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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 (r.{v@h,dV  
    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! \6Ze H  
     
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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 @mJN  
    function z = zernfun(n,m,r,theta,nflag) kFM'?L&  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. cT0utR&  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N g@Ni!U"_c  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ;$&-c/]F#  
    %   unit circle.  N is a vector of positive integers (including 0), and &OhKx  
    %   M is a vector with the same number of elements as N.  Each element .4!N #'  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) fe37T@  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, {C]M]b*F6(  
    %   and THETA is a vector of angles.  R and THETA must have the same ;wQWt_OtuJ  
    %   length.  The output Z is a matrix with one column for every (N,M) EJWMr`zdn  
    %   pair, and one row for every (R,THETA) pair. 6eDIS|/  
    %  29sgi"  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike z}bnw2d]  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), FOk @W&  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral uaPBM<  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, )i_FU~ LRq  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized NNl/'ge <\  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. zK-hNDFL{  
    % Etu>z+P!  
    %   The Zernike functions are an orthogonal basis on the unit circle. ^ Nsl5  
    %   They are used in disciplines such as astronomy, optics, and CY>NU  
    %   optometry to describe functions on a circular domain. mLk Z4OZ  
    % 4G>|It  
    %   The following table lists the first 15 Zernike functions. P/I{q s  
    % %o"Rcw|  
    %       n    m    Zernike function           Normalization 7t04!dD}  
    %       -------------------------------------------------- 6ZG)`u".("  
    %       0    0    1                                 1 #dpt=  
    %       1    1    r * cos(theta)                    2 |~HlNUPR  
    %       1   -1    r * sin(theta)                    2 2 !;4mij,  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ;n;^f&;sJ  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 68HX,t  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) \PLV]%3,  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 9>i6oF]Oq  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) $k`8Zx w  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 7 YK+TGmU^  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) \4j+pU  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) j&Hn`G  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *c AoE l  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) (j~V  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) YjX=@  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) sN C?o[9l!  
    %       -------------------------------------------------- &1h3o^K  
    % "qj[[L Q  
    %   Example 1: `U g.c  
    % kH&ZPAI  
    %       % Display the Zernike function Z(n=5,m=1) %UQ{'JW?K  
    %       x = -1:0.01:1; "T&uS1+=c  
    %       [X,Y] = meshgrid(x,x); @qC:% |>  
    %       [theta,r] = cart2pol(X,Y); 0wkLM-lN  
    %       idx = r<=1; N/%#GfXx  
    %       z = nan(size(X)); z;/'OJ[.  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); .u*].As=  
    %       figure zl:D|h77  
    %       pcolor(x,x,z), shading interp $1?X%8V  
    %       axis square, colorbar <=inogf  
    %       title('Zernike function Z_5^1(r,\theta)') o(``7A@7a  
    % g \-3c=X  
    %   Example 2: p&4n3%(R@  
    % Nb#7&_f=  
    %       % Display the first 10 Zernike functions V1:3  
    %       x = -1:0.01:1; P\ s+2/  
    %       [X,Y] = meshgrid(x,x); Eo Urc9G2  
    %       [theta,r] = cart2pol(X,Y); :7ngVc  
    %       idx = r<=1; 1ZL_;k  
    %       z = nan(size(X)); cLU*Tx\  
    %       n = [0  1  1  2  2  2  3  3  3  3]; -$)Et|  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; if}]8  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; *i{.@RX?  
    %       y = zernfun(n,m,r(idx),theta(idx)); zrew:5*uZ  
    %       figure('Units','normalized') U9 59=e  
    %       for k = 1:10 cA%U  
    %           z(idx) = y(:,k); VjqdKQeVq  
    %           subplot(4,7,Nplot(k)) BLH=:zb5  
    %           pcolor(x,x,z), shading interp U(N$6{i_  
    %           set(gca,'XTick',[],'YTick',[]) 8e@JvAaa$  
    %           axis square Qyjuzfmz  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 5lxq-E3  
    %       end Z]S0AB.Z@  
    % _cw ^5  
    %   See also ZERNPOL, ZERNFUN2. "J5Pwvs-  
    nTU~M~gky  
    %   Paul Fricker 11/13/2006 y|Y3,s  
    WHZng QmY  
    B%@!\ D#  
    % Check and prepare the inputs: -HsBV>C  
    % -----------------------------  y:OywIi(  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Hm* vKFhz  
        error('zernfun:NMvectors','N and M must be vectors.') 6h_k`z  
    end ++!E9GU{  
    %gMpV  
    if length(n)~=length(m) R{o*O_qX  
        error('zernfun:NMlength','N and M must be the same length.') #=H}6!18  
    end )Zf}V0!?+  
    B ^(rUR  
    n = n(:); Kg`x9._2  
    m = m(:); IVzA>Vd  
    if any(mod(n-m,2)) jN} 7Bb X  
        error('zernfun:NMmultiplesof2', ... 87(^P3;@  
              'All N and M must differ by multiples of 2 (including 0).') un^IQMIh  
    end -fx88  
    ]XG n2U\  
    if any(m>n) 4D8yb|o  
        error('zernfun:MlessthanN', ... DsW`V~ T  
              'Each M must be less than or equal to its corresponding N.') PBs<8xBx^  
    end c;rp@_ULG?  
    0bxvM  
    if any( r>1 | r<0 ) A4Q)YY9~  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') .(1j!B4^  
    end !en F8a  
    +R#`j r"  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) pu$XUt  
        error('zernfun:RTHvector','R and THETA must be vectors.') ?SOF n  
    end 6>BDA?  
    @*"<U]  
    r = r(:); X_({};mz  
    theta = theta(:); T:S{3  
    length_r = length(r); _Q}RElA  
    if length_r~=length(theta) `;qv}  
        error('zernfun:RTHlength', ... ms\/=96F  
              'The number of R- and THETA-values must be equal.') Bb[0\Hs7  
    end #Wm@&|U  
    i)=89?8  
    % Check normalization: khN:+V|  
    % -------------------- ]6%%X+$7  
    if nargin==5 && ischar(nflag) `{|}LFS>  
        isnorm = strcmpi(nflag,'norm'); @oqi@&L'C  
        if ~isnorm h NOYFH  
            error('zernfun:normalization','Unrecognized normalization flag.') x\bRj>%(  
        end F}B/-".^  
    else S[hJ{0V  
        isnorm = false; D@(M+u9/%  
    end T3t~=b>&L  
    LB*#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /yw\(|T  
    % Compute the Zernike Polynomials t6%xit+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% aBVEk2 p  
    C|d!'"p  
    % Determine the required powers of r: tD~PvUJ  
    % ----------------------------------- svq9@!go  
    m_abs = abs(m); K]pKe" M  
    rpowers = []; $|cp;~ 1  
    for j = 1:length(n) R3{*v =ov  
        rpowers = [rpowers m_abs(j):2:n(j)]; 9{UP)17  
    end -90ZI1O`  
    rpowers = unique(rpowers); k|$"TFXx;  
    8/>wgY  
    % Pre-compute the values of r raised to the required powers, 2 .Eu+*UC  
    % and compile them in a matrix: itC *Z6^  
    % ----------------------------- b?2X>QJ  
    if rpowers(1)==0  lGnql1(  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Q 9gFTLQ  
        rpowern = cat(2,rpowern{:}); yrE,,N%I  
        rpowern = [ones(length_r,1) rpowern]; Dmm r]~  
    else @1/}-.(n  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); L=$?q/=-  
        rpowern = cat(2,rpowern{:}); y800(z  
    end .i3lG( YG  
    H81.p  
    % Compute the values of the polynomials: CDnR  
    % -------------------------------------- pRiH,:\  
    y = zeros(length_r,length(n)); {glqWFT  
    for j = 1:length(n) "doU.U&u  
        s = 0:(n(j)-m_abs(j))/2; Pi"~/MGP$  
        pows = n(j):-2:m_abs(j); u_p7Mcb  
        for k = length(s):-1:1 #GY&$8.u*  
            p = (1-2*mod(s(k),2))* ... |>IUtUg\  
                       prod(2:(n(j)-s(k)))/              ... rAlh& ?X  
                       prod(2:s(k))/                     ... FD XWFJ  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... /H%<oAjp6  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); e\^g|60f_  
            idx = (pows(k)==rpowers); aJy>  
            y(:,j) = y(:,j) + p*rpowern(:,idx); z)ft3(!  
        end da9*9yN  
         }DDVGs[  
        if isnorm +3/k/W  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); [V> :`?  
        end daA47`+d  
    end "RV`L[(P*k  
    % END: Compute the Zernike Polynomials *l>[`U+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L@1,7@  
    O?nPxa<  
    % Compute the Zernike functions: j.=UI-&m  
    % ------------------------------ D0Vyh"ua  
    idx_pos = m>0; i14[3bPLk!  
    idx_neg = m<0; 9S:{  
    C@TN5?Z  
    z = y; ,YP1$gj  
    if any(idx_pos) ba(arGZ+{  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); .%x"t>]  
    end Sc;iAi (  
    if any(idx_neg) )(:+q(m  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); *Fa )\.XX  
    end `G=ztL!gq  
    {h/OnBwG  
    % 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) Ujce |>Wn  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. BA~a?"HS  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated  /6+1{p  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive zW*}`S "  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 0Y2\n-`z  
    %   and THETA is a vector of angles.  R and THETA must have the same eV[`P&j_C  
    %   length.  The output Z is a matrix with one column for every P-value, JP{UgcaF  
    %   and one row for every (R,THETA) pair. 9 9Ba{qj  
    % cZNi~  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 0lX)Cl  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) pyUNRqp  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) j2RRSz&9  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 vS7/~:C  
    %   for all p. |HrM_h<X  
    % a$w},= `E  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 )>(L{y|uYX  
    %   Zernike functions (order N<=7).  In some disciplines it is u9TzZ  
    %   traditional to label the first 36 functions using a single mode lp,\]]  
    %   number P instead of separate numbers for the order N and azimuthal Cw 1 9y  
    %   frequency M. `d75@0:  
    % 285_|!.Y  
    %   Example: +O)]^"TG  
    % FJO"|||Y'|  
    %       % Display the first 16 Zernike functions 3LKB;  
    %       x = -1:0.01:1; ykxbX  
    %       [X,Y] = meshgrid(x,x); w+q?T  
    %       [theta,r] = cart2pol(X,Y); , p_G/ OU  
    %       idx = r<=1; VBu8}}Ql  
    %       p = 0:15; ~}h^38  
    %       z = nan(size(X)); fJX\'Rc\  
    %       y = zernfun2(p,r(idx),theta(idx)); y)6,0K {k  
    %       figure('Units','normalized') ?Q@L-H`  
    %       for k = 1:length(p) J#0GlK@"  
    %           z(idx) = y(:,k); N*~_\x  
    %           subplot(4,4,k) Z`rK\Bc  
    %           pcolor(x,x,z), shading interp p lK=D#)  
    %           set(gca,'XTick',[],'YTick',[]) pJd0k"{  
    %           axis square yc*<:(p  
    %           title(['Z_{' num2str(p(k)) '}']) ?$3r5sx  
    %       end 6^Ph '  
    %  VJ3hC[  
    %   See also ZERNPOL, ZERNFUN. +W6Hva.  
    ;P3>>DZ  
    %   Paul Fricker 11/13/2006 #e*X0;m  
    j8pFgnQ  
    fCB:733H  
    % Check and prepare the inputs: 8)sg_JC  
    % ----------------------------- C*7!dW6  
    if min(size(p))~=1 Wdo#?@m  
        error('zernfun2:Pvector','Input P must be vector.') wa" uFW  
    end &ik$L!iX  
    M:_!w[NiLp  
    if any(p)>35 +O'vj  
        error('zernfun2:P36', ... Qu`n&  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... VMx%1^/(  
               '(P = 0 to 35).']) 3w Z(+<4i  
    end I0DM=V>;  
    \k;U}Te<  
    % Get the order and frequency corresonding to the function number: /KAlK5<  
    % ---------------------------------------------------------------- Uh.Sc:trA  
    p = p(:); ;+ G9-  
    n = ceil((-3+sqrt(9+8*p))/2); s;J\Kc?"|  
    m = 2*p - n.*(n+2); va5FxF*%  
    4b4QbJ$  
    % Pass the inputs to the function ZERNFUN: CN/IH   
    % ---------------------------------------- ;W0]66&  
    switch nargin  Vu [:A  
        case 3 84'?u m  
            z = zernfun(n,m,r,theta); Y;,Hzmbs6w  
        case 4 ~Eq\DK  
            z = zernfun(n,m,r,theta,nflag); ('t kZt%8  
        otherwise "x&3Z@q7  
            error('zernfun2:nargin','Incorrect number of inputs.') JvkL37^ n:  
    end . |uLt J  
    YdI0E   
    % 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) #dc1pfL!y{  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M.  ]>Si0%  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ''S&e  
    %   order N and frequency M, evaluated at R.  N is a vector of 5h8o4  
    %   positive integers (including 0), and M is a vector with the TY.FpW  
    %   same number of elements as N.  Each element k of M must be a 0Q~@F3N-\>  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) z856 nl  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is =x0No*#|'  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix $%PVJs  
    %   with one column for every (N,M) pair, and one row for every r7 VXeoX  
    %   element in R. u,7zFg)H  
    % 5+P@s D  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- :EZQ'3X  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is \Hwg) Uc{  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to \iU]s\{).  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 hazq#J!  
    %   for all [n,m]. Z0ReWrl;`  
    % ['[KR BJL  
    %   The radial Zernike polynomials are the radial portion of the 8$vK5Dnn8  
    %   Zernike functions, which are an orthogonal basis on the unit :eOR-}p'  
    %   circle.  The series representation of the radial Zernike u@=?#a$$  
    %   polynomials is ( 9$"#o  
    % Ht[{ryTxu  
    %          (n-m)/2 Dag`>|my  
    %            __ ;GsQR+en  
    %    m      \       s                                          n-2s ALGg AX3t  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ],lV}Mlg*  
    %    n      s=0 [B^G-  
    % !e3YnlE  
    %   The following table shows the first 12 polynomials. [vTMS2  
    % s_eOcm  
    %       n    m    Zernike polynomial    Normalization 0JY WrPR  
    %       --------------------------------------------- zgKY4R{V  
    %       0    0    1                        sqrt(2) v27Ja .tA  
    %       1    1    r                           2 iOqk*EL_r\  
    %       2    0    2*r^2 - 1                sqrt(6) 0a2@b"l  
    %       2    2    r^2                      sqrt(6) I,O#X)O|i  
    %       3    1    3*r^3 - 2*r              sqrt(8) Cca0](R*&  
    %       3    3    r^3                      sqrt(8) (/h5zCc/v  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) w([$@1]  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) q\$6F)ha3  
    %       4    4    r^4                      sqrt(10) >UHa  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) t FgX\4  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) N[U9d}Zv  
    %       5    5    r^5                      sqrt(12) /_\W+^fE  
    %       --------------------------------------------- N/~N7MwJj  
    % x#8w6@iPQ  
    %   Example: \cuS>G  
    % 'Qn~H[$/p  
    %       % Display three example Zernike radial polynomials UQT'6* !  
    %       r = 0:0.01:1; uDayBaR  
    %       n = [3 2 5]; Kt/)pc  
    %       m = [1 2 1]; :&VcB$  
    %       z = zernpol(n,m,r); nr2r8u9r  
    %       figure @CQb[!9C  
    %       plot(r,z) =%qEf   
    %       grid on 4]&<?"LSK  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') q'S =Eav8  
    % u oVNK  
    %   See also ZERNFUN, ZERNFUN2. H+Z SPHs  
    >q7 %UK]&  
    % A note on the algorithm. tRfm+hqRZ  
    % ------------------------ y' xF0  
    % The radial Zernike polynomials are computed using the series :q+N&j'3  
    % representation shown in the Help section above. For many special ?Zcj}e.r  
    % functions, direct evaluation using the series representation can w+AuMc  
    % produce poor numerical results (floating point errors), because #a9_~\s  
    % the summation often involves computing small differences between $v27]"]  
    % large successive terms in the series. (In such cases, the functions 3/goCg  
    % are often evaluated using alternative methods such as recurrence ;+Y i.Q/\  
    % relations: see the Legendre functions, for example). For the Zernike =-jD~rN4;P  
    % polynomials, however, this problem does not arise, because the 8cVzFFQP  
    % polynomials are evaluated over the finite domain r = (0,1), and py<_HyJ  
    % because the coefficients for a given polynomial are generally all k%Vv?{g  
    % of similar magnitude. raB+,Oi$G  
    % 3$p#;a:=n  
    % ZERNPOL has been written using a vectorized implementation: multiple (ku5WWJ  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] NHaMo*xQ  
    % values can be passed as inputs) for a vector of points R.  To achieve ;b%{ilx:  
    % this vectorization most efficiently, the algorithm in ZERNPOL XutF"9u  
    % involves pre-determining all the powers p of R that are required to :FQ1[X1 xm  
    % compute the outputs, and then compiling the {R^p} into a single D`o<,Y  
    % matrix.  This avoids any redundant computation of the R^p, and \54}T 4R  
    % minimizes the sizes of certain intermediate variables. |V&G81sM  
    % 3h=8"lRc  
    %   Paul Fricker 11/13/2006 pyB~M9Bp/  
    Cmd329AH  
     46,j9x  
    % Check and prepare the inputs: KL3<Iz]  
    % ----------------------------- "Pc$\zJm;  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) O!R"v'  
        error('zernpol:NMvectors','N and M must be vectors.') ')#,X^   
    end VWf %v  
    Dy9\O77>  
    if length(n)~=length(m)  Ewo~9 4{  
        error('zernpol:NMlength','N and M must be the same length.') cCdX0@hY  
    end 4zc<GL3[  
    a/:XXy |  
    n = n(:); m *X7T  
    m = m(:); WS0JS'  
    length_n = length(n); Ex(3D[WmMW  
    ;Ss$2V'a  
    if any(mod(n-m,2)) jX */piSq  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ;4~U,+Av  
    end nkY@_N  
    ;+_8&wbqW  
    if any(m<0) 3C^1f rF  
        error('zernpol:Mpositive','All M must be positive.') j$@tK0P  
    end _a'A~JY  
    8b&uU [  
    if any(m>n) H7&xLYQ2  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') (H#M<N  
    end FxU'LN<;HY  
    *i%quMv  
    if any( r>1 | r<0 ) %FA@)?~  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 6Z! y  
    end T0e<Slo~C  
    < K %j  
    if ~any(size(r)==1) O.+9,4A(  
        error('zernpol:Rvector','R must be a vector.') Y94MI1O5$  
    end w G!u+  
    4u1KF:g  
    r = r(:); !`wW_W  
    length_r = length(r); Z!{UWegun  
    n^9  ?~  
    if nargin==4 *"9<TSU%m  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm');  tFh|V pB  
        if ~isnorm tk?UX7F  
            error('zernpol:normalization','Unrecognized normalization flag.') >P(`MSc  
        end M?@p N<|  
    else ;=;JfNnbm  
        isnorm = false; `/:cfP\  
    end  Y7Gs7  
    &O1v,$}'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 46XN3r  
    % Compute the Zernike Polynomials f(Hu {c5yV  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8jnz}aBd  
    e3(<8]`b[  
    % Determine the required powers of r: k W-81  
    % ----------------------------------- 8l)  
    rpowers = []; .iv3q?8.b  
    for j = 1:length(n) @.9I3E-=  
        rpowers = [rpowers m(j):2:n(j)]; 7_)|I? =0d  
    end '"xiS$b(  
    rpowers = unique(rpowers); =3'(A14C=  
    Fdt}..H%  
    % Pre-compute the values of r raised to the required powers, ImkrV{,e  
    % and compile them in a matrix: T9jw X:n  
    % ----------------------------- b|?;h21rG  
    if rpowers(1)==0 {B{i(6C(  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5P ke8K  
        rpowern = cat(2,rpowern{:}); p@I9< ^"  
        rpowern = [ones(length_r,1) rpowern]; Cvn$]bt/s  
    else 9([6d.`~  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); b9("DZW;  
        rpowern = cat(2,rpowern{:}); 2*OxA%QELM  
    end T\sNtdF`:  
    '}{?AUDx  
    % Compute the values of the polynomials: km 5E)_]  
    % -------------------------------------- Q7a(P  
    z = zeros(length_r,length_n); oXGZK5w<l  
    for j = 1:length_n F)ci9-b@  
        s = 0:(n(j)-m(j))/2; P6Xp<^%E  
        pows = n(j):-2:m(j); ]VjLKFb~U  
        for k = length(s):-1:1 c> ~:dcy  
            p = (1-2*mod(s(k),2))* ... q=0 pQ1>  
                       prod(2:(n(j)-s(k)))/          ... &]NZvqdj.]  
                       prod(2:s(k))/                 ... GU6 qIz|  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... E(Gr0#8  
                       prod(2:((n(j)+m(j))/2-s(k))); 5~}!@yzc  
            idx = (pows(k)==rpowers); |]ucHV  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 0m=(W^c  
        end x_:hii?6V  
         04JT@s"o  
        if isnorm e\N0@   
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 5*j:K&R-.K  
        end ;`jU_  
    end GX38~pq  
    A ,<@m2  
    % 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)  GR_p1 C\  
    Z={UM/6w  
    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。 +D6-m  
    t=AE7  
    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)