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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 S4r-s;U-v/  
    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! pZpAb+  
     
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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 SxAZ2|/-  
    function z = zernfun(n,m,r,theta,nflag) kYwV0xQ  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~>j5z&:&  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1FkS$ j8:  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ~d9R:t1  
    %   unit circle.  N is a vector of positive integers (including 0), and M, uQ8SZA[  
    %   M is a vector with the same number of elements as N.  Each element W7\s=t\  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ;ui=7[ Us  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, /t4#-vz  
    %   and THETA is a vector of angles.  R and THETA must have the same ZxDh94w/  
    %   length.  The output Z is a matrix with one column for every (N,M) KOYU'hw  
    %   pair, and one row for every (R,THETA) pair. 1N3qMm^  
    % w=|"{-ijo  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ;5ANw"Dq  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), lRy^Wp  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral bL6, fUS  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, E8`AU<  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized vv  F:  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. !4?QR  
    % B1u.aa$  
    %   The Zernike functions are an orthogonal basis on the unit circle. JBvMe H5  
    %   They are used in disciplines such as astronomy, optics, and r+yl{  
    %   optometry to describe functions on a circular domain. $,s"c(pv[,  
    % p+ki1! Ed  
    %   The following table lists the first 15 Zernike functions. 'yIz<o  
    % )0tq&  
    %       n    m    Zernike function           Normalization h)~i ?bq!/  
    %       -------------------------------------------------- (^U 8wit/  
    %       0    0    1                                 1 ,; 81FK  
    %       1    1    r * cos(theta)                    2 zfm-v U  
    %       1   -1    r * sin(theta)                    2 Omkpjr(1  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) `S&.gPE2  
    %       2    0    (2*r^2 - 1)                    sqrt(3) n _H]*~4F  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) Klv~#9Si  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) GIs *;ps7w  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) $K'A_G^  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8)  ~T'!.^/  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) D.ajO^[  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) JKJ+RkXf3  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) JvI6+[  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 9 M<3m  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) vgyv~Px]AW  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) :JI&ngWK  
    %       -------------------------------------------------- MODi:jsl  
    % }zE Qrfl  
    %   Example 1: an<loL W  
    % yE3l%<;q  
    %       % Display the Zernike function Z(n=5,m=1) v"~0 3-SX  
    %       x = -1:0.01:1; sf(2~BMQI  
    %       [X,Y] = meshgrid(x,x); N H$!<ffz  
    %       [theta,r] = cart2pol(X,Y); V\=QAN^  
    %       idx = r<=1; V=+wsc  
    %       z = nan(size(X)); v;_k*y[VV$  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); BT3X7Cx  
    %       figure |PY*"Ul  
    %       pcolor(x,x,z), shading interp :tTP3 t5  
    %       axis square, colorbar  FTk`Mq  
    %       title('Zernike function Z_5^1(r,\theta)') 920 o]Dh=t  
    % wV& UB@  
    %   Example 2: ` yXJaTbo  
    % Mu>WS)1lS  
    %       % Display the first 10 Zernike functions /z(;1$Ld6{  
    %       x = -1:0.01:1; ndB [f  
    %       [X,Y] = meshgrid(x,x); FKVf_Ncf%  
    %       [theta,r] = cart2pol(X,Y); 4^>FN"Ve`B  
    %       idx = r<=1; T=- $ok`G  
    %       z = nan(size(X)); c c^I9g~  
    %       n = [0  1  1  2  2  2  3  3  3  3]; >AUj4d  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; !92zC._  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Ic,V ,#my  
    %       y = zernfun(n,m,r(idx),theta(idx)); $Lf-Gi  
    %       figure('Units','normalized') &nXa /XIZ_  
    %       for k = 1:10 u,f$cR  
    %           z(idx) = y(:,k); 5Y}=,v*h}  
    %           subplot(4,7,Nplot(k)) ] 1:pnd  
    %           pcolor(x,x,z), shading interp !}$,) ~<+H  
    %           set(gca,'XTick',[],'YTick',[]) zo{WmV7[|  
    %           axis square $SAk|  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) So^;5tG  
    %       end Y7L1`<SC  
    % + NpH k  
    %   See also ZERNPOL, ZERNFUN2. q n2X._`  
    =w#sCy  
    %   Paul Fricker 11/13/2006 c7[+gc5}  
    gb,X"ODq  
    `N,q~@gL  
    % Check and prepare the inputs: zK@DQ5  
    % ----------------------------- m@2;9  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) d0"Xlle ld  
        error('zernfun:NMvectors','N and M must be vectors.') Jd0I!L  
    end *|F ;An.N^  
    {;0+N -U  
    if length(n)~=length(m) Bl6>y/  
        error('zernfun:NMlength','N and M must be the same length.') zwEZ?m!  
    end  E qc,/  
    {WYHT6Z  
    n = n(:); n\x@~ SzrX  
    m = m(:); cf7UV6D g  
    if any(mod(n-m,2)) ,f(:i^iz!  
        error('zernfun:NMmultiplesof2', ... ^vQ,t*Uj=  
              'All N and M must differ by multiples of 2 (including 0).') i[\`]C{gf  
    end 8F#z)>q~  
    hDsSOpj  
    if any(m>n) LaolAqU  
        error('zernfun:MlessthanN', ... w]ZE('3%W  
              'Each M must be less than or equal to its corresponding N.') )kl(}.9X  
    end +LEU|#  
    dRXEF6G  
    if any( r>1 | r<0 ) y~ZYI]` J  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') E2Jmo5yJR  
    end =,4iMENm!  
    =Co[pt  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 1[&V6=n  
        error('zernfun:RTHvector','R and THETA must be vectors.') {*jo,<4ee  
    end 0qPbmLMK  
    zP(UaSXz/  
    r = r(:); %Uz 5Ve  
    theta = theta(:); ^zs]cFN#%  
    length_r = length(r); 6bXP{,}Gp  
    if length_r~=length(theta) bWe_<'N  
        error('zernfun:RTHlength', ... /`b(} m  
              'The number of R- and THETA-values must be equal.') *Mg. * N  
    end ]LE  
    `YinhO:Z  
    % Check normalization: 1m5 =Nu  
    % -------------------- c%bGVRhE  
    if nargin==5 && ischar(nflag) S# 9EBw7  
        isnorm = strcmpi(nflag,'norm'); 3cH`>#c  
        if ~isnorm 4EZl (v"f`  
            error('zernfun:normalization','Unrecognized normalization flag.') F6$QEiDu@  
        end `c )//o  
    else ?;dfA/  
        isnorm = false; AzmISm  
    end eInx\/  
    k-`5T mW  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6S2u%-]  
    % Compute the Zernike Polynomials 4-wCk=I  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pg4J)<t#  
    *co=<g]4KY  
    % Determine the required powers of r: XC D&Im  
    % ----------------------------------- r{Cbx#;  
    m_abs = abs(m); <Z -d5D>  
    rpowers = []; (i"@{[IP  
    for j = 1:length(n) l1utk8'-  
        rpowers = [rpowers m_abs(j):2:n(j)]; e7cqm*Qi  
    end "kHQ}#6r  
    rpowers = unique(rpowers); TO|&}sDh  
    ycr\vn t  
    % Pre-compute the values of r raised to the required powers, b;;C><  
    % and compile them in a matrix: g3`:d)|  
    % -----------------------------  @o g&l;  
    if rpowers(1)==0 6u'+#nm  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); :k"VR,riF  
        rpowern = cat(2,rpowern{:}); +frkC| .  
        rpowern = [ones(length_r,1) rpowern]; f.~-31  
    else ?<l,a!V'6  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); !}TZmwf'  
        rpowern = cat(2,rpowern{:}); O'OVj  
    end *_aeK~du.  
    eVVm"96Q.;  
    % Compute the values of the polynomials: "/O`#Do/  
    % -------------------------------------- \"X<\3z2  
    y = zeros(length_r,length(n)); w[A$bqz   
    for j = 1:length(n) <![]=~z $  
        s = 0:(n(j)-m_abs(j))/2; 20O\@}2q2M  
        pows = n(j):-2:m_abs(j); BM@:=>ypQ  
        for k = length(s):-1:1 B}(+\Q$I  
            p = (1-2*mod(s(k),2))* ... C_RxJWka  
                       prod(2:(n(j)-s(k)))/              ... nisW<Q`uB  
                       prod(2:s(k))/                     ... 6^Q Bol  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Wd R~  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); _I&];WM\  
            idx = (pows(k)==rpowers); rTgCmr'&  
            y(:,j) = y(:,j) + p*rpowern(:,idx); [KT'aGK$  
        end ZP]l%6\.  
         U1Z.#ETnM  
        if isnorm !@r1B`]j+"  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); t81}jD  
        end SXA`o<Ma  
    end Td7=La0   
    % END: Compute the Zernike Polynomials }=+J&cR  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }! jk  
    >A+0"5+_p  
    % Compute the Zernike functions: ^Ia:e ?)W  
    % ------------------------------ c']3N  
    idx_pos = m>0; 6zJ<27  
    idx_neg = m<0; sn4wd:b7%  
    u+&t"B  
    z = y; g.& n X/  
    if any(idx_pos) {GTOHJ2  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 4490l"  
    end (sXR@Ce$  
    if any(idx_neg) (4hCT*  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Y6>@zznk  
    end  2]$ 7  
    Jj_ t0"  
    % 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) l"I G;qO.  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. rv75R}.6R^  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ;^Q - 1  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive j~|pSu.<  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, N^)\+*tf1  
    %   and THETA is a vector of angles.  R and THETA must have the same z qM:'x*  
    %   length.  The output Z is a matrix with one column for every P-value, w?r   
    %   and one row for every (R,THETA) pair. 'zEmg}  
    % KA=cIm  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike deRnP$u0  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) $jpAnZR- /  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) J=%(f1X<W  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 lK}W%hzU  
    %   for all p. TqvgCk-  
    % 0|RFsJ"  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 |#y+iXTJ   
    %   Zernike functions (order N<=7).  In some disciplines it is kw%vO6"q(  
    %   traditional to label the first 36 functions using a single mode C J@G8>  
    %   number P instead of separate numbers for the order N and azimuthal l7+[Zn/v *  
    %   frequency M. 7Fg-}lJAC  
    % -<Wv7FNpD  
    %   Example: 0[f8Gb3  
    % lURL;h  
    %       % Display the first 16 Zernike functions 0Gq}x;8H&  
    %       x = -1:0.01:1; ]r|nz~Aa$  
    %       [X,Y] = meshgrid(x,x); _P^ xX'v  
    %       [theta,r] = cart2pol(X,Y); wM]j#  
    %       idx = r<=1; ^}F@*A;o  
    %       p = 0:15;  QB/H  
    %       z = nan(size(X)); F2B9Q_>P  
    %       y = zernfun2(p,r(idx),theta(idx)); d0b`qk @4  
    %       figure('Units','normalized') Vy- kogVt  
    %       for k = 1:length(p) jm~qD T,  
    %           z(idx) = y(:,k); uxxS."~  
    %           subplot(4,4,k) rZ|!y ~S|  
    %           pcolor(x,x,z), shading interp 'S[&-D%(3  
    %           set(gca,'XTick',[],'YTick',[]) L.%N   
    %           axis square ;Q1/53Y<  
    %           title(['Z_{' num2str(p(k)) '}']) A6D@#(D  
    %       end -0 e&>H%  
    % =b{!p|  
    %   See also ZERNPOL, ZERNFUN. ogOUrJ}P  
    =GP~h*5es  
    %   Paul Fricker 11/13/2006 2[O\"a%  
    j06Xz\c  
    _ ?\4k{ET  
    % Check and prepare the inputs: (_9cL,v  
    % ----------------------------- gz,x6mnQ  
    if min(size(p))~=1 ug|'}\LY  
        error('zernfun2:Pvector','Input P must be vector.') 7%%FYHMO:  
    end UC u4S >  
    AUan^Om  
    if any(p)>35 H.n+CR  
        error('zernfun2:P36', ... _#kjiJj *  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... |t1ij'N  
               '(P = 0 to 35).']) ?HsQ417.H  
    end qv+8wJ((  
    hj8S".A_  
    % Get the order and frequency corresonding to the function number: voD0 u  
    % ---------------------------------------------------------------- TB8a#bK4  
    p = p(:); Nydhal00  
    n = ceil((-3+sqrt(9+8*p))/2); Z7Y+rP[l  
    m = 2*p - n.*(n+2); 8|*=p4_fn  
    [i  ]  
    % Pass the inputs to the function ZERNFUN: A3!xYG=+  
    % ---------------------------------------- WgV'T#*  
    switch nargin AXQG  
        case 3 aS7%x>.A!  
            z = zernfun(n,m,r,theta); 0zL7$Q#c  
        case 4 jOE~?{8m  
            z = zernfun(n,m,r,theta,nflag); #nzVgV]  
        otherwise ff1Em.  
            error('zernfun2:nargin','Incorrect number of inputs.') U,Duq^l~s  
    end f<Co&^A  
    PCx] >&  
    % 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) p?#cn   
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Y XBU9T{r  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Za&.sg3RG  
    %   order N and frequency M, evaluated at R.  N is a vector of B F,rZZL  
    %   positive integers (including 0), and M is a vector with the +( *;F4>  
    %   same number of elements as N.  Each element k of M must be a v)TFpV6b{p  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 2u> [[U1:  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is mp\`9j+{  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 7n_'2qY  
    %   with one column for every (N,M) pair, and one row for every ub#>kCL9  
    %   element in R. @`hnp:  
    % Yy_o*Ozq  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- #4iiY6  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is *>ilT5q  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to |.s#m^"  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 K?4/x4p@  
    %   for all [n,m]. Dn}Wsd=  
    % e2onR~Cf  
    %   The radial Zernike polynomials are the radial portion of the S!/N lSr<  
    %   Zernike functions, which are an orthogonal basis on the unit $?dAO}f3O)  
    %   circle.  The series representation of the radial Zernike :*{>=BD  
    %   polynomials is CQLh;W`Dc  
    % 0%m}tfQ5  
    %          (n-m)/2 '+ 8.nN  
    %            __ "DW; 6<m  
    %    m      \       s                                          n-2s ?^# h|aUp.  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r !A6l\_  
    %    n      s=0 e^Ds|}{V  
    % {O"?_6',  
    %   The following table shows the first 12 polynomials. V&' :S{i  
    % zeXMi:X  
    %       n    m    Zernike polynomial    Normalization Hko(@z  
    %       --------------------------------------------- >/kwy2  
    %       0    0    1                        sqrt(2) w'Kc#2  
    %       1    1    r                           2 mNvK|bTUT  
    %       2    0    2*r^2 - 1                sqrt(6) P p}N-me>_  
    %       2    2    r^2                      sqrt(6) 05|,-S  
    %       3    1    3*r^3 - 2*r              sqrt(8) PR&D67:Jy  
    %       3    3    r^3                      sqrt(8) Ul<'@A8  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) I& `>6=)  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) "/EE$eU  
    %       4    4    r^4                      sqrt(10) $rZ:$d.C  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) `f@VX :aL}  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Y'.WO[dgf  
    %       5    5    r^5                      sqrt(12) :\0q\2e[<  
    %       --------------------------------------------- `%2e?"OOJ  
    % -*?Y4}mK  
    %   Example: %Jrdr`<  
    % K|H&x"t  
    %       % Display three example Zernike radial polynomials $ljgFmR_  
    %       r = 0:0.01:1; 3b_tK^|'  
    %       n = [3 2 5]; DIk\=[{2q  
    %       m = [1 2 1]; -zeodv7  
    %       z = zernpol(n,m,r); Z66b>.<8  
    %       figure :Rs% (Z  
    %       plot(r,z) xLE+"6;W  
    %       grid on  V/0?0VKG  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 0I.9m[<Fc  
    % ZOFhX$I  
    %   See also ZERNFUN, ZERNFUN2. ,RkL|'1l  
    b}G4eXkuj  
    % A note on the algorithm. ^J8sR4p#  
    % ------------------------ u@`)u#  
    % The radial Zernike polynomials are computed using the series }OeEv@^  
    % representation shown in the Help section above. For many special [;c'o5M&  
    % functions, direct evaluation using the series representation can I5"ew=x#  
    % produce poor numerical results (floating point errors), because  c|N!ZYJI  
    % the summation often involves computing small differences between iA~b[20&  
    % large successive terms in the series. (In such cases, the functions z.H*"r  
    % are often evaluated using alternative methods such as recurrence ASuxty  
    % relations: see the Legendre functions, for example). For the Zernike 8ycmvpJ  
    % polynomials, however, this problem does not arise, because the {__Z\D2I  
    % polynomials are evaluated over the finite domain r = (0,1), and /H)K_H#|;  
    % because the coefficients for a given polynomial are generally all 4>4*4!KR}  
    % of similar magnitude. 7Q<uk[d0  
    % Yx_[vLm  
    % ZERNPOL has been written using a vectorized implementation: multiple q8:Z.<%8  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] K_V44f1f  
    % values can be passed as inputs) for a vector of points R.  To achieve PmtBu`OkV  
    % this vectorization most efficiently, the algorithm in ZERNPOL vqLC?{i+  
    % involves pre-determining all the powers p of R that are required to o7feH 6Sh  
    % compute the outputs, and then compiling the {R^p} into a single `Z{kJMS  
    % matrix.  This avoids any redundant computation of the R^p, and bQ-5uFe~$B  
    % minimizes the sizes of certain intermediate variables. 5Wj+ey^ ^w  
    % PN{l)&K2.  
    %   Paul Fricker 11/13/2006 oZ O 6J-ea  
    28[dTsd%  
    R v9?<]  
    % Check and prepare the inputs: o~ .[sn5l-  
    % ----------------------------- oZ1#.o{  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) r}i<cyL  
        error('zernpol:NMvectors','N and M must be vectors.') %/dYSC  
    end }>JFO:v&  
    D4yJ:ATO&  
    if length(n)~=length(m) [y y D-  
        error('zernpol:NMlength','N and M must be the same length.') TB] %?L:  
    end JMu|$"o&{  
    Q? a&q0f  
    n = n(:); B$k<F8!%  
    m = m(:); ^e$;I8l  
    length_n = length(n); O6P0Am7s  
    Q p7|p  
    if any(mod(n-m,2)) c'_-jdi`>_  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') lKs*KwG  
    end T0WB  
    /)SwQgK#  
    if any(m<0) B&0^3iKFi  
        error('zernpol:Mpositive','All M must be positive.') 65VnH=  
    end oC>QJ(o,8  
    [ADr _  
    if any(m>n) A)En25,X  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') lTPo2-j/eK  
    end /%Bc*k=ox  
    ?7@Y=7BS4  
    if any( r>1 | r<0 ) i^(_Gk  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') (veGztt  
    end vq *N  
    lIO.LF3  
    if ~any(size(r)==1) $}<+~JpGfP  
        error('zernpol:Rvector','R must be a vector.') N<+ ><>9  
    end 3%m2$\  
    s+>""yi  
    r = r(:); L)VEA8}  
    length_r = length(r); 9|T%q2O  
    i TY4X:x  
    if nargin==4 PYqx&om  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); WO$PW`k  
        if ~isnorm `pF|bZ?v  
            error('zernpol:normalization','Unrecognized normalization flag.') s)"C~w^  
        end %'j)~  
    else Y((s<]7  
        isnorm = false; Zo`'xg  
    end / Dj6Bj }  
    gF1q Z=<  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% o=?sMq1<  
    % Compute the Zernike Polynomials 7/NXb  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% aksyr$d0V<  
    Ev IL[\Dy  
    % Determine the required powers of r: .ps'{rl8  
    % ----------------------------------- l2>G +t(,  
    rpowers = []; u&`7 C  
    for j = 1:length(n) b9[;qqq@'  
        rpowers = [rpowers m(j):2:n(j)]; >/1N#S#9  
    end 6}  !n0  
    rpowers = unique(rpowers); ZAzn-n  
    CJk$o K{Q  
    % Pre-compute the values of r raised to the required powers, `@ULG>   
    % and compile them in a matrix: +vaz gO<u  
    % ----------------------------- $x 6Rmd{  
    if rpowers(1)==0 bCg {z b#  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); aOfL;I  
        rpowern = cat(2,rpowern{:}); oC>e'_6_b  
        rpowern = [ones(length_r,1) rpowern]; y%k\=:m  
    else VYkOJAEBg  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); /HgdTyR)  
        rpowern = cat(2,rpowern{:}); {bL6%._C  
    end #_y#sDfzh  
    *}Zd QJL  
    % Compute the values of the polynomials: v0|A N  
    % -------------------------------------- rH8^Fl&jT  
    z = zeros(length_r,length_n); d7qY(!&  
    for j = 1:length_n }N(-e$88  
        s = 0:(n(j)-m(j))/2; jNB|98NN  
        pows = n(j):-2:m(j); R_\{a*lV0  
        for k = length(s):-1:1 pj&vnX6O^  
            p = (1-2*mod(s(k),2))* ... LYNd^}  
                       prod(2:(n(j)-s(k)))/          ... )6iY9[@tN  
                       prod(2:s(k))/                 ... #9}E@GGs  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... \-N 4G1  
                       prod(2:((n(j)+m(j))/2-s(k))); {&8-OoH ~  
            idx = (pows(k)==rpowers); _ 0%sYkUc  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Jf@M>BT^A  
        end 6+BR5Nr  
         'YQ"Lf  
        if isnorm ,i#]&f`c;5  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); f:\jPkf'  
        end Ev%4}GwO4  
    end L=3^A'|  
    h<\o[n7j  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  &cztUM(  
    H$TYp  
    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。 #Ki(9oWd  
    n0g,r/  
    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)