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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 (u:^4,Z  
    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! QMAineO  
     
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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 u8Ak2:   
    function z = zernfun(n,m,r,theta,nflag) XT%\Ce!  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. OaeX:r+&Q  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N j@u]( nf  
    %   and angular frequency M, evaluated at positions (R,THETA) on the E*AI}:or;  
    %   unit circle.  N is a vector of positive integers (including 0), and C2}f'  
    %   M is a vector with the same number of elements as N.  Each element 38E %]*5F  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 8yDe{  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, c4V%>A  
    %   and THETA is a vector of angles.  R and THETA must have the same yQ!I`T>a  
    %   length.  The output Z is a matrix with one column for every (N,M) c]%~X&Tg`  
    %   pair, and one row for every (R,THETA) pair. U[EZ, 7n8  
    % ?Gqq]ozm  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike :Xi&H.k)p  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), NH'Dz6K5  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral \@B 'f  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, V| &->9"  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized SceK$  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. r#'ug^^k$X  
    % dt||nF  
    %   The Zernike functions are an orthogonal basis on the unit circle. B",;z)(%  
    %   They are used in disciplines such as astronomy, optics, and 6o d^+>U  
    %   optometry to describe functions on a circular domain. + l hJ8&  
    % LU $=j  
    %   The following table lists the first 15 Zernike functions. p?2^JJpUb  
    % = 6'Fm$R  
    %       n    m    Zernike function           Normalization 8I[=iU7]l  
    %       -------------------------------------------------- 4$+1&+@ ]  
    %       0    0    1                                 1 < Dt/JA(p  
    %       1    1    r * cos(theta)                    2 ZM16 ~k  
    %       1   -1    r * sin(theta)                    2 XR_Gsb%l  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) *3\*GatJ  
    %       2    0    (2*r^2 - 1)                    sqrt(3) $f?GD<}?7r  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) Ozg,6&3ji  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) |*$0~mA  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) FBxg^g%PB@  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) m+Kl   
    %       3    3    r^3 * sin(3*theta)             sqrt(8) _#K?yP?  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) />n!2'!  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ON9L+"vqv0  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ;ObrBN,Fu  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "H#pN;)+   
    %       4    4    r^4 * sin(4*theta)             sqrt(10) uJ`:@Z^J  
    %       -------------------------------------------------- 7M)<Sv  
    % xz Hb+1+p  
    %   Example 1: f?$yxMw:@  
    % h~lps?.#b  
    %       % Display the Zernike function Z(n=5,m=1) Z!-V&H.  
    %       x = -1:0.01:1; A0,h 7<i  
    %       [X,Y] = meshgrid(x,x); ,bzC| AK  
    %       [theta,r] = cart2pol(X,Y); UD=[::##  
    %       idx = r<=1; jO-T1P']Y  
    %       z = nan(size(X)); ~BiLzT1,  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); OS-k_l L  
    %       figure 8*;>:g  
    %       pcolor(x,x,z), shading interp 2@W`OW Njm  
    %       axis square, colorbar EU7nS3K)O~  
    %       title('Zernike function Z_5^1(r,\theta)') E W`3$J;  
    % 5"y)<VLJX  
    %   Example 2: T+q5~~\d  
    % zs6rd83#  
    %       % Display the first 10 Zernike functions B@v (ZY  
    %       x = -1:0.01:1; orOq5?3  
    %       [X,Y] = meshgrid(x,x); aLl=L_  
    %       [theta,r] = cart2pol(X,Y); +|Izjx]ZV  
    %       idx = r<=1; Tm$8\c4V:*  
    %       z = nan(size(X)); n-g#nEc:  
    %       n = [0  1  1  2  2  2  3  3  3  3]; +p[O|[z  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; W[R`],x`  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; jvxCCYXR  
    %       y = zernfun(n,m,r(idx),theta(idx)); 0{ _6le]  
    %       figure('Units','normalized') |ZC'a!  
    %       for k = 1:10 +IMt$}7[  
    %           z(idx) = y(:,k); fR?'HsQg  
    %           subplot(4,7,Nplot(k)) k<x7\T  
    %           pcolor(x,x,z), shading interp \u04m}h]  
    %           set(gca,'XTick',[],'YTick',[]) YC$>D? FW  
    %           axis square #0?3RP  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 3xN_z?Rg  
    %       end m#ig.z|A  
    % U&43/;<,  
    %   See also ZERNPOL, ZERNFUN2. ?gBFfi  
    -g:i'e  
    %   Paul Fricker 11/13/2006 %g^:0me`  
    _DAqL@5n  
    "_2;+@+  
    % Check and prepare the inputs: 65nK1W`i  
    % ----------------------------- -?l`LbD  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) rp^:{6O  
        error('zernfun:NMvectors','N and M must be vectors.') Rn`DUYg  
    end -p%cw0*Y]C  
    :;c`qO4  
    if length(n)~=length(m) bN6i*) }  
        error('zernfun:NMlength','N and M must be the same length.') qQIX:HWDKZ  
    end YI;MS:Qj  
    c$lZ\r"  
    n = n(:); < 2fy(9y  
    m = m(:); kGL3*x  
    if any(mod(n-m,2)) r i)`e  
        error('zernfun:NMmultiplesof2', ... pFV~1W:  
              'All N and M must differ by multiples of 2 (including 0).') qu^~K.I"  
    end a_]l?t  
    }#2(WHf =<  
    if any(m>n) F(ZczwvR  
        error('zernfun:MlessthanN', ...  3bJ|L3G  
              'Each M must be less than or equal to its corresponding N.') 'vYt_T  
    end q: X^V$`  
    sCmN|Q  
    if any( r>1 | r<0 ) t BG 9Mn  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') d^v.tYM$N  
    end `~_H\_JpO  
    ^w&!}f+  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) TA8  
        error('zernfun:RTHvector','R and THETA must be vectors.') ur7S K(#  
    end f@$kK?c?  
    u.*}'C>^^v  
    r = r(:); h(GSM'v  
    theta = theta(:); #3{{[i(;i  
    length_r = length(r); gzy|K%K  
    if length_r~=length(theta) #_|O93HN'  
        error('zernfun:RTHlength', ... B#}EYY  
              'The number of R- and THETA-values must be equal.') G{O{ p  
    end ~w9`l8/0  
    <r(D\rmD  
    % Check normalization: g>#}(u!PH  
    % -------------------- KfPgj  
    if nargin==5 && ischar(nflag) B9Wd '  
        isnorm = strcmpi(nflag,'norm'); G'';VoW=   
        if ~isnorm I~Qi):&x  
            error('zernfun:normalization','Unrecognized normalization flag.') |7Ab_  
        end NxDVU?@p*  
    else yjq|8.L[ G  
        isnorm = false; .JJ50p  
    end [0]J 2  
    Vg :''!4t2  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kY6_n4  
    % Compute the Zernike Polynomials 8J- ?bo  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SG1AYUs V  
    =fdW H4  
    % Determine the required powers of r: 0%Y}CDn_  
    % ----------------------------------- F\ GNLi  
    m_abs = abs(m); l8 $.k5X  
    rpowers = []; fC[~X[H  
    for j = 1:length(n) &Vu-*?  
        rpowers = [rpowers m_abs(j):2:n(j)]; ,7DyTeMpN  
    end O3%#Q3c>3  
    rpowers = unique(rpowers); vS[\ j  
    8rFP*K9  
    % Pre-compute the values of r raised to the required powers, Fey^hx w =  
    % and compile them in a matrix: jGo\_O<of  
    % ----------------------------- 1'iQlnMO@  
    if rpowers(1)==0 ( z F_<  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); g!r) yzK  
        rpowern = cat(2,rpowern{:}); TZ3gJ6 Cb  
        rpowern = [ones(length_r,1) rpowern]; M'oZK  
    else <^'IC9D]  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); m(EV C}Y  
        rpowern = cat(2,rpowern{:}); SQ]M"&\{y  
    end 52,'8` ]  
    fY #Yn  
    % Compute the values of the polynomials: Q`4I a<5B  
    % -------------------------------------- y@7CY-1  
    y = zeros(length_r,length(n)); + Okw+v  
    for j = 1:length(n) TDWD8??e  
        s = 0:(n(j)-m_abs(j))/2; ,^ dpn  
        pows = n(j):-2:m_abs(j); 4d}n0b\d  
        for k = length(s):-1:1 tB4yj_ZF  
            p = (1-2*mod(s(k),2))* ... &OEBAtc/  
                       prod(2:(n(j)-s(k)))/              ... Uyeo0B"  
                       prod(2:s(k))/                     ... G `B=:s]  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... L|1~'Fz#w  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); <]|!quY<*  
            idx = (pows(k)==rpowers); =NnG[#n%  
            y(:,j) = y(:,j) + p*rpowern(:,idx); qSD3]Dv"  
        end Ir*{IVvej  
         gw%L M7yQR  
        if isnorm a1[J>  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); yJlRW!@&:  
        end )KkV<$  
    end N pQOLX/<?  
    % END: Compute the Zernike Polynomials ] \!,yiVeU  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v |pHbX  
    ]"YXa~b  
    % Compute the Zernike functions: &Fjyi"8(r  
    % ------------------------------ a5d_= :S ;  
    idx_pos = m>0; :<0lCj  
    idx_neg = m<0; kGakdLl  
    Qv;b$by3  
    z = y; >?G!>kw  
    if any(idx_pos) wAzaxeV=  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); +%~me?  
    end nLPd]%78>  
    if any(idx_neg) Y$j !-l5z  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); zzh7 "M3Qn  
    end nr( C*E  
    }g|9P SbJ  
    % 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) u09D`QPP]  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. KF!d?  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Q7UQwAN'  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive AP4s_X+=  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, W3^^aD-  
    %   and THETA is a vector of angles.  R and THETA must have the same L}_VT J  
    %   length.  The output Z is a matrix with one column for every P-value, q6%m .X7  
    %   and one row for every (R,THETA) pair. }>3jHWxLc  
    % ORXH<;^0y  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike rsw= a_S  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) vNZ"x)?  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) uBC#4cX`D*  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 tn(6T^u  
    %   for all p. - &)  
    % "avG#rsH  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 q%)."10}]  
    %   Zernike functions (order N<=7).  In some disciplines it is NpbZt;%t  
    %   traditional to label the first 36 functions using a single mode gl2l%]=\'  
    %   number P instead of separate numbers for the order N and azimuthal W&3,XFnI_  
    %   frequency M. -KG1"g,2  
    % wNNg"}&P  
    %   Example: aqQ  U7  
    % zU4*FXt  
    %       % Display the first 16 Zernike functions (&_^1  
    %       x = -1:0.01:1; r $YEq5  
    %       [X,Y] = meshgrid(x,x); ?f!&M  
    %       [theta,r] = cart2pol(X,Y); >{Xyl):  
    %       idx = r<=1; H6KBXMYO  
    %       p = 0:15; fN9uSnu  
    %       z = nan(size(X)); ^.*zBrFx  
    %       y = zernfun2(p,r(idx),theta(idx)); "1p, r&}  
    %       figure('Units','normalized') OL@$RTh  
    %       for k = 1:length(p) 9tmnx')_  
    %           z(idx) = y(:,k); 4ZYywDwn  
    %           subplot(4,4,k) ^ 7)H;$  
    %           pcolor(x,x,z), shading interp 8\PI1U  
    %           set(gca,'XTick',[],'YTick',[]) tCu.Fc@  
    %           axis square Zl.,pcL  
    %           title(['Z_{' num2str(p(k)) '}']) S]4!uv^y  
    %       end wawJZ+V  
    % gXY]NWI  
    %   See also ZERNPOL, ZERNFUN. kp6&e  
    $2Bll5!]  
    %   Paul Fricker 11/13/2006 'S9jMyZrZ  
    fEGnI\  
    #;;A~d:V  
    % Check and prepare the inputs: "wxyY^"  
    % ----------------------------- _!?a9  
    if min(size(p))~=1 $wl_  
        error('zernfun2:Pvector','Input P must be vector.') gTd r  
    end 3wPUP+)c7  
    c68,,rJO]i  
    if any(p)>35 }1.'2.<Y  
        error('zernfun2:P36', ... 3]7j, 1^  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... @jZ1WHS_a  
               '(P = 0 to 35).']) A3J=,aRI_v  
    end UunZ/A$]m  
    .B!  Z0  
    % Get the order and frequency corresonding to the function number: -"x@V7X  
    % ---------------------------------------------------------------- A yOy&]g  
    p = p(:); 8}Q 2!,9Q  
    n = ceil((-3+sqrt(9+8*p))/2); MaF4lFmS  
    m = 2*p - n.*(n+2); E[FE-{B#  
    1`~.!yd8(  
    % Pass the inputs to the function ZERNFUN: L3s"L.G  
    % ---------------------------------------- hK %FpGYA  
    switch nargin s+h}O}RV  
        case 3 Bt(nm> Ng  
            z = zernfun(n,m,r,theta); uu/2C \n}  
        case 4 AH:0h X6+  
            z = zernfun(n,m,r,theta,nflag); m<J:6^H@  
        otherwise \]3[Xw-$  
            error('zernfun2:nargin','Incorrect number of inputs.') E+$D$a  
    end e^N}(Kpy  
    y<l(F?_  
    % 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) ~"8r=8|  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. y<c7RK]  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of gSe3S-Lt  
    %   order N and frequency M, evaluated at R.  N is a vector of ^zfs8]QSf  
    %   positive integers (including 0), and M is a vector with the ~_ wSB[z  
    %   same number of elements as N.  Each element k of M must be a 7j88^59  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) {+EnJ"  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ?}(B8^  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix RNt9Qdr4y  
    %   with one column for every (N,M) pair, and one row for every 3u< ntx ><  
    %   element in R. S F da?>  
    % fm!\**Q1  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- `v)ZOw9&  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is F45-M[z  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 20I/En  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 pnXwE-c_  
    %   for all [n,m]. jsP+,brO  
    % ')Y1c O  
    %   The radial Zernike polynomials are the radial portion of the kM(m$Oo.  
    %   Zernike functions, which are an orthogonal basis on the unit r;+a%?P  
    %   circle.  The series representation of the radial Zernike (O& HCT|  
    %   polynomials is 8is QL  
    % R*2F)e\|  
    %          (n-m)/2 ex66GJQe1  
    %            __ lbC,*U^  
    %    m      \       s                                          n-2s Z*=$n_ G  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 3 rR1/\  
    %    n      s=0 ;s-@m<  
    % +=sw&DH  
    %   The following table shows the first 12 polynomials. \ _?d?:#RD  
    % 9Q'[>P=1  
    %       n    m    Zernike polynomial    Normalization ,sT5TS q  
    %       --------------------------------------------- Zay%QNsb  
    %       0    0    1                        sqrt(2) RE1M4UV.  
    %       1    1    r                           2 *,~L_)vWO  
    %       2    0    2*r^2 - 1                sqrt(6) c0;rvw7  
    %       2    2    r^2                      sqrt(6) (6b0rqPF  
    %       3    1    3*r^3 - 2*r              sqrt(8) @Gy.p5J8  
    %       3    3    r^3                      sqrt(8) amQTPNI  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) }Kn l  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) /!qP=ngw9  
    %       4    4    r^4                      sqrt(10) 7 D{%  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) c e; zn\  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 0& ?L%Y  
    %       5    5    r^5                      sqrt(12) #T@k(Bz{L  
    %       --------------------------------------------- Ul}<@d9: B  
    % lS#^v#uS  
    %   Example: Ey=}bBx  
    % 5>ktr)]  
    %       % Display three example Zernike radial polynomials B{p74 >  
    %       r = 0:0.01:1; >]o>iOz;]  
    %       n = [3 2 5]; wuW{ 2+)B  
    %       m = [1 2 1]; [ako8  
    %       z = zernpol(n,m,r); c _!!DEe7  
    %       figure c2?VjuB0  
    %       plot(r,z) be$']}cP  
    %       grid on 4YR{ *  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') >0uj\5h)I]  
    % X5)(,036  
    %   See also ZERNFUN, ZERNFUN2. L\t_zf_0  
    |o'r?"  
    % A note on the algorithm. j(k}NWPH  
    % ------------------------ ) .KMZ]  
    % The radial Zernike polynomials are computed using the series p#_ 5w  
    % representation shown in the Help section above. For many special Zo }^"u  
    % functions, direct evaluation using the series representation can ]Qa|9G,b  
    % produce poor numerical results (floating point errors), because vVrM[0*c  
    % the summation often involves computing small differences between eTay/i<-  
    % large successive terms in the series. (In such cases, the functions lJdYR'/Wd  
    % are often evaluated using alternative methods such as recurrence yH>C7M7 t  
    % relations: see the Legendre functions, for example). For the Zernike YBR)S_C$_  
    % polynomials, however, this problem does not arise, because the $15H_X*!  
    % polynomials are evaluated over the finite domain r = (0,1), and R[)bGl6#  
    % because the coefficients for a given polynomial are generally all ?%Ww3cU+J  
    % of similar magnitude. UEhFId  
    % c{KJNH%7  
    % ZERNPOL has been written using a vectorized implementation: multiple (E,Ibz2G:e  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] s`0IyQXVU  
    % values can be passed as inputs) for a vector of points R.  To achieve $R NHRA.  
    % this vectorization most efficiently, the algorithm in ZERNPOL \ 9iiS(e  
    % involves pre-determining all the powers p of R that are required to +r9neS.l  
    % compute the outputs, and then compiling the {R^p} into a single E.+%b;Eqe  
    % matrix.  This avoids any redundant computation of the R^p, and T7Y}v,+-  
    % minimizes the sizes of certain intermediate variables. w=a$]`  
    % WuFBt=%  
    %   Paul Fricker 11/13/2006 _:WNk(  
    3^xq+{\)  
    w7&.U qjf  
    % Check and prepare the inputs: Pai8r%Zfu  
    % ----------------------------- >EZZEd   
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ^!0z+M:>^  
        error('zernpol:NMvectors','N and M must be vectors.') M ?AX:0  
    end /oLY\>pD  
    8n-Xt7z  
    if length(n)~=length(m)  K+XUC  
        error('zernpol:NMlength','N and M must be the same length.') e-[PuJ  
    end k7;i^$@c  
    T,rRE7  
    n = n(:); r4DHALu#)  
    m = m(:); <RZqs  
    length_n = length(n); xUCq%r_  
    ^8J`*R8CL  
    if any(mod(n-m,2)) )rt%.`  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') xI~A Z:m  
    end nMfR< %r  
    k_ywwkG9lU  
    if any(m<0) &43c/T Sb  
        error('zernpol:Mpositive','All M must be positive.') +6 =lN[b  
    end 8DY:a['-d  
    YOj&1ymBZ  
    if any(m>n) odC"#Rb  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') \7>*ULP  
    end ^y KkWB*  
    9V[}#(f$  
    if any( r>1 | r<0 ) i3Bpim.  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ",J&UTUh  
    end N)AlQ'Lwx  
    &;)B qqXc  
    if ~any(size(r)==1) `JpFqZ'58  
        error('zernpol:Rvector','R must be a vector.') ey,f igjd.  
    end Yf~Kzv1]*  
    lX)AbK]nb  
    r = r(:); 3\ ,t_6}  
    length_r = length(r); P:N> #G~z  
    TI/RJF b  
    if nargin==4 D,[Nn_N  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ,(zV~-:9  
        if ~isnorm 2f@Cy+W'[  
            error('zernpol:normalization','Unrecognized normalization flag.') 8ih_S2Cd  
        end HKI\i)c  
    else Ry"4v_e9  
        isnorm = false; S50}]5K  
    end WZPj?ou`G  
    qtozMa  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s%`l>#H  
    % Compute the Zernike Polynomials cz /cY:o)  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !<HMMf,-D  
    .%e>>U>F  
    % Determine the required powers of r: q5=,\S3=  
    % ----------------------------------- (a8iCci:   
    rpowers = []; r|DIf28MIq  
    for j = 1:length(n) CfP-oFHoQ  
        rpowers = [rpowers m(j):2:n(j)]; !ehjLFS?_  
    end eOF *|9  
    rpowers = unique(rpowers); .5o~^  
    f\_PNZCc  
    % Pre-compute the values of r raised to the required powers, EPH" 5$8  
    % and compile them in a matrix: l9="ccM  
    % ----------------------------- #jG?{j3;?  
    if rpowers(1)==0 D&2NO/ R  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); adIrrK  
        rpowern = cat(2,rpowern{:}); T 4p}5ew'  
        rpowern = [ones(length_r,1) rpowern]; X' 5R4j  
    else n8=D zv0  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); jll:Rh(b  
        rpowern = cat(2,rpowern{:}); g3&nxZ  
    end n7K%lj-.P  
    9T5 F0?qd  
    % Compute the values of the polynomials: ^>Z_3 {s:$  
    % -------------------------------------- zPqJeYK  
    z = zeros(length_r,length_n); fW+ "Kuw  
    for j = 1:length_n a{Y|`*7y  
        s = 0:(n(j)-m(j))/2; .'_}:~  
        pows = n(j):-2:m(j); }u3|w0~c)  
        for k = length(s):-1:1 y*{zX=]l<  
            p = (1-2*mod(s(k),2))* ... y&5 O)  
                       prod(2:(n(j)-s(k)))/          ... h8 N|m0W  
                       prod(2:s(k))/                 ... z EtsMU  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... UE"v+GH  
                       prod(2:((n(j)+m(j))/2-s(k))); G~(\N?2  
            idx = (pows(k)==rpowers); [r8[lkR  
            z(:,j) = z(:,j) + p*rpowern(:,idx); K-#d1+P+  
        end hk:>*B}  
         gDLS)4^w  
        if isnorm ^@RvCJ+  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); U'(zKqC   
        end %sOY:>  
    end (?R!y -  
    pC,[!>0g8  
    % 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)  U=KUx  
    S>T ;`,  
    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。 8*/;W&7y  
    Axb=1_--  
    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)