[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， aK@ Y) Ju'  
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， ?
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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？ 9*ZlNZ  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ /[\g8U{5B}  
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function z = zernfun(n,m,r,theta,nflag) prz COw  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. x"NQatdq  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N U{M3QOF  
%   and angular frequency M, evaluated at positions (R,THETA) on the ?{B5gaU9F  
%   unit circle.  N is a vector of positive integers (including 0), and %gAT\R_f  
%   M is a vector with the same number of elements as N.  Each element A?!RF7v  
%   k of M must be a positive integer, with possible values M(k) = -N(k) {>msE }L  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, fPUr O  
%   and THETA is a vector of angles.  R and THETA must have the same j7kX"nz  
%   length.  The output Z is a matrix with one column for every (N,M) il@>b  
%   pair, and one row for every (R,THETA) pair. 6` TwP\!$/  
% =zK4jiM1  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike [B)!  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), wb?k  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral YxJQ^D
`  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ;6[6~L%K}  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized NF6xKwRU]_  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. lsOv#X-bE  
% /ta}12Z  
%   The Zernike functions are an orthogonal basis on the unit circle. '%[ Y  
%   They are used in disciplines such as astronomy, optics, and
jo<xrn\  
%   optometry to describe functions on a circular domain. bAZoi0LR  
% ;98b SR/  
%   The following table lists the first 15 Zernike functions. EpMxq7*  
% 9Sxr9FLW~  
%       n    m    Zernike function           Normalization :)lG}c  
%       -------------------------------------------------- xBTx`+%WS  
%       0    0    1                                 1 aX;>XL4  
%       1    1    r * cos(theta)                    2 .k]`z>
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%       1   -1    r * sin(theta)                    2 )0exGx+:  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) nZ(]WPIN"  
%       2    0    (2*r^2 - 1)                    sqrt(3) v7 *L3Ol  
%       2    2    r^2 * sin(2*theta)             sqrt(6) qsep9z.  
%       3   -3    r^3 * cos(3*theta)             sqrt(8)
'@.6Rd 8  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) #:gl+  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) & mOn]  
%       3    3    r^3 * sin(3*theta)             sqrt(8) ,X^3.ILz  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) OlRXgJ  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `5?0yXK  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ITw *m3
 
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Zpkd8@g@  
%       4    4    r^4 * sin(4*theta)             sqrt(10) lK=Is v+  
%       -------------------------------------------------- iF^qbh%%E  
% 8c)GUx  
%   Example 1: H%vfRl3rB  
% l[$GOLeS  
%       % Display the Zernike function Z(n=5,m=1) ]i.N'O<p  
%       x = -1:0.01:1; O>]i?  
%       [X,Y] = meshgrid(x,x); FAdTm#tgW]  
%       [theta,r] = cart2pol(X,Y); l2St)`K8  
%       idx = r<=1; -a)1L'R  
%       z = nan(size(X)); )Ri!  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); ]{6/6jl  
%       figure f%o[eW#  
%       pcolor(x,x,z), shading interp 6U*CR=4  
%       axis square, colorbar 'cpm 4mT  
%       title('Zernike function Z_5^1(r,\theta)') Q`9c/vPU  
% MRt"#CO  
%   Example 2: =m2_:&@0x  
% (`dz37@*  
%       % Display the first 10 Zernike functions (NLw#)?  
%       x = -1:0.01:1;  \nEMj,)  
%       [X,Y] = meshgrid(x,x); x!_5/  
%       [theta,r] = cart2pol(X,Y); E,6|-V;?  
%       idx = r<=1; kFp^?+WI%H  
%       z = nan(size(X)); >SDQ@63E?  
%       n = [0  1  1  2  2  2  3  3  3  3]; w/*G!o-<  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; @\?ubF  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; B8Fb$  
%       y = zernfun(n,m,r(idx),theta(idx)); ,6{z  
%       figure('Units','normalized') :1*E5pX0n  
%       for k = 1:10 #4bT8kq  
%           z(idx) = y(:,k); ev;&n@k_I  
%           subplot(4,7,Nplot(k)) F9j@KC(yg  
%           pcolor(x,x,z), shading interp xA Ez1  
%           set(gca,'XTick',[],'YTick',[]) ~x,_A>a  
%           axis square bs
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%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ^5E9p@d"J  
%       end 3LETzsJ  
% v ^h:E  
%   See also ZERNPOL, ZERNFUN2. g9" wX?*  
[ *Dj:A)V^  
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%   Paul Fricker 11/13/2006 !ddyJJ^a  
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% Check and prepare the inputs: */M`KPW  
% ----------------------------- nnj<k5  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) S9l,P-X`  
    error('zernfun:NMvectors','N and M must be vectors.') s<{ Hu0K$  
end $$m0mK
  
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if length(n)~=length(m) \+V"JIStUj  
    error('zernfun:NMlength','N and M must be the same length.') !vf:mMo  
end CKn2ZL  
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n = n(:); Bn]=T  
m = m(:); i=#`7pt%'a  
if any(mod(n-m,2)) ~".@mubt1$  
    error('zernfun:NMmultiplesof2', ... vT EqT  
          'All N and M must differ by multiples of 2 (including 0).') D:Q#%wJ  
end 3
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if any(m>n) ;<garDf  
    error('zernfun:MlessthanN', ... h}@wPP{  
          'Each M must be less than or equal to its corresponding N.') ? #rXc%F  
end >Y08/OAI.2  
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if any( r>1 | r<0 ) .JLJ(WM  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') \eKXsO"d  
end +4%~.,<_to  
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) )  ,RR{Y-  
    error('zernfun:RTHvector','R and THETA must be vectors.') /iO"4%v  
end "BSY1?k{  
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r = r(:); <5E'`T  
theta = theta(:); ^!S4?<v  
length_r = length(r); {*O%A  
if length_r~=length(theta) w317]-n  
    error('zernfun:RTHlength', ... !tTv$L>  
          'The number of R- and THETA-values must be equal.') &tZIWV1&  
end }3: mn  
ltuV2.$  
 9Do75S{(  
% Check normalization: qkhre3  
% -------------------- Em&3g  
if nargin==5 && ischar(nflag) f DXK<v)  
    isnorm = strcmpi(nflag,'norm'); :o^ioX.J  
    if ~isnorm [nxYfER7  
        error('zernfun:normalization','Unrecognized normalization flag.') @HbRfD/!  
    end GPHb-  
else >`03EsU  
    isnorm = false; *}89.kCBF  
end z|3v~,  
@LI;q  
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~}b0zL  
% Compute the Zernike Polynomials ,Sgo_bC/|  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }BM`4/  
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% Determine the required powers of r: b~&cYk'  
% -----------------------------------  d\#yWY  
m_abs = abs(m);
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rpowers = []; ` 1+*-g^r  
for j = 1:length(n) W\Pd:t  
    rpowers = [rpowers m_abs(j):2:n(j)]; N-2#-poDe  
end /rZk^/'  
rpowers = unique(rpowers); u;9iuc`*  
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% Pre-compute the values of r raised to the required powers, "}_J"%  
% and compile them in a matrix: 5b rM..  
% ----------------------------- liYsUmjZ=  
if rpowers(1)==0 3Y#  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); H&ek"nP_  
    rpowern = cat(2,rpowern{:}); 'G65zz  
    rpowern = [ones(length_r,1) rpowern]; !X7z y9  
else =*'yGB[x)  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 4Vi*Qa_,y  
    rpowern = cat(2,rpowern{:}); \{<ml n  
end )*}\fmOv{  
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% Compute the values of the polynomials: ?x",VA  
% -------------------------------------- fZf>>mu@r'  
y = zeros(length_r,length(n)); #8t=vb3  
for j = 1:length(n) gtH^'vFZ  
    s = 0:(n(j)-m_abs(j))/2; e/Z{{FP%6  
    pows = n(j):-2:m_abs(j); BD]J/o  
    for k = length(s):-1:1 !KXcg9e  
        p = (1-2*mod(s(k),2))* ... ;sA 5&a>!  
                   prod(2:(n(j)-s(k)))/              ... L$c 1<7LU  
                   prod(2:s(k))/                     ... N_:!uR  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... w;@v#<q6  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); Fb<'L5
}i  
        idx = (pows(k)==rpowers); =?Ry,^=b  
        y(:,j) = y(:,j) + p*rpowern(:,idx); pWzYC@_W  
    end Mm8_EjMp  
     ?W ^`Fa)]o  
    if isnorm %62|dhl6  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); LT{g^g  
    end RQ|K?^k v  
end R{brf6,  
% END: Compute the Zernike Polynomials &O+S[~  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t@lTA>;U@  
t_I-6`8o]  
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% Compute the Zernike functions: sifjmNP  
% ------------------------------ (~\HizSl  
idx_pos = m>0; =Cf@!wZ^  
idx_neg = m<0; w`boQ_Ir  
6@0?~  
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z = y; w@P86'< v  
if any(idx_pos) l{rHXST|  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); nUq@`G  
end g[b;1$  
if any(idx_neg) G@rh/b<$  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); D_F1<q  
end X..M!3
W  
( q*/=u  
?jO<<@*2S  
% EOF zernfun Q.4+"JoG  

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)  @k2nID^>  

]2B=@V t,  
DDE还是手动输入的呢？ uYwJ[1C  
xyTjK.N  
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。