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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 J8?6G&0H  
    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! ;Owu:}   
     
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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 nGgc~E$j  
    function z = zernfun(n,m,r,theta,nflag) .FRF<_`^  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 2Wf qgR[3  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 6="&K_Q7  
    %   and angular frequency M, evaluated at positions (R,THETA) on the at]Q4  
    %   unit circle.  N is a vector of positive integers (including 0), and o(NyOC  
    %   M is a vector with the same number of elements as N.  Each element ?s} E<Kr  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) |aJ6363f.  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Ic!83-  
    %   and THETA is a vector of angles.  R and THETA must have the same Qf(e'e  
    %   length.  The output Z is a matrix with one column for every (N,M) 0BE^qe  
    %   pair, and one row for every (R,THETA) pair. <OfzE5  
    % BXw,Rz }  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike )K3 vzX  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), <qY>d,+E'  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral #%tL8/K*  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, [4rMUS7-m"  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ;]x5;b9`  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Qs X59d  
    % 0-f-  
    %   The Zernike functions are an orthogonal basis on the unit circle. (gB=!1/|G  
    %   They are used in disciplines such as astronomy, optics, and $%8n,FJ[  
    %   optometry to describe functions on a circular domain. K"$ky,tU  
    % .3&OFM  
    %   The following table lists the first 15 Zernike functions. >*xzSd? \  
    % U%\2drM&]  
    %       n    m    Zernike function           Normalization iquGLwJ  
    %       -------------------------------------------------- *tPY  
    %       0    0    1                                 1 { F8,^+b|  
    %       1    1    r * cos(theta)                    2 6ng g*kE<  
    %       1   -1    r * sin(theta)                    2 XPTB,1g+f  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) rqJj!{<B  
    %       2    0    (2*r^2 - 1)                    sqrt(3) jk}PucV  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) <qt%MM [Y  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) &B7KWvAy  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 4\es@2q  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) O G}&%NgH  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) bA,D]  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) \>7-<7+I6  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) N6%q%7F.:  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) *OcptmY<  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) l= S_#  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ?7a[| -  
    %       -------------------------------------------------- s>I}-=.(Q  
    % qrYeh`Mv  
    %   Example 1: ?=rh=#  
    % +t{FF!mL  
    %       % Display the Zernike function Z(n=5,m=1) -~ Q3T9+  
    %       x = -1:0.01:1; '#6DI"vJ  
    %       [X,Y] = meshgrid(x,x); [~S0b  
    %       [theta,r] = cart2pol(X,Y); !W^II>Y  
    %       idx = r<=1; x%&V!L  
    %       z = nan(size(X)); -v@^6bQVp  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); j,jUg}b  
    %       figure n//a;m  
    %       pcolor(x,x,z), shading interp O v6=|]cW  
    %       axis square, colorbar 8;3FTF  
    %       title('Zernike function Z_5^1(r,\theta)') r'?&VS-Cj  
    % -H]O&u3'c  
    %   Example 2: qChPT:a  
    % 9z}kkYk  
    %       % Display the first 10 Zernike functions R!CUR~F  
    %       x = -1:0.01:1; -E"o)1Pj6C  
    %       [X,Y] = meshgrid(x,x); li^E$9oWC  
    %       [theta,r] = cart2pol(X,Y); w2GY,,R  
    %       idx = r<=1; HjD= .Q  
    %       z = nan(size(X)); 6}2Lt[>O  
    %       n = [0  1  1  2  2  2  3  3  3  3]; zv@o- R$l  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; / KM+PeO  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; :+$_(* Z  
    %       y = zernfun(n,m,r(idx),theta(idx)); v)EJ|2`  
    %       figure('Units','normalized') E;0"1 P|S  
    %       for k = 1:10 C?k4<B7V  
    %           z(idx) = y(:,k); 7lu;lAAP  
    %           subplot(4,7,Nplot(k)) u}_q'=<\  
    %           pcolor(x,x,z), shading interp a8TE  
    %           set(gca,'XTick',[],'YTick',[]) [MG:Ym).2`  
    %           axis square n2~rrQ \/p  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) NunT2JP.  
    %       end X3vrD{uNU  
    % 1|CO>)*D  
    %   See also ZERNPOL, ZERNFUN2. qm@hD>W+  
    up6LO7drW/  
    %   Paul Fricker 11/13/2006 s!Vtw p9  
    9UX-)!  
    $2 0*&4y^  
    % Check and prepare the inputs: UQ y+ &;#5  
    % ----------------------------- $[e*0!e  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) J u7AxTf~  
        error('zernfun:NMvectors','N and M must be vectors.') e2v,#3Q\  
    end ZN^Q!v  
    '|.u*M,b  
    if length(n)~=length(m) r38CPdE;}  
        error('zernfun:NMlength','N and M must be the same length.') %' Fc%3  
    end fpUX @b  
    ~mU#u\r(*  
    n = n(:); klKt^h-  
    m = m(:); SBA;p7^"  
    if any(mod(n-m,2)) DpAuI w7|  
        error('zernfun:NMmultiplesof2', ... %* 8QLI  
              'All N and M must differ by multiples of 2 (including 0).') #PGExN3e  
    end EP @=i  
    mz''-1YY$  
    if any(m>n) ~W4<M:R  
        error('zernfun:MlessthanN', ... R?k1)n   
              'Each M must be less than or equal to its corresponding N.') F-t-d1w6  
    end SU^/qF%8  
    <W1!n$V ]  
    if any( r>1 | r<0 ) 3ul  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') azSS:=A  
    end f|EWu  
    Sc(2c.HO*  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Ty5\zxC|  
        error('zernfun:RTHvector','R and THETA must be vectors.') y}|zH  
    end @/~41\=e  
    h&XyMm9C  
    r = r(:); 'RhMzPmY>  
    theta = theta(:); }x+{=%~N  
    length_r = length(r); h^4oy^9  
    if length_r~=length(theta) OT zh=Z^r  
        error('zernfun:RTHlength', ... LY"/ Q  
              'The number of R- and THETA-values must be equal.') {.sF&(e   
    end vwg\qKqSM  
    )g-*fSa  
    % Check normalization: ky*-_  
    % -------------------- 2>mDT  
    if nargin==5 && ischar(nflag) "8N]1q:$4  
        isnorm = strcmpi(nflag,'norm'); hFKYRZtP.8  
        if ~isnorm r$+9grm<  
            error('zernfun:normalization','Unrecognized normalization flag.') YEGXhn5E  
        end m{' q(w}  
    else GXwV>)!x  
        isnorm = false; n '&WIf3  
    end {It4=I)M  
    StE4n0V  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }[1I_)  
    % Compute the Zernike Polynomials P5Fm<f8\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :f `1  
    }/6jom9U?  
    % Determine the required powers of r: 6(wpf^br2  
    % ----------------------------------- yjr!8L:m  
    m_abs = abs(m); D[<8(~VP  
    rpowers = []; 7Y_S%B:F  
    for j = 1:length(n) Qv8Z64#  
        rpowers = [rpowers m_abs(j):2:n(j)]; K@h v[4  
    end upWq=_  
    rpowers = unique(rpowers); =U?"#   
    FG'1;x!  
    % Pre-compute the values of r raised to the required powers, yNO5h]o  
    % and compile them in a matrix: Yx,  
    % ----------------------------- e-Eoe_k  
    if rpowers(1)==0 @o8\`G  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); D:f0W v  
        rpowern = cat(2,rpowern{:}); a7ZPV1k  
        rpowern = [ones(length_r,1) rpowern]; :.@gd7T  
    else W8\K_M}  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); !Y5O3^I=u  
        rpowern = cat(2,rpowern{:}); R# gip  
    end G|.>p<q   
    &K}!R$[,:P  
    % Compute the values of the polynomials: s`&8tP  
    % -------------------------------------- #b:8-Lt:M  
    y = zeros(length_r,length(n)); fAJQ8nb{@]  
    for j = 1:length(n) a(bgPkPP  
        s = 0:(n(j)-m_abs(j))/2; NoV2<m$  
        pows = n(j):-2:m_abs(j); @ %kCe>r  
        for k = length(s):-1:1 .aF+>#V=Q  
            p = (1-2*mod(s(k),2))* ... d!8`}L:=M  
                       prod(2:(n(j)-s(k)))/              ... U nGG%  
                       prod(2:s(k))/                     ... R}BHRmSQ  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... faThXq8B  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); \9!W^i[+  
            idx = (pows(k)==rpowers); m"NZ;*d'  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 9"oc.ue.2D  
        end OLlNCb#t  
         <kt,aMw[*  
        if isnorm {3'z}q  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); kE=}.  
        end F+|zCEc  
    end GYZzWN}U  
    % END: Compute the Zernike Polynomials !|hv49!H  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2BEF8o]Np  
    In5' (UHW:  
    % Compute the Zernike functions: }_Jr[iaB  
    % ------------------------------ byoDGUv  
    idx_pos = m>0; q B5cF_  
    idx_neg = m<0; cOq^}Ohan  
    \_qiUvPf\  
    z = y; \2@OS6LUe  
    if any(idx_pos) Y;4nIWe JL  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); x)h5W+$  
    end `A])4q$  
    if any(idx_neg) 8" XbW7^o  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); (@>X!]{$  
    end =EgiV<6vcH  
    tUH#%  
    % 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) :Em[> XA  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Ol"*(ea-TX  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated HNu/b)-Rb  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 8HS1^\~(6l  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, K\v1o  
    %   and THETA is a vector of angles.  R and THETA must have the same WgF Xv@Jjt  
    %   length.  The output Z is a matrix with one column for every P-value, l1 fP@|  
    %   and one row for every (R,THETA) pair. :)_Ap{9J  
    % ~m2tWi@  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike dq?{?~3  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) X!KjRP\\  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) a=>PGriL  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 epqX2`!V  
    %   for all p. O'a Srjl  
    % 6&5p3G{%0  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 TL lR"L5  
    %   Zernike functions (order N<=7).  In some disciplines it is r~N0P|Tq  
    %   traditional to label the first 36 functions using a single mode bX23F?  
    %   number P instead of separate numbers for the order N and azimuthal GSj04-T"  
    %   frequency M. tG+ E'OP  
    % ?{ns1nW:  
    %   Example: u2,V34b-  
    % ]~iOO %&R  
    %       % Display the first 16 Zernike functions ;"l>HL:^  
    %       x = -1:0.01:1; 1A^~gYr  
    %       [X,Y] = meshgrid(x,x); _1S^A0ft  
    %       [theta,r] = cart2pol(X,Y); z'GYU=  
    %       idx = r<=1; )>abB?RZ  
    %       p = 0:15; O:3LA-vA  
    %       z = nan(size(X)); ]U.1z  
    %       y = zernfun2(p,r(idx),theta(idx)); 1$vsw  
    %       figure('Units','normalized') K"B2 SsC  
    %       for k = 1:length(p) =QXLr+ y@  
    %           z(idx) = y(:,k); D^V0kC p!F  
    %           subplot(4,4,k) PZmg7N  
    %           pcolor(x,x,z), shading interp `&xo;Vnc  
    %           set(gca,'XTick',[],'YTick',[]) u?6L.^Op  
    %           axis square gI a/sD2m>  
    %           title(['Z_{' num2str(p(k)) '}']) c~bi ~ f  
    %       end sJu^deX  
    % / V}>v  
    %   See also ZERNPOL, ZERNFUN. 4 qMO@E_  
    O CIWQ/ P  
    %   Paul Fricker 11/13/2006  JsAl;w  
    huVw+vAA  
    frV *+  
    % Check and prepare the inputs: 6 B>1"h%Wf  
    % ----------------------------- HF>Gf2- C  
    if min(size(p))~=1 =g| e- XC  
        error('zernfun2:Pvector','Input P must be vector.') ~$xLR/{y  
    end #~<cp)!3  
    e%. Xya#\  
    if any(p)>35 r:Uqtqxh  
        error('zernfun2:P36', ... [gI;;GW  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... } m5AO4:  
               '(P = 0 to 35).']) 6apK]PT  
    end H 6 i4>U*  
    $h"Ht2/ J  
    % Get the order and frequency corresonding to the function number: v|r\kr k  
    % ---------------------------------------------------------------- U,Py+c6  
    p = p(:); ;{'{*g[  
    n = ceil((-3+sqrt(9+8*p))/2); R(_UR)G0 @  
    m = 2*p - n.*(n+2); XwWp4`Fd  
    ~gU.z6us  
    % Pass the inputs to the function ZERNFUN: Ws2SD6!4`  
    % ---------------------------------------- |KEq-  
    switch nargin ~a@O1MB  
        case 3 R&Mv|R   
            z = zernfun(n,m,r,theta); K6"#&0  
        case 4 c *<"&  
            z = zernfun(n,m,r,theta,nflag); {@j0?s  
        otherwise : V16bRpjL  
            error('zernfun2:nargin','Incorrect number of inputs.') m2&"}bI{  
    end 5cLq6[uO  
    Y JzKE7%CO  
    % 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) bpq2TgFj  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. ^@W98_bd;  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ERSo&8  
    %   order N and frequency M, evaluated at R.  N is a vector of :W]IJ mI\  
    %   positive integers (including 0), and M is a vector with the )na 8a!  
    %   same number of elements as N.  Each element k of M must be a fwvPh&U&  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ^(,qkq'u D  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 'EF\=o)^Y  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix GS),rNBur  
    %   with one column for every (N,M) pair, and one row for every `LD#fg*  
    %   element in R. Yr9>ATR  
    % B I9~% dm  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- dOm`p W^  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ?,Z[)5 ZN  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to f5)4H  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 w]n ,`r^  
    %   for all [n,m]. #is1y3yh  
    % (dSf>p r2  
    %   The radial Zernike polynomials are the radial portion of the R7'a/  
    %   Zernike functions, which are an orthogonal basis on the unit Sw##C l#  
    %   circle.  The series representation of the radial Zernike ^A9D;e6!-  
    %   polynomials is ^a9v5hu  
    % 'EsN{.l?  
    %          (n-m)/2 z'cK,psq(  
    %            __ h@nNm30i  
    %    m      \       s                                          n-2s 8J60+2Wa  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r -w8c;5X  
    %    n      s=0 i21ybXA=Z  
    % K@Z K@++  
    %   The following table shows the first 12 polynomials. &zVF!xNy&  
    % ( e> .hfrs  
    %       n    m    Zernike polynomial    Normalization Dx<">4   
    %       ---------------------------------------------  VlGg?  
    %       0    0    1                        sqrt(2) x,kZ>^]&b  
    %       1    1    r                           2 Z<j(ZVO  
    %       2    0    2*r^2 - 1                sqrt(6) M>Y ge~3  
    %       2    2    r^2                      sqrt(6) :mwNkT2et  
    %       3    1    3*r^3 - 2*r              sqrt(8) lTNfTO^  
    %       3    3    r^3                      sqrt(8) `1I@tz|  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) gQpF(P  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 7"L`|O?8)  
    %       4    4    r^4                      sqrt(10) Vq7L:,N9  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) %m8;Lh- X  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) eURy]  
    %       5    5    r^5                      sqrt(12) eBZ^YY<*g  
    %       --------------------------------------------- B?}ZAw>  
    % ^QX3p,Y  
    %   Example: UNc!6Q-.  
    % a-I3#3VJ@  
    %       % Display three example Zernike radial polynomials @S3G>i  
    %       r = 0:0.01:1; x50,4J%J'r  
    %       n = [3 2 5]; d1=kHU4_9  
    %       m = [1 2 1]; E1,Sr?'  
    %       z = zernpol(n,m,r); &p\fdR4e  
    %       figure +-=o16*{ !  
    %       plot(r,z) r[P5 ufy2]  
    %       grid on cO$ PK  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ]6wo]nV[P  
    % }m6zu'CV  
    %   See also ZERNFUN, ZERNFUN2. aL63=y  
    IvLo&6swW  
    % A note on the algorithm. *W()|-[V3  
    % ------------------------ z6B(}(D  
    % The radial Zernike polynomials are computed using the series "^A4!.  
    % representation shown in the Help section above. For many special &<</[h/B/F  
    % functions, direct evaluation using the series representation can vB0O3]  
    % produce poor numerical results (floating point errors), because MPt:bf#  
    % the summation often involves computing small differences between ,3^gB,ka  
    % large successive terms in the series. (In such cases, the functions Vc!` BiH  
    % are often evaluated using alternative methods such as recurrence `N 0Mm7  
    % relations: see the Legendre functions, for example). For the Zernike *&VH!K#@{  
    % polynomials, however, this problem does not arise, because the A?{ X5` y  
    % polynomials are evaluated over the finite domain r = (0,1), and )wU.|9o]M  
    % because the coefficients for a given polynomial are generally all &Nx'Nq9y  
    % of similar magnitude. xFZA1 8  
    % >YPC &@9   
    % ZERNPOL has been written using a vectorized implementation: multiple hdB.u^!  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 8nOMyNpy~M  
    % values can be passed as inputs) for a vector of points R.  To achieve QN=a{  
    % this vectorization most efficiently, the algorithm in ZERNPOL :Mz$~o<  
    % involves pre-determining all the powers p of R that are required to p7 b`Z>}  
    % compute the outputs, and then compiling the {R^p} into a single ,o(7z^1Pe;  
    % matrix.  This avoids any redundant computation of the R^p, and lSw9e<jYO  
    % minimizes the sizes of certain intermediate variables. L(tA~Z"k  
    % 57/9i> @  
    %   Paul Fricker 11/13/2006 n-m+@jRz  
    odxsF(Q0p  
    qx0RCP /s  
    % Check and prepare the inputs: w*.q t<rH)  
    % ----------------------------- + QcgLq  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) u|m>h(O  
        error('zernpol:NMvectors','N and M must be vectors.') 5m,{?M`  
    end y74Ph:^ k  
    QG\lXY,  
    if length(n)~=length(m) 7_r$zEP6  
        error('zernpol:NMlength','N and M must be the same length.') ZA@QP1  
    end !6_lD 0  
    C2GF N1i  
    n = n(:); 5>.)7D%  
    m = m(:); 8>.l4:`  
    length_n = length(n); bSmF"H0cP  
     V"n0"\k,  
    if any(mod(n-m,2)) 8zew8I~s  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 9g3J{pKcZ  
    end (fON\)l  
    T:+%3+;a  
    if any(m<0) HA%% WSuf  
        error('zernpol:Mpositive','All M must be positive.') mG[S"?C  
    end @vWC "W  
    *ayn<Vlh`^  
    if any(m>n) M/GQQG;  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') J%']t$ AR  
    end T1bPI/  
    H!U\;ny  
    if any( r>1 | r<0 ) ]_NN,m>z  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') `_Bvae j?,  
    end }J}a;P4  
    /3D!,V,  
    if ~any(size(r)==1) MCHRNhb9  
        error('zernpol:Rvector','R must be a vector.') XHu Y'\;-  
    end 910Ym!\{:  
    z)Xf6&  
    r = r(:); ;+]9KIa_Pq  
    length_r = length(r); 7sECbbJT  
    6|U0"C#]  
    if nargin==4 *_d+cG  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); /sY(/ J E  
        if ~isnorm Q+|8|V}w  
            error('zernpol:normalization','Unrecognized normalization flag.') BCB"& :}  
        end LO@.aJpp  
    else <,qJ% kc  
        isnorm = false; U,"lOG'  
    end %zE_Q  
    <{@?c  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8cn)ox|J[  
    % Compute the Zernike Polynomials WTPp/Nq'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ko6 tp9G  
    `Z]Tp1U  
    % Determine the required powers of r: 7|3Qcn7P)@  
    % ----------------------------------- Q\G8R^9j p  
    rpowers = []; bF %#KSVw  
    for j = 1:length(n) }OO(uC2  
        rpowers = [rpowers m(j):2:n(j)]; &T?>Kx  
    end g~_cYy  
    rpowers = unique(rpowers); |D)NP N&  
    j"o`K}C  
    % Pre-compute the values of r raised to the required powers, \Ku=a{Ne  
    % and compile them in a matrix: rP.qCl+J  
    % ----------------------------- jI@0jxF  
    if rpowers(1)==0 3 #R~>c2  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); "~x\bSY  
        rpowern = cat(2,rpowern{:}); B]dHMLzl  
        rpowern = [ones(length_r,1) rpowern]; 8[(eV.  
    else :@w ;no>=*  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 6Uq@v8mh  
        rpowern = cat(2,rpowern{:}); ) Ph.  
    end =O~1L m;  
    {snLiCl  
    % Compute the values of the polynomials: GL&ri!,  
    % -------------------------------------- ~/1kCZB  
    z = zeros(length_r,length_n); j>~^jz:  
    for j = 1:length_n \{J gjd  
        s = 0:(n(j)-m(j))/2; i'J.c4  
        pows = n(j):-2:m(j); B&A4-w v  
        for k = length(s):-1:1 &,+G}  
            p = (1-2*mod(s(k),2))* ... p,}-8#K[  
                       prod(2:(n(j)-s(k)))/          ... & Sy0Of  
                       prod(2:s(k))/                 ... B9|!8V  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... `),7*gn*)  
                       prod(2:((n(j)+m(j))/2-s(k))); %Rv&VFg  
            idx = (pows(k)==rpowers); Gxv@a   
            z(:,j) = z(:,j) + p*rpowern(:,idx); | Q:$G!/  
        end XG ]yfux`  
         =]E(iR_&  
        if isnorm p?X.I]=vRv  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); +B^ / =3P  
        end e/lfT?J\  
    end I9N?zmH  
    s$6zA j!  
    % 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)  ]7O)iq%  
    WfBA5  
    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。 j*x8K,fN  
    T.<er iv  
    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)