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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 6B!j(R  
    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! ~t.WwxY+  
     
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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 R{OE{8;  
    function z = zernfun(n,m,r,theta,nflag) jcv1z v.  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. AZ9\>U@hD  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1f pS"_}  
    %   and angular frequency M, evaluated at positions (R,THETA) on the g0:4zeL  
    %   unit circle.  N is a vector of positive integers (including 0), and ";S*[d.2tA  
    %   M is a vector with the same number of elements as N.  Each element ch,Zk )y:_  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) N>nvt.`P  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ?lwQne8/  
    %   and THETA is a vector of angles.  R and THETA must have the same EDidg"0p  
    %   length.  The output Z is a matrix with one column for every (N,M) kFIB lPV  
    %   pair, and one row for every (R,THETA) pair. vb"dX0)<  
    % .dKRIFo  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike FG5c:Ep  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )Y,?r[4{  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Va |9)m  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, xjhAAM  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized %}ApO{  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. gM5p1?E  
    % =u3@ Dhw  
    %   The Zernike functions are an orthogonal basis on the unit circle. L5k>;|SA  
    %   They are used in disciplines such as astronomy, optics, and "k1Tsd-  
    %   optometry to describe functions on a circular domain. yDkDtO`K  
    % F)5B[.ce  
    %   The following table lists the first 15 Zernike functions. 4@mXtA  
    % $@qs(Xwr  
    %       n    m    Zernike function           Normalization k-ex<el)#  
    %       -------------------------------------------------- f~"V  
    %       0    0    1                                 1 4bFVyv  
    %       1    1    r * cos(theta)                    2 o(>-:l i0  
    %       1   -1    r * sin(theta)                    2 V&+$V q  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Oc/_ T>  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 1DlcO>#@  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) eZod}~J8  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ^.1VhTB  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) hC, -9c  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) v{{2<,l  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) @Rb1)$~#  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) s^?sJUj  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .q9|XDqQc  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) |UDD/e  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) %FWfiFV|<  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ]|La MMD  
    %       -------------------------------------------------- T!xy^n]}  
    % '-]BSU  
    %   Example 1: 8!%"/*P$  
    % kbT-Oz  2  
    %       % Display the Zernike function Z(n=5,m=1) JX0_UU  
    %       x = -1:0.01:1; U9fF;[g  
    %       [X,Y] = meshgrid(x,x); U>-#('  
    %       [theta,r] = cart2pol(X,Y); pL/.JzB  
    %       idx = r<=1; jG(~9P7  
    %       z = nan(size(X)); PW//8lsR  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 6N+)LF}P b  
    %       figure P5xmLefng  
    %       pcolor(x,x,z), shading interp |wb(rua  
    %       axis square, colorbar @gjdyz  
    %       title('Zernike function Z_5^1(r,\theta)')  wY_-  
    % EbYH?hPo  
    %   Example 2: *^+xcG  
    % ,Ve@=<  
    %       % Display the first 10 Zernike functions n9/0W%X>  
    %       x = -1:0.01:1; R|$`MX}'z  
    %       [X,Y] = meshgrid(x,x); N5Mz=UgB  
    %       [theta,r] = cart2pol(X,Y); @OY-(cW  
    %       idx = r<=1; BI^]juH-c  
    %       z = nan(size(X)); T_%]#M  
    %       n = [0  1  1  2  2  2  3  3  3  3]; _%TeTNY#  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; *=9#tYn~  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 71&+dC  
    %       y = zernfun(n,m,r(idx),theta(idx)); (<JDD]J  
    %       figure('Units','normalized') 3 DHA^9<q  
    %       for k = 1:10 `DllW{l  
    %           z(idx) = y(:,k); <a[8;YQC  
    %           subplot(4,7,Nplot(k)) M>gZVB,eP>  
    %           pcolor(x,x,z), shading interp Jv.R?1;8i  
    %           set(gca,'XTick',[],'YTick',[]) d@f2Vxe7  
    %           axis square F-,{+B66  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) dTQvz9C  
    %       end T`ZJ=gv  
    % "[S 6w  
    %   See also ZERNPOL, ZERNFUN2. AR6vc  
    g2<S4  
    %   Paul Fricker 11/13/2006 l{o{=]x1  
    }F`2$ Q+CW  
    -?1J+}?  
    % Check and prepare the inputs: y]4 `d  
    % ----------------------------- "$pg mf2  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Ht^2)~e~:  
        error('zernfun:NMvectors','N and M must be vectors.') 5w{pX1z1  
    end *Y0,d`  
    mM{v>Em2K#  
    if length(n)~=length(m) ucPMT0k  
        error('zernfun:NMlength','N and M must be the same length.') $QBUnLOek&  
    end `2+e\%f/0  
    g9Gy3zk=  
    n = n(:); '\\Cpc_g  
    m = m(:); BQ0\+  
    if any(mod(n-m,2)) Ka\b_P&  
        error('zernfun:NMmultiplesof2', ... %\&dFwb  
              'All N and M must differ by multiples of 2 (including 0).') xumv I{  
    end qDd/wR,44  
    #e>MNc 'z  
    if any(m>n) J3^ZPW  
        error('zernfun:MlessthanN', ... -JK4-Hg  
              'Each M must be less than or equal to its corresponding N.') |raQ]b@t&  
    end  F]#fl%  
    yLOLv6g~e  
    if any( r>1 | r<0 ) fGWK&nONyk  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Z@/5~p  
    end 2<@!m @  
    Y{tuaBzD  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) V <pjR@  
        error('zernfun:RTHvector','R and THETA must be vectors.') kk+8NwM1  
    end ZhaOH5{9  
    y<d#sv(s  
    r = r(:); w/6@R 4)p  
    theta = theta(:); 'FFc"lqj  
    length_r = length(r); <U pjAuG8  
    if length_r~=length(theta) Fsj[JE  
        error('zernfun:RTHlength', ... %([H*sLX  
              'The number of R- and THETA-values must be equal.') xR`2+t&t  
    end 26K~m@  
    k"{U}Y/}  
    % Check normalization: {?hjx+v[  
    % -------------------- cpnwx1q@  
    if nargin==5 && ischar(nflag) c%.& F  
        isnorm = strcmpi(nflag,'norm'); oH"N>@Vl  
        if ~isnorm Ntiz-qW  
            error('zernfun:normalization','Unrecognized normalization flag.') G3?z.5 ,Q  
        end c$fM6M }  
    else -;"l 5oX  
        isnorm = false; ),,vu  
    end `,d7_#9'  
    u`|fmVI  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <-}\V!@E!  
    % Compute the Zernike Polynomials Q#KjX;No  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% oD\+ 5[x  
    }*.*{I  
    % Determine the required powers of r: 'DQyB`V2y  
    % ----------------------------------- UI;{3Bn  
    m_abs = abs(m); BUyA]  
    rpowers = []; m.1BLN[9  
    for j = 1:length(n) 6~>k]G  
        rpowers = [rpowers m_abs(j):2:n(j)]; I#U44+c  
    end eVXbYv=gJ@  
    rpowers = unique(rpowers); {8RGW0 Y  
    9l]IE,u  
    % Pre-compute the values of r raised to the required powers, :TI1tJS~*  
    % and compile them in a matrix: 8F1!9W7  
    % ----------------------------- mM.&c5U  
    if rpowers(1)==0 =w-H )  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); F}>`3//u  
        rpowern = cat(2,rpowern{:}); (xL=X%6a  
        rpowern = [ones(length_r,1) rpowern]; |=s3a5sl  
    else :f;|^(]"  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); aDuanGC/V  
        rpowern = cat(2,rpowern{:}); Ji q[VeLe  
    end 4+Y5u4 `t  
    Cq~Ir*"  
    % Compute the values of the polynomials:  7I|Mq  
    % -------------------------------------- bAp`lmFI  
    y = zeros(length_r,length(n)); GWKefH  
    for j = 1:length(n) rY}ofq7b  
        s = 0:(n(j)-m_abs(j))/2; F1>,^qyG6  
        pows = n(j):-2:m_abs(j); :cTi$n  
        for k = length(s):-1:1 T*m21<  
            p = (1-2*mod(s(k),2))* ... t ,$)PV  
                       prod(2:(n(j)-s(k)))/              ... 1CbC|q  
                       prod(2:s(k))/                     ... soF^G21N  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... k1J}9HNYR  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 2uIAnbW]M  
            idx = (pows(k)==rpowers); l<0V0R(  
            y(:,j) = y(:,j) + p*rpowern(:,idx); }g?]B+0  
        end pjFgIG2=9  
         X!Q"p$D4(  
        if isnorm 7Y/_/t~Y  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); f$|v  
        end >nX'RE|F  
    end zVu}7v()  
    % END: Compute the Zernike Polynomials V 6F,X`7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q9Q4F  
    ;q Z2V  
    % Compute the Zernike functions: xpz Jt2S  
    % ------------------------------ [z\*Zg  
    idx_pos = m>0; 1a<~Rmcil  
    idx_neg = m<0; \B)<<[ $  
    J3=jC5=J4  
    z = y; w]_a0{Uh  
    if any(idx_pos) ?=/l@d  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); i+}M#Y-O  
    end e 6*=Si}V  
    if any(idx_neg) ''G @n*  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); aC*J=_9o #  
    end _),@^^&x  
    Go4l#6  
    % 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) "'%x|nB  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 3hkA`YSYt  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated n}2}4^  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive ZP1EO Z  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, R0Qp*&AL  
    %   and THETA is a vector of angles.  R and THETA must have the same \C#Vh7z"2&  
    %   length.  The output Z is a matrix with one column for every P-value, tZJKB1#WbP  
    %   and one row for every (R,THETA) pair. |$Td-M^)  
    % yDPek*#^"q  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 46b.= }  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) -szvO_UP  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) uaiG (O   
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 B$)KZR(u  
    %   for all p. k,2% %m  
    % t^q/'9Ai&J  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 jsuQ R  
    %   Zernike functions (order N<=7).  In some disciplines it is )[A}h'J)  
    %   traditional to label the first 36 functions using a single mode aD.A +es  
    %   number P instead of separate numbers for the order N and azimuthal  BzDS  
    %   frequency M. >6Q-e$GS@  
    %  A/9 wr  
    %   Example: *Nv<,Br,F  
    % p0Vw@R=  
    %       % Display the first 16 Zernike functions FK->|  
    %       x = -1:0.01:1; 6uDNqq  
    %       [X,Y] = meshgrid(x,x); g%K3ah v  
    %       [theta,r] = cart2pol(X,Y); IlH*s/  
    %       idx = r<=1; Q~jUZ-qN  
    %       p = 0:15; iKu5K0x{>I  
    %       z = nan(size(X)); ,$*$w<  
    %       y = zernfun2(p,r(idx),theta(idx)); >scEdeM  
    %       figure('Units','normalized') !O*\|7A(  
    %       for k = 1:length(p) Yt&^ i(  
    %           z(idx) = y(:,k); +?%huJYK,  
    %           subplot(4,4,k) /MA4Er r  
    %           pcolor(x,x,z), shading interp nfc&.(6x<  
    %           set(gca,'XTick',[],'YTick',[]) X"yLo8y8$  
    %           axis square <MoWS9s!yb  
    %           title(['Z_{' num2str(p(k)) '}']) S^3I"B  
    %       end zH.7!jeE  
    % a4c~ThbI  
    %   See also ZERNPOL, ZERNFUN. }psJ'aiG*  
    nM@S`"  
    %   Paul Fricker 11/13/2006 Uc.K6%iI  
     1D6iJ  
     E;k'bz  
    % Check and prepare the inputs: }=NjFK_6  
    % ----------------------------- -|g~--@Q  
    if min(size(p))~=1 cO' \s  
        error('zernfun2:Pvector','Input P must be vector.') Wfp[)MM;  
    end yKi* 8N"e<  
    A%pcPzG;  
    if any(p)>35 XJGOX n$/  
        error('zernfun2:P36', ... gz8<&*2  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... :hZYh.y\l  
               '(P = 0 to 35).']) /.P*%'g  
    end oGa8}Vtc  
    rlgp1>89  
    % Get the order and frequency corresonding to the function number: Or0=:?4`  
    % ---------------------------------------------------------------- ;:f.a(~c  
    p = p(:); >a[)F  
    n = ceil((-3+sqrt(9+8*p))/2); 5EM(3eY^q  
    m = 2*p - n.*(n+2); ,'[0tl}8K  
    0X.pI1jCO  
    % Pass the inputs to the function ZERNFUN: Dr&2q X!  
    % ---------------------------------------- S-GcH  
    switch nargin SZNM$X|T  
        case 3 = oQ-I  
            z = zernfun(n,m,r,theta); .cjSgK1  
        case 4 h49|x&03  
            z = zernfun(n,m,r,theta,nflag); .hXxh)F  
        otherwise k68\ _NUL  
            error('zernfun2:nargin','Incorrect number of inputs.') }/Pz1,/  
    end "1t%J7c_  
    wUv Zc  
    % 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) !`JaYUL[e  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. x4Q*~,n  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of a"@k11  
    %   order N and frequency M, evaluated at R.  N is a vector of $hXhq*5|c  
    %   positive integers (including 0), and M is a vector with the [@(M%  
    %   same number of elements as N.  Each element k of M must be a (R5n ND  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) J'jwRn  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is O`<id+rx  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix `7[z%cuK  
    %   with one column for every (N,M) pair, and one row for every tJ;<=.n  
    %   element in R. .SzP ig  
    % pUi|&F K">  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ya.!zGH  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is c6VfFt6p  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to LlKvi_z  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 4>x]v!d  
    %   for all [n,m]. ;6P #V`u  
    % PT4Wox9U  
    %   The radial Zernike polynomials are the radial portion of the 2:3-mWE  
    %   Zernike functions, which are an orthogonal basis on the unit %&w 8E[  
    %   circle.  The series representation of the radial Zernike LX;w~fRr.  
    %   polynomials is &3~lZa;D  
    % $R6iG\V5  
    %          (n-m)/2 IYAvO%~  
    %            __ qz[qjGdHg  
    %    m      \       s                                          n-2s )IQ5Qu  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r rKxIOJ,T  
    %    n      s=0 sF-{ (  
    % }81eef4$S  
    %   The following table shows the first 12 polynomials. HGIPz{/5U  
    % ])Rs.Y{Q5  
    %       n    m    Zernike polynomial    Normalization @/jLN  
    %       --------------------------------------------- 742 sqHx  
    %       0    0    1                        sqrt(2) 3h N?l :/b  
    %       1    1    r                           2 [Ume^  
    %       2    0    2*r^2 - 1                sqrt(6) %8C,9q  
    %       2    2    r^2                      sqrt(6) c$fi3O  
    %       3    1    3*r^3 - 2*r              sqrt(8) YvA@I|..~  
    %       3    3    r^3                      sqrt(8) +pMa-{  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) dZ^(e0& :H  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) o^ Z/~N  
    %       4    4    r^4                      sqrt(10) }#'I,?_k  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ygI81\ D  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 4PdJ  
    %       5    5    r^5                      sqrt(12) $r>$ u  
    %       --------------------------------------------- y-C=_v_X  
    % xwvg @  
    %   Example: Yvmo%.oU  
    % TgC8EcLr  
    %       % Display three example Zernike radial polynomials 3QM;K^$  
    %       r = 0:0.01:1; i*ibx;s-  
    %       n = [3 2 5]; -!_\4  
    %       m = [1 2 1]; "&9L  
    %       z = zernpol(n,m,r); ayGYVYi  
    %       figure Q#gzk%jL@  
    %       plot(r,z) +`jI z'+  
    %       grid on VT@,RlB0  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 8nV#\J9  
    % WC0gJy  
    %   See also ZERNFUN, ZERNFUN2. A8|DB@ Bi  
    MawWgd*  
    % A note on the algorithm. [-Xz:  
    % ------------------------ I| V yv  
    % The radial Zernike polynomials are computed using the series mE>v (JY  
    % representation shown in the Help section above. For many special BIJlU(aF  
    % functions, direct evaluation using the series representation can ioJ~k[T  
    % produce poor numerical results (floating point errors), because p-CBsm5P  
    % the summation often involves computing small differences between (v/mKGyg  
    % large successive terms in the series. (In such cases, the functions l(Y U9dp  
    % are often evaluated using alternative methods such as recurrence 1&7~.S;km  
    % relations: see the Legendre functions, for example). For the Zernike O4c[,Uq8~  
    % polynomials, however, this problem does not arise, because the  H8lh.K  
    % polynomials are evaluated over the finite domain r = (0,1), and fCMFPhF  
    % because the coefficients for a given polynomial are generally all Ire+r "am  
    % of similar magnitude. >0XB7sC  
    % M'(4{4rC  
    % ZERNPOL has been written using a vectorized implementation: multiple cu |S|]g  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ?@@BIg-  
    % values can be passed as inputs) for a vector of points R.  To achieve "J.7@\^ h/  
    % this vectorization most efficiently, the algorithm in ZERNPOL 4hQ.RO  
    % involves pre-determining all the powers p of R that are required to &7 0o4~Fr  
    % compute the outputs, and then compiling the {R^p} into a single 'L k& iph  
    % matrix.  This avoids any redundant computation of the R^p, and jWUpzf)q=T  
    % minimizes the sizes of certain intermediate variables. t%k1=Ow5i  
    % :Qc[>:N  
    %   Paul Fricker 11/13/2006 ,)svSzR  
    J 7/)XS  
     <u=k X  
    % Check and prepare the inputs: 7>@g)%",  
    % ----------------------------- 0`H)c) pP  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) >du _/*8:  
        error('zernpol:NMvectors','N and M must be vectors.') iHYvH   
    end $ap6Vxjr  
    S7(Vc H  
    if length(n)~=length(m) !kpnBgmU  
        error('zernpol:NMlength','N and M must be the same length.') Qnr7Qnb  
    end e5z U`R  
    th4yuDPuA  
    n = n(:); )ZHo7X  
    m = m(:); [(81-j1v  
    length_n = length(n); 8T]x4JQ0  
    bMCy=5  
    if any(mod(n-m,2)) <H]1 6  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') }#bX{?f  
    end \9Yc2$dY  
    $qp,7RW  
    if any(m<0) Qzh`x-S  
        error('zernpol:Mpositive','All M must be positive.') jmkVolz  
    end 9)~Ha iVB  
    O_~vl m<#  
    if any(m>n) 3W#f Fy  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') &pk&8_=f  
    end WU:r:m+ >  
    i0jR~vF {B  
    if any( r>1 | r<0 ) FV];od&c  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') J;R1OJs S  
    end QE\t}>  
    dH[TnqJn  
    if ~any(size(r)==1) ;*AK eI2  
        error('zernpol:Rvector','R must be a vector.') ^6?)EM#  
    end I5$]{:L|9  
    ]A#K;AW{U  
    r = r(:); [m+):q^  
    length_r = length(r); FVo_=O)  
    %9HL "  
    if nargin==4 ;5.S"  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); <<iwJ U%:  
        if ~isnorm /t $J<bU  
            error('zernpol:normalization','Unrecognized normalization flag.') v "Yo  
        end :,~]R,tJQ  
    else >I&'Rj&Mc  
        isnorm = false; xkPH_+4i8  
    end Ug~ ]!L  
    h!4jl0 oX]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g/_j"Nn  
    % Compute the Zernike Polynomials O&vE 5%x  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yr"BeTrS.  
    &40# _>W7  
    % Determine the required powers of r: fa:V8xa  
    % ----------------------------------- 7#G8qh<  
    rpowers = []; K4`)srd  
    for j = 1:length(n) [>#@?@x`P  
        rpowers = [rpowers m(j):2:n(j)]; 9`8D Ga  
    end ~IhLjE  
    rpowers = unique(rpowers);  tOEY|  
    !\(j[d#  
    % Pre-compute the values of r raised to the required powers, Zk .V   
    % and compile them in a matrix: +jifbf-  
    % ----------------------------- ZPO+ #,  
    if rpowers(1)==0 BJ,D1E  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Z H1UAf  
        rpowern = cat(2,rpowern{:}); $bd tiD  
        rpowern = [ones(length_r,1) rpowern]; !STa}wl  
    else }s>.Fh  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);  A&8{0  
        rpowern = cat(2,rpowern{:}); _=*ph0nu  
    end J}M_Ka  
    }r _d{nhi  
    % Compute the values of the polynomials: a"Q>K7K  
    % -------------------------------------- .r4M]1Of  
    z = zeros(length_r,length_n); 1v2pPUH\  
    for j = 1:length_n ^{+,j}V_H  
        s = 0:(n(j)-m(j))/2; ]\ DIJ>JZ  
        pows = n(j):-2:m(j); 3Y6W)$ Q  
        for k = length(s):-1:1 3`%U)gCT5  
            p = (1-2*mod(s(k),2))* ... 7!kbe2/]'  
                       prod(2:(n(j)-s(k)))/          ... "'zVwU  
                       prod(2:s(k))/                 ... 079mn/8;  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... pGHn   
                       prod(2:((n(j)+m(j))/2-s(k))); 1nAAs;`'  
            idx = (pows(k)==rpowers); fk!P#  
            z(:,j) = z(:,j) + p*rpowern(:,idx); jYJRG<*e  
        end YI!@ ,t  
         PZB_6!}2[F  
        if isnorm iLR^V!  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); s^6"qhTa  
        end [:xpz,  
    end Bq 9 Eu1  
    o%1dbbh  
    % 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)  ^/k`URQ  
    N8s2v W  
    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。 @aX$}  
    w&F/P]1  
    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)