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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, ~qe%Yq  
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, H~ZV *[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? RrU BpqA  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? qTZFPfyU  
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function z = zernfun(n,m,r,theta,nflag) -*5Rnx|Y{  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. F}Vr:~  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N *5w{8  
%   and angular frequency M, evaluated at positions (R,THETA) on the Z{&cuo.@<]  
%   unit circle.  N is a vector of positive integers (including 0), and D}8EERb  
%   M is a vector with the same number of elements as N.  Each element Eu"_MgD  
%   k of M must be a positive integer, with possible values M(k) = -N(k)  N?Lb  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, rZ8`sIWQt  
%   and THETA is a vector of angles.  R and THETA must have the same Y0eE-5F,  
%   length.  The output Z is a matrix with one column for every (N,M) PkI:*\R  
%   pair, and one row for every (R,THETA) pair. dy_:-2S  
% MSf;ZB  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 8@so"d2e  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), {s.=)0V  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral w$JvB5O  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, N('&jHF  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized >EY3/Go>  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. TB0 5?F  
% jy-{~xdg[  
%   The Zernike functions are an orthogonal basis on the unit circle. I? ,>DHUX  
%   They are used in disciplines such as astronomy, optics, and Lemui)  
%   optometry to describe functions on a circular domain. M4as  
%  w@,zFV  
%   The following table lists the first 15 Zernike functions. E>l~-PaZY  
% 98^V4maR:  
%       n    m    Zernike function           Normalization 7uzk p&+:  
%       -------------------------------------------------- #%DE;  
%       0    0    1                                 1 / m=HG^!  
%       1    1    r * cos(theta)                    2 2}8v(%s p  
%       1   -1    r * sin(theta)                    2 |1j["u1  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) dAuJXGo  
%       2    0    (2*r^2 - 1)                    sqrt(3) j]`PSl+w  
%       2    2    r^2 * sin(2*theta)             sqrt(6) HTG%t/S  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) 41&\mx  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) KCs[/]  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) B_.%i+ZZ  
%       3    3    r^3 * sin(3*theta)             sqrt(8) ~@}Bi@*  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) a\r\PBi  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) M3.do^ss  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) s0vDHkf8  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) E>K!Vrh-L  
%       4    4    r^4 * sin(4*theta)             sqrt(10) ov, hI>0!D  
%       -------------------------------------------------- q<M2,YrbAI  
% hIT+gnhh  
%   Example 1: 79;<_(Y  
% $&=S#_HQS  
%       % Display the Zernike function Z(n=5,m=1) Hm*/C4B`  
%       x = -1:0.01:1; uA< n  
%       [X,Y] = meshgrid(x,x); Hl,W=2N  
%       [theta,r] = cart2pol(X,Y); m;,N)<~  
%       idx = r<=1; ?32&]iM oW  
%       z = nan(size(X)); FYpzQ6s~  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); [@.!~E)P  
%       figure ~A\GT$  
%       pcolor(x,x,z), shading interp +L;e^#>d  
%       axis square, colorbar |!4K!_y  
%       title('Zernike function Z_5^1(r,\theta)') +{oG|r3L  
% z:wutqru  
%   Example 2: wfH^<jY)E  
% a^I\ /&aw'  
%       % Display the first 10 Zernike functions XuFYYx~ ^3  
%       x = -1:0.01:1; BI%$c~wS  
%       [X,Y] = meshgrid(x,x); {N+$Q'  
%       [theta,r] = cart2pol(X,Y); @u6B;)'l  
%       idx = r<=1; p;>ec:z3M  
%       z = nan(size(X)); XwJ7|cB  
%       n = [0  1  1  2  2  2  3  3  3  3]; EFM5,gB.m  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Y^wW2-,m  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; %WjXg:R  
%       y = zernfun(n,m,r(idx),theta(idx)); A PEE ~  
%       figure('Units','normalized') JkbQyn  
%       for k = 1:10 = %TWX[w  
%           z(idx) = y(:,k); nWw":K<@Q_  
%           subplot(4,7,Nplot(k)) + R~'7*EI  
%           pcolor(x,x,z), shading interp ZbdZ rE$  
%           set(gca,'XTick',[],'YTick',[]) I=`U7Bis"  
%           axis square pOIJH =#  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) g,!L$,/F  
%       end #V~me  
% vg32y /l]S  
%   See also ZERNPOL, ZERNFUN2. X}Ai -D  
[M=7M}f;  
!$gR{XH$]  
%   Paul Fricker 11/13/2006 Yi.N&&o  
I&x=;   
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% Check and prepare the inputs: KW pVw!  
% ----------------------------- Swig;`  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) tEvut=k'  
    error('zernfun:NMvectors','N and M must be vectors.') iP ->S\  
end &]|?o_p3W  
mn"G_I  
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if length(n)~=length(m) vXrx{5gz  
    error('zernfun:NMlength','N and M must be the same length.') (c=6yV@  
end Z>k#n'm^z  
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*&^Pj%DX  
n = n(:); SJn;{X>)q  
m = m(:); +V ;l6D  
if any(mod(n-m,2)) `ts$(u.w  
    error('zernfun:NMmultiplesof2', ... =(j1rW!  
          'All N and M must differ by multiples of 2 (including 0).') >* f-Wde  
end *K8$eDNZ  
LM<qT-/qs  
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if any(m>n) K7_UP&`=J  
    error('zernfun:MlessthanN', ... uP)'FI  
          'Each M must be less than or equal to its corresponding N.') 4yy>jXDG  
end /$Nsd  
-">;-3,K  
Q%`@0#"]Sv  
if any( r>1 | r<0 ) ~ D j8 z+^  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') Cn34b_Sbd  
end ^eY!U%.  
RLjc&WhzXu  
@alK;\  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) {H>gtpVy  
    error('zernfun:RTHvector','R and THETA must be vectors.') %v M-mbX  
end 5uGq%(24  
X]TG<r  
m]6mGp  
r = r(:); <`=j^LU  
theta = theta(:); "<N*"euH  
length_r = length(r); lgL%u K)  
if length_r~=length(theta) N)X3XTY  
    error('zernfun:RTHlength', ... g wRZ%.Cn  
          'The number of R- and THETA-values must be equal.') YoNDf39  
end X 'Xx"M  
-=Q*Ml#I  
`Y$4 H,8L  
% Check normalization: kS);xA8s]  
% -------------------- b{&)6M)zo  
if nargin==5 && ischar(nflag) d7;um<%zn  
    isnorm = strcmpi(nflag,'norm'); cOJo3p;&  
    if ~isnorm NH4#  
        error('zernfun:normalization','Unrecognized normalization flag.') IM'r8 V  
    end oJz^|dW  
else JX;G<lev  
    isnorm = false; Q Z  
end 77f9(~ZnT  
5|s\* bV`  
E{`fF8]K  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *] ) `z8Ox  
% Compute the Zernike Polynomials +I|vzz`ZVr  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7Ovi{xd@  
m<Dy<((_I  
LU%E:i|  
% Determine the required powers of r: #q=Efn'  
% ----------------------------------- 0S!K{xyR  
m_abs = abs(m); @qAS*3j  
rpowers = []; }S-O& Z  
for j = 1:length(n) yvB.&<]No  
    rpowers = [rpowers m_abs(j):2:n(j)]; mz0X3  
end d\8l`Krs[_  
rpowers = unique(rpowers); on `3&0,.  
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1oGw4kD^x  
% Pre-compute the values of r raised to the required powers, S.NPZ39}ZE  
% and compile them in a matrix: |@d\S[~^G  
% ----------------------------- .aQ \jA  
if rpowers(1)==0 u^  ~W+  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 83#mB:^R  
    rpowern = cat(2,rpowern{:}); eng'X-x  
    rpowern = [ones(length_r,1) rpowern]; U>N1Od4vTO  
else MQ6KN(?\ZL  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); $ddCTS^  
    rpowern = cat(2,rpowern{:}); }5"u[Z.  
end }BP;1y6-r  
^ [@ ,  
i9x+A/ o[  
% Compute the values of the polynomials: 6=Otq=WH  
% -------------------------------------- G4"F+%.  
y = zeros(length_r,length(n)); fz "Y CHe  
for j = 1:length(n) utV_W&  
    s = 0:(n(j)-m_abs(j))/2; =T7.~W  
    pows = n(j):-2:m_abs(j); uwGc@xOgg,  
    for k = length(s):-1:1 Qo|\-y-#  
        p = (1-2*mod(s(k),2))* ... >XfbP]  
                   prod(2:(n(j)-s(k)))/              ... 'm$L Ij?@  
                   prod(2:s(k))/                     ... H<+TR6k<  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... vnuN6M{  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); EfT=?  
        idx = (pows(k)==rpowers); TB31- ()  
        y(:,j) = y(:,j) + p*rpowern(:,idx); dk^~;m#iN  
    end N8df8=.kw  
     493*{  
    if isnorm 4#Jg9o   
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); q!@4~plz  
    end d&>^&>?$zh  
end V !~wj  
% END: Compute the Zernike Polynomials 6_B]MN!(  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B%68\  
]6j{@z?{  
kyV8K#}%8  
% Compute the Zernike functions: Zv{'MIv&v  
% ------------------------------ 1_G^w qk  
idx_pos = m>0; ~wdGd+ez  
idx_neg = m<0; (/$^uWj  
{oL>1h,%3?  
\Vk:93OH21  
z = y; 7zj{wp!  
if any(idx_pos) s5. CFA  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 0> \sQ,T  
end yB!dp;gM{  
if any(idx_neg) k;Y5BB  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); /x *3}oI  
end "<gOzXpa  
8{ I|$*nB  
rvM{M/4  
% EOF zernfun m4Zk\,1m.|  
phoenixzqy 2012-04-23 20:38
慢慢研究,这个专业性很强的。用的人又少。
sansummer 2012-04-27 10:22
这个太牛了,我目前只能把zygo中的zernike的36项参数带入到zemax中,但是我目前对其结果的可信性表示质疑,以后多交流啊
jssylttc 2012-05-14 11:28
sansummer:这个太牛了,我目前只能把zygo中的zernike的36项参数带入到zemax中,但是我目前对其结果的可信性表示质疑,以后多交流啊 (2012-04-27 10:22)  ,"ql5Q4  
*lb<$E]="!  
DDE还是手动输入的呢? S]{oPc[7  
@o.I;}*N  
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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