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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 :6jh*,OHZl  
    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! &cZl2ynPi  
     
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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 (,D:6(R7t  
    function z = zernfun(n,m,r,theta,nflag) g(dReC  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ) uTFId  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ,olP}  
    %   and angular frequency M, evaluated at positions (R,THETA) on the '7 t:.88  
    %   unit circle.  N is a vector of positive integers (including 0), and Mc{-2  
    %   M is a vector with the same number of elements as N.  Each element "V`5 $ur  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ;_#<a*f  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, G7%f| Y  
    %   and THETA is a vector of angles.  R and THETA must have the same 1 %8JMq\  
    %   length.  The output Z is a matrix with one column for every (N,M) JHa\"h  
    %   pair, and one row for every (R,THETA) pair. ?\$6"c<G  
    % EMzJyGt7  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike nP_)PDTFp  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), [1e]_9)p  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral C!U$<_I\2  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 6+4SMf3  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized gxmY^" Jy  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. N@X(YlO  
    %  54#P  
    %   The Zernike functions are an orthogonal basis on the unit circle. c7D{^$L9 v  
    %   They are used in disciplines such as astronomy, optics, and kK:U+`+  
    %   optometry to describe functions on a circular domain. Py#TXzEcC  
    % "c+$GS  
    %   The following table lists the first 15 Zernike functions. Z1_F)5pn  
    % x@3cZd0j#  
    %       n    m    Zernike function           Normalization Q CO,f  
    %       -------------------------------------------------- $HCgawQ  
    %       0    0    1                                 1 1A[(RT]  
    %       1    1    r * cos(theta)                    2 VaA.J  
    %       1   -1    r * sin(theta)                    2 @VQ<X4 Za  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) f)mOeD*u|  
    %       2    0    (2*r^2 - 1)                    sqrt(3) =1y~Qlu  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) qWJHb Dd  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) MT6"b  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8)  dZX;k0  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Oh%p1$H  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Bj GfUQ  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 5fRrd;  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *4%%^*g.I  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) jig3M N  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) \/b[V3<"  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ^7yaM B!  
    %       -------------------------------------------------- Bo\~PV[  
    % *5{1.7  
    %   Example 1: eAStpG"*  
    % Tv6y +l  
    %       % Display the Zernike function Z(n=5,m=1) N6`U)=2o>h  
    %       x = -1:0.01:1; 2A:&Cqo  
    %       [X,Y] = meshgrid(x,x); _l+C0lQl=  
    %       [theta,r] = cart2pol(X,Y); eL.WP`Lz  
    %       idx = r<=1; )+ 'r-AF*  
    %       z = nan(size(X)); t+K1ArQc  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); d2TIG<6/  
    %       figure kP'm$+1or  
    %       pcolor(x,x,z), shading interp z$L e,+  
    %       axis square, colorbar p{:y?0pGN  
    %       title('Zernike function Z_5^1(r,\theta)') T8&eaAoo  
    % Q @[gj:w  
    %   Example 2: zszmG^W{  
    % }9glr]=  
    %       % Display the first 10 Zernike functions jo3(\Bq  
    %       x = -1:0.01:1; OMM5ALc(F  
    %       [X,Y] = meshgrid(x,x); w=3 j'y{f  
    %       [theta,r] = cart2pol(X,Y); sPVE_n  
    %       idx = r<=1; \hn$-'=4  
    %       z = nan(size(X)); 1;'-$K`}  
    %       n = [0  1  1  2  2  2  3  3  3  3]; eoXbZ  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; V.6pfL  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; *?$M=tH  
    %       y = zernfun(n,m,r(idx),theta(idx)); 5SZa, +]  
    %       figure('Units','normalized') "Q:h[)a  
    %       for k = 1:10 ~ch%mI~  
    %           z(idx) = y(:,k); Ke=+D'=  
    %           subplot(4,7,Nplot(k)) 9gglyoZ%  
    %           pcolor(x,x,z), shading interp Gs,e8ri!  
    %           set(gca,'XTick',[],'YTick',[]) f/s"2r  
    %           axis square k"C'8<T)'  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) M< .1U?_#  
    %       end NqGSoOjIO2  
    % I>##iiKN  
    %   See also ZERNPOL, ZERNFUN2. Od ^Sr4C  
    Z`86YYGK  
    %   Paul Fricker 11/13/2006 [S8*b^t4  
    S4?WR+:h  
    U=7nz|  
    % Check and prepare the inputs: @rA V;D%  
    % ----------------------------- aC%Q.+-t  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) aEh9 za  
        error('zernfun:NMvectors','N and M must be vectors.') KU*aJl_n,  
    end .gzfaxi  
    G "!v)o  
    if length(n)~=length(m) SH#*Lc   
        error('zernfun:NMlength','N and M must be the same length.') M] +.xo+A  
    end vU5}E\Ny  
    ;<thEWH;Y  
    n = n(:); KV$4}{  
    m = m(:); }S3  oX$  
    if any(mod(n-m,2)) F3]VSI6^E,  
        error('zernfun:NMmultiplesof2', ... "^!y>]j#A  
              'All N and M must differ by multiples of 2 (including 0).') p Pag@L  
    end k`A39ln7wu  
    (x?Tjyzw  
    if any(m>n) (vX< B h  
        error('zernfun:MlessthanN', ... U djYRfk  
              'Each M must be less than or equal to its corresponding N.') u"m(a:jQ  
    end |$e'y x6j  
    jWV}U a  
    if any( r>1 | r<0 ) -ucgET`  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') UV0[S8A  
    end `'sD(e  
    NdSuOkwwt  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) PgGUs4[  
        error('zernfun:RTHvector','R and THETA must be vectors.') a@ <-L  
    end ;gSRpTS:  
    >C!^%e;m  
    r = r(:); Hk@Gkx_  
    theta = theta(:); { V[}#Mf  
    length_r = length(r); tq3Rc}  
    if length_r~=length(theta) %2\tly!{ %  
        error('zernfun:RTHlength', ... M?L$xE_&  
              'The number of R- and THETA-values must be equal.') _MLf58  
    end A_9J ~3  
    CswKT 9  
    % Check normalization: a!-J=\>9  
    % -------------------- :$,MAQ'9  
    if nargin==5 && ischar(nflag) {>9ED.t  
        isnorm = strcmpi(nflag,'norm'); FKz5,PeL  
        if ~isnorm 2 \}J*0  
            error('zernfun:normalization','Unrecognized normalization flag.') Cl9nmyf   
        end ]VLseF  
    else O^row1D_  
        isnorm = false; rf:H$\yw  
    end B5|\<CF  
    Cp"7R&s  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,&WwADZ-s  
    % Compute the Zernike Polynomials Cd"{7<OyM4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ] 2qKc  
    \rzMgR$/rj  
    % Determine the required powers of r: >20dK  
    % ----------------------------------- [i ~qVn2vT  
    m_abs = abs(m); Pap6JR{7  
    rpowers = []; h )5S4)  
    for j = 1:length(n) (H !iK,R  
        rpowers = [rpowers m_abs(j):2:n(j)]; !p/?IW+  
    end E KV[cq  
    rpowers = unique(rpowers); 9%iQ~   
    !Vw1w1  
    % Pre-compute the values of r raised to the required powers, n*=#jL  
    % and compile them in a matrix: {#k[-\|;  
    % ----------------------------- s{yw1:  
    if rpowers(1)==0 o?hr>b  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); U>f'j;5  
        rpowern = cat(2,rpowern{:}); ~Q]5g7k=&  
        rpowern = [ones(length_r,1) rpowern]; cS9jGD92  
    else -"dt3$ju  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); /0XMQy  
        rpowern = cat(2,rpowern{:}); pLtw|S'4  
    end +)"Rv%.  
    3EICdC  
    % Compute the values of the polynomials: ]>VG}e~b  
    % -------------------------------------- ~s'tr&+  
    y = zeros(length_r,length(n)); znwKwc8,  
    for j = 1:length(n) % (y{Sca  
        s = 0:(n(j)-m_abs(j))/2; n%7?G=_kj  
        pows = n(j):-2:m_abs(j); c:Nm!+5_(  
        for k = length(s):-1:1 }ARA K^%  
            p = (1-2*mod(s(k),2))* ... (jDz[b#OPz  
                       prod(2:(n(j)-s(k)))/              ... ?l^Xauk4Pj  
                       prod(2:s(k))/                     ... 7}UG&t{  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... daI_@kY"  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ~! -JN}H m  
            idx = (pows(k)==rpowers); R; c9)>8L  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ?zf3Fn2y  
        end ?Z7QD8N  
         7*{f*({  
        if isnorm -9i7Ja  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); nm,LKS7  
        end 4}uOut  
    end |j`73@6   
    % END: Compute the Zernike Polynomials Km8aHc]O~  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z4/D38_  
    +45SKu=  
    % Compute the Zernike functions: 4x)vy -y  
    % ------------------------------ JY CMW! ~  
    idx_pos = m>0; O;RBK&P  
    idx_neg = m<0; HU>>\t?d  
    j2,sI4  
    z = y; bss2<mqlH  
    if any(idx_pos) 5c::U=  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 'EIe5O p  
    end Q$5 t~*$`  
    if any(idx_neg) l jK?2z>  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); qj _0 td$  
    end nRBS&&V  
    OS#aYER~/  
    % 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) ^|Z'}p|&  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. _<f%== I'  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated yJ!26  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive C0f%~UMwd  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, )] q Qgc&  
    %   and THETA is a vector of angles.  R and THETA must have the same 2?7ID~\  
    %   length.  The output Z is a matrix with one column for every P-value, PsUO8g'\  
    %   and one row for every (R,THETA) pair. i ed 1+H  
    % a k5D  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 8F>9CO:&N  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) z&H.fsL  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) !IR cv a  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 #+D][LH4  
    %   for all p. *e [*  
    % 1f+A_k/@  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Z]uc *Ed  
    %   Zernike functions (order N<=7).  In some disciplines it is NB<8M!X/  
    %   traditional to label the first 36 functions using a single mode v CsE|eMP  
    %   number P instead of separate numbers for the order N and azimuthal y2+f)Xp_.C  
    %   frequency M. ggPGKY-b=  
    % O$,  
    %   Example: F#|y,<}<  
    % &v0]{)PO  
    %       % Display the first 16 Zernike functions ?J2A.x5` a  
    %       x = -1:0.01:1; pBe1:  
    %       [X,Y] = meshgrid(x,x); NpGi3>5  
    %       [theta,r] = cart2pol(X,Y); `scW.Vem  
    %       idx = r<=1; sT1k]duT  
    %       p = 0:15; =XQGg`8<LB  
    %       z = nan(size(X)); 4J[zNB]  
    %       y = zernfun2(p,r(idx),theta(idx)); KKk~vwW  
    %       figure('Units','normalized') .BJoY <P*  
    %       for k = 1:length(p) yHE\Q  
    %           z(idx) = y(:,k); 07>m*1G  
    %           subplot(4,4,k) +mBS&FK  
    %           pcolor(x,x,z), shading interp &i3SB[|  
    %           set(gca,'XTick',[],'YTick',[]) |e!Y C iU  
    %           axis square (&79}IEd  
    %           title(['Z_{' num2str(p(k)) '}']) (7<G1$:z=  
    %       end g~-IT&O  
    % e=h-}XRC  
    %   See also ZERNPOL, ZERNFUN. T\Xf0|y  
    FWeUZI+  
    %   Paul Fricker 11/13/2006 7l-MV n_8  
    MzB.Vvsy%9  
    #@-dT,t  
    % Check and prepare the inputs: r{?qvl!q  
    % ----------------------------- BYdG K@ouk  
    if min(size(p))~=1 KW'nW  
        error('zernfun2:Pvector','Input P must be vector.') D8! Y0  
    end VXZYRr3F  
    {Pe&J2 +  
    if any(p)>35 q'7.lrKwa>  
        error('zernfun2:P36', ... Q'jGNWep  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 6H2Bf*i  
               '(P = 0 to 35).']) |8{ k,!P'K  
    end mEYfsO  
    Ln6emXqw  
    % Get the order and frequency corresonding to the function number: "4xo,JUf  
    % ---------------------------------------------------------------- XBX`L"0  
    p = p(:); 4/{pz$  
    n = ceil((-3+sqrt(9+8*p))/2); lE%KzX?&  
    m = 2*p - n.*(n+2); Chl^LEN:  
    13 L&f\b  
    % Pass the inputs to the function ZERNFUN: jQ7;-9/~N  
    % ---------------------------------------- %VB4/~ "  
    switch nargin   +fM8  
        case 3  7Oe$Ou  
            z = zernfun(n,m,r,theta); }s?w-u+(c6  
        case 4 +MHIZI  
            z = zernfun(n,m,r,theta,nflag); ;uc3_J]  
        otherwise OR+_s @Yg  
            error('zernfun2:nargin','Incorrect number of inputs.') WU=EJY}#n  
    end id+EBVHAd  
     l #]#_  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) ^o>WCU=  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. IDF0nx]  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of _7u&.l<;  
    %   order N and frequency M, evaluated at R.  N is a vector of ~ m vv :u  
    %   positive integers (including 0), and M is a vector with the bUy!hS;s  
    %   same number of elements as N.  Each element k of M must be a SR.xI:}4  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) H/ ejO_{  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is S{ F\_'%  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix K&{ _s  
    %   with one column for every (N,M) pair, and one row for every I4ZbMnO  
    %   element in R. v5 p`=Z@%  
    % ldU ><xc2  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- #~m^RoE  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is B k\K G  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to GHLFn~z@XJ  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 i&m6;>?`  
    %   for all [n,m]. l{?9R.L  
    % "p+oi@  
    %   The radial Zernike polynomials are the radial portion of the Z/GSR$@lI  
    %   Zernike functions, which are an orthogonal basis on the unit Ap"%%D^{:  
    %   circle.  The series representation of the radial Zernike >e.vUUQ{  
    %   polynomials is j5n"LC+oz  
    % {Z!t:'x8  
    %          (n-m)/2 #/9Y}2G|]  
    %            __ <jFov`^  
    %    m      \       s                                          n-2s ['>r tV  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r A=ez,87  
    %    n      s=0 1W0[|Hf2v*  
    % qKeR}&b  
    %   The following table shows the first 12 polynomials. sGiK S,.K  
    % 8eh3K8tL#  
    %       n    m    Zernike polynomial    Normalization N5#j}tT  
    %       --------------------------------------------- ^I6Vz?0Jl  
    %       0    0    1                        sqrt(2) o9M r7  
    %       1    1    r                           2 ,0nrSJED  
    %       2    0    2*r^2 - 1                sqrt(6) m4~~q[t  
    %       2    2    r^2                      sqrt(6) {fEb>  
    %       3    1    3*r^3 - 2*r              sqrt(8) .W;cz8te  
    %       3    3    r^3                      sqrt(8)  N`X|z  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) &>Y.$eW_  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) &0ymAf5R  
    %       4    4    r^4                      sqrt(10)  cFjD*r-  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ^*y 1Fn0  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) +#UawYLJ  
    %       5    5    r^5                      sqrt(12) lFA-T I&  
    %       --------------------------------------------- ?mVSc/  
    % _\@zq*E  
    %   Example: U? U3?Y-k`  
    % Mxd7X<\$  
    %       % Display three example Zernike radial polynomials % bdBg  
    %       r = 0:0.01:1; SuV3$-);z  
    %       n = [3 2 5]; ?caHS2%?ae  
    %       m = [1 2 1]; ?%Q=l;W.  
    %       z = zernpol(n,m,r); QR-pji y  
    %       figure sQr M"i0Y>  
    %       plot(r,z) \SgBI/L^  
    %       grid on j,i)ecZ>  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') "9vL+Hh  
    % &`]T# ">  
    %   See also ZERNFUN, ZERNFUN2. W^;4t3eQf  
    #c/K.?  
    % A note on the algorithm. j@UE#I|h  
    % ------------------------ *|6vCR  
    % The radial Zernike polynomials are computed using the series g>b{hkIXg  
    % representation shown in the Help section above. For many special \x\(36\u  
    % functions, direct evaluation using the series representation can [%Z{Mp'g  
    % produce poor numerical results (floating point errors), because J-klpr#  
    % the summation often involves computing small differences between PCH$)F4^  
    % large successive terms in the series. (In such cases, the functions +n &8" )  
    % are often evaluated using alternative methods such as recurrence 0}:Wh&g  
    % relations: see the Legendre functions, for example). For the Zernike ET*:iioP  
    % polynomials, however, this problem does not arise, because the 7~N4~KAUS  
    % polynomials are evaluated over the finite domain r = (0,1), and {8;}y[R  
    % because the coefficients for a given polynomial are generally all .%0a  
    % of similar magnitude. Qo(<>d  
    % z 7cA5'c  
    % ZERNPOL has been written using a vectorized implementation: multiple cxPOO#  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] h]Gvt 5  
    % values can be passed as inputs) for a vector of points R.  To achieve -0k{O@l"  
    % this vectorization most efficiently, the algorithm in ZERNPOL %bG\  
    % involves pre-determining all the powers p of R that are required to ?l|&JgJ$  
    % compute the outputs, and then compiling the {R^p} into a single Xoq -  
    % matrix.  This avoids any redundant computation of the R^p, and ),%@X  
    % minimizes the sizes of certain intermediate variables. T$DFTr\\  
    % *u[@C  
    %   Paul Fricker 11/13/2006 NUuIhB+  
    W_ ;b e  
    g\49[U}[~F  
    % Check and prepare the inputs: +t]Ge >S  
    % ----------------------------- cwHbm%  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) x97L>>|  
        error('zernpol:NMvectors','N and M must be vectors.') 'OU3-K  
    end zCS }i_ p  
    G}dq ft5"  
    if length(n)~=length(m) j97K\]tQ  
        error('zernpol:NMlength','N and M must be the same length.') .\ vrBf  
    end Pvm pWa  
    2c*}1 _  
    n = n(:); 95/;II  
    m = m(:); J0t_wM Ja  
    length_n = length(n); ]4FAbY2'h  
    +R 8dy  
    if any(mod(n-m,2)) <c,iu{:  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') @cG+ D  
    end TyR@3H  
    M~9IL\J^G  
    if any(m<0) VAdUd {  
        error('zernpol:Mpositive','All M must be positive.') iQiXwEAi[  
    end 4~d:@Gmk&  
    I3ugBLxVC3  
    if any(m>n) A`I1G9s  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') |B2>}Y/  
    end -(#`JT8  
    btDTC 9O  
    if any( r>1 | r<0 ) 6>b#nFVJ  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') vhU#<59a1  
    end ?uF3Q)rCk  
    {JfL7%  
    if ~any(size(r)==1) k/Q8:qA  
        error('zernpol:Rvector','R must be a vector.') <yw=+hz[u  
    end M'NOM>8  
    E7<l^/<2S+  
    r = r(:); ciCQe]fS  
    length_r = length(r); vU#>3[aC  
    }fhGofN$e  
    if nargin==4 4Ub7T=LG  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ~KxK+ 6[ :  
        if ~isnorm 'SWK{t \4  
            error('zernpol:normalization','Unrecognized normalization flag.') :{-/b  
        end +4[Je$qYa  
    else ji5Nq+S2  
        isnorm = false; }AS3]Lub@  
    end V#~.n ;d  
    ?nM]eUAP  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6ziBGU#.-  
    % Compute the Zernike Polynomials <CS,v)4,nH  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :;&3"-  
    U@}P]'`'f  
    % Determine the required powers of r: t+Qx-sW  
    % ----------------------------------- -_>E8PhM  
    rpowers = []; z E\~Oa;  
    for j = 1:length(n) cfc=a  
        rpowers = [rpowers m(j):2:n(j)]; X09i+/ICK  
    end %(r.`I$  
    rpowers = unique(rpowers); }l5Q0'  
    aOmQ<N]a  
    % Pre-compute the values of r raised to the required powers, w!Ii   
    % and compile them in a matrix: 85:mh\@-G  
    % ----------------------------- ,,h>_IA  
    if rpowers(1)==0 22EI`}"J  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 8HWEObRY  
        rpowern = cat(2,rpowern{:}); NLev(B:OQH  
        rpowern = [ones(length_r,1) rpowern]; H<l0]-S{  
    else W: 3fLXk+  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); TTSq}sb}  
        rpowern = cat(2,rpowern{:}); tG 0 &0`  
    end <t,lq  
    6PMu*-Nv!j  
    % Compute the values of the polynomials: Gr~J-#a3~D  
    % -------------------------------------- x!fRT.,}  
    z = zeros(length_r,length_n); u/` t+-A  
    for j = 1:length_n |#22pq?RP  
        s = 0:(n(j)-m(j))/2; W"D>>]$|u  
        pows = n(j):-2:m(j); pH396GFIW  
        for k = length(s):-1:1 R1Q~UX]d=  
            p = (1-2*mod(s(k),2))* ... q)RTy|NJ^  
                       prod(2:(n(j)-s(k)))/          ... 9lqD~H.  
                       prod(2:s(k))/                 ... OB+QVYk"  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... L#MMNc+  
                       prod(2:((n(j)+m(j))/2-s(k))); X1&c?T1 %[  
            idx = (pows(k)==rpowers); Z07SK ' U  
            z(:,j) = z(:,j) + p*rpowern(:,idx); \*30E<;C_  
        end 0He^r &c3  
         Qt=OiKZ  
        if isnorm ^:ehG9  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); %p^`,b}  
        end -8sB\E  
    end c,5yH  
    F7`[r9 $  
    % 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)  qD,/Qu62  
    D_ XOYzN}  
    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。 _=HNcpDA;0  
    {TaYkuWS  
    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)