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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 ^? }-x  
    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! ZwM(H[iqL  
     
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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 Q p7h|<  
    function z = zernfun(n,m,r,theta,nflag) L I*=T   
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. bFjH* ~ P  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N .do8\  
    %   and angular frequency M, evaluated at positions (R,THETA) on the S4\a"WYg  
    %   unit circle.  N is a vector of positive integers (including 0), and `*6|2  
    %   M is a vector with the same number of elements as N.  Each element ClG\Kpi rh  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) JR8|!Of@B  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, X$e*s\4  
    %   and THETA is a vector of angles.  R and THETA must have the same eSQkW  
    %   length.  The output Z is a matrix with one column for every (N,M) ^hXm=r4ozR  
    %   pair, and one row for every (R,THETA) pair. "}MP{/  
    % NOg/rDs'{  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike {0~\T[qm  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), `WIZY33V  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral \3OEC`  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ~UJ.A<>Fh  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ~7 `,}) d  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. "AU.Eh"-1  
    % -0UR%R7q  
    %   The Zernike functions are an orthogonal basis on the unit circle. R2v9gz;W  
    %   They are used in disciplines such as astronomy, optics, and >TMd1? ,  
    %   optometry to describe functions on a circular domain. ;plBo%EBV  
    % $C.a@gm  
    %   The following table lists the first 15 Zernike functions. EsGf+-}|!0  
    % ((C|&$@M  
    %       n    m    Zernike function           Normalization 58XZ]Mc0  
    %       -------------------------------------------------- ^3[_4av  
    %       0    0    1                                 1 }4p)UX>aWT  
    %       1    1    r * cos(theta)                    2 fX]`vjM{  
    %       1   -1    r * sin(theta)                    2 Q7rBc wm5  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) \_WR:?l  
    %       2    0    (2*r^2 - 1)                    sqrt(3) EjL]#,QR  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) f";pfu_FZ  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Vm|KL3}NRv  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) iLch3[p%  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) )7 q"l3e"u  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) >MJ#|vO  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) / cb`%"Z  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +}O -WX?  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) T? Kh '  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?HJh;96B  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) S=ZZ[E_~S  
    %       -------------------------------------------------- s]% C z\  
    % ~v%6*9  
    %   Example 1: 4^uSW&`;/  
    % r[4n2Mys  
    %       % Display the Zernike function Z(n=5,m=1) (IBT|K  
    %       x = -1:0.01:1; @QV0l]H0+  
    %       [X,Y] = meshgrid(x,x); GA[Ebzi  
    %       [theta,r] = cart2pol(X,Y); "Yh;3tI4*  
    %       idx = r<=1; Rjq Xz6  
    %       z = nan(size(X)); & y5"0mA  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); &nP0T-T5y  
    %       figure &EqLF  
    %       pcolor(x,x,z), shading interp +9w[/n^,G  
    %       axis square, colorbar JD#x+~pb,8  
    %       title('Zernike function Z_5^1(r,\theta)') iP0m1  
    % #h?I oB7  
    %   Example 2: UB.1xcI  
    % 4d`YZNvZW/  
    %       % Display the first 10 Zernike functions B~w$j/sWU  
    %       x = -1:0.01:1; iqvLu{  
    %       [X,Y] = meshgrid(x,x); *[{j'7*cc  
    %       [theta,r] = cart2pol(X,Y); 9a=Ll]=\  
    %       idx = r<=1; nd]SI;<  
    %       z = nan(size(X)); aOH|[  
    %       n = [0  1  1  2  2  2  3  3  3  3]; l)9IgJ|<b  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; M@R"-$Z  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; j:h}ka/!p  
    %       y = zernfun(n,m,r(idx),theta(idx)); zbmC? 2$  
    %       figure('Units','normalized') r }lGcG)  
    %       for k = 1:10 eAfi!!Z<  
    %           z(idx) = y(:,k); @ j^R+F  
    %           subplot(4,7,Nplot(k)) x="Wqcnj{  
    %           pcolor(x,x,z), shading interp =p8uP5H  
    %           set(gca,'XTick',[],'YTick',[]) tw_o?9  
    %           axis square r,Uk)xa/^  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) kJJT`Ba&/  
    %       end TI'v /=;)  
    % _K o#36.S  
    %   See also ZERNPOL, ZERNFUN2. eR$@Q  
    j(=w4Sd_W  
    %   Paul Fricker 11/13/2006 XVqOiv)  
    HU'Mi8xxy  
    f' ?/P~[  
    % Check and prepare the inputs: {V6&((E8  
    % ----------------------------- Ca|egQv  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) |}z)>E  
        error('zernfun:NMvectors','N and M must be vectors.') wM+1/[7  
    end /W/e%.  
    Co1d44Q  
    if length(n)~=length(m) X:oOp=y]|  
        error('zernfun:NMlength','N and M must be the same length.') oX|T&"&  
    end G:<f(Gy  
    <rBW6o7  
    n = n(:); Y;/@[AwF  
    m = m(:); fB8, )&  
    if any(mod(n-m,2)) J].Oxch&y  
        error('zernfun:NMmultiplesof2', ... Ix-Mp   
              'All N and M must differ by multiples of 2 (including 0).') 'X;cgAq8(  
    end  >Uw:cq  
    AELj"=RA  
    if any(m>n) dHy9 wU  
        error('zernfun:MlessthanN', ... o;$xN3f,  
              'Each M must be less than or equal to its corresponding N.') iFd !ED  
    end 1&|]8=pG7  
    UzxL" `^7  
    if any( r>1 | r<0 ) PVIOe}N  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') mtmC,jnD  
    end }bb,Iib  
    .9bi%=hP  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) #EH=tJgO|J  
        error('zernfun:RTHvector','R and THETA must be vectors.') \ %Mcvb.?  
    end duaF?\vv  
    32wtN8kx  
    r = r(:); MgeC-XQM  
    theta = theta(:); KN}#8.'>3  
    length_r = length(r); x3q^}sj%  
    if length_r~=length(theta) Rl Oy,/-<  
        error('zernfun:RTHlength', ... !"N,w9MbD  
              'The number of R- and THETA-values must be equal.') 39v Bsc  
    end 7hHID>,o9%  
    (!* l+}  
    % Check normalization: `?z('FV  
    % -------------------- }9^:(ty2A  
    if nargin==5 && ischar(nflag) _%e8GWf  
        isnorm = strcmpi(nflag,'norm'); =A'>1N  
        if ~isnorm t%:7W[_s  
            error('zernfun:normalization','Unrecognized normalization flag.') v \:AOY'  
        end 7m2iL#5[  
    else c,a8#Og  
        isnorm = false; 0Y8gUpe3P6  
    end o%_-u +  
    1dN/H)]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 74([~Qs _M  
    % Compute the Zernike Polynomials L]=]/>jQ6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% cfTT7O#Dc  
    w){B$X  
    % Determine the required powers of r: }b456J  
    % ----------------------------------- $MR1 *_\V  
    m_abs = abs(m); y!b"Cj  
    rpowers = []; SY,ns*>1F  
    for j = 1:length(n) o@)Fy51DD  
        rpowers = [rpowers m_abs(j):2:n(j)]; SoziFI  
    end Ti? "Hr<W  
    rpowers = unique(rpowers); A?MM9Y}K  
    P.Nt jz/B  
    % Pre-compute the values of r raised to the required powers, aT,WXW*  
    % and compile them in a matrix: ;P S4@,  
    % ----------------------------- sPNm.W$_  
    if rpowers(1)==0 /nO_ e  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); e|tx`yA  
        rpowern = cat(2,rpowern{:}); $n<1D -0!r  
        rpowern = [ones(length_r,1) rpowern]; I#OZ:g^  
    else <WUgH6"  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); f#l9rV"@g  
        rpowern = cat(2,rpowern{:}); tR!C8:u  
    end !j$cBf4  
    a4s't% P  
    % Compute the values of the polynomials: cxR.:LD}  
    % -------------------------------------- ef'kG"1  
    y = zeros(length_r,length(n)); H,D5)1Uu  
    for j = 1:length(n) Qb {[xmc  
        s = 0:(n(j)-m_abs(j))/2; 7&id(&y/  
        pows = n(j):-2:m_abs(j); 6w%n$tiX  
        for k = length(s):-1:1 &k'<xW?x  
            p = (1-2*mod(s(k),2))* ... t^&hG7L_m,  
                       prod(2:(n(j)-s(k)))/              ... .s\lfBo9  
                       prod(2:s(k))/                     ... H^'%$F?Ss  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 1tY+0R  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); oaj.5hM  
            idx = (pows(k)==rpowers); >a975R*g  
            y(:,j) = y(:,j) + p*rpowern(:,idx); #H6YI3 `G  
        end |Ua);B~F  
         Fx!D:.)/G  
        if isnorm N_92,xI#  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ;gL{*gR]S  
        end `%\CO `  
    end ,x\qYz+7|  
    % END: Compute the Zernike Polynomials jTS8 qu  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *C55DO^w  
    oLkzLJ  
    % Compute the Zernike functions: #e.x]v:  
    % ------------------------------ )"?'~5A  
    idx_pos = m>0; %f<>Kwr`2  
    idx_neg = m<0; X0L \Ewm  
    0:Bpvl5  
    z = y; /S J><  
    if any(idx_pos) B9,39rG/7+  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ^Zvb3RJg  
    end p"P+8"`  
    if any(idx_neg) [Q:mq=<Z%  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); F=Xb_Gd`  
    end 0to`=;JI  
    </'n={+q  
    % 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) VsTgK  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. +wz1kPRs  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated _<]0hC  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Syseiw  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, khjdTq\\  
    %   and THETA is a vector of angles.  R and THETA must have the same <r <{4\%}  
    %   length.  The output Z is a matrix with one column for every P-value, ..Dm@m}  
    %   and one row for every (R,THETA) pair. 0Sk~m4fj(  
    % iOfO+3'Z_U  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike rMVcoO@3  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) Q\zaa9P  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ;ZuHv {=  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 W\-`}{B_/  
    %   for all p. =p5]r:9W  
    % ,){#J"W  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 T*@o?U  
    %   Zernike functions (order N<=7).  In some disciplines it is #qk=R7" Q  
    %   traditional to label the first 36 functions using a single mode MA_YMxP.'  
    %   number P instead of separate numbers for the order N and azimuthal ?f9M59(l  
    %   frequency M. Q_p&~PNy5  
    % =}tomN(F~[  
    %   Example: Kn3Xn`P?  
    % 3=U#v<  
    %       % Display the first 16 Zernike functions DZmVm['l  
    %       x = -1:0.01:1; q{G8 Po$z'  
    %       [X,Y] = meshgrid(x,x); ~-NSIV:f  
    %       [theta,r] = cart2pol(X,Y); NRG06M  
    %       idx = r<=1; g?|Z/eVJ  
    %       p = 0:15; (;=|2N>7  
    %       z = nan(size(X)); I@z@s}x>  
    %       y = zernfun2(p,r(idx),theta(idx)); {/)i}V#RE  
    %       figure('Units','normalized') @f"[*7Q`/  
    %       for k = 1:length(p) b00$3,L   
    %           z(idx) = y(:,k); zOA~<fhT  
    %           subplot(4,4,k) }|/A &c  
    %           pcolor(x,x,z), shading interp 6:S, {@G  
    %           set(gca,'XTick',[],'YTick',[]) (X^,.qy  
    %           axis square sqpo5~  
    %           title(['Z_{' num2str(p(k)) '}']) 8ZbXGQ  
    %       end ,_H H8[&  
    % HCrQ+r{g  
    %   See also ZERNPOL, ZERNFUN. .|u`s,\  
    BUwL?  
    %   Paul Fricker 11/13/2006 doTbol?+  
    $?!]?{K  
    @D*PO-s9  
    % Check and prepare the inputs: 2gklGDJD  
    % ----------------------------- F{QOu0$cA4  
    if min(size(p))~=1 I74Rw*fB  
        error('zernfun2:Pvector','Input P must be vector.') 5m'AT]5Tn_  
    end KF(y`(8f  
    8a@k6OZ  
    if any(p)>35 {HM[ )t0  
        error('zernfun2:P36', ... :sK4mRF  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... I6;6x  
               '(P = 0 to 35).']) r aOuD3  
    end {hOS0).(w7  
    )N~ p4kp  
    % Get the order and frequency corresonding to the function number: e(0 cz6  
    % ---------------------------------------------------------------- $Bncdf  
    p = p(:); :qqG%RB  
    n = ceil((-3+sqrt(9+8*p))/2); k7@QFw4 j  
    m = 2*p - n.*(n+2); ha;fxM]  
    oV['%Z'  
    % Pass the inputs to the function ZERNFUN: 0+qC_ISns  
    % ---------------------------------------- H-&27?s^  
    switch nargin oB!Y)f6H1  
        case 3 0U/[hG"DKN  
            z = zernfun(n,m,r,theta); &qPezyt  
        case 4 451.VI}MR  
            z = zernfun(n,m,r,theta,nflag); RLL ph  
        otherwise ?[bE/Ya+S  
            error('zernfun2:nargin','Incorrect number of inputs.') <]%6x[  
    end /kyO,g$9  
    h*y+qk-!\g  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) *szs"mQ/  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 4P)#\$d:  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of W3Ee3  
    %   order N and frequency M, evaluated at R.  N is a vector of 6y Muj<L  
    %   positive integers (including 0), and M is a vector with the t {1 [Ip  
    %   same number of elements as N.  Each element k of M must be a 2/t;}pw8  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) i Pr(X  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is }OnU32P  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Y R~e_cA:  
    %   with one column for every (N,M) pair, and one row for every OW=3t#"7Kp  
    %   element in R. D9P,[:"  
    % ,KM%/;1Dm  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- b@4UR<  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is .eVX/6,  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to eJ<P  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 iJ*Wsp  
    %   for all [n,m]. 3k>#z%//  
    % :epB:r  
    %   The radial Zernike polynomials are the radial portion of the e~)4v  
    %   Zernike functions, which are an orthogonal basis on the unit 5QXU"kWH  
    %   circle.  The series representation of the radial Zernike QaEiPn~  
    %   polynomials is I*o6Bn |D  
    % ]Z\W%'q+  
    %          (n-m)/2 ZBY}Mz$  
    %            __ UJp'v_hN  
    %    m      \       s                                          n-2s # SCLU9-  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Rl0"9D87z  
    %    n      s=0 .j,xh )v"  
    % y_W?7 S  
    %   The following table shows the first 12 polynomials. X#0yOSR  
    % T>1#SWQ/9  
    %       n    m    Zernike polynomial    Normalization !.V_?aYi8  
    %       --------------------------------------------- cy mC?8<  
    %       0    0    1                        sqrt(2) ,3}+t6O"  
    %       1    1    r                           2 ,-EN{ed  
    %       2    0    2*r^2 - 1                sqrt(6) BH^*K/ ^  
    %       2    2    r^2                      sqrt(6) v_%6Ly  
    %       3    1    3*r^3 - 2*r              sqrt(8) RaTNA W)v>  
    %       3    3    r^3                      sqrt(8) : Gi8Jo  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) G.XxlI}  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 7|dm"%@  
    %       4    4    r^4                      sqrt(10) 4mp)v*z  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) =&"pG` x  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) \,p?pL<'  
    %       5    5    r^5                      sqrt(12) G='`*_$  
    %       --------------------------------------------- Citumc)E  
    % G] tT=X[  
    %   Example: \j)c?1*$  
    % g]44|9x(W  
    %       % Display three example Zernike radial polynomials B&59c*K  
    %       r = 0:0.01:1; buzpmRoN)  
    %       n = [3 2 5]; *1b0IQ$g  
    %       m = [1 2 1]; ? B|i  
    %       z = zernpol(n,m,r); Dn#5H{D-d  
    %       figure x7l}u`N4  
    %       plot(r,z) tQ'R(H`  
    %       grid on 3kGg;z6  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') h \`(  
    % !(Y|Vm'   
    %   See also ZERNFUN, ZERNFUN2. c; .y  
    ';v2ld 9  
    % A note on the algorithm. Mx93D   
    % ------------------------ fWfhs}_  
    % The radial Zernike polynomials are computed using the series :Zq?V`+M  
    % representation shown in the Help section above. For many special }/NjZ*u  
    % functions, direct evaluation using the series representation can {nA+-=T  
    % produce poor numerical results (floating point errors), because {#z47Rz  
    % the summation often involves computing small differences between 5gx;Bp^_  
    % large successive terms in the series. (In such cases, the functions :|I"Em3R  
    % are often evaluated using alternative methods such as recurrence :nnch?J_  
    % relations: see the Legendre functions, for example). For the Zernike =r`E%P:  
    % polynomials, however, this problem does not arise, because the q(s0dkrj  
    % polynomials are evaluated over the finite domain r = (0,1), and w\Q(wH'  
    % because the coefficients for a given polynomial are generally all bfJ<~ss/  
    % of similar magnitude. 'X&"(M  
    % y\iECdPU  
    % ZERNPOL has been written using a vectorized implementation: multiple 1T~`$zS7  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Bgsi$2hI  
    % values can be passed as inputs) for a vector of points R.  To achieve /N/jwLr  
    % this vectorization most efficiently, the algorithm in ZERNPOL 8BS Nm  
    % involves pre-determining all the powers p of R that are required to 7I(QTc)*  
    % compute the outputs, and then compiling the {R^p} into a single $V<fJpA  
    % matrix.  This avoids any redundant computation of the R^p, and +W[{UC4b  
    % minimizes the sizes of certain intermediate variables. 8rU| Oh  
    % 8193d%Wb  
    %   Paul Fricker 11/13/2006 0H}O6kU  
    W Kd:O)J  
    y?}<SnjP:  
    % Check and prepare the inputs: Dg ~k"Ice  
    % ----------------------------- -=1>t3~\  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) XL1x8IB  
        error('zernpol:NMvectors','N and M must be vectors.') nM8'="$  
    end Ve"M8-{oKk  
    R >[G6LOG  
    if length(n)~=length(m) 3ox|Mz<aZX  
        error('zernpol:NMlength','N and M must be the same length.') [Q8vS;.  
    end li')U  
    ##] `  
    n = n(:); \Q?#^<O  
    m = m(:); yzNDXA.  
    length_n = length(n); KAr5>^<zw  
    V3 ~&R:Z9e  
    if any(mod(n-m,2)) v&66F`  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 4*q6#=G  
    end #N97  
    7.yCs[Z  
    if any(m<0) eM7 F8j  
        error('zernpol:Mpositive','All M must be positive.') &y3;`A7,  
    end #V[Os!ns  
    Fl==k  
    if any(m>n) 1)-VlQK p  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') NeewV=[%  
    end 7$L*nf  
    `P;3,@ e  
    if any( r>1 | r<0 ) b^P\Kky  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') @_#]7  
    end d=HD! e  
    [/J(E\9  
    if ~any(size(r)==1) B-$ps=G+z  
        error('zernpol:Rvector','R must be a vector.') j#VR>0oC]\  
    end 9J}^{AA  
    \&v)#w  
    r = r(:); W=K+kB  
    length_r = length(r); 4)snt3k  
     |L  <  
    if nargin==4 JWxSN9.X  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); J@q!N;eh|  
        if ~isnorm j'SGZnsy*  
            error('zernpol:normalization','Unrecognized normalization flag.') > mP([]  
        end <+<,$jGC-  
    else WsmP]i^Q  
        isnorm = false; 2<_|1%C  
    end }A<fCm7  
    @ `SlOKz!=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $h1pL>^J  
    % Compute the Zernike Polynomials ~ #P` 7G  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &:=[\Ws R  
    xI5zP? _v  
    % Determine the required powers of r: ^%33&<mB}  
    % ----------------------------------- 2 3A)^j  
    rpowers = []; 2cv=7!K4Uv  
    for j = 1:length(n) jXyK[q&O&  
        rpowers = [rpowers m(j):2:n(j)]; 7I:<i$)V  
    end `{nzw$  
    rpowers = unique(rpowers); 4+N9Ylh  
    +Jq~39  
    % Pre-compute the values of r raised to the required powers, [g lhru=+  
    % and compile them in a matrix: |OBZSk1jp  
    % ----------------------------- KC-@2,c9V  
    if rpowers(1)==0 ru*}lDJ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); %wmbFj}  
        rpowern = cat(2,rpowern{:}); )KN]"<jB  
        rpowern = [ones(length_r,1) rpowern]; ].x`Fq3  
    else l`EKL2n  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); k NUNh[  
        rpowern = cat(2,rpowern{:}); -lI6!a^  
    end =K6{AmG$  
    ']>/$[!  
    % Compute the values of the polynomials: 1lHBg  
    % -------------------------------------- }vX/55  
    z = zeros(length_r,length_n); #Gu(h(Z s  
    for j = 1:length_n e>_Il']Mb  
        s = 0:(n(j)-m(j))/2; Z}r9jM  
        pows = n(j):-2:m(j); #D8u#8Dz  
        for k = length(s):-1:1 G -RE  
            p = (1-2*mod(s(k),2))* ... @Yzb6@g"  
                       prod(2:(n(j)-s(k)))/          ... D~f[Rg  
                       prod(2:s(k))/                 ... x^!LA,`j  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... T=T1?@2C  
                       prod(2:((n(j)+m(j))/2-s(k))); (L7%V !  
            idx = (pows(k)==rpowers); 7V;wCm#b  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ]=sGLd^)E  
        end j:J7  
         ZTi KU)  
        if isnorm qf B!)Y  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); Q+^"v]V`d  
        end T|h'"3'  
    end [kPF Jf  
    ?lQ-HOAw  
    % 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)  s`Z'5J;S  
    d0MF\yxh  
    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。 =& .KKr  
    9XSZD93L  
    07年就写过这方面的计算程序了。
    2024年6月28-30日于上海组织线下成像光学设计培训,欢迎报名参加。请关注子在川上光学公众号。详细内容请咨询13661915143(同微信号)