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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 0VrsbkS  
    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! 8{]nS8i  
     
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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 <ShA_+Nd  
    function z = zernfun(n,m,r,theta,nflag) me{u~9&  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. uoOUgNwGg  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N xpO;V}M|  
    %   and angular frequency M, evaluated at positions (R,THETA) on the +&S6se4  
    %   unit circle.  N is a vector of positive integers (including 0), and [>r0 (x&.  
    %   M is a vector with the same number of elements as N.  Each element `Fo/RZOW  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 4bp})>}jB  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, \lm]G7h  
    %   and THETA is a vector of angles.  R and THETA must have the same fqY'Uq$=  
    %   length.  The output Z is a matrix with one column for every (N,M) 4oH ,_sr  
    %   pair, and one row for every (R,THETA) pair. :UP8nq  
    % /5/gnp C  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike z'$1$~I  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), =EMB~i  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral }mK,Bi?bj  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, aA52Li  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 6 iMJ0  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5qW>#pTFVV  
    % A9 g%>  
    %   The Zernike functions are an orthogonal basis on the unit circle. ] uyp i#[  
    %   They are used in disciplines such as astronomy, optics, and YS){ N=g&'  
    %   optometry to describe functions on a circular domain. .?Y"o3  
    % xlJWCA*>  
    %   The following table lists the first 15 Zernike functions. &Q;sbI}  
    % ~=iH*AQR  
    %       n    m    Zernike function           Normalization CX{6  
    %       -------------------------------------------------- Dqii60  
    %       0    0    1                                 1 D?"P\b[/  
    %       1    1    r * cos(theta)                    2 .kg 3>*  
    %       1   -1    r * sin(theta)                    2 <7F-WR/2n  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) aK - x{  
    %       2    0    (2*r^2 - 1)                    sqrt(3) B+U:=591  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ^7gKs2M  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) oC49c~`8  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ]#^v754X^T  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) rG6G~ |mS  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) _Iav2= 0Wi  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) gee~>l  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?..BA&zRk  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) th[v"qD9G  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) t~j 6wsx;  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) $1|E(d1  
    %       -------------------------------------------------- bV&9>fC  
    % `qs}L  
    %   Example 1: r4X}U|s!0  
    % \8QOZjy  
    %       % Display the Zernike function Z(n=5,m=1) .cQO?UKK  
    %       x = -1:0.01:1; <JWU@A-.y  
    %       [X,Y] = meshgrid(x,x); jBYv Oy*$Q  
    %       [theta,r] = cart2pol(X,Y); v;o1c44;  
    %       idx = r<=1; pN5kcvQ  
    %       z = nan(size(X)); 2vjkThh`I  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ~ W8X g)  
    %       figure >lUPOc  
    %       pcolor(x,x,z), shading interp "nu]3zcd  
    %       axis square, colorbar ;un@E:  
    %       title('Zernike function Z_5^1(r,\theta)') S \]O8#OX  
    % "4\  
    %   Example 2: EwN{|34C  
    % h>\C2Q  
    %       % Display the first 10 Zernike functions s<F*kLib  
    %       x = -1:0.01:1; d'ZNp2L  
    %       [X,Y] = meshgrid(x,x); j@z IJ  
    %       [theta,r] = cart2pol(X,Y); Mww^  
    %       idx = r<=1; /Rq\Mgb  
    %       z = nan(size(X)); >pfeP"[(3  
    %       n = [0  1  1  2  2  2  3  3  3  3]; K9k!P8Rd  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ~h3G}EH  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; {V QGfN  
    %       y = zernfun(n,m,r(idx),theta(idx)); ]A=\P,D  
    %       figure('Units','normalized') OA3J(4!"W  
    %       for k = 1:10 -[-oz0`Sl{  
    %           z(idx) = y(:,k); (V6bX]<  
    %           subplot(4,7,Nplot(k)) BjvQ6M{Y"+  
    %           pcolor(x,x,z), shading interp SKH}!Id}n  
    %           set(gca,'XTick',[],'YTick',[]) (^}t  
    %           axis square JK =A=  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]64}Xob87_  
    %       end c}qpmWF  
    % V )<>W_g  
    %   See also ZERNPOL, ZERNFUN2. ,]2?S5R  
    c{/R?<  
    %   Paul Fricker 11/13/2006 n]IF`kYQV  
    dRJ ](Gw  
    XMI*obS'z  
    % Check and prepare the inputs: /@ @F nQ++  
    % ----------------------------- n;Oe-+oSC  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) dw<i)P^   
        error('zernfun:NMvectors','N and M must be vectors.') s0?'mC+p  
    end DPzW,aIgv  
    r V%6 8x9  
    if length(n)~=length(m) C{J5:ak  
        error('zernfun:NMlength','N and M must be the same length.') hUlRtt  
    end AfTm#-R  
    et 1HbX  
    n = n(:); o7!A(Eu  
    m = m(:); IEy$2f>Ns  
    if any(mod(n-m,2)) 3$ ! QP N  
        error('zernfun:NMmultiplesof2', ... dA h cA.  
              'All N and M must differ by multiples of 2 (including 0).') })-V,\  
    end y]jx-w c3O  
    6LDZ|K@  
    if any(m>n) iP(MDVg  
        error('zernfun:MlessthanN', ... ~DK.Y   
              'Each M must be less than or equal to its corresponding N.') 5qnei\~  
    end plWNuEW  
    #,#_"  
    if any( r>1 | r<0 ) 8?nn4]P  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Z2]0brV  
    end FFw(`[A_  
    .:j{d}p}  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) XS&Pc  
        error('zernfun:RTHvector','R and THETA must be vectors.') 8UjIC4'  
    end w PR Ns9^  
    \XB,)XDB  
    r = r(:); X <xM '  
    theta = theta(:); 8`*5[ L~~/  
    length_r = length(r); 1-p#}VX  
    if length_r~=length(theta) #a}w&O";  
        error('zernfun:RTHlength', ... -KGJr  
              'The number of R- and THETA-values must be equal.') M$EF 8   
    end m-O*t$6  
    t`JT  
    % Check normalization: PL= v,NB  
    % -------------------- ^ `yhN  
    if nargin==5 && ischar(nflag) bDvGFSAH  
        isnorm = strcmpi(nflag,'norm'); U^7hw(}me  
        if ~isnorm ~},H+A!?  
            error('zernfun:normalization','Unrecognized normalization flag.') AJ/Hw>>$?m  
        end %_E5B6xi{  
    else pA.orx  
        isnorm = false; ^N<aHFF  
    end  GhfhR^P  
    .$-;`&0cZ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ye On   
    % Compute the Zernike Polynomials !6|_`l>G,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2*D2jw  
    \5}PF+)|  
    % Determine the required powers of r: 1^$hbRq  
    % ----------------------------------- Q I";[  
    m_abs = abs(m); hXI[FICQU{  
    rpowers = []; ZiR}S  
    for j = 1:length(n) _(f@b1O~  
        rpowers = [rpowers m_abs(j):2:n(j)]; $CB&>?~  
    end h's[) t  
    rpowers = unique(rpowers); ]xvhUv!G  
    l#cVQ_^"  
    % Pre-compute the values of r raised to the required powers, P7}w^#x  
    % and compile them in a matrix: :j+E]|d(~6  
    % ----------------------------- \)28,`  
    if rpowers(1)==0 *=@8t^fa86  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ek)rsxf1A  
        rpowern = cat(2,rpowern{:}); GThGV"  
        rpowern = [ones(length_r,1) rpowern]; Q3ZGN1aX<  
    else kVtP~  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); E~q3o*  
        rpowern = cat(2,rpowern{:}); \_.'/<aQ  
    end yzfiH4  
    ;VCV%=W<  
    % Compute the values of the polynomials: 1<@lM8&.kO  
    % -------------------------------------- Lb$Uba-_  
    y = zeros(length_r,length(n)); s8(Z&pQ  
    for j = 1:length(n) hRuiuGC  
        s = 0:(n(j)-m_abs(j))/2; ZOqA8#\  
        pows = n(j):-2:m_abs(j); ^e "4@O"  
        for k = length(s):-1:1 jR1^e$  
            p = (1-2*mod(s(k),2))* ... AIl`>ac  
                       prod(2:(n(j)-s(k)))/              ... rMG[,:V  
                       prod(2:s(k))/                     ... W9gQho%9b  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... u^C\aujg  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); >}.~Y#Ge  
            idx = (pows(k)==rpowers); XKp(31])  
            y(:,j) = y(:,j) + p*rpowern(:,idx); EO'+r[Y  
        end 2O(k@M5E?  
         TS=%iMa  
        if isnorm gz'{l[  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); \l(}8;5}  
        end si%V63^lN  
    end T:Q+ Z }v+  
    % END: Compute the Zernike Polynomials q:vN3#=^qf  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fc:87ZR{K  
    6/QWzw.0c  
    % Compute the Zernike functions: w2 (}pz:  
    % ------------------------------ .nr%c*JUp  
    idx_pos = m>0; ?>=vKU5  
    idx_neg = m<0; 0*^f EoV  
     svo%NQ  
    z = y; ,EH-Sf2Cb  
    if any(idx_pos) zGO_S\  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); #/(L.5d[  
    end pkIQ,W{Ke  
    if any(idx_neg) tm34Z''.>  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); +7"UF) ~k  
    end B$ =1@  
    /;TD n>lq  
    % 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) }/xdHt  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 1{xkAy0  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated &@&^k$du8q  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive GO#eI]>/r  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 8r.MODZG/  
    %   and THETA is a vector of angles.  R and THETA must have the same CZv^,O(M?2  
    %   length.  The output Z is a matrix with one column for every P-value, 2JHF*zvO-  
    %   and one row for every (R,THETA) pair. TTxSl p2=;  
    % kvN6K6  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike v<} $d.&*  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) Q!fk|D+j  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) )/v`k>E  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 d D^?%,a  
    %   for all p. ,(yaWd6  
    % e $5s],,n  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 xUs1-O1i  
    %   Zernike functions (order N<=7).  In some disciplines it is KC\W6|NtGj  
    %   traditional to label the first 36 functions using a single mode ~r]$(V n  
    %   number P instead of separate numbers for the order N and azimuthal kdm@1x  
    %   frequency M. 0pe*DbYP5  
    % ^[ >  
    %   Example: Ac,bf 8C  
    % DV bY   
    %       % Display the first 16 Zernike functions cmLI!"RLe  
    %       x = -1:0.01:1; 6}mSA@4&  
    %       [X,Y] = meshgrid(x,x); ^6^A/]v  
    %       [theta,r] = cart2pol(X,Y); A%#."2vq~  
    %       idx = r<=1; ~!dO2\X+  
    %       p = 0:15; k:E+]5  
    %       z = nan(size(X)); t9kqX(!  
    %       y = zernfun2(p,r(idx),theta(idx)); 4vPKDd  
    %       figure('Units','normalized') ?Qh[vcF7`  
    %       for k = 1:length(p) +3;[1dpgf  
    %           z(idx) = y(:,k); rOq>jvy  
    %           subplot(4,4,k) r%oXO]X  
    %           pcolor(x,x,z), shading interp 771r(X?Fa  
    %           set(gca,'XTick',[],'YTick',[]) '~Gk{'Nx"  
    %           axis square w3oe.hWP3N  
    %           title(['Z_{' num2str(p(k)) '}']) 1\Vp[^#Vx  
    %       end 0bMbM^xV6  
    % yCye3z.  
    %   See also ZERNPOL, ZERNFUN. Zv1/J}+  
    v%2Jm!i+  
    %   Paul Fricker 11/13/2006 Nxt z1  
    y +c 3#  
    cX-) ]D  
    % Check and prepare the inputs: xXc3#n  
    % ----------------------------- "/U~j4O  
    if min(size(p))~=1 ;!N_8{ 7r  
        error('zernfun2:Pvector','Input P must be vector.') xHdv?69,  
    end qgLj^{  
    |/p2DU2  
    if any(p)>35 &l1CE1 9<  
        error('zernfun2:P36', ... ~bk+JK- >  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... s av  
               '(P = 0 to 35).']) )SFy Q  
    end %L;'C v  
    R a?0jcSQ$  
    % Get the order and frequency corresonding to the function number: Q" an6ht|  
    % ---------------------------------------------------------------- ~ f>km|Q{u  
    p = p(:); H;eOrX {GT  
    n = ceil((-3+sqrt(9+8*p))/2); 9 l9|w4YJs  
    m = 2*p - n.*(n+2); ZvO,1B  
    J#& C&S 2  
    % Pass the inputs to the function ZERNFUN: N,NEg4 q[  
    % ---------------------------------------- S~LT Lv:>  
    switch nargin 0xg6  
        case 3 5%Q[X  
            z = zernfun(n,m,r,theta); /WKp\r(Hp  
        case 4 !NFP=m1  
            z = zernfun(n,m,r,theta,nflag); u9%)_Q!14  
        otherwise VjVL/SO/  
            error('zernfun2:nargin','Incorrect number of inputs.') |F#L{=B  
    end JmK[7t  
    DN;An0 {MK  
    % 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) ( nH3  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. a_0G4@=T  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ;tF7 GjEp  
    %   order N and frequency M, evaluated at R.  N is a vector of <2a7>\74E0  
    %   positive integers (including 0), and M is a vector with the jreY'y:  
    %   same number of elements as N.  Each element k of M must be a iE$/ Rcp  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) tCdgtZm  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is {s=$.Kg  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix =K)au$BE|  
    %   with one column for every (N,M) pair, and one row for every 4V`ypFme  
    %   element in R. 85fBKpEe  
    % `_;VD?")*l  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- Wh)QCp0|n  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is i+V4_`  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 2Xm\;7  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 F*_mHYa;  
    %   for all [n,m]. q)uq?sZe  
    % F_$K+6  
    %   The radial Zernike polynomials are the radial portion of the l%^'K%'b  
    %   Zernike functions, which are an orthogonal basis on the unit S/ Y1NH  
    %   circle.  The series representation of the radial Zernike %HJ_0qg  
    %   polynomials is :B<lDcFKJ  
    %  nO~TW  
    %          (n-m)/2 N(]>(S o  
    %            __ B%L0g.D"  
    %    m      \       s                                          n-2s ba3_5 5]  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r k.DDfuKN  
    %    n      s=0 :LL>C)(f  
    % ^6~CA  
    %   The following table shows the first 12 polynomials. ^AUmIyf_  
    % m(Cn'@i`"0  
    %       n    m    Zernike polynomial    Normalization  [;D4,@A  
    %       --------------------------------------------- m.MOn3n]  
    %       0    0    1                        sqrt(2) V$wf;v0d(  
    %       1    1    r                           2 }Jgz#d  
    %       2    0    2*r^2 - 1                sqrt(6) rBP!RSl1  
    %       2    2    r^2                      sqrt(6) ]OoqU-q  
    %       3    1    3*r^3 - 2*r              sqrt(8) 1e;^Mz B"  
    %       3    3    r^3                      sqrt(8) ~h;c3#wuc  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) %cD7}o:u  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) >Z?3dM~[  
    %       4    4    r^4                      sqrt(10) J*8fGR%  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) /0 ,#c2aq  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ?R0sY ?u  
    %       5    5    r^5                      sqrt(12) `?Wak =]g  
    %       --------------------------------------------- B_[^<2_  
    % H;<hmbN?d  
    %   Example: ' hL\xf{  
    % 6ZVJ2xs[%  
    %       % Display three example Zernike radial polynomials +gTnq")wnI  
    %       r = 0:0.01:1; aF$HF;-y  
    %       n = [3 2 5]; Z^AACKME  
    %       m = [1 2 1]; Q^8C*ekfg!  
    %       z = zernpol(n,m,r); 5isejR{r  
    %       figure ,|b<as@X  
    %       plot(r,z) "qb3\0O  
    %       grid on c{j0A;XMS  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') NrTQ}_3)  
    % ^zQ;8)ng  
    %   See also ZERNFUN, ZERNFUN2. (?&X<=|"  
    8J%^gy>m]  
    % A note on the algorithm. oa+Rr&t'  
    % ------------------------ _ 9Tv*@  
    % The radial Zernike polynomials are computed using the series LN?f w  
    % representation shown in the Help section above. For many special ekfD+X  
    % functions, direct evaluation using the series representation can p SASMc@  
    % produce poor numerical results (floating point errors), because mz[rB|v"/7  
    % the summation often involves computing small differences between }(dhXOf\q  
    % large successive terms in the series. (In such cases, the functions `Y/DttjL  
    % are often evaluated using alternative methods such as recurrence `w]=x e  
    % relations: see the Legendre functions, for example). For the Zernike C]-Z+9Vvv  
    % polynomials, however, this problem does not arise, because the :M1S*"&:  
    % polynomials are evaluated over the finite domain r = (0,1), and yjpV71!M  
    % because the coefficients for a given polynomial are generally all IyYC).wU}  
    % of similar magnitude. 'rU [V+  
    % !tFs(![  
    % ZERNPOL has been written using a vectorized implementation: multiple H(!)]dO  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] X #-U  
    % values can be passed as inputs) for a vector of points R.  To achieve yuk64o2QE  
    % this vectorization most efficiently, the algorithm in ZERNPOL PV>-"2n  
    % involves pre-determining all the powers p of R that are required to ) ]U-7  
    % compute the outputs, and then compiling the {R^p} into a single /_?Ly$>'  
    % matrix.  This avoids any redundant computation of the R^p, and nvxftbfE^D  
    % minimizes the sizes of certain intermediate variables. N/Z3 EF_  
    % k#ED#']N  
    %   Paul Fricker 11/13/2006 7Rk eV  
    S?*v p=  
    >"B95$x5  
    % Check and prepare the inputs: >tqLwC."'  
    % ----------------------------- #5N#^#r"  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) zhU)bb[A  
        error('zernpol:NMvectors','N and M must be vectors.') b-@VR  
    end .3A66 O~zT  
    Ej ip%m  
    if length(n)~=length(m) 0eQyzn*98  
        error('zernpol:NMlength','N and M must be the same length.') "N"9PTX  
    end GrW+P[j9  
    1Qtojph  
    n = n(:); U"A]b(54  
    m = m(:); pA+W 8v#*  
    length_n = length(n); %w,  
    y1(smZU  
    if any(mod(n-m,2)) Xp{+){Iu  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') b"t!nfgo  
    end Ja| ! fT  
     "C B*  
    if any(m<0) WsTbqR)W%  
        error('zernpol:Mpositive','All M must be positive.') T#Qn\ 8  
    end eR D?O  
    vL`wn=  
    if any(m>n) A}FEM[2  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') OnC|9  
    end f:GZb?Wyd  
    B8'" ^a^&-  
    if any( r>1 | r<0 ) :z56!qU  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') KO<Yc`Fs  
    end ddmTMfH  
    Q_euNoA0  
    if ~any(size(r)==1) .ZFs+8qU>  
        error('zernpol:Rvector','R must be a vector.') YW}q@AY7  
    end [iwn"e  
    h)"PPI  
    r = r(:); VJS|H!CH  
    length_r = length(r); j#"?Oe{_1  
    t;T MD\BU  
    if nargin==4 &7!&]kA+  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); p)N=  
        if ~isnorm **d3uc4y  
            error('zernpol:normalization','Unrecognized normalization flag.') S " R]i  
        end 5*xk8*  
    else Y'&A~/Adf  
        isnorm = false; r]sv50Fy  
    end SG2s!Ht  
    -LJbx<'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (GJ)FWen0"  
    % Compute the Zernike Polynomials M%7{g"J*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SEq_37  
    <->Nex  
    % Determine the required powers of r: J4#t1P@Na  
    % ----------------------------------- 8C#R  
    rpowers = []; %g{m12  
    for j = 1:length(n) ^\Nsx)Y;  
        rpowers = [rpowers m(j):2:n(j)]; r`pg`ChHv  
    end aB;syl{  
    rpowers = unique(rpowers); 8 sZ~3  
    UM(tM9  
    % Pre-compute the values of r raised to the required powers, !jL|HwlA  
    % and compile them in a matrix: "~0m_brf  
    % ----------------------------- Q"n*`#Yt'  
    if rpowers(1)==0 Jp=eh   
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); OW- [#r  
        rpowern = cat(2,rpowern{:}); Q0i.gEwe  
        rpowern = [ones(length_r,1) rpowern]; `9n%Dy<  
    else Fd*)1FQKT  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); U8KB @E  
        rpowern = cat(2,rpowern{:}); NK2Kw{c"iI  
    end m_\w)  
    o7)<pfif  
    % Compute the values of the polynomials: Gkv<)}G  
    % -------------------------------------- 1O>wXq7q  
    z = zeros(length_r,length_n); 0f}zm8p7.  
    for j = 1:length_n 1Y:lFGoe  
        s = 0:(n(j)-m(j))/2; l)< '1dqe  
        pows = n(j):-2:m(j); CpNnywDRwU  
        for k = length(s):-1:1 U~n>k<`sr  
            p = (1-2*mod(s(k),2))* ... (xf_  
                       prod(2:(n(j)-s(k)))/          ... Gx ci  
                       prod(2:s(k))/                 ... \Y&*sfQ  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... u[q1]]   
                       prod(2:((n(j)+m(j))/2-s(k))); ;UTT>j  
            idx = (pows(k)==rpowers); 6!bp;iLKy  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Fk@A;22N  
        end +b"RZ:tKp  
         +e?mKLw14  
        if isnorm d0$dQg  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); P9= L?t.  
        end U]tbV<m%  
    end 2`hc0 IE  
    7cV9xIe^  
    % 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)  arf`%9M  
    `5}XmSJ?5  
    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。 cy=,Dr9O  
    =x "N0p  
    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)