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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 T3u%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! JBtcl# |  
     
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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 D^yZ!}Kl  
    function z = zernfun(n,m,r,theta,nflag) /{vv n  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. t}>6"^}U  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N `CA-s  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 6*8"?S'  
    %   unit circle.  N is a vector of positive integers (including 0), and |Wd]:ijJ  
    %   M is a vector with the same number of elements as N.  Each element _U(b  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) fDt#<f 4;  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 8!2NZOZOS  
    %   and THETA is a vector of angles.  R and THETA must have the same p \A^kX^5  
    %   length.  The output Z is a matrix with one column for every (N,M) 43-mv1>.  
    %   pair, and one row for every (R,THETA) pair. DXu#07\  
    % j&,,~AZm  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ?$i`K|  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), uCO-f<b  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral W+36"?*k3  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 0h/gqlTK1  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized `T7gfb%1-3  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. R_ymTB}<t(  
    % A:PQIcR;V  
    %   The Zernike functions are an orthogonal basis on the unit circle. ^ZV1Ev8T6  
    %   They are used in disciplines such as astronomy, optics, and H^z6.!$m  
    %   optometry to describe functions on a circular domain. JJ`RF   
    % d2`m0U  
    %   The following table lists the first 15 Zernike functions. Oya:{d&=  
    % piKYO+;W'  
    %       n    m    Zernike function           Normalization 4>eY/~odq]  
    %       -------------------------------------------------- RnC96"";R.  
    %       0    0    1                                 1 z(b0U6)qQ  
    %       1    1    r * cos(theta)                    2 0NrUB  
    %       1   -1    r * sin(theta)                    2 2z+Vt_%  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) *"Yz"PK  
    %       2    0    (2*r^2 - 1)                    sqrt(3) {:BAh 5e|  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) XgL-t~_  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Z BjyQ4h  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) e/hA>  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) |s[kY  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Gu[G_^>  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) &XAG| #  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;D.a |(Q  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) h6J0b_3h4  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ey<vvZ  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ~,[-pZ <  
    %       -------------------------------------------------- [Q+8Ku  
    % S. my" j  
    %   Example 1: _RI`I}&9Z  
    % XTboFrf  
    %       % Display the Zernike function Z(n=5,m=1) wJ#fmQXKJ5  
    %       x = -1:0.01:1; Mh [TZfV  
    %       [X,Y] = meshgrid(x,x); u&{}hv&FY  
    %       [theta,r] = cart2pol(X,Y); EGpN@  
    %       idx = r<=1; (Z(O7X(/  
    %       z = nan(size(X)); r:pS[f|4\  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); XG_h\NIL  
    %       figure |dNJx<-  
    %       pcolor(x,x,z), shading interp c#o(y6  
    %       axis square, colorbar Itq248+Ci  
    %       title('Zernike function Z_5^1(r,\theta)') dJyf.VJ  
    % [R V_{F:'  
    %   Example 2: ,liFo.kT8%  
    % H'2&3v  
    %       % Display the first 10 Zernike functions o[Ojl .r<  
    %       x = -1:0.01:1; J)(KGdk  
    %       [X,Y] = meshgrid(x,x); Rdb[{Ruxb  
    %       [theta,r] = cart2pol(X,Y); 99 W-sV  
    %       idx = r<=1; 9vIqGz-o  
    %       z = nan(size(X)); } U <T>0  
    %       n = [0  1  1  2  2  2  3  3  3  3]; BG ] w2=  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; W)F<<B,  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; `zf,$67>1  
    %       y = zernfun(n,m,r(idx),theta(idx)); $ZnLYuGb  
    %       figure('Units','normalized') Dsq_}6l{  
    %       for k = 1:10 ^G :}%4  
    %           z(idx) = y(:,k); ^n! j"  
    %           subplot(4,7,Nplot(k)) %DyukUJ  
    %           pcolor(x,x,z), shading interp aqL#g18  
    %           set(gca,'XTick',[],'YTick',[]) i/Zv@GF  
    %           axis square Vyy;mEBg  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 5:S=gARz  
    %       end tc-pVw:TV  
    % o>Fc.$ngZ  
    %   See also ZERNPOL, ZERNFUN2. bcx,K b  
    </xz V<Pi  
    %   Paul Fricker 11/13/2006 n*(9:y=l1  
    RbOEXH*]  
    h"C7l#u  
    % Check and prepare the inputs: ~H<oqk:O-  
    % ----------------------------- =*paa  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) d7, ZpHt  
        error('zernfun:NMvectors','N and M must be vectors.') *[VO03  
    end Myj 5qh  
    j?c"BF.  
    if length(n)~=length(m) qKt*<KGeY  
        error('zernfun:NMlength','N and M must be the same length.') d6(R-k#B  
    end g+( Cs  
    {@1;kG  
    n = n(:); uGXN ciEp`  
    m = m(:); -4 *94<  
    if any(mod(n-m,2)) XK*55W &og  
        error('zernfun:NMmultiplesof2', ... c#)!-5E~H  
              'All N and M must differ by multiples of 2 (including 0).') J\06j%d,  
    end u92);1R  
    qu8!fFQjYL  
    if any(m>n) J#1-Le8@  
        error('zernfun:MlessthanN', ... ot%^FvQ[c  
              'Each M must be less than or equal to its corresponding N.') "+0Yhr?  
    end ON,sN  
    vJ +sdG  
    if any( r>1 | r<0 ) S['rfD>9  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') yT$CImP73  
    end d#tqa`@~  
    \*a7o GyH>  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) QD8.C=2R  
        error('zernfun:RTHvector','R and THETA must be vectors.') %yyvB5Y^  
    end w}20l F  
    `j#zwgUs  
    r = r(:); pA%}CmrMq  
    theta = theta(:); TTDcVG_}  
    length_r = length(r); Pv#Oea?  
    if length_r~=length(theta) l1M %   
        error('zernfun:RTHlength', ... I ~U1vtgp  
              'The number of R- and THETA-values must be equal.') R^p'gQc$   
    end k^H&IS!  
    #oYPe:8|m  
    % Check normalization: 'VMov  
    % -------------------- c 5%uiv]  
    if nargin==5 && ischar(nflag) (yJY/|  
        isnorm = strcmpi(nflag,'norm'); N1',`L5  
        if ~isnorm =~DQX\  
            error('zernfun:normalization','Unrecognized normalization flag.') L2sUh+'|  
        end "^froQ{"T  
    else MQ#nP_i  
        isnorm = false; yv;KKQ   
    end JI3x^[(Z  
    ?lPn{oB9"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7Mj:bm&9  
    % Compute the Zernike Polynomials P Nf_{4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /!i`K{  
    YAdk3y~pL  
    % Determine the required powers of r: k4E2OyCFoJ  
    % ----------------------------------- 3>'TYXs-  
    m_abs = abs(m); ?~:4O}5Ax  
    rpowers = []; mG*ER^Y@D  
    for j = 1:length(n) IDY2X+C#U  
        rpowers = [rpowers m_abs(j):2:n(j)]; 6(1S_b=a  
    end $eq*@5B  
    rpowers = unique(rpowers); /ucS*m:<x  
    Oxp!G7qfo  
    % Pre-compute the values of r raised to the required powers, cr`NHl/XF  
    % and compile them in a matrix: @* <`*W  
    % ----------------------------- ]3\%i2NM  
    if rpowers(1)==0 si,)!%b  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); zl3GWj|?\7  
        rpowern = cat(2,rpowern{:}); $~,J8?)(z  
        rpowern = [ones(length_r,1) rpowern]; h}U>K4BJ  
    else \zT{zO&!  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); u9*7Buou^  
        rpowern = cat(2,rpowern{:}); fq[1|Q  
    end -`A+Qp)  
    R*`=Bk0+  
    % Compute the values of the polynomials: /8? u2 q  
    % -------------------------------------- 6QYHPz  
    y = zeros(length_r,length(n)); 96d&vm~m1  
    for j = 1:length(n) Djr/!j  
        s = 0:(n(j)-m_abs(j))/2; $vLGX>H  
        pows = n(j):-2:m_abs(j); ,@]*Xgt=  
        for k = length(s):-1:1 KIGMWS^^  
            p = (1-2*mod(s(k),2))* ... d6XdN  
                       prod(2:(n(j)-s(k)))/              ... Y D,<]q%  
                       prod(2:s(k))/                     ... D=jtXQF  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... <dKHZ4  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 7NMy1'-q  
            idx = (pows(k)==rpowers); s}<i[hY>  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 2w;Cw~<=d  
        end Y_FQB K U  
         UZ;FrQ(l{  
        if isnorm tPb<*{eG  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); (XNd]G  
        end B.4Or]  
    end o&)v{q  
    % END: Compute the Zernike Polynomials N5 b^  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #OwxxUeZ  
    _/NPXDL  
    % Compute the Zernike functions: ?BR Z){)  
    % ------------------------------ .1f!w!ltVR  
    idx_pos = m>0; ?P;=_~X  
    idx_neg = m<0; @ek8t2??x  
    m>^vr7  
    z = y; ()ww9L2  
    if any(idx_pos) pD]2.O  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); pN{XGkX.  
    end .umN>/o[  
    if any(idx_neg) ?!u9=??  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); tP89gN^PA|  
    end i8!err._  
    tN;^{O-(V  
    % 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) k)zBw(wr  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. w{riXOjS4  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated >#y1(\e  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive +I@2,T(eG  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 1U(!%},  
    %   and THETA is a vector of angles.  R and THETA must have the same F(`Q62o@  
    %   length.  The output Z is a matrix with one column for every P-value, BkB9u&s^  
    %   and one row for every (R,THETA) pair. *, R ~[g  
    % _ucixM#  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike OI:T#uk5  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) e 8^%}\F  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) f[q_eY  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 4'.] -u  
    %   for all p. jX,A.  
    % MfraTUxIo/  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Uv(}x 7e)  
    %   Zernike functions (order N<=7).  In some disciplines it is GuF-HP}xM  
    %   traditional to label the first 36 functions using a single mode b/4gs62{k  
    %   number P instead of separate numbers for the order N and azimuthal bd3>IWihp  
    %   frequency M. `FK qVd  
    % z=4E#y `?U  
    %   Example: @h5Q?I  
    % z'zC  
    %       % Display the first 16 Zernike functions `F~Fb S  
    %       x = -1:0.01:1; kdMB.~(K=  
    %       [X,Y] = meshgrid(x,x); u@aM8Na  
    %       [theta,r] = cart2pol(X,Y); _+gpdQq\p  
    %       idx = r<=1; UJ`%uLR~  
    %       p = 0:15; M#yUdl7d  
    %       z = nan(size(X)); iHWt;]  
    %       y = zernfun2(p,r(idx),theta(idx)); :~p_(rE  
    %       figure('Units','normalized') BbI),iP  
    %       for k = 1:length(p) cGW L'r)P  
    %           z(idx) = y(:,k); 17 VNw/Y  
    %           subplot(4,4,k) &# @"^(} 6  
    %           pcolor(x,x,z), shading interp &A^2hPe}  
    %           set(gca,'XTick',[],'YTick',[]) xG(:O@  
    %           axis square K,*IfHi6[  
    %           title(['Z_{' num2str(p(k)) '}']) x!onan  
    %       end th=45y"C  
    % UHDcheeRD  
    %   See also ZERNPOL, ZERNFUN. '=IuwCB|;  
    "?YpF2pD  
    %   Paul Fricker 11/13/2006 V.[b${  
    DE?@8k  
    QYWl`Yqf  
    % Check and prepare the inputs: n4zns,:)/  
    % ----------------------------- N mN:x&/  
    if min(size(p))~=1 dTVM !=  
        error('zernfun2:Pvector','Input P must be vector.') * =O@D2g0  
    end u[!Ex=9W  
    Q?%v b  
    if any(p)>35 tI2p-d9B  
        error('zernfun2:P36', ... $P&27  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... m>b i$Y  
               '(P = 0 to 35).']) S*rcXG6Q^  
    end !v3wl0  
    H{;8i7%  
    % Get the order and frequency corresonding to the function number: q5w)i  
    % ---------------------------------------------------------------- <7X+-%yb;  
    p = p(:); D7$xY\0r  
    n = ceil((-3+sqrt(9+8*p))/2); yNQ 9~P2  
    m = 2*p - n.*(n+2); xX])IZ D  
    ;}k_2mr~  
    % Pass the inputs to the function ZERNFUN: "2@Ys* e  
    % ---------------------------------------- PvdR)ZE m  
    switch nargin %P]-wBJw  
        case 3 5TdI  
            z = zernfun(n,m,r,theta); o-t!z'\lO  
        case 4 ?/s=E+  
            z = zernfun(n,m,r,theta,nflag); # /pZ#ny  
        otherwise 1'* {Vm M  
            error('zernfun2:nargin','Incorrect number of inputs.') 2qkC{klC^M  
    end ,<-a 6  
    )5bdWJ>l  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) _'oy C(:}  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Sq==)$G  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of g@"6QAP  
    %   order N and frequency M, evaluated at R.  N is a vector of VVje|T^{Z  
    %   positive integers (including 0), and M is a vector with the ,@ Cru=  
    %   same number of elements as N.  Each element k of M must be a u]c nbm  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) G8?<(.pi@  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is  Q-&]Vg  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Qi7^z;  
    %   with one column for every (N,M) pair, and one row for every QX~*aqS3s8  
    %   element in R. `ionMTZY  
    % Xl*-A|:j  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- bvR*sT#rg  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is V2]S{!p}k  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to @;,O V&XYn  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 /ADxHw`k  
    %   for all [n,m]. 0KT{K(  
    % 9e :E% 2  
    %   The radial Zernike polynomials are the radial portion of the A?|cJ"N  
    %   Zernike functions, which are an orthogonal basis on the unit HNuwq\w  
    %   circle.  The series representation of the radial Zernike qc'tK6=jp  
    %   polynomials is "x$S%:p  
    % ?3z+|;t6C  
    %          (n-m)/2 <p0$Q!^dK=  
    %            __ |H_)u  
    %    m      \       s                                          n-2s (\/HGxv  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r @7^#_772  
    %    n      s=0 8rp-Xi W  
    % pmW=l/6+V3  
    %   The following table shows the first 12 polynomials. Nyqm0C6m^  
    % ZJ[ Uz_%W  
    %       n    m    Zernike polynomial    Normalization A#  M  
    %       --------------------------------------------- RLHe;-*b]I  
    %       0    0    1                        sqrt(2) F5<{-{Ky  
    %       1    1    r                           2 4l`gAE$  
    %       2    0    2*r^2 - 1                sqrt(6) {M~!?# <K  
    %       2    2    r^2                      sqrt(6) N[+dX_h  
    %       3    1    3*r^3 - 2*r              sqrt(8) WNYLQ=;  
    %       3    3    r^3                      sqrt(8) \+AH>I;vO  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) };!c]/,  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) P/PS(`  
    %       4    4    r^4                      sqrt(10) \!V6` @0KC  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ;W*$<~_  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) =W|Q0|U  
    %       5    5    r^5                      sqrt(12) ,6buo~?W:  
    %       --------------------------------------------- GKd>AP_  
    % `( a^=e5  
    %   Example: ^ KjqS\<  
    % #129 i2  
    %       % Display three example Zernike radial polynomials 86I*  
    %       r = 0:0.01:1; YWZF*,4  
    %       n = [3 2 5]; Go67VqJr  
    %       m = [1 2 1]; O46/[{p+8  
    %       z = zernpol(n,m,r); kv4J@  
    %       figure B&$89]gs|  
    %       plot(r,z) 8Z!ea3kAT  
    %       grid on E37@BfpO3  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 2Ls<OO  
    % 5nn*)vK {  
    %   See also ZERNFUN, ZERNFUN2. v: 0i5h&M  
    } R/  
    % A note on the algorithm. 3U`.:w`  
    % ------------------------ rh l5r"%  
    % The radial Zernike polynomials are computed using the series _# F'rl6'  
    % representation shown in the Help section above. For many special m#eD v*  
    % functions, direct evaluation using the series representation can *j* WE\  
    % produce poor numerical results (floating point errors), because #?=cg]v_  
    % the summation often involves computing small differences between D{l((t3=T  
    % large successive terms in the series. (In such cases, the functions z,7^dlT  
    % are often evaluated using alternative methods such as recurrence MnI $%  
    % relations: see the Legendre functions, for example). For the Zernike .2P?1HpK  
    % polynomials, however, this problem does not arise, because the Vwqfn4sx?i  
    % polynomials are evaluated over the finite domain r = (0,1), and ^Bb_NcU  
    % because the coefficients for a given polynomial are generally all !!86Sv  
    % of similar magnitude. W<L6,  
    % Dn3~8  
    % ZERNPOL has been written using a vectorized implementation: multiple dW`D?$(@,  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 0R]CI  
    % values can be passed as inputs) for a vector of points R.  To achieve %ze1ZWO{  
    % this vectorization most efficiently, the algorithm in ZERNPOL KV*:,>  
    % involves pre-determining all the powers p of R that are required to QBy*y $  
    % compute the outputs, and then compiling the {R^p} into a single \d+HYLAJn  
    % matrix.  This avoids any redundant computation of the R^p, and l}2WW1b(  
    % minimizes the sizes of certain intermediate variables. f)x}_dw%  
    % 9-^p23.@[j  
    %   Paul Fricker 11/13/2006 ka3 Z5  
    20qVzXi  
     o %%fO  
    % Check and prepare the inputs: hI8C XG  
    % ----------------------------- NLl~/smMS  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) G~L?q~b  
        error('zernpol:NMvectors','N and M must be vectors.') WL Lv a<{  
    end Fc~w`~tv  
    srLr~^$j[  
    if length(n)~=length(m) g8"7wf`0k  
        error('zernpol:NMlength','N and M must be the same length.') 0Y 2^}u@5  
    end V2,WP  
    ~a%hRJg  
    n = n(:); rk|(BA  
    m = m(:); ,<^HB+{Wo  
    length_n = length(n); B,833Azi  
    |q2lTbJ  
    if any(mod(n-m,2)) g4~qc I=a  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ek)(pJ(+#  
    end }YP7x|  
    rb'GveW[  
    if any(m<0) \ZRoTh  
        error('zernpol:Mpositive','All M must be positive.') Z D%_PgiT  
    end YX VJJd$U  
    'kvFU_)  
    if any(m>n) eF^"{a3b  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') Q; /F0JDH  
    end U]0)$OH5e  
    PAU+C_P  
    if any( r>1 | r<0 ) !(K{*7|h  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ;-GzGDc~0  
    end TrU@mYnE  
    _ D9@<+MS*  
    if ~any(size(r)==1) o}+Uy  
        error('zernpol:Rvector','R must be a vector.') vfUfrk@D~  
    end Lu39eO6  
    V55J[s*6!  
    r = r(:); c dbSv=r  
    length_r = length(r); bxSKe6l  
    m8j-lNu  
    if nargin==4 o`mIi  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); qd~98FS  
        if ~isnorm aiZo{j<6  
            error('zernpol:normalization','Unrecognized normalization flag.') NJf(,Mr*|  
        end -5v.1y=!L  
    else uQ]]]Z(H'  
        isnorm = false; #S%Y; ilq  
    end `uZv9I"  
    +`zi>=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YOV4)P"  
    % Compute the Zernike Polynomials C'czXZtn  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C!{AnWf  
    ~po%GoH(K  
    % Determine the required powers of r: xY'qm8V  
    % ----------------------------------- G7A bhb,  
    rpowers = []; nH>V Da  
    for j = 1:length(n) ^I<T+X+<  
        rpowers = [rpowers m(j):2:n(j)]; E< CxKY9  
    end qD;v/,?  
    rpowers = unique(rpowers); B UQn+;be  
    f\);HJbg  
    % Pre-compute the values of r raised to the required powers, #} ~p^ 0  
    % and compile them in a matrix: (vAv^A*i}  
    % ----------------------------- =Xy`"i{`(  
    if rpowers(1)==0 [TK? P0  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); bV$8 >[`  
        rpowern = cat(2,rpowern{:}); Rw}2*5#y  
        rpowern = [ones(length_r,1) rpowern]; 6{+_T  
    else 5Z6-R}uXk  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3P#+) F~  
        rpowern = cat(2,rpowern{:}); ]LBvYjMY  
    end *$L z2 ]  
    i=1 }lk q  
    % Compute the values of the polynomials: nl'J.dJe  
    % -------------------------------------- Q6.*"`  
    z = zeros(length_r,length_n); }or2 $\>m  
    for j = 1:length_n JC&6q >$  
        s = 0:(n(j)-m(j))/2; U8K &Q4^  
        pows = n(j):-2:m(j); &#-|Yh/  
        for k = length(s):-1:1 r'd:SaU+  
            p = (1-2*mod(s(k),2))* ... Vo9>o@FlLM  
                       prod(2:(n(j)-s(k)))/          ... R] Disljq  
                       prod(2:s(k))/                 ... w.D4dv_H  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 0ck&kpL:9  
                       prod(2:((n(j)+m(j))/2-s(k))); nGx ~) T  
            idx = (pows(k)==rpowers); ~)wwX:;B_  
            z(:,j) = z(:,j) + p*rpowern(:,idx); =s0g2Zv"\  
        end Q)G!Y (g\  
         B9LSxB  
        if isnorm K=tx5{V  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); J&63Z  
        end U+.PuC[3  
    end W1?!iE~tO  
    gHvW e  
    % 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)  `\wUkmH  
    $iwIF7,\P  
    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。 D 0  
    3i6h"Wu`n  
    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)