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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 }>< v7  
    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! 0mi$_Ld+  
     
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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 S?<hs,  
    function z = zernfun(n,m,r,theta,nflag) #b wGDF  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 9>T5~C'*  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N .)Zs:5 0l  
    %   and angular frequency M, evaluated at positions (R,THETA) on the z=yE- I{  
    %   unit circle.  N is a vector of positive integers (including 0), and kcG_ n  
    %   M is a vector with the same number of elements as N.  Each element L6Io u  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) @RXkj-,eC#  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ;DX g  
    %   and THETA is a vector of angles.  R and THETA must have the same ]8/g[Ii  
    %   length.  The output Z is a matrix with one column for every (N,M) 6<mlx'  
    %   pair, and one row for every (R,THETA) pair. 7(l>Ck3B#  
    % TX).*%f [r  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike }*?,&9/_)  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), X+kgx!u'y  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral \- 8S"  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, > PK 6CR  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized %00cC~}4  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. wDDNB1_ E  
    % W5,&*mo  
    %   The Zernike functions are an orthogonal basis on the unit circle. 5 BLAa1  
    %   They are used in disciplines such as astronomy, optics, and =z[$ o9  
    %   optometry to describe functions on a circular domain. &a.A8v)  
    % M@+Pq/f:  
    %   The following table lists the first 15 Zernike functions. l 1vI  
    % 6{!Cx9V  
    %       n    m    Zernike function           Normalization {"c`k4R  
    %       -------------------------------------------------- qL4s@<|~  
    %       0    0    1                                 1 FxmHy{JG  
    %       1    1    r * cos(theta)                    2 j]C}S*`"  
    %       1   -1    r * sin(theta)                    2 ==AmL]*  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) nh*6`5yj  
    %       2    0    (2*r^2 - 1)                    sqrt(3) #Q'#/\5  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) xVk5%  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 3,?LpdTS  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8)  0*E_D  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) XK&G`cJ[  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) )H(i)$I  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 055C1RV%  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .$fSWlM;  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) {2k]$|  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) X0Wx\xDg[  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Zc'^iDAY  
    %       -------------------------------------------------- /@B2-.w  
    % Qk >9o  
    %   Example 1: C8x9 Jrc  
    % z69u@  
    %       % Display the Zernike function Z(n=5,m=1) /cT6X]o8  
    %       x = -1:0.01:1; ?dPr HSy  
    %       [X,Y] = meshgrid(x,x); Xdf4%/Op  
    %       [theta,r] = cart2pol(X,Y); -^3uQa<zN^  
    %       idx = r<=1; !jvl"+_FV  
    %       z = nan(size(X)); IhRdn1&  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 6-z(34&N  
    %       figure )-0+O=v  
    %       pcolor(x,x,z), shading interp 0SQrz$y  
    %       axis square, colorbar udXzsY9Ng  
    %       title('Zernike function Z_5^1(r,\theta)') '{-Ic?F<P  
    % <4n"LJ9  
    %   Example 2: {Fqwr>e  
    % >qs/o$+t}  
    %       % Display the first 10 Zernike functions H+Aidsn  
    %       x = -1:0.01:1; &u~Pp=kv  
    %       [X,Y] = meshgrid(x,x); 4/%Y@Z5  
    %       [theta,r] = cart2pol(X,Y);  AGm=0Om  
    %       idx = r<=1; uW [yNwM  
    %       z = nan(size(X)); zU0SlRFu  
    %       n = [0  1  1  2  2  2  3  3  3  3]; #m17cDL  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ]&N>F8.L+  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; F 2Y!aR  
    %       y = zernfun(n,m,r(idx),theta(idx)); KY}H-  
    %       figure('Units','normalized') ik #Wlz`4  
    %       for k = 1:10 F]t=5 -O<  
    %           z(idx) = y(:,k); xZ6x`BET-  
    %           subplot(4,7,Nplot(k)) ~sZ$`t  
    %           pcolor(x,x,z), shading interp wqi0%Cu*  
    %           set(gca,'XTick',[],'YTick',[]) 7377g'jL  
    %           axis square ?J,,RK.  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) e"_kH_7sv  
    %       end ANNVE},  
    % I$MlIz$l v  
    %   See also ZERNPOL, ZERNFUN2. _-3n'i8  
    .cHkh^EDY  
    %   Paul Fricker 11/13/2006 ^/6P~iK'  
    YWs?2I  
    b kc*it  
    % Check and prepare the inputs: 6ypLE@Mk  
    % ----------------------------- K7([Gc9  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ;b:'i& r  
        error('zernfun:NMvectors','N and M must be vectors.') D6H?*4f]  
    end R7U%v"F>`  
    9K#3JyW*  
    if length(n)~=length(m) -cijLlz%+  
        error('zernfun:NMlength','N and M must be the same length.') reNf?7G+m  
    end V[uSo$k+>  
    vS)>g4  
    n = n(:); L~^5Ez6U  
    m = m(:); Dk>6PBl  
    if any(mod(n-m,2)) "l9aBBiu  
        error('zernfun:NMmultiplesof2', ... +wJ!zab`  
              'All N and M must differ by multiples of 2 (including 0).') JSi0-S[Y{  
    end N'WC!K.e  
    vg5_@7  
    if any(m>n) RgA"`p7{  
        error('zernfun:MlessthanN', ... [61*/=gWe  
              'Each M must be less than or equal to its corresponding N.') "TJ*mN.i{}  
    end hxK;f  
    fBctG~CJH  
    if any( r>1 | r<0 ) n=bdV(?4  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ruGeN  
    end R"9w VM;*c  
    huS*1xl  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) jS~Pdz  
        error('zernfun:RTHvector','R and THETA must be vectors.') PkI+z_  
    end p7@R+F\.};  
    Y*PfU +y~  
    r = r(:); QxdC[t$Lp  
    theta = theta(:); ($kw*H{Ah^  
    length_r = length(r); ?h&?`WO (  
    if length_r~=length(theta) )S(Ly.  
        error('zernfun:RTHlength', ... "I)zi]vk  
              'The number of R- and THETA-values must be equal.') 8\!E )M|4  
    end Y}v3J(l  
    <JH,B91  
    % Check normalization: z-606g  
    % -------------------- bY=[ USgps  
    if nargin==5 && ischar(nflag) )?UoF&c/  
        isnorm = strcmpi(nflag,'norm'); 3_Xu3hNH!  
        if ~isnorm O"+0 b|  
            error('zernfun:normalization','Unrecognized normalization flag.') $q)YC.5$  
        end UJSIbb5  
    else -]HZ?@  
        isnorm = false; sHc-xnd  
    end Lr D@QBT  
    jt on\9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {V2"Pym?  
    % Compute the Zernike Polynomials 5C9b*]-#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =I546($  
    kuy?n-1g  
    % Determine the required powers of r: B(++*#T!^m  
    % ----------------------------------- ZQ_6I}i")  
    m_abs = abs(m); T5."3i  
    rpowers = []; Ly+UY.v"  
    for j = 1:length(n) JRo/ HY+  
        rpowers = [rpowers m_abs(j):2:n(j)]; ^0}ma*gi~  
    end +h4W<YnW  
    rpowers = unique(rpowers); BZ?Ck[E]Z  
    #mw !_]  
    % Pre-compute the values of r raised to the required powers, oNyYx6q:Q  
    % and compile them in a matrix: hOU H1m.  
    % ----------------------------- eMC^ORdY  
    if rpowers(1)==0 31a,i2Q4  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); "mW'tm1+  
        rpowern = cat(2,rpowern{:}); p35=CX`T.  
        rpowern = [ones(length_r,1) rpowern]; <.QaOLD  
    else hr fF1 >A  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %-540V{q  
        rpowern = cat(2,rpowern{:}); #f2k*8"eAF  
    end !%,7*F(  
    bTc >-e,  
    % Compute the values of the polynomials: B2ln8NF#Q  
    % -------------------------------------- u^tQ2&?O!P  
    y = zeros(length_r,length(n)); /{i~-DVME  
    for j = 1:length(n) Nrr}) g  
        s = 0:(n(j)-m_abs(j))/2; sv% X8  
        pows = n(j):-2:m_abs(j); 7Ed0BJTa  
        for k = length(s):-1:1 xo_STLAw  
            p = (1-2*mod(s(k),2))* ... {Ya$Q#l  
                       prod(2:(n(j)-s(k)))/              ... +Y sGH~jX  
                       prod(2:s(k))/                     ... 9j>2C  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &-yRa45?  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); J[ Gpd  
            idx = (pows(k)==rpowers); ;\mX=S|a  
            y(:,j) = y(:,j) + p*rpowern(:,idx); mrP48#Y+l  
        end _Sr7b#)o  
         X3:z=X&Zd  
        if isnorm 1_] X  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 9&eY<'MgP  
        end [<R haZz  
    end L1SKOM$  
    % END: Compute the Zernike Polynomials N>H@vt~  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% STW?0B'Jr  
    [Km{6L&  
    % Compute the Zernike functions: L7C ;l,ot  
    % ------------------------------ 2<EV iP9  
    idx_pos = m>0; :2? g_  
    idx_neg = m<0; Vke<; k-  
    */;7Uv7  
    z = y; ttsR`R1.k  
    if any(idx_pos) `q*[fd1u.  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); k<<x}=  
    end !cyrt<  
    if any(idx_neg)  ##rkyd  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); S; % &X  
    end gZ,h9 5'  
     cca g8LC  
    % 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) Z3S+")^  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ( s3k2Z  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated GTdoUSUq  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive HOP*QX8C%  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, FCE y1^u  
    %   and THETA is a vector of angles.  R and THETA must have the same m)Plv+R}  
    %   length.  The output Z is a matrix with one column for every P-value, JsJP%'^/R  
    %   and one row for every (R,THETA) pair. qbv\uYow3k  
    % kUd]8Ff!  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike FiUQ2w4  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) -5<[oBL;  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) a^R?w|zCX  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 a^ %iAe  
    %   for all p. Ehx9-*]  
    % bJ^h{]  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 iOk ;o=  
    %   Zernike functions (order N<=7).  In some disciplines it is )E^S+ps  
    %   traditional to label the first 36 functions using a single mode PQ&*(G  
    %   number P instead of separate numbers for the order N and azimuthal *S,~zOYN  
    %   frequency M. VQ9A/DH/  
    % 6'#5Dqw"r  
    %   Example: hne@I1  
    % ;,f\Wf"BW  
    %       % Display the first 16 Zernike functions C"I jr=w  
    %       x = -1:0.01:1; ;{ifLI0#  
    %       [X,Y] = meshgrid(x,x); y:;.r:  
    %       [theta,r] = cart2pol(X,Y); Scrj%h%[  
    %       idx = r<=1; 6("_}9ZOc  
    %       p = 0:15; mLhM_=  
    %       z = nan(size(X)); f^F;`;z  
    %       y = zernfun2(p,r(idx),theta(idx)); rwP#Yj[BK+  
    %       figure('Units','normalized') -<#) ]um  
    %       for k = 1:length(p) P[3i!"O>  
    %           z(idx) = y(:,k);  !VGG2N8  
    %           subplot(4,4,k) /WN YS  
    %           pcolor(x,x,z), shading interp =-U0r$sK+F  
    %           set(gca,'XTick',[],'YTick',[]) gb_Y]U  
    %           axis square y!FO  
    %           title(['Z_{' num2str(p(k)) '}']) i7Qb~RW  
    %       end 6<lo0PQ"Z  
    %  /qLO/Mim  
    %   See also ZERNPOL, ZERNFUN. EvT$|#FY  
    P 9?cp{*  
    %   Paul Fricker 11/13/2006 1VJ${\H]  
    &6sF wK  
    Sm'Tz&!  
    % Check and prepare the inputs: 0S{23L4C  
    % ----------------------------- =5|7S&{  
    if min(size(p))~=1 2K}49*  
        error('zernfun2:Pvector','Input P must be vector.') QEyL/#Q  
    end ~i.*fL_Y  
    ]+D@E2E  
    if any(p)>35 $k~TVm Yex  
        error('zernfun2:P36', ... 7e"}ojt$  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... :;{M0  
               '(P = 0 to 35).']) ]oGd,v X  
    end ^')8-aF .  
    N5?bflY  
    % Get the order and frequency corresonding to the function number: <[dcIw<7  
    % ---------------------------------------------------------------- [^hW>O=@TN  
    p = p(:); !5ps,+o  
    n = ceil((-3+sqrt(9+8*p))/2); z!}E2j_9P  
    m = 2*p - n.*(n+2); dFz"wvu` o  
    z CLaHx!  
    % Pass the inputs to the function ZERNFUN: 5JzvT JMx  
    % ---------------------------------------- 6`e{l+c=F  
    switch nargin j`_S%E%X  
        case 3 a'VQegP(f\  
            z = zernfun(n,m,r,theta); DDrR9}k  
        case 4 CS[]T9|_  
            z = zernfun(n,m,r,theta,nflag); \YvG+7a  
        otherwise >=@-]X2%j  
            error('zernfun2:nargin','Incorrect number of inputs.') +x%u?ZR  
    end e9LX0=  
    gFaZ ._  
    % 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) u({^8: AYu  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. T@W:@,34  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 2C S9v  
    %   order N and frequency M, evaluated at R.  N is a vector of LK'(OZ  
    %   positive integers (including 0), and M is a vector with the Q>1BOH1by  
    %   same number of elements as N.  Each element k of M must be a XM]m%I  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) K,S4  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 8Ths"zwn  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix yY$^ R|t  
    %   with one column for every (N,M) pair, and one row for every /zIG5RK>  
    %   element in R. PD&e6;rj;  
    % //@6w;P  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- o0r&w;!  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is A]bb*a1  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to #0AyC.\  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 6P{bUom?  
    %   for all [n,m]. !U`4  
    % x;vfmgty  
    %   The radial Zernike polynomials are the radial portion of the w{tA{{  
    %   Zernike functions, which are an orthogonal basis on the unit Fs]N9],=I  
    %   circle.  The series representation of the radial Zernike |V34;}\4  
    %   polynomials is A'EI1_3{  
    % I0 t#{i  
    %          (n-m)/2 /d&m#%9Up]  
    %            __ MHwfJ{"zo  
    %    m      \       s                                          n-2s t 24`*'  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r dS1HA>c)O  
    %    n      s=0 7C|AiSH  
    % P& 1$SWNyW  
    %   The following table shows the first 12 polynomials. - (s0f  
    % ;@; a eu  
    %       n    m    Zernike polynomial    Normalization 2Bt/co-~4  
    %       --------------------------------------------- 2IYzc3Z{9  
    %       0    0    1                        sqrt(2) )G48,. "  
    %       1    1    r                           2 Yc#Uu8f-  
    %       2    0    2*r^2 - 1                sqrt(6) p[4 +`8  
    %       2    2    r^2                      sqrt(6) ~(GvjB/C8  
    %       3    1    3*r^3 - 2*r              sqrt(8) :hICe+2ca  
    %       3    3    r^3                      sqrt(8) )"TVR{I%B  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) =z}PR1X!  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) a?gF;AYk  
    %       4    4    r^4                      sqrt(10) &g?GF\Y  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) uzp\V 39  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) hWly8B[I  
    %       5    5    r^5                      sqrt(12) SS/vw%  
    %       --------------------------------------------- e=LrgRy+  
    % {t;o^pUF  
    %   Example: Oti;wf G7o  
    % P#TPI*qw  
    %       % Display three example Zernike radial polynomials ~ZafTCa;  
    %       r = 0:0.01:1; jI,[(Z>  
    %       n = [3 2 5]; ,!> ~izB  
    %       m = [1 2 1]; \]>821r  
    %       z = zernpol(n,m,r); Mmz; uy_  
    %       figure *k(FbZ  
    %       plot(r,z) bqn(5)%{  
    %       grid on e"866vc,  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') jwwRejNV  
    % mc]+j,d  
    %   See also ZERNFUN, ZERNFUN2. 1V,@uY)s  
    J@>|`9T9$  
    % A note on the algorithm. [>NMuwtG  
    % ------------------------ @>2]zMFf  
    % The radial Zernike polynomials are computed using the series ?q6#M&|j/I  
    % representation shown in the Help section above. For many special C-edQWbcP  
    % functions, direct evaluation using the series representation can co,0@.i  
    % produce poor numerical results (floating point errors), because feX o"J  
    % the summation often involves computing small differences between * o1US  
    % large successive terms in the series. (In such cases, the functions jNxTy UU  
    % are often evaluated using alternative methods such as recurrence ?EUg B\  
    % relations: see the Legendre functions, for example). For the Zernike \zU<o~gs  
    % polynomials, however, this problem does not arise, because the !W XV1S  
    % polynomials are evaluated over the finite domain r = (0,1), and 0^*,E/}P&  
    % because the coefficients for a given polynomial are generally all Q7y' 0s  
    % of similar magnitude. D@p{EH  
    % fDYTupKXH  
    % ZERNPOL has been written using a vectorized implementation: multiple POk5+^  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 9o,Eq x4J  
    % values can be passed as inputs) for a vector of points R.  To achieve n]6 '!Eo  
    % this vectorization most efficiently, the algorithm in ZERNPOL QP~["%}T  
    % involves pre-determining all the powers p of R that are required to E$lbm>jsb$  
    % compute the outputs, and then compiling the {R^p} into a single @Yt394gA%\  
    % matrix.  This avoids any redundant computation of the R^p, and uWx<J3~q.  
    % minimizes the sizes of certain intermediate variables. glC,E>  
    % r!b>!  
    %   Paul Fricker 11/13/2006 yoGG[l2k>s  
    'LoWp} f9  
    ,~7~ S"  
    % Check and prepare the inputs: r]6+&K  
    % ----------------------------- ~AWn 1vFc  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) #i~P])%gNP  
        error('zernpol:NMvectors','N and M must be vectors.') H%vgPQ8  
    end iUz?mt;k  
    h,y_ ^cf  
    if length(n)~=length(m) ,|O6<u9  
        error('zernpol:NMlength','N and M must be the same length.') `(j~b=PP  
    end wYe;xk`>  
    Yv=L'0K&  
    n = n(:); .hckZx /  
    m = m(:); 2aTq?ZR|8A  
    length_n = length(n); v,opyTwG|  
    C_3,|Zq?|  
    if any(mod(n-m,2)) T0A=vh;S  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') mm_)=Ipj>  
    end /w?zO,!  
    K 91O$'J  
    if any(m<0) F&`%L#s|  
        error('zernpol:Mpositive','All M must be positive.') j#3IF *"  
    end .Ao _c x  
    5OPvy,e6  
    if any(m>n) 'hu'}F{  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') +PBl3  
    end 1 jLQij  
    cRs\()W  
    if any( r>1 | r<0 ) p%iZ6H>G  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') E<D^j^T  
    end  N MkOx$  
    Eve,*ATI  
    if ~any(size(r)==1) 3w>1R>7  
        error('zernpol:Rvector','R must be a vector.') KtJc9dnX  
    end EPwU{*F  
    zk1]?  
    r = r(:); tSni[,4Kq  
    length_r = length(r); w^cQL%  
    <8~c7kT'  
    if nargin==4 <k3KCt  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); TPx`qyW  
        if ~isnorm PDH|=meXM  
            error('zernpol:normalization','Unrecognized normalization flag.') 8B+C[Q:+'  
        end H/*slqL  
    else 3-AOB3](  
        isnorm = false; _s<BXj  
    end } PL{i  
    `*0VN(gf'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zr A3bWs  
    % Compute the Zernike Polynomials b7+(g [O  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N,8.W"fV  
    \d.\M  
    % Determine the required powers of r: -|u yJh  
    % ----------------------------------- 5{!"}  
    rpowers = []; C \5yo  
    for j = 1:length(n) ffI=Bt]t  
        rpowers = [rpowers m(j):2:n(j)]; CX2qtI8N?  
    end %S`Wu|y  
    rpowers = unique(rpowers); vo:h"ti  
    W@z xGH$z>  
    % Pre-compute the values of r raised to the required powers, ,c`Wmp^AY  
    % and compile them in a matrix: Jw;G_dQ[  
    % ----------------------------- .i )n1  
    if rpowers(1)==0 E:B<_  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); }4piZ ch  
        rpowern = cat(2,rpowern{:}); BbCW3!(  
        rpowern = [ones(length_r,1) rpowern]; xY.?OHgG/  
    else 9:3`LY3wW  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); (]?M=?0\  
        rpowern = cat(2,rpowern{:}); JbitRV@a  
    end x1Z'_Qw  
    s^.tj41Gx}  
    % Compute the values of the polynomials: ;*+H&  
    % -------------------------------------- :)4c_51 `  
    z = zeros(length_r,length_n); _V8;dv8  
    for j = 1:length_n \R-'<kN.*  
        s = 0:(n(j)-m(j))/2; ugj I$u  
        pows = n(j):-2:m(j); Q t>|TGz  
        for k = length(s):-1:1 q-@&n6PEOZ  
            p = (1-2*mod(s(k),2))* ... B7Zi|-F  
                       prod(2:(n(j)-s(k)))/          ... 4$mtc*tzT  
                       prod(2:s(k))/                 ... Gr}NgyT<!D  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... K:VZ#U(_  
                       prod(2:((n(j)+m(j))/2-s(k))); A42!%>PB  
            idx = (pows(k)==rpowers); _d^d1Q}V  
            z(:,j) = z(:,j) + p*rpowern(:,idx); \J#&]o)Y  
        end FI$ -."F  
         xDPR^xY  
        if isnorm Hj`\Fm*A  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 7 _"G@h  
        end $*:$-  
    end e_l|32#/  
    rf`xY4I\  
    % 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)  EA9.?F  
    g>w {{G  
    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。 x2r.4  
    ?$uF(>LD  
    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)