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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 ?w:\0j5 ~  
    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! [U&k"s?  
     
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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 .5\@G b.8  
    function z = zernfun(n,m,r,theta,nflag) 2D:/.9= 8v  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. )xVf3l pQ  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N nReIi;pi  
    %   and angular frequency M, evaluated at positions (R,THETA) on the P].Eb7I  
    %   unit circle.  N is a vector of positive integers (including 0), and S:z|"u:+  
    %   M is a vector with the same number of elements as N.  Each element r`- 8+"P  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) !Ge;f/@  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 6?x F!VIL  
    %   and THETA is a vector of angles.  R and THETA must have the same 1L`V{\_0s  
    %   length.  The output Z is a matrix with one column for every (N,M) 5@RcAQb:  
    %   pair, and one row for every (R,THETA) pair. 3[Q7'\  
    % .-YE(}^  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike GG%;~4#2  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), >K'dgJ245  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 0:Bpvl5  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, @q!T,({kx  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized B9,39rG/7+  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ^Zvb3RJg  
    % C[fefV9g2  
    %   The Zernike functions are an orthogonal basis on the unit circle. sSh." H  
    %   They are used in disciplines such as astronomy, optics, and - "zW"v)\  
    %   optometry to describe functions on a circular domain. ;%0kzIvP  
    % ;39b.v\^  
    %   The following table lists the first 15 Zernike functions. v2tVq_\AMx  
    % b~UWFX#U  
    %       n    m    Zernike function           Normalization :^W}$7$T  
    %       -------------------------------------------------- :2KPvp 7?  
    %       0    0    1                                 1 .RmFYV0,  
    %       1    1    r * cos(theta)                    2 Y*#xo7#B  
    %       1   -1    r * sin(theta)                    2 [4xZy5V  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ts<\n-f  
    %       2    0    (2*r^2 - 1)                    sqrt(3) +\["HS7+'0  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) @_t=0Rc  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 0e&&k  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 6&]Z'nW0k  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) y_>DszRN`u  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) HY_>sD  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) \s[L=^!  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +@uA  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) q~#>MB}".  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !e<5JO;c  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) #)n$Q^9&  
    %       -------------------------------------------------- 0,-]O=   
    % 9_==C"F  
    %   Example 1: {Y/0BS2D  
    % r]-n,  
    %       % Display the Zernike function Z(n=5,m=1) xtCMK1# x  
    %       x = -1:0.01:1; ]gX8z#*k  
    %       [X,Y] = meshgrid(x,x); _"x%s  
    %       [theta,r] = cart2pol(X,Y); t{B@k[|  
    %       idx = r<=1; J0vQqTaT  
    %       z = nan(size(X)); P0; y  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); >VZxDJ$R  
    %       figure ~)#E?:h5  
    %       pcolor(x,x,z), shading interp =}tomN(F~[  
    %       axis square, colorbar Kn3Xn`P?  
    %       title('Zernike function Z_5^1(r,\theta)') vn*K\,  
    % DZmVm['l  
    %   Example 2: s)E8}-v  
    % YJ6:O{AL1  
    %       % Display the first 10 Zernike functions Y5 ;a  
    %       x = -1:0.01:1; )?OdD7gd  
    %       [X,Y] = meshgrid(x,x); cQxUEY('+  
    %       [theta,r] = cart2pol(X,Y); uX!6: v]  
    %       idx = r<=1; O Lt0Q.{  
    %       z = nan(size(X)); "6IZf>N@#  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Z&?4<-@6\p  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; y|+5R5}K  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; m+8:_0x "  
    %       y = zernfun(n,m,r(idx),theta(idx)); [;aM8N  
    %       figure('Units','normalized') b 1.S21  
    %       for k = 1:10 LN (\B:wAY  
    %           z(idx) = y(:,k); ";`jS&"=  
    %           subplot(4,7,Nplot(k)) 1!V[fPJ  
    %           pcolor(x,x,z), shading interp ah<p_qe9|  
    %           set(gca,'XTick',[],'YTick',[]) |5`ecjb.  
    %           axis square r [^.\&-  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) \z6UWZ  
    %       end uWClT):  
    % D=vw0Q_3Y3  
    %   See also ZERNPOL, ZERNFUN2. A@_>9;   
    DazoY&AWE  
    %   Paul Fricker 11/13/2006 ?fP3R':s  
     wT19m  
    'hWA&Xx +  
    % Check and prepare the inputs: 8a@k6OZ  
    % ----------------------------- 8EkzSe  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) \tvL<U"'  
        error('zernfun:NMvectors','N and M must be vectors.') 6/3E!8  
    end lb9?Uc@  
    lijT L-3  
    if length(n)~=length(m) #?r|6<4X  
        error('zernfun:NMlength','N and M must be the same length.') 1yz%ud-l  
    end [ *It' J^  
    NwOV2E6@OW  
    n = n(:); y@$E5sz  
    m = m(:); 0+1!-Wo  
    if any(mod(n-m,2)) zJ(DO>,p&  
        error('zernfun:NMmultiplesof2', ... At<MY`ka  
              'All N and M must differ by multiples of 2 (including 0).') ZY7-.  
    end !^y;|9?O  
    9XQE5^  
    if any(m>n) @i(9k  
        error('zernfun:MlessthanN', ... e0TxJ*  
              'Each M must be less than or equal to its corresponding N.') QsxvA;7%  
    end mzM95yQ^Z  
    2G-"HOG  
    if any( r>1 | r<0 ) iex%$> "  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') F4-rPv  
    end 12L`Gi  
    [G|(E  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 3B%7SX  
        error('zernfun:RTHvector','R and THETA must be vectors.') h4K Mhr  
    end Kv1~,j6  
    k ?6d\Q  
    r = r(:); Hc<@T_h+2  
    theta = theta(:); IQC[ewk  
    length_r = length(r); ^{IZpT3  
    if length_r~=length(theta) 'l!\2Wv2  
        error('zernfun:RTHlength', ... %X\A|V&  
              'The number of R- and THETA-values must be equal.') S]%,g%6i  
    end SX'NFdY  
    C[%&;\3S@  
    % Check normalization: Va.TUz4  
    % -------------------- =$bF[3D  
    if nargin==5 && ischar(nflag) CDtL.a\  
        isnorm = strcmpi(nflag,'norm'); RuVk>(?WK%  
        if ~isnorm 1Zp/EYWa{  
            error('zernfun:normalization','Unrecognized normalization flag.') A9SL|9Q  
        end YWd2bRb  
    else ,)d`_AD+5  
        isnorm = false; w0nbL^f  
    end gn/]1NNfR  
    z<!A;.iD  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b p?TO]LH  
    % Compute the Zernike Polynomials saZK+kD4I  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mYJ8O$  
    JBw2#ry  
    % Determine the required powers of r: sl$y&C-  
    % ----------------------------------- ,#;`f=aqTG  
    m_abs = abs(m); IMdp"  
    rpowers = []; G6>sAOf  
    for j = 1:length(n) 2P'Vp7f6 Y  
        rpowers = [rpowers m_abs(j):2:n(j)]; :O@n6%pSL  
    end bxxLAWQ(  
    rpowers = unique(rpowers); S?i^ ~  
    ?(B}w*G~  
    % Pre-compute the values of r raised to the required powers, Gl w|*{$  
    % and compile them in a matrix: $4ZV(j]  
    % ----------------------------- sVP\EF8PY  
    if rpowers(1)==0 Ufi#y<dP  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); O,^s)>c  
        rpowern = cat(2,rpowern{:}); Oz_CEMcy  
        rpowern = [ones(length_r,1) rpowern]; nIBeZof  
    else ' ZTRl+  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Ho/tCU|w  
        rpowern = cat(2,rpowern{:}); b0h\l#6  
    end ;}S_PnwC@  
    H@zv-{}T8  
    % Compute the values of the polynomials: mM/#(Ghl  
    % -------------------------------------- t xnH~;(  
    y = zeros(length_r,length(n)); r^"sZk#  
    for j = 1:length(n) qR2cRepV  
        s = 0:(n(j)-m_abs(j))/2; x%@M*4:&  
        pows = n(j):-2:m_abs(j); |8k^jq  
        for k = length(s):-1:1 5Y`4%*$  
            p = (1-2*mod(s(k),2))* ... }lPWA/  
                       prod(2:(n(j)-s(k)))/              ... MU] F'6V  
                       prod(2:s(k))/                     ... o8E<_rei  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... d@*dbECG  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); x2I|iA=  
            idx = (pows(k)==rpowers); twldwuN  
            y(:,j) = y(:,j) + p*rpowern(:,idx); BOvJEs!UX  
        end V?^qW#AG  
         #LR6wEk  
        if isnorm KdHkX+-R  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); hTby:$aCg  
        end BBX/&d8n  
    end c"`HKfL  
    % END: Compute the Zernike Polynomials (qc <'$o  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% OH n~DL2  
    *qL2=2  
    % Compute the Zernike functions: K]>4*)A:  
    % ------------------------------ 9,Dw;|A]  
    idx_pos = m>0; MGw XZ7?E  
    idx_neg = m<0; Wx;%W"a  
    <daH0l0  
    z = y; H)*%eG~  
    if any(idx_pos) \KpJIHkBRy  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 4TU\SP8sM  
    end '5T:*Yh  
    if any(idx_neg) #X!seQ7a  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 5c%Fb :BW=  
    end ,T  3M  
     d*([!!i  
    % 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) iY sQ:3s  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 7Fa1utV I  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated "x_G6JE4tv  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 11fV|b%  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, V5jy,Qi)  
    %   and THETA is a vector of angles.  R and THETA must have the same m5?t<H~  
    %   length.  The output Z is a matrix with one column for every P-value, ]nxSVKE4p  
    %   and one row for every (R,THETA) pair. pnv)D}"  
    % G [yI[7=d  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike fE>JoQs38  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ?6MUyH]a  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) *'n=LB8R  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 yWH!v]S  
    %   for all p. 4>HQ2S{t  
    % J}@.f-W\j  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ?FZ) LZM  
    %   Zernike functions (order N<=7).  In some disciplines it is <\ ".6=E#W  
    %   traditional to label the first 36 functions using a single mode YcSPU(  
    %   number P instead of separate numbers for the order N and azimuthal eM7 F8j  
    %   frequency M. &y3;`A7,  
    % #V[Os!ns  
    %   Example: Fl==k  
    % 1)-VlQK p  
    %       % Display the first 16 Zernike functions O`>u70  
    %       x = -1:0.01:1; =}+xD|T  
    %       [X,Y] = meshgrid(x,x); e:QH3|'y  
    %       [theta,r] = cart2pol(X,Y); '?mky,:HT  
    %       idx = r<=1; [F27i#'I]  
    %       p = 0:15; >(Wt  
    %       z = nan(size(X)); 0SZ:C(]  
    %       y = zernfun2(p,r(idx),theta(idx)); Z+FhI^  
    %       figure('Units','normalized') \ tU[,3  
    %       for k = 1:length(p) "@xL9[d  
    %           z(idx) = y(:,k); 2.a{,d  
    %           subplot(4,4,k) r,GgMk  
    %           pcolor(x,x,z), shading interp ~^g*cA t}  
    %           set(gca,'XTick',[],'YTick',[]) p<RIvSqM  
    %           axis square 'L%)B-,n  
    %           title(['Z_{' num2str(p(k)) '}']) kxY9[#:<fB  
    %       end -ozcK  
    % hi ),PfAV  
    %   See also ZERNPOL, ZERNFUN. gp^xl>E  
    7z+Ngt' !  
    %   Paul Fricker 11/13/2006 gYw=Z_z  
    G/_8xmsU  
    "(;t`,F  
    % Check and prepare the inputs: |@MGGAk  
    % ----------------------------- TI^X gl~  
    if min(size(p))~=1 2$joM`j$  
        error('zernfun2:Pvector','Input P must be vector.') cN: ek|r  
    end 5pxw[c53#  
    l27J  
    if any(p)>35 B!1L W4^  
        error('zernfun2:P36', ... ~:UAL}b{\~  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... @.v{hkM`  
               '(P = 0 to 35).']) +Jq~39  
    end U]iZ3^8VT  
    Gwfi  
    % Get the order and frequency corresonding to the function number: <TEDs4 C  
    % ---------------------------------------------------------------- ru*}lDJ  
    p = p(:); %wmbFj}  
    n = ceil((-3+sqrt(9+8*p))/2); )KN]"<jB  
    m = 2*p - n.*(n+2); J~5+=V7OV  
    ?Aky!43  
    % Pass the inputs to the function ZERNFUN: D{]9s  
    % ---------------------------------------- P)06<n1">Z  
    switch nargin 9P-I)ZqL  
        case 3 6p~8(-nG  
            z = zernfun(n,m,r,theta); wg KM6?  
        case 4 X0]5I0YP  
            z = zernfun(n,m,r,theta,nflag); 7pZd?-6M^  
        otherwise pRh)DM#9  
            error('zernfun2:nargin','Incorrect number of inputs.') {&)E$ M  
    end D{iPsH6};5  
    x /?w1  
    % 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>7a;  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. -ob1_0  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of hXP'NS`iv  
    %   order N and frequency M, evaluated at R.  N is a vector of Ve|=<7%%S  
    %   positive integers (including 0), and M is a vector with the "v5jYz5M  
    %   same number of elements as N.  Each element k of M must be a $ud\CU:r  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 6vebGf  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is v\Xyz )  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 5^GrG|~  
    %   with one column for every (N,M) pair, and one row for every Gbc2\A\  
    %   element in R. ]*pro|  
    % , Y cF~  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- C4t@;U=x  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ](sT,'  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to V=5v7Y3( j  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 _[u fH*  
    %   for all [n,m]. |C.[eHe&D  
    % ?wbf)fbq  
    %   The radial Zernike polynomials are the radial portion of the `>skcvkm  
    %   Zernike functions, which are an orthogonal basis on the unit tJZ3P@ L  
    %   circle.  The series representation of the radial Zernike 'jd fUB  
    %   polynomials is 5jK9cF$>  
    % 5SwQ9#  
    %          (n-m)/2 >`D$Jz,  
    %            __ CC{{@  
    %    m      \       s                                          n-2s x_BnWFP  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Ox}a\B8  
    %    n      s=0 !ZTBiC5R  
    % 9|1J pb  
    %   The following table shows the first 12 polynomials. w2o5+G=  
    %  B/ACU  
    %       n    m    Zernike polynomial    Normalization Rkz[x  
    %       --------------------------------------------- zT"W(3  
    %       0    0    1                        sqrt(2) E|hW{oX3  
    %       1    1    r                           2 &{H LYxh   
    %       2    0    2*r^2 - 1                sqrt(6) ]R8JBnA  
    %       2    2    r^2                      sqrt(6) m<| *  
    %       3    1    3*r^3 - 2*r              sqrt(8) 56^#x  
    %       3    3    r^3                      sqrt(8) ?cD2EX%(  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) cuo'V*nWQ  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ])uhm)U@  
    %       4    4    r^4                      sqrt(10) I 7s}{pG  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) O2C&XeB:4  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Vrx3%_NkQ  
    %       5    5    r^5                      sqrt(12) C9%2}E3Z$)  
    %       --------------------------------------------- qQx5n  
    % Z2hIoCT  
    %   Example: |sklY0?l(  
    % k1Thjt  
    %       % Display three example Zernike radial polynomials Ob>M]udn  
    %       r = 0:0.01:1; Iji9N!Yx  
    %       n = [3 2 5]; 2C_/T8  
    %       m = [1 2 1]; 7\sRf/  
    %       z = zernpol(n,m,r); Mg76v<mv<  
    %       figure bO\E)%zp  
    %       plot(r,z) e!JC5Al7  
    %       grid on :~{x'`czJ  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') bf1EMai"  
    % >pq= .)X}  
    %   See also ZERNFUN, ZERNFUN2. U CF'%R  
    mj9r#v3.  
    % A note on the algorithm. P{_Xg,Z  
    % ------------------------ ; E]^7T  
    % The radial Zernike polynomials are computed using the series r&?i>.Kz8  
    % representation shown in the Help section above. For many special |$aTJ9 Iq:  
    % functions, direct evaluation using the series representation can NM:\T1  
    % produce poor numerical results (floating point errors), because [[ ;vZ  
    % the summation often involves computing small differences between dyMj=e  
    % large successive terms in the series. (In such cases, the functions l&LrcM  
    % are often evaluated using alternative methods such as recurrence ?RVY%s;g  
    % relations: see the Legendre functions, for example). For the Zernike rLzN #Zoi  
    % polynomials, however, this problem does not arise, because the /agX! E4s  
    % polynomials are evaluated over the finite domain r = (0,1), and oD>j2 6Q  
    % because the coefficients for a given polynomial are generally all iF1E 5{dH  
    % of similar magnitude. 3ZEV*=+T5  
    % kz+OUA@~  
    % ZERNPOL has been written using a vectorized implementation: multiple ^wm>\o;  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] :M'V**A(  
    % values can be passed as inputs) for a vector of points R.  To achieve $^;b 1bnO  
    % this vectorization most efficiently, the algorithm in ZERNPOL gV-x1s+  
    % involves pre-determining all the powers p of R that are required to 8#&axg?a  
    % compute the outputs, and then compiling the {R^p} into a single WC<K(PP  
    % matrix.  This avoids any redundant computation of the R^p, and fF*{\  
    % minimizes the sizes of certain intermediate variables. DsZBhjCB  
    % FG[YH5  
    %   Paul Fricker 11/13/2006 ,]UCq?YW)T  
    ;)nV  
    q" aUA_}\  
    % Check and prepare the inputs: bO2s'!x  
    % ----------------------------- A:-r 2;xB  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7@*l2edXm+  
        error('zernpol:NMvectors','N and M must be vectors.') )lsR8Hi8  
    end X|iWnz+^  
    i^zncDMA  
    if length(n)~=length(m) >PBP:s1f4>  
        error('zernpol:NMlength','N and M must be the same length.') $xl>YYEBMH  
    end d|RqS`h ]  
    S[zX@3eZV  
    n = n(:); qB0F9[U  
    m = m(:); +.u)\'r;h  
    length_n = length(n); Z+G.v=2q<  
    RZg8y+jM  
    if any(mod(n-m,2)) #uzp  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 8UahoNrSt  
    end =K ctAR;  
    l9eCsVQ~V  
    if any(m<0) c4'k-\JvT  
        error('zernpol:Mpositive','All M must be positive.') BlrZ<\-/  
    end h/a|-V}m&  
    --}5%6  
    if any(m>n) s.p4+K J  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') n8dJ6"L<"  
    end #2~-I  
    AsOkOS3  
    if any( r>1 | r<0 ) -J<{NF  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') p]-\\o}  
    end ,sqx xq  
    vTdJe  
    if ~any(size(r)==1) $k|:V&6SV  
        error('zernpol:Rvector','R must be a vector.') [10y13  
    end `3CdW  
    -xXz}2S4  
    r = r(:); ZHWxU  
    length_r = length(r); G'M;]R9EP  
    nH=8I~jp  
    if nargin==4 ls9 28  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); Uf ?._&:  
        if ~isnorm %]m/fo4b  
            error('zernpol:normalization','Unrecognized normalization flag.') XJ3p<  
        end abW[hp  
    else ,L.*95 ,  
        isnorm = false; `v|w&ty*  
    end N-9Vx#i  
    3;hztCZj  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nN]GO}  
    % Compute the Zernike Polynomials Sz_{#-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t6+c"=P#  
    KS3>c7  
    % Determine the required powers of r: 9[5qN!P;y  
    % ----------------------------------- fK %${   
    rpowers = []; K|{IX^3)V  
    for j = 1:length(n) 5x=aJl;G  
        rpowers = [rpowers m(j):2:n(j)]; E<~Fi .M;\  
    end 8?za&v  
    rpowers = unique(rpowers); j^V r!y  
    +~x'1*A_  
    % Pre-compute the values of r raised to the required powers, e` QniTkT  
    % and compile them in a matrix: p" ;5J+?(  
    % ----------------------------- <*/IV<  
    if rpowers(1)==0 pXy'Ss@y  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); <?&Y_  
        rpowern = cat(2,rpowern{:}); Y_gMoo  
        rpowern = [ones(length_r,1) rpowern]; vR)f'+_Nz  
    else 3b d(.he2u  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 0'QX*xfa>  
        rpowern = cat(2,rpowern{:}); GI[TD?s  
    end 9Ev<t \B  
    O)jpnNz  
    % Compute the values of the polynomials: "K9/^S_  
    % -------------------------------------- aob+_9o  
    z = zeros(length_r,length_n); ^vHh*Ub  
    for j = 1:length_n ~O c:b>~  
        s = 0:(n(j)-m(j))/2; IW{}l=D/  
        pows = n(j):-2:m(j); cEK<CV  
        for k = length(s):-1:1 u^4$<fd  
            p = (1-2*mod(s(k),2))* ... lM |}K-2  
                       prod(2:(n(j)-s(k)))/          ... _ g8CvH)?!  
                       prod(2:s(k))/                 ... 9zL(PkC%\  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... !S3^{l-  
                       prod(2:((n(j)+m(j))/2-s(k))); q?* z<)#  
            idx = (pows(k)==rpowers); N8E  
            z(:,j) = z(:,j) + p*rpowern(:,idx); KL<,avC/  
        end J85S'cwZZ  
         2bnYYQ14:  
        if isnorm )yt_i'D}  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); +e87/\5  
        end .}c&" L;W  
    end zCe[+F  
    \V_ Tc`  
    % 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)  *NG\3%}%|@  
    e=l:!E10  
    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。 )o<^6Ic%7  
    ' '(rC38  
    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)