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    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

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    离线niuhelen
     
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    只看楼主 倒序阅读 楼主  发表于: 2011-03-12
    小弟不是学光学的,所以想请各位大侠指点啊!谢谢啦 K~ gt=NH  
    就是我用ansys计算出了镜面的面型的数据,怎样可以得到zernike多项式系数,然后用zemax各阶得到像差!谢谢啦! vfAR^*7e  
     
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    离线phility
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    只看该作者 1楼 发表于: 2011-03-12
    可以用matlab编程,用zernike多项式进行波面拟合,求出zernike多项式的系数,拟合的算法有很多种,最简单的是最小二乘法,你可以查下相关资料,挺简单的
    离线phility
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    只看该作者 2楼 发表于: 2011-03-12
    泽尼克多项式的前9项对应象差的
    离线niuhelen
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    只看该作者 3楼 发表于: 2011-03-12
    回 2楼(phility) 的帖子
    非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 x]o~ %h$  
    function z = zernfun(n,m,r,theta,nflag) KS%LXc('  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 5tUp[/]pl  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N &u) R+7bl,  
    %   and angular frequency M, evaluated at positions (R,THETA) on the /c 3A>  
    %   unit circle.  N is a vector of positive integers (including 0), and aOZSX3;wg  
    %   M is a vector with the same number of elements as N.  Each element x=(y  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) .OI&Zm-  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, J8Bz|.@Q  
    %   and THETA is a vector of angles.  R and THETA must have the same AwrW!)n }  
    %   length.  The output Z is a matrix with one column for every (N,M) Y'tPD#|r  
    %   pair, and one row for every (R,THETA) pair. 1FC'DH!  
    % Cx(|ZD^  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 82ay("ZY  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), F)dJws7-  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral o%dKi]  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ;"/[gFD5u  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 7,0^|P  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. &&Ruy(&]I  
    % tQz=_;jy  
    %   The Zernike functions are an orthogonal basis on the unit circle. 3ZRi@=kWz  
    %   They are used in disciplines such as astronomy, optics, and }pk)\^/w/  
    %   optometry to describe functions on a circular domain. n.+%eYM<  
    % 1.p2{  
    %   The following table lists the first 15 Zernike functions. ]o}g~Xn  
    % :&*Y Io  
    %       n    m    Zernike function           Normalization yV`H_iC  
    %       -------------------------------------------------- ^5j+O.zgN  
    %       0    0    1                                 1 -E, d)O`;$  
    %       1    1    r * cos(theta)                    2 V`*N2ztSL  
    %       1   -1    r * sin(theta)                    2 39 D!e&  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) MR$R#  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 88%7  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 45g:q  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 7K"{}:  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) @~t^zI1  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) VRe7Q0  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) (9gL  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) qfJi[8".  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) bs_>!H1  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 1< gY  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ]B8`b  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) %t&   
    %       -------------------------------------------------- 7X+SK&PX  
    % m/ D ~D~  
    %   Example 1: mab921-n  
    % b)+nNqY|  
    %       % Display the Zernike function Z(n=5,m=1) awYnlE/Z1  
    %       x = -1:0.01:1; O9%`G  
    %       [X,Y] = meshgrid(x,x); ;U+4!N  
    %       [theta,r] = cart2pol(X,Y); l(&3s:Ud  
    %       idx = r<=1; (2 nSZRB  
    %       z = nan(size(X)); S*"uXTS  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); FA5|`  
    %       figure jh7-Fl`  
    %       pcolor(x,x,z), shading interp h2k"iO }  
    %       axis square, colorbar 80(Olf@PE  
    %       title('Zernike function Z_5^1(r,\theta)') [)efh9P*  
    % FM{^ND9x  
    %   Example 2: 1 8*M  
    % &m{SWV+   
    %       % Display the first 10 Zernike functions S10"yhn(-t  
    %       x = -1:0.01:1; YK xkO  
    %       [X,Y] = meshgrid(x,x); sd5%Szx  
    %       [theta,r] = cart2pol(X,Y); AW{"9f4  
    %       idx = r<=1; G5MoIC  
    %       z = nan(size(X)); =()Vrk|uK  
    %       n = [0  1  1  2  2  2  3  3  3  3]; }4Q~<2  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; |DUWB;  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; B3AWJ1o  
    %       y = zernfun(n,m,r(idx),theta(idx)); 9w)W|9  
    %       figure('Units','normalized') sej$$m R  
    %       for k = 1:10 /)+V(Jlu  
    %           z(idx) = y(:,k); rXh*nC  
    %           subplot(4,7,Nplot(k)) \&!qw[;O  
    %           pcolor(x,x,z), shading interp >O;V[H2[  
    %           set(gca,'XTick',[],'YTick',[]) `jHbA#sO  
    %           axis square :P'M|U  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) G'#f*) f  
    %       end \\R$C  
    % }Qu kn  
    %   See also ZERNPOL, ZERNFUN2. 7.mYzl-F(  
    MpNgp )%>  
    %   Paul Fricker 11/13/2006 R$|"eb5  
    ,1K`w:uhS  
    L'?7~Cdls  
    % Check and prepare the inputs: {sOWDM5  
    % ----------------------------- * :kMv;9  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 634OH*6  
        error('zernfun:NMvectors','N and M must be vectors.') ^RI& `5g  
    end -){aBMOv3  
    3< 'bi}{  
    if length(n)~=length(m) {P {h|+;  
        error('zernfun:NMlength','N and M must be the same length.') h_>DcVNIx  
    end K[q{)>,9  
    ln1!%B;  
    n = n(:); dZWO6k9[H  
    m = m(:); N^Hj%5  
    if any(mod(n-m,2)) ''Y'ZsQ;  
        error('zernfun:NMmultiplesof2', ... % lK/2-  
              'All N and M must differ by multiples of 2 (including 0).') xqQLri}  
    end >vPv 4e7&3  
    cM_!_8o  
    if any(m>n) +RBX2$kB  
        error('zernfun:MlessthanN', ... *|4/XHi  
              'Each M must be less than or equal to its corresponding N.') vojXo|c  
    end Oq9E$0JW  
    X,A]<$ACu%  
    if any( r>1 | r<0 ) ;F;Vm$  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 0-Ga2Go9  
    end &cp `? k  
    p]eVby"  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) i5 0c N<o  
        error('zernfun:RTHvector','R and THETA must be vectors.') HO_!/4hrU  
    end G' '9eV$  
    *x-@}WY$U  
    r = r(:); z -c1,GOD  
    theta = theta(:); Qv W vS9]  
    length_r = length(r); j  Gp&P  
    if length_r~=length(theta) ]iY O}JuX  
        error('zernfun:RTHlength', ... a@S{ A5j  
              'The number of R- and THETA-values must be equal.') Bra}HjHO  
    end AM0CIRX$  
    9RPZj>ezjA  
    % Check normalization: %M,^)lRP  
    % -------------------- u[ E0jI  
    if nargin==5 && ischar(nflag) LzQOzl@z  
        isnorm = strcmpi(nflag,'norm'); K (,MtY*  
        if ~isnorm ,m Nd#  
            error('zernfun:normalization','Unrecognized normalization flag.') JT! Cb$!  
        end I {%Y0S  
    else 60G(jO14  
        isnorm = false; \iRmGvT  
    end !l-Q.=yw  
    l@0${&n  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% / K(l[M  
    % Compute the Zernike Polynomials m}(M{^\|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d=0{vsrB  
    Z<iK(?@O  
    % Determine the required powers of r: " ?Ux\)*  
    % ----------------------------------- P=aYwmC  
    m_abs = abs(m); SyI\ulmL  
    rpowers = []; m al?3*x/  
    for j = 1:length(n) D}`MY\H  
        rpowers = [rpowers m_abs(j):2:n(j)]; ZPG~@lU  
    end |'``pq/}_  
    rpowers = unique(rpowers); "/y SHB[  
    kX2Z@ w`  
    % Pre-compute the values of r raised to the required powers, ..R JHa6B  
    % and compile them in a matrix: 3Rhoul[S  
    % ----------------------------- n/{ pQ&B  
    if rpowers(1)==0 ,e^~(ITaq  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ^-Rqlr,F;  
        rpowern = cat(2,rpowern{:}); {9FL}Jrt  
        rpowern = [ones(length_r,1) rpowern]; u+O"c  
    else IF  cre  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ~3Za"q*0s  
        rpowern = cat(2,rpowern{:}); zE Ly1v\"  
    end AyNpY_B0c  
    >`l^ C  
    % Compute the values of the polynomials: 'ka}x~EF  
    % -------------------------------------- <G0Ut6J>  
    y = zeros(length_r,length(n)); <iBn-EG l>  
    for j = 1:length(n) |~@yXc5a  
        s = 0:(n(j)-m_abs(j))/2; kCALJRf~d  
        pows = n(j):-2:m_abs(j); IML.6<,(Z  
        for k = length(s):-1:1 F1S0C>N?5  
            p = (1-2*mod(s(k),2))* ... w9StW9 4p  
                       prod(2:(n(j)-s(k)))/              ... I/%L,XyRI  
                       prod(2:s(k))/                     ... /#z"c]#  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... WL|<xNL  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); kxR!hA8wv4  
            idx = (pows(k)==rpowers); bXeJk]#y  
            y(:,j) = y(:,j) + p*rpowern(:,idx); k[}WYs+r  
        end } lXor~_i  
         H& $M/`  
        if isnorm H|$ *HQm  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); hE E1i  
        end Cd ]g+R}j  
    end A)gSOC{3F)  
    % END: Compute the Zernike Polynomials e _(';Lk  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PI@?I&Bo  
    E`.:V<KW/  
    % Compute the Zernike functions: IEd?-L  
    % ------------------------------ AiL80W^=d)  
    idx_pos = m>0; K%W;-W*'  
    idx_neg = m<0; )H`V\ H[0P  
    \=P(?!v  
    z = y; i8KoJY"  
    if any(idx_pos) &^w "  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Q{5.;{/eC  
    end Y78DYbU.  
    if any(idx_neg) $ce*W 9`  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); _#Lq~02 %  
    end $=X>5B  
    @LFB}B  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) )*R';/zaI  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. y(/5l   
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated WJ)4rQ$o  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive IlwHHt;njp  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, a#G3dY>  
    %   and THETA is a vector of angles.  R and THETA must have the same _mk@1ft  
    %   length.  The output Z is a matrix with one column for every P-value, f`*VNB`  
    %   and one row for every (R,THETA) pair. miTff[hsMa  
    % Y@< j vH1  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike $iMLT8U  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ~{);Ab.9+  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) #qUGc`  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ?Ok&,\F@E  
    %   for all p. T@`Al('  
    % f& \ Bs8la  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 DajN1}]  
    %   Zernike functions (order N<=7).  In some disciplines it is bo@ ?`5  
    %   traditional to label the first 36 functions using a single mode Q&+)Kp]A  
    %   number P instead of separate numbers for the order N and azimuthal R#.H&#  
    %   frequency M. s&'FaqE  
    % 7 , _b  
    %   Example: T$AVMVq  
    % ]T&d_~l   
    %       % Display the first 16 Zernike functions 49<t2^1q  
    %       x = -1:0.01:1; hSXJDT2  
    %       [X,Y] = meshgrid(x,x); a1Q%Gn@R  
    %       [theta,r] = cart2pol(X,Y); l]#=I7 6  
    %       idx = r<=1; s[dIWYs#  
    %       p = 0:15; H'7s`^- >I  
    %       z = nan(size(X)); _<DOA:'v  
    %       y = zernfun2(p,r(idx),theta(idx)); m2YsE  j7  
    %       figure('Units','normalized') Vp0_R9oQ  
    %       for k = 1:length(p) %3|/t-US  
    %           z(idx) = y(:,k); ~)`\ j  
    %           subplot(4,4,k) `m8WLj  
    %           pcolor(x,x,z), shading interp n)Cr<^j  
    %           set(gca,'XTick',[],'YTick',[]) M# -E  
    %           axis square RHpjJZUV  
    %           title(['Z_{' num2str(p(k)) '}']) v`jHd*&6)  
    %       end $o;c:Kh$$  
    % g oyQ',+  
    %   See also ZERNPOL, ZERNFUN. >dJ~  
    '*&dP"  
    %   Paul Fricker 11/13/2006 ]NI CQ9  
    >}Bcv%zZ  
    "vQ%` Q  
    % Check and prepare the inputs: -meY[!"X  
    % ----------------------------- ^W9O_5\g4a  
    if min(size(p))~=1 diVg|Z3T  
        error('zernfun2:Pvector','Input P must be vector.') L;y BZLM  
    end _Y/*e<bU  
    $$W2{vr7+  
    if any(p)>35 ~tV7yY|zr  
        error('zernfun2:P36', ... 'RF`XX  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 12hD*,A5j  
               '(P = 0 to 35).']) q8-hbWNm4  
    end -Ah&|!/  
    8O(L;&h  
    % Get the order and frequency corresonding to the function number: @D=%J!!*  
    % ---------------------------------------------------------------- t+q;}ZvG  
    p = p(:); : ir3u  
    n = ceil((-3+sqrt(9+8*p))/2); :g' 'GqGZ  
    m = 2*p - n.*(n+2); Y'bz>@1(  
    K/*"U*9Kv  
    % Pass the inputs to the function ZERNFUN: ^k$Bx_{  
    % ---------------------------------------- ,EVPnH[F~  
    switch nargin ' Q(kx*;  
        case 3 SdYb T)y  
            z = zernfun(n,m,r,theta); WiB~sIp  
        case 4 /DyeMCY-  
            z = zernfun(n,m,r,theta,nflag); QxxPImubB  
        otherwise g6P^JW}.  
            error('zernfun2:nargin','Incorrect number of inputs.') QG~6mvD  
    end Njr;Wa.r+  
    Zlh 2qq  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) i>[xN[U(  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. m[Ihte->  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 1#7|au%:)  
    %   order N and frequency M, evaluated at R.  N is a vector of WAR!#E#J7  
    %   positive integers (including 0), and M is a vector with the mAGD qz>f  
    %   same number of elements as N.  Each element k of M must be a X=Ar"Dx}}s  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) DNqV]N_W  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Q&w_kz.  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix DEhR\Z!  
    %   with one column for every (N,M) pair, and one row for every %e0X-tXcmX  
    %   element in R. ]v),[]Xs  
    % ?V+\E2  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- kONn7Itbu  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is =T26vu   
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to eq8faC5  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 bma.RCyY<  
    %   for all [n,m]. ]3,  
    % -!qjBK,`X  
    %   The radial Zernike polynomials are the radial portion of the u9~Ncz  
    %   Zernike functions, which are an orthogonal basis on the unit F%&lM[N%  
    %   circle.  The series representation of the radial Zernike @NL<v-t  
    %   polynomials is IDw`k[k  
    % `v)'(R7){  
    %          (n-m)/2 Mt`LOdiC_  
    %            __ qLb~^'<iD  
    %    m      \       s                                          n-2s `8O Bw  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r :@P6ibcX  
    %    n      s=0 !\d~9H%`B  
    % ,30lu a  
    %   The following table shows the first 12 polynomials. :E|Jqi\  
    % islHtX VE  
    %       n    m    Zernike polynomial    Normalization >R6mI  
    %       --------------------------------------------- SSla^,MHef  
    %       0    0    1                        sqrt(2) ~,KrL(jC  
    %       1    1    r                           2 &[j9Up'   
    %       2    0    2*r^2 - 1                sqrt(6) w6h83m 3  
    %       2    2    r^2                      sqrt(6) Q(aNa!  
    %       3    1    3*r^3 - 2*r              sqrt(8) ,xrA2  
    %       3    3    r^3                      sqrt(8) B6TE9IoSb8  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) y4|<+9<7  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ):Z #!O<  
    %       4    4    r^4                      sqrt(10) v?6*n >R  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) GYb&'#F~t  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) /U!B2%vq_  
    %       5    5    r^5                      sqrt(12) "s]  
    %       --------------------------------------------- 4I2:"CK06  
    % $8&Y(`  
    %   Example: P*K"0[\n  
    % <A|z   
    %       % Display three example Zernike radial polynomials cfv: Ld m  
    %       r = 0:0.01:1; g@s`PBF7`  
    %       n = [3 2 5]; C@]D*k  
    %       m = [1 2 1]; ntPj9#lf  
    %       z = zernpol(n,m,r); +e*C`uP!  
    %       figure <& +jl($"  
    %       plot(r,z) B<-("P(q  
    %       grid on SB('Nqih  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 0F_hXy@K  
    % )16+Pm8  
    %   See also ZERNFUN, ZERNFUN2. 1'(";  0I  
    ;~$Q;m 1  
    % A note on the algorithm. F;_o `h  
    % ------------------------ TJW8l[M  
    % The radial Zernike polynomials are computed using the series N TDmOS\,  
    % representation shown in the Help section above. For many special {` bX*]  
    % functions, direct evaluation using the series representation can eFf9T@  
    % produce poor numerical results (floating point errors), because ]}l.*v\uK  
    % the summation often involves computing small differences between \h s7>5O^K  
    % large successive terms in the series. (In such cases, the functions ujBm"p_|  
    % are often evaluated using alternative methods such as recurrence >FHx],  
    % relations: see the Legendre functions, for example). For the Zernike j r .{M  
    % polynomials, however, this problem does not arise, because the ZBx,'ph}4  
    % polynomials are evaluated over the finite domain r = (0,1), and iR{@~JN=)  
    % because the coefficients for a given polynomial are generally all #?D[WTV  
    % of similar magnitude. /y4A?*w6  
    % KQ6][2-  
    % ZERNPOL has been written using a vectorized implementation: multiple ?6ssSjR}  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] NYg&8s.  
    % values can be passed as inputs) for a vector of points R.  To achieve cL.>e=x$  
    % this vectorization most efficiently, the algorithm in ZERNPOL {E)tzBI;^  
    % involves pre-determining all the powers p of R that are required to 33eOM(`D[  
    % compute the outputs, and then compiling the {R^p} into a single  lFcHE c  
    % matrix.  This avoids any redundant computation of the R^p, and @gf <%>  
    % minimizes the sizes of certain intermediate variables. /u9 0)x  
    % !blGc$kC  
    %   Paul Fricker 11/13/2006 S5F5Tr;TN  
    @cxM#N8e  
    *KiY+_8>  
    % Check and prepare the inputs: :@E^oNKa0  
    % ----------------------------- HfP<hQmN'  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) %" mki>  
        error('zernpol:NMvectors','N and M must be vectors.') + `'wY?  
    end | a i#rU  
    }uaFmXy3  
    if length(n)~=length(m) =^*EM<WG)  
        error('zernpol:NMlength','N and M must be the same length.') "7Kw]8mRR  
    end 8::y5Yv]  
    )>Z@')Uk:  
    n = n(:); ?*kB>U9e  
    m = m(:); K%t&a RjS  
    length_n = length(n); SJLs3iz_)  
    n[y^S3}%;  
    if any(mod(n-m,2)) I~p*~mLh'  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 2Q%M2Ua  
    end x N>\t& c  
    _(io8zqe{j  
    if any(m<0) $/JXI?K  
        error('zernpol:Mpositive','All M must be positive.') fo/sA9  
    end 2Z<S^9O9  
    0v1~#KCm  
    if any(m>n) pK_zq  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ;"9Ks.  
    end Rw[!Jq  
    XS^du{ai  
    if any( r>1 | r<0 ) U Lq`!1{   
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') uV}GUE%W  
    end "la0@/n  
    GuL0:,  
    if ~any(size(r)==1) S}0-2T[  
        error('zernpol:Rvector','R must be a vector.') )G]J@36  
    end g3%x"SlIU  
    8<Yv:8%B6  
    r = r(:); 0lYP!\J3]%  
    length_r = length(r); >k=@YLj  
    )ytP$,r![S  
    if nargin==4 }y+a )2  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 4-'0# a  
        if ~isnorm &lzCRRnvt  
            error('zernpol:normalization','Unrecognized normalization flag.') UN;U+5,t  
        end ^n4aoj  
    else hmb=_W  
        isnorm = false; ;r]! qv:  
    end +[S<"}ls7  
    l#+@!2z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vt(n: Xk  
    % Compute the Zernike Polynomials o ?.VW/"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i{Q,>Rt  
    +Bt%W%_X  
    % Determine the required powers of r: \sW>Y#9]  
    % ----------------------------------- J]48th0,  
    rpowers = []; ~G^+.>j  
    for j = 1:length(n) w`#9Re  
        rpowers = [rpowers m(j):2:n(j)]; L!ms{0rJ  
    end 0BjP|API  
    rpowers = unique(rpowers); LT,zk)5  
    P$clSJW  
    % Pre-compute the values of r raised to the required powers, 1O)m(0tb[  
    % and compile them in a matrix: 76c:* bZ  
    % ----------------------------- 'q8:1i9\[  
    if rpowers(1)==0 Y~lOkH[z  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Yc5) ^v  
        rpowern = cat(2,rpowern{:}); 1mfB6p1Z(  
        rpowern = [ones(length_r,1) rpowern]; `VglE?M  
    else = P$7 "  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); R-f('[u  
        rpowern = cat(2,rpowern{:}); EWVn*xl?  
    end EzCi%>q  
    oMq:4W,  
    % Compute the values of the polynomials: p8&rl|z|  
    % -------------------------------------- >DzW  OB  
    z = zeros(length_r,length_n); l]IQjjJ`  
    for j = 1:length_n YQO9$g0% ~  
        s = 0:(n(j)-m(j))/2; *;T HD>  
        pows = n(j):-2:m(j); Q b5vyV `  
        for k = length(s):-1:1 {qSYe!`  
            p = (1-2*mod(s(k),2))* ... H3ob 8+J  
                       prod(2:(n(j)-s(k)))/          ... ET6}V"UD  
                       prod(2:s(k))/                 ... o1 &Oug  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... OF! n}.O(  
                       prod(2:((n(j)+m(j))/2-s(k))); +6<g N[  
            idx = (pows(k)==rpowers); s=#[>^?  
            z(:,j) = z(:,j) + p*rpowern(:,idx); >lO]/3j1  
        end lOIf4  
         R}OjSiS\  
        if isnorm dW|S\S'&  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); >- CNHb  
        end h~&5;  
    end f kdJgK  
    ?SoRi</1  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  ^;]Q,*Q  
    c O[Hr  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 h1`u-tc2x  
    BVk&TGa;[$  
    07年就写过这方面的计算程序了。
    让光学不再神秘,让光学变得容易,快速实现客户关于光学的设想与愿望。