  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, eFS;+?bu  
%   and THETA is a vector of angles.  R and THETA must have the same Y5 e6|b|  
%   length.  The output Z is a matrix with one column for every (N,M) k2DT+}u7G  
%   pair, and one row for every (R,THETA) pair. [F{q.mZj  
% gBb+Q,  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 5:v"^"Sz  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), NF+^  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral %_C!3kKv~  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, W,dqk=n  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized ,?g}->ZB  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. {#"[h1  
% >KXSb@  
%   The Zernike functions are an orthogonal basis on the unit circle. W@U<GF1  
%   They are used in disciplines such as astronomy, optics, and I?c "\Fe  
%   optometry to describe functions on a circular domain. mTXeIng?  
% |^p7:)cy  
%   The following table lists the first 15 Zernike functions. 6S7 =+>  
% @H[)U/.  
%       n    m    Zernike function           Normalization + |(-7 "  
%       -------------------------------------------------- t;X  !+  
%       0    0    1                                 1 =yo?]ZS  
%       1    1    r * cos(theta)                    2 2VO bj7F  
%       1   -1    r * sin(theta)                    2 r5y p jT^  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) 9>,$q"M}?  
%       2    0    (2*r^2 - 1)                    sqrt(3) Xm,w.|dx  
%       2    2    r^2 * sin(2*theta)             sqrt(6) 6t@kft>Nv  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) ajB4 Lj,:r  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8)  _0^f  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) eT 8(O36%  
%       3    3    r^3 * sin(3*theta)             sqrt(8) NvCq5B$C  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) *b#00)d  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 2_i/ F)W  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5)  g=W1y  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) vzDoF0Ts*p  
%       4    4    r^4 * sin(4*theta)             sqrt(10) PNVYW?l  
%       -------------------------------------------------- qQ\&]  
% x[XN;W&  
%   Example 1: O*% 1   
% qy@v, a  
%       % Display the Zernike function Z(n=5,m=1) R%l6+Okr  
%       x = -1:0.01:1; "Z xM,kI  
%       [X,Y] = meshgrid(x,x); 8K(3{\J[V  
%       [theta,r] = cart2pol(X,Y); cTlitf9  
%       idx = r<=1; xZ2^lsY  
%       z = nan(size(X)); ,au-g)IFZ  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); ]M2<b:yo  
%       figure YT:])[gVV  
%       pcolor(x,x,z), shading interp xF|P6GXg  
%       axis square, colorbar G.Z4h/1<  
%       title('Zernike function Z_5^1(r,\theta)') ^\|Hz\"*  
% [fVtQ@-S!  
%   Example 2: & !0[T   
% X{2))t%  
%       % Display the first 10 Zernike functions _g{*;?mS  
%       x = -1:0.01:1; lJZ-*"9V  
%       [X,Y] = meshgrid(x,x); W>jgsR79M  
%       [theta,r] = cart2pol(X,Y); {zGM[A  
%       idx = r<=1; 4n1-@qTPF~  
%       z = nan(size(X)); gN"Abc  
%       n = [0  1  1  2  2  2  3  3  3  3]; P|M#S9^]  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; :.xdG>\n3  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; x.gRTR`7(  
%       y = zernfun(n,m,r(idx),theta(idx)); 8Ter]0M&  
%       figure('Units','normalized') /eFudMl  
%       for k = 1:10 +[W_J z  
%           z(idx) = y(:,k); Fh)`A5#  
%           subplot(4,7,Nplot(k)) 5Z (1&  
%           pcolor(x,x,z), shading interp 42 6l:>D(  
%           set(gca,'XTick',[],'YTick',[]) JjO="Cmk/  
%           axis square |n9q 4*dN  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) PH,MZ"Z%  
%       end R 2.y=P8N  
% E]Wnl\Be  
%   See also ZERNPOL, ZERNFUN2. <<Zt.!hS  
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%   Paul Fricker 11/13/2006 ZAK NyA2  
/K+GM8rtE  
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% Check and prepare the inputs: Z{' .fq2A  
% ----------------------------- FPg5!O%  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) N\ Nwmx  
    error('zernfun:NMvectors','N and M must be vectors.') ]J`yh$a  
end 52RFB!Z[  
=aL=SC+  
hu=b ,  
if length(n)~=length(m) h~\bJ*Zp  
    error('zernfun:NMlength','N and M must be the same length.') 49/j9#hr  
end :)cn&'l(S  
K/^70;/!.  
D7'P^*4_B  
n = n(:); FNQR sNi  
m = m(:); K9-?7X  
if any(mod(n-m,2)) dV~yIxD}C*  
    error('zernfun:NMmultiplesof2', ... BK+(Uf;g  
          'All N and M must differ by multiples of 2 (including 0).') !21#NCw  
end ,F4 _ps?(  
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x+mf QcSD&  
if any(m>n) R78=im7  
    error('zernfun:MlessthanN', ... oM')NIW@  
          'Each M must be less than or equal to its corresponding N.') O&ur |&v  
end rSGt`#E-s.  
M=HP!hn  
4nIs+  
if any( r>1 | r<0 ) k@,&'imx  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') wK0= I\WN9  
end E`^?2dv+/  
R^nkcLFb/q  
8ec6J*b  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) #fF~6wopV  
    error('zernfun:RTHvector','R and THETA must be vectors.') nWrkn m  
end k!%[W,*  
.H.#W1`  
"q-,140_  
r = r(:); yUZ;keQ_Tw  
theta = theta(:); &7gL&AY8  
length_r = length(r); !W^b:qjJ  
if length_r~=length(theta) ?2;gmZd7  
    error('zernfun:RTHlength', ... %Q)3*L  
          'The number of R- and THETA-values must be equal.') - %ul9}.  
end }w,^]fC:  
Z(' iZ'55F  
3IrmDT  
% Check normalization: zsQhydTR  
% -------------------- |'C {nTX  
if nargin==5 && ischar(nflag) Ym)8L.  
    isnorm = strcmpi(nflag,'norm'); x{$~u2|  
    if ~isnorm W?*]' 0  
        error('zernfun:normalization','Unrecognized normalization flag.') ]A;{D~X^w  
    end I 0/enL  
else %*>ee[^L ,  
    isnorm = false; `ViFY   
end 9c /&+j  
#i#4h<R  
,mu=#}a@}  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~|LlT^C  
% Compute the Zernike Polynomials H;&^A5  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ciq'fy  
?1r>t"e5  
>&1MD}  
% Determine the required powers of r: hXvg<Rf  
% ----------------------------------- $@[`/Uh   
m_abs = abs(m); tkN5 |95  
rpowers = []; B/*`u  
for j = 1:length(n) kJ;fA|(I  
    rpowers = [rpowers m_abs(j):2:n(j)]; 1T{A(<:o$  
end n1X.]|6'  
rpowers = unique(rpowers); rv(Qz|K@  
7~t,Pt)  
mP1EWh|  
% Pre-compute the values of r raised to the required powers, t+R8{9L-  
% and compile them in a matrix: S{ v [65  
% ----------------------------- -SZW[T<N"  
if rpowers(1)==0 \2F$FRWo  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); '>GZB  
    rpowern = cat(2,rpowern{:}); 9~6FWBt  
    rpowern = [ones(length_r,1) rpowern]; !y8/El  
else wKjL}1.k  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); (6xrs_ea  
    rpowern = cat(2,rpowern{:}); kW v)+  
end tMWDKatb  
g3p*OYf  
~*Fbs! ;,  
% Compute the values of the polynomials: LuM[*_8  
% -------------------------------------- qusX]Tst z  
y = zeros(length_r,length(n)); {b|:q>Be8  
for j = 1:length(n) vgfLI}|5  
    s = 0:(n(j)-m_abs(j))/2; $'SWH+G  
    pows = n(j):-2:m_abs(j); wnf'-dw]  
    for k = length(s):-1:1 ryd*Ha">I  
        p = (1-2*mod(s(k),2))* ... [LwmzmV+F  
                   prod(2:(n(j)-s(k)))/              ... *c\:ogd  
                   prod(2:s(k))/                     ... 9-<EeV_/  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... mk)F3[ ke  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); vOb=>  
        idx = (pows(k)==rpowers); Iz'*^{Ssm  
        y(:,j) = y(:,j) + p*rpowern(:,idx); O-rHfIxY  
    end &E@8 z&  
     k<mfBNvuo  
    if isnorm I}5#!s< {&  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); =.@{ uu;  
    end vT*z3  
end ,a N8`M  
% END: Compute the Zernike Polynomials gbP]!d:I  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kPN:m ow  
[4V{~`sF  
`0@onDQVc=  
% Compute the Zernike functions: 5*.JXx E;U  
% ------------------------------ `QH-VR\_  
idx_pos = m>0; Z.a`S~U  
idx_neg = m<0; PgNg1  
oDY $F%  
|hsg= LX  
z = y; HZp}<7NR(7  
if any(idx_pos) 2}Ga   
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); I]HrtI  
end !d@qT.  
if any(idx_neg) o6JCy\Bx  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ]>E)0<t  
end y be:u  
;T!w$({V0z  
@dl{ .,J  
% EOF zernfun [TUy><Z  
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)  uc>u=kEue  
s L;  
DDE还是手动输入的呢? "eqNd"~  
j2@19YXe@  
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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