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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, <_o).hE{  
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, mE|?0mRA %  
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? "s$$M\)T  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? {WYJQKs8  
cdBD.sg  
GkOZ =ej  
]~YY#I":  
l2Gtw*i_I  
function z = zernfun(n,m,r,theta,nflag) yYdow.b!  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle.  S2;u!f  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N aEL^N0\d  
%   and angular frequency M, evaluated at positions (R,THETA) on the #N?VbDK9_  
%   unit circle.  N is a vector of positive integers (including 0), and xF/u('A  
%   M is a vector with the same number of elements as N.  Each element ?M<q95pL  
%   k of M must be a positive integer, with possible values M(k) = -N(k) >}"9heF  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, W(gOid KKz  
%   and THETA is a vector of angles.  R and THETA must have the same yi29+T7j4S  
%   length.  The output Z is a matrix with one column for every (N,M) -Lo3@:2i  
%   pair, and one row for every (R,THETA) pair. !_yWe  
% b.N$eJlQ&  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike (dH "b *  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), lG1\41ZxB  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral (aeS+d x  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, r5> 1n/+6  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized k1.h|&JJN  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. n|p(Cb#G  
% Wb1?>q  
%   The Zernike functions are an orthogonal basis on the unit circle. ~;V5*t  
%   They are used in disciplines such as astronomy, optics, and SsY :gp_  
%   optometry to describe functions on a circular domain. prk@uYCa =  
% ^t 2b`n60  
%   The following table lists the first 15 Zernike functions. :{g;J  
% 99KW("C1F  
%       n    m    Zernike function           Normalization +u[^@>_I0  
%       -------------------------------------------------- ]jB`"to*}  
%       0    0    1                                 1 (:9=M5d  
%       1    1    r * cos(theta)                    2 2FE13{+f  
%       1   -1    r * sin(theta)                    2 +jPJv[W  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) (zmL MG(R  
%       2    0    (2*r^2 - 1)                    sqrt(3) ~WW!P_wI,  
%       2    2    r^2 * sin(2*theta)             sqrt(6) A)5;ae  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) p0|PVn.^h  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ,6EFJVu \  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) x@p1(V.  
%       3    3    r^3 * sin(3*theta)             sqrt(8) 6)h~9iK  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) qlNB\~HCe  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) YXlaE=9bn  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) L!c.1Rf_  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) VE $Kdo^  
%       4    4    r^4 * sin(4*theta)             sqrt(10) H "; !A=0  
%       -------------------------------------------------- .',d*H))E7  
% GzN /0:b  
%   Example 1: .gJv})Vi  
% <9/?+)  
%       % Display the Zernike function Z(n=5,m=1) >4^,[IO/  
%       x = -1:0.01:1; _qf$dGqc  
%       [X,Y] = meshgrid(x,x); M/abd 7q  
%       [theta,r] = cart2pol(X,Y); KKRj#m(:!  
%       idx = r<=1; s}93nv*ez  
%       z = nan(size(X)); {5NE jUu{j  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); Q>yO,H|  
%       figure ?X'l&k>  
%       pcolor(x,x,z), shading interp L2Z-seE  
%       axis square, colorbar ?Z2_y-  
%       title('Zernike function Z_5^1(r,\theta)') ZWb\^N  
% GTocN1,Z~a  
%   Example 2: qCI0[U@  
% E5X#9;U8E"  
%       % Display the first 10 Zernike functions ($X2SIZh  
%       x = -1:0.01:1; *G"}m/j-  
%       [X,Y] = meshgrid(x,x); ?58*#'r  
%       [theta,r] = cart2pol(X,Y); 89YG `  
%       idx = r<=1; zL Sha\X  
%       z = nan(size(X)); ]^6r7nfR6|  
%       n = [0  1  1  2  2  2  3  3  3  3]; =KW~k7TaN  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; !F08F>@D  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; h @2.D|c)g  
%       y = zernfun(n,m,r(idx),theta(idx)); 87-z=>IU  
%       figure('Units','normalized') Q- }cB  
%       for k = 1:10 P_F0lO  
%           z(idx) = y(:,k); cq4sgQ?sW  
%           subplot(4,7,Nplot(k)) p1']+4r%  
%           pcolor(x,x,z), shading interp Rebo.6rG  
%           set(gca,'XTick',[],'YTick',[]) v m.%)F#@  
%           axis square ?2<V./2F  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])  q!as~{!  
%       end j-k]|0ea}  
% `G<|5pe  
%   See also ZERNPOL, ZERNFUN2. T( CTU/a-,  
A,;[9J2\&  
@ [<B:Tqo  
%   Paul Fricker 11/13/2006 5gZ *  
ja%IGaH;s  
3Lm7{s?=Z-  
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=GL^tAUJ  
% Check and prepare the inputs: +<^c2diX  
% ----------------------------- S.*.nv  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) %T DY &@i=  
    error('zernfun:NMvectors','N and M must be vectors.') =PmIrvr'[5  
end UJ^-T+fut  
yUX<W'-Hev  
P] Xl  
if length(n)~=length(m) '=(@3ggA:  
    error('zernfun:NMlength','N and M must be the same length.') L[. )!c8k  
end 0F%V+Y\R  
yC9~X='D  
v4W<_ 7L_  
n = n(:); 13MB1n  
m = m(:); ;*>':-4  
if any(mod(n-m,2)) l*|m(7s  
    error('zernfun:NMmultiplesof2', ... [w}KjV/yi  
          'All N and M must differ by multiples of 2 (including 0).') xX\A& 9m  
end hEfFMi=a`  
3 Bn9Ce=  
QV_Ep8  
if any(m>n) )'e9(4[V1  
    error('zernfun:MlessthanN', ... 7KZ>x*o  
          'Each M must be less than or equal to its corresponding N.') AxiCpAS;J  
end X~rHNRIU  
PaBqv]  
F=V_ACU  
if any( r>1 | r<0 ) ke5_lr(  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') C''[[sw'K  
end zF_aJ+i:~  
r=ht:+m  
! 345  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) K~jN"ev  
    error('zernfun:RTHvector','R and THETA must be vectors.') csms8J  
end QUi=ZD1  
3.D|xE]g  
+KHk`2{y~  
r = r(:); G-G\l?R(  
theta = theta(:); H >1mi_1  
length_r = length(r); .ot[_*A.FD  
if length_r~=length(theta) 6a*OQ{8  
    error('zernfun:RTHlength', ... rtk1 8U-  
          'The number of R- and THETA-values must be equal.') o;J_"' kP  
end C:P.+AU"`  
~n9-  
<dX7{="&  
% Check normalization:  nCSXvd/  
% -------------------- Yc~c(1VRz  
if nargin==5 && ischar(nflag) U66zm9 3&  
    isnorm = strcmpi(nflag,'norm'); 0?\d%J!"S  
    if ~isnorm ]?j[P=\  
        error('zernfun:normalization','Unrecognized normalization flag.') Avo"jN*<d  
    end vV /fTO  
else a3(q;^v  
    isnorm = false; = ms o1  
end S0-/9h  
UZ3oc[#D=]  
'/K-i.8F  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GeCyq%dN  
% Compute the Zernike Polynomials A]mXV4RmI  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2a[_^v $v  
t 4tXLI;'  
PU{7s  
% Determine the required powers of r: +}@6V4BRn  
% ----------------------------------- {;Ispx0m  
m_abs = abs(m); T0Zv.  
rpowers = []; 'UL"yM  
for j = 1:length(n) $XO#qOW  
    rpowers = [rpowers m_abs(j):2:n(j)]; Tq=OYJq5U  
end B;mt11M  
rpowers = unique(rpowers); uZ7~E._  
$ h<l  
Y]!{ n W  
% Pre-compute the values of r raised to the required powers, a]u1_ $)  
% and compile them in a matrix: %$.]g  
% ----------------------------- @Zd/>'  
if rpowers(1)==0 U,)@+?U+h  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); < &~KYu\r  
    rpowern = cat(2,rpowern{:}); jM  DG  
    rpowern = [ones(length_r,1) rpowern]; ; \N${YIn  
else 8I*WVa$l  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); rezH5d6z62  
    rpowern = cat(2,rpowern{:}); C!r9+z)<  
end sVJwe\!  
'dTg\ Qv  
<!M ab}  
% Compute the values of the polynomials: !O~5<tA[#1  
% -------------------------------------- N#? Ohz  
y = zeros(length_r,length(n)); 4)=\5wJDg1  
for j = 1:length(n) :6Oh?y@  
    s = 0:(n(j)-m_abs(j))/2; 7ZVW7%,zF  
    pows = n(j):-2:m_abs(j); =7WE   
    for k = length(s):-1:1 56R)631]p  
        p = (1-2*mod(s(k),2))* ... ]rP'\a  
                   prod(2:(n(j)-s(k)))/              ... MGzuQrl{H  
                   prod(2:s(k))/                     ... y $K#M  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... \.7O0Q{  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); E6NrBPm  
        idx = (pows(k)==rpowers); R^=)Ucj  
        y(:,j) = y(:,j) + p*rpowern(:,idx); "L p"o  
    end G~\ SI.  
     xRx8E;Q@h?  
    if isnorm H _%yh,L  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); eVYUJ,  
    end z a^s%^:yK  
end (YJ]}J^  
% END: Compute the Zernike Polynomials uBe1{Z  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i+z;tF`  
? <.U,  
TdAHw @(  
% Compute the Zernike functions: "?~u*5  
% ------------------------------  FGP~^Dr/  
idx_pos = m>0; ] EzX$T  
idx_neg = m<0; 3)J0f+M>dv  
)@]Y1r4U  
'0!IF&p'  
z = y; 'h6Vj6  
if any(idx_pos) *qLOr6  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); %7$oig\wE  
end \5wC&|WEB  
if any(idx_neg) !PfIe94{`  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); !%x=o&  
end ,DT =(  
&x(^=sTHI  
Z-!W#   
% EOF zernfun nFn@Z'T$N  
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)  Cm;WQuv@  
rs@,<DV)u  
DDE还是手动输入的呢? 7oPBe1P,K+  
T8.@ }a  
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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