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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[nyW3MZ  
    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! >e\9Bf_  
     
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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 c%=IL M4  
    function z = zernfun(n,m,r,theta,nflag) YW{C} NA  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. wE~V]bmtW  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ,yd?gP-O  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ANgw"&&>(  
    %   unit circle.  N is a vector of positive integers (including 0), and i&VsW7  
    %   M is a vector with the same number of elements as N.  Each element kT;S4B  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) S#+h$UVh  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, {GC?SaK  
    %   and THETA is a vector of angles.  R and THETA must have the same r#XT3qp$d  
    %   length.  The output Z is a matrix with one column for every (N,M) @|\}.M<e*)  
    %   pair, and one row for every (R,THETA) pair. L:FoSCN Y(  
    % Uw 47LP  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ?Wz8[u  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), R~Ne|V2  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ztw@Y|<2  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ,T2G~^0  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized TA{\PKA)  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. b,C aWg  
    % *hw\35%P`?  
    %   The Zernike functions are an orthogonal basis on the unit circle. J>\B`E  
    %   They are used in disciplines such as astronomy, optics, and Z,=7Tu bR#  
    %   optometry to describe functions on a circular domain. -{ H0g]  
    % xXM{pd  
    %   The following table lists the first 15 Zernike functions. [ i]Ub0Dh7  
    % ,lyb!k8  
    %       n    m    Zernike function           Normalization X-wf:h?i  
    %       -------------------------------------------------- P'`r  
    %       0    0    1                                 1 wu)w   
    %       1    1    r * cos(theta)                    2 ^/r7@:  
    %       1   -1    r * sin(theta)                    2 .FLy;_f+  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) sQ fFu  
    %       2    0    (2*r^2 - 1)                    sqrt(3) gM _hi  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) rnF/H=I/  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) <kCU@SK  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Y*UA, <-  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Z:*76PP,  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) (2=Zm@Zp f  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) l g-X:Z.  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) L|,!?cSAT  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) +u3=dj"[  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9T1ZL5  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) :3I@(k\PY  
    %       --------------------------------------------------  Y*14v~\'  
    % f\jLqZY  
    %   Example 1: kOed ]>H  
    % *FMMjz  
    %       % Display the Zernike function Z(n=5,m=1) }b-g*dn]5  
    %       x = -1:0.01:1; (_"*NY0  
    %       [X,Y] = meshgrid(x,x); og kD^   
    %       [theta,r] = cart2pol(X,Y); w'UVKpG+  
    %       idx = r<=1; /bi}'H+#  
    %       z = nan(size(X)); }yz (xH  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); +1D+]*t_?[  
    %       figure L>3x9  
    %       pcolor(x,x,z), shading interp 3J5!oF{H  
    %       axis square, colorbar w$Rro)?}7  
    %       title('Zernike function Z_5^1(r,\theta)') 9_ d pR.  
    % h]TQn)X]  
    %   Example 2: H(Z88.OM  
    % ;NHt7p8SE  
    %       % Display the first 10 Zernike functions MP>dW nl  
    %       x = -1:0.01:1; 6=fSE=]DY  
    %       [X,Y] = meshgrid(x,x); nYX@J6!  
    %       [theta,r] = cart2pol(X,Y); 1#ft#-g}  
    %       idx = r<=1; ^Gqt+K%  
    %       z = nan(size(X)); v^1pN>#%g  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 7BJzM lJ1Y  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; c5u@pvSP  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; kYjGj,m"  
    %       y = zernfun(n,m,r(idx),theta(idx)); 9;B0Mq py  
    %       figure('Units','normalized') [_Qa9e  
    %       for k = 1:10 v uoQz\  
    %           z(idx) = y(:,k); J{k79v  
    %           subplot(4,7,Nplot(k)) ;oy-#p>N%  
    %           pcolor(x,x,z), shading interp L{8xlx`  
    %           set(gca,'XTick',[],'YTick',[]) 28UU60  
    %           axis square o !vE~  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) MpF$xzh  
    %       end )3>hhuaa  
    % K5xX)oV  
    %   See also ZERNPOL, ZERNFUN2. .n~M(59  
    id1s3b;  
    %   Paul Fricker 11/13/2006 /!3@]xz*  
    w.\&9]P3~  
    D?NbW @]  
    % Check and prepare the inputs: N19({0+i2  
    % ----------------------------- `|ASx8_!  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) "$2 y-|  
        error('zernfun:NMvectors','N and M must be vectors.') {o.FlX  
    end uA#P'?  
    ,f[>L|?e  
    if length(n)~=length(m) @ < Q|5  
        error('zernfun:NMlength','N and M must be the same length.') 5nKj )RH7M  
    end rV T{90,  
    34Kw!  
    n = n(:); ]hFW 73FV  
    m = m(:); F;7dt@5;  
    if any(mod(n-m,2)) TzNn^ir=HX  
        error('zernfun:NMmultiplesof2', ... H*$jc\ dC  
              'All N and M must differ by multiples of 2 (including 0).') Qm Ce>+  
    end Ht&:-F+dm  
    % a@>_  
    if any(m>n) V7Ek-2M  
        error('zernfun:MlessthanN', ... TX&Jt%  
              'Each M must be less than or equal to its corresponding N.') ! q M=a3  
    end kNobl  
    '|Kmq5)  
    if any( r>1 | r<0 ) ]Ccg`AR{  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') MP4z-4Y  
    end .K p  
    <w)r`D6  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) jhb6T ?}  
        error('zernfun:RTHvector','R and THETA must be vectors.') B4i!/@0s  
    end $Z\.-QE\  
    B(n{e53 9f  
    r = r(:); CTZh0 x  
    theta = theta(:);  y"H*%]  
    length_r = length(r); +h r@#n4A  
    if length_r~=length(theta) /XzH?n/{R  
        error('zernfun:RTHlength', ... v33dxZ'  
              'The number of R- and THETA-values must be equal.') ;;:-l99  
    end ~;#Y9>7\\'  
    8q,6}mV  
    % Check normalization: V;:jZpG  
    % -------------------- tavpq.0O  
    if nargin==5 && ischar(nflag) G"Sd@%W(  
        isnorm = strcmpi(nflag,'norm'); s#)5h0t#du  
        if ~isnorm Zf65`K3  
            error('zernfun:normalization','Unrecognized normalization flag.') S|]X'f  
        end Zw ^kmSL"  
    else iX2]VRNxl  
        isnorm = false; +ayos[<0#  
    end ?MgUY)X  
    a{qM2P(S  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a *ushB  
    % Compute the Zernike Polynomials =Q+= f  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% bqnNLs<N  
    L.jh   
    % Determine the required powers of r: /oR<A  
    % ----------------------------------- 'Pn3%&O$  
    m_abs = abs(m); 7:)n$,31FW  
    rpowers = []; 8p@Piy{p  
    for j = 1:length(n) TiO"xMX  
        rpowers = [rpowers m_abs(j):2:n(j)]; $0lD>yu  
    end qT`k*i?  
    rpowers = unique(rpowers); JSTuXW  
    P#XID 2;  
    % Pre-compute the values of r raised to the required powers, 06N}k<10O  
    % and compile them in a matrix: EuyXgK>g  
    % ----------------------------- ZRg;/sX]  
    if rpowers(1)==0 RWg No #<  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); :QB<?HaS'  
        rpowern = cat(2,rpowern{:}); Od %"B\  
        rpowern = [ones(length_r,1) rpowern]; PSZL2iGj9V  
    else yl1gx  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); or';A'k  
        rpowern = cat(2,rpowern{:}); H=Y{rq@  
    end }A7j/uy}s  
    f,:9N5Z  
    % Compute the values of the polynomials: Db1pW=66:  
    % -------------------------------------- /5:bvg+  
    y = zeros(length_r,length(n)); i-6F:\;  
    for j = 1:length(n) 2|}+T6_q  
        s = 0:(n(j)-m_abs(j))/2; -U/c\-~fU  
        pows = n(j):-2:m_abs(j); fH >NJK;  
        for k = length(s):-1:1 \3S8 62B7  
            p = (1-2*mod(s(k),2))* ... <\}KT*Xp  
                       prod(2:(n(j)-s(k)))/              ... ,~OwLWi-|X  
                       prod(2:s(k))/                     ... Ko&>C_N  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ZfgJ.<<  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 'zGo?a  
            idx = (pows(k)==rpowers); m|:_]/*qE  
            y(:,j) = y(:,j) + p*rpowern(:,idx); &qr;IL7'  
        end Gch[Otq]%  
         @>)r}b  
        if isnorm vWf; 'j  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); KNLfp1!  
        end ek}a}.3 {  
    end A?t%e  
    % END: Compute the Zernike Polynomials R5 9S@MsuD  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kZerKP  
    %^>ju;i^O  
    % Compute the Zernike functions: ktdW`R\+  
    % ------------------------------ ~ ArP9 K "  
    idx_pos = m>0; 26k LhFS  
    idx_neg = m<0; /O^RF}  
    2g>SHS@1>  
    z = y; Oms. e  
    if any(idx_pos) tGl;@V@Qj  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); O2BDL1o  
    end X6mqi;+  
    if any(idx_neg) 66I"=:  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Y5FbU  
    end `/ q|@B7  
    .b-f9qc=  
    % 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) \a6^LD}B  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. h0g:@ae%&  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated sh`s /JRf  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive N.]qU d  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ]):<ZsT  
    %   and THETA is a vector of angles.  R and THETA must have the same qLT>Mz)$ %  
    %   length.  The output Z is a matrix with one column for every P-value, Eg2[k.{P  
    %   and one row for every (R,THETA) pair. k'IYA#T6  
    % S<WdZ=8sA  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike krC{ed  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ~D5\O6mU-  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) k25WucQ  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 =|%Cu&  
    %   for all p. $n+w$CI)  
    % 5c^Z/ Jl$c  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 3`B6w$z>(  
    %   Zernike functions (order N<=7).  In some disciplines it is *IY*yR6  
    %   traditional to label the first 36 functions using a single mode 4)"n RjGg  
    %   number P instead of separate numbers for the order N and azimuthal "E8zh|m o  
    %   frequency M. a(9L,v#?  
    % vt;{9\Y  
    %   Example: 3&[>u;Bp  
    % j|/]#@Yr  
    %       % Display the first 16 Zernike functions 9v }G{mQ#  
    %       x = -1:0.01:1; 7A\~)U @  
    %       [X,Y] = meshgrid(x,x); MwR 0@S}*  
    %       [theta,r] = cart2pol(X,Y); bV8!"{  
    %       idx = r<=1; ywb4LKD  
    %       p = 0:15; E !a|Xp  
    %       z = nan(size(X)); -#2)?NkeE  
    %       y = zernfun2(p,r(idx),theta(idx)); q*7zx_ o  
    %       figure('Units','normalized') %ix)8+Eb  
    %       for k = 1:length(p) }p!HT6 tZ  
    %           z(idx) = y(:,k); )~+e`q  
    %           subplot(4,4,k) =7C%P%yt  
    %           pcolor(x,x,z), shading interp mXUGe:e8  
    %           set(gca,'XTick',[],'YTick',[]) NLrPSqz  
    %           axis square VGceD$<  
    %           title(['Z_{' num2str(p(k)) '}']) -GqT7`:(H4  
    %       end BVr0Gk  
    % l411a9o  
    %   See also ZERNPOL, ZERNFUN. H~+l7OhV  
    *+\S yO  
    %   Paul Fricker 11/13/2006 P#_sg0oJF  
    gx&Tt  
    >layJt  
    % Check and prepare the inputs: kH4Ai3#g  
    % ----------------------------- Q"t<3-"  
    if min(size(p))~=1 Mnz!nWhk  
        error('zernfun2:Pvector','Input P must be vector.') g-e #!(  
    end ;Y; qg  
    T[sDVkCbxf  
    if any(p)>35 Pp| *J^U 4  
        error('zernfun2:P36', ... .9"Y_/0   
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 3nu^l'WQ  
               '(P = 0 to 35).']) qWx][D"  
    end @EDs~ lPv  
    RgGyoZ  
    % Get the order and frequency corresonding to the function number: m <w "T7  
    % ---------------------------------------------------------------- `8I&7c  
    p = p(:); TP"1\O  
    n = ceil((-3+sqrt(9+8*p))/2); >9f%@uSM$3  
    m = 2*p - n.*(n+2); s7l;\XBy  
    sPpsq  
    % Pass the inputs to the function ZERNFUN: !O#dV1wAa  
    % ---------------------------------------- 3T e^  
    switch nargin u&9 r2R959  
        case 3 V`RNM%Y  
            z = zernfun(n,m,r,theta); ^RP)>d9Xp{  
        case 4 A5H3%o(6k  
            z = zernfun(n,m,r,theta,nflag); h?f>X"*|(  
        otherwise svuq gSn  
            error('zernfun2:nargin','Incorrect number of inputs.') rS?pWTg"8  
    end $ KRI'4  
    h^*4}GU  
    % 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) ^<:sdv>Y5  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. k $f Gom  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Q3 eM2i8Y  
    %   order N and frequency M, evaluated at R.  N is a vector of e+]6OV&+  
    %   positive integers (including 0), and M is a vector with the h@@nR(<i  
    %   same number of elements as N.  Each element k of M must be a Fk6x<^Q<w  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 3VUWX5K?  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is #CnHf  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix AxZD-|.  
    %   with one column for every (N,M) pair, and one row for every #!9S}b$  
    %   element in R. &tZG @  
    % oP2fX_v1x  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- gxT4PQDy  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is R@*O!bD  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Y'u7 IX}  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 cDoo*  
    %   for all [n,m]. Maqf[ Vky  
    % /Ux*u#  
    %   The radial Zernike polynomials are the radial portion of the oM2UzB{(  
    %   Zernike functions, which are an orthogonal basis on the unit ZKz,|+X0G  
    %   circle.  The series representation of the radial Zernike r,]#b[:.s|  
    %   polynomials is K 9kUS  
    % `?y<>m*  
    %          (n-m)/2 ^@"H1  
    %            __ Pe_!?:vF  
    %    m      \       s                                          n-2s ooj~&fu  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r mC z,2K|^~  
    %    n      s=0 9~0^PzTA  
    % `Rd m-[&  
    %   The following table shows the first 12 polynomials. jS!`2li?{  
    % G|m1.=DJm  
    %       n    m    Zernike polynomial    Normalization YxGcFjJ  
    %       --------------------------------------------- /oL;YIoQX  
    %       0    0    1                        sqrt(2) }%-t+Tf,  
    %       1    1    r                           2 ;@nFVy>U  
    %       2    0    2*r^2 - 1                sqrt(6) gUAxyV  
    %       2    2    r^2                      sqrt(6) ~aXqU#8  
    %       3    1    3*r^3 - 2*r              sqrt(8) E_1="&p  
    %       3    3    r^3                      sqrt(8) : 5U"XY x@  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) nq1 9Q)  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) R|P_GN6 >  
    %       4    4    r^4                      sqrt(10) M('d-Q{B7L  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12)  2T)sXBu  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) hAqg Iu*  
    %       5    5    r^5                      sqrt(12) DOQc"+  
    %       --------------------------------------------- 2`a q**}  
    % "{E q hR~  
    %   Example: G2#d $  
    % -.<k~71  
    %       % Display three example Zernike radial polynomials 3S BZ>  
    %       r = 0:0.01:1; = pIy  
    %       n = [3 2 5]; }4>JO""  
    %       m = [1 2 1]; 46h@j>/K  
    %       z = zernpol(n,m,r); AY SSa 1}  
    %       figure + zkm(  
    %       plot(r,z) qUo-Dq>  
    %       grid on w# * 1/N  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') FZH\Q~IUV  
    % .5Q:Xp  
    %   See also ZERNFUN, ZERNFUN2. ]feyJLF  
    t=R6mjb  
    % A note on the algorithm. ^# A.@  
    % ------------------------ nPkZHIxuD  
    % The radial Zernike polynomials are computed using the series ~JuKV&&}K  
    % representation shown in the Help section above. For many special cE{ =(OQ  
    % functions, direct evaluation using the series representation can 6`$[Ini  
    % produce poor numerical results (floating point errors), because (shK  
    % the summation often involves computing small differences between &s)0z)mR8&  
    % large successive terms in the series. (In such cases, the functions \Xt) E[  
    % are often evaluated using alternative methods such as recurrence [ B0K  
    % relations: see the Legendre functions, for example). For the Zernike 15zrrU~D  
    % polynomials, however, this problem does not arise, because the !RlC~^ -  
    % polynomials are evaluated over the finite domain r = (0,1), and uO >x:*^8  
    % because the coefficients for a given polynomial are generally all Ae?e 70bY  
    % of similar magnitude. }wSy  
    % l SkEuN  
    % ZERNPOL has been written using a vectorized implementation: multiple 4S L_-Hm.  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] |z^pL1Z]5  
    % values can be passed as inputs) for a vector of points R.  To achieve (\dK4JJ  
    % this vectorization most efficiently, the algorithm in ZERNPOL L|^o7 1t|  
    % involves pre-determining all the powers p of R that are required to ~ E=\t9r  
    % compute the outputs, and then compiling the {R^p} into a single 3]n0 &MZAR  
    % matrix.  This avoids any redundant computation of the R^p, and )9P&=  
    % minimizes the sizes of certain intermediate variables. ,fnsE^}.U  
    % LQ-6vrbs  
    %   Paul Fricker 11/13/2006 cCxi{a1uo  
    u{bL-a8}  
    .dI)R40L/\  
    % Check and prepare the inputs: nd+?O7~}(  
    % ----------------------------- Y5-kj,CB  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) mjEs5XCC"  
        error('zernpol:NMvectors','N and M must be vectors.') djT. 1(  
    end |,}E0G.  
    +=8X8<Pu  
    if length(n)~=length(m) ggou*;'  
        error('zernpol:NMlength','N and M must be the same length.') XLTD;[jO  
    end b Dg9P^<n  
    4R+P  
    n = n(:); o@d y:AR  
    m = m(:); acOJ]]  
    length_n = length(n); ]@SU4  
    _2jw,WKr  
    if any(mod(n-m,2)) pIVq("&  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') {TL +7kiX/  
    end 4YJ=q% G  
    j;2<-{  
    if any(m<0) bV3lE6z  
        error('zernpol:Mpositive','All M must be positive.') +$(0w35V5  
    end 3$"/>g/  
    ';/84j-3F  
    if any(m>n) lIuXo3  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') { (\(m/!Z  
    end KtMbze  
    3C"_$?y"  
    if any( r>1 | r<0 ) fr#Qz{  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') k!doIMj  
    end tF`MT%{Va  
    KzkgWMM  
    if ~any(size(r)==1) >%c*Xe  
        error('zernpol:Rvector','R must be a vector.') \n@V-b  
    end +{6`F1MO  
    b~W)S/wF$P  
    r = r(:); / Dw@d,&[  
    length_r = length(r); ogeRYq,g  
    (/fT]6(  
    if nargin==4 +t>XxYScx  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 0VIZ=-e  
        if ~isnorm 79z)C35~  
            error('zernpol:normalization','Unrecognized normalization flag.') >Zdi5') 5  
        end d_iY&-gq/  
    else pAg$oe#  
        isnorm = false; l.7d$8'\  
    end pb$fb  
    n{=7 yK  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ih!~G5Xi9i  
    % Compute the Zernike Polynomials )nnCCR S6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% E!@/NE\-  
    MW]8;`|jC  
    % Determine the required powers of r: C*O ,rm}  
    % ----------------------------------- Y*\6o7  
    rpowers = []; 6z1\a  
    for j = 1:length(n) C|$L6n>DR6  
        rpowers = [rpowers m(j):2:n(j)]; \[T{M!s  
    end f N0bIE Y  
    rpowers = unique(rpowers); t{=i=K 3  
     ,F}r@  
    % Pre-compute the values of r raised to the required powers, =z-5  
    % and compile them in a matrix: SKJW%(|3  
    % ----------------------------- Tc,$TCF  
    if rpowers(1)==0 %|jzEBz@  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); (+x]##Q  
        rpowern = cat(2,rpowern{:}); &>V/X{>$`K  
        rpowern = [ones(length_r,1) rpowern]; jI Z+d;1  
    else ~T&% VvI  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); H`?* bG  
        rpowern = cat(2,rpowern{:}); lO_c/o$  
    end {Ve D@  
    [Gf{f\O  
    % Compute the values of the polynomials: ,$BgR2^  
    % -------------------------------------- #~1wv^  
    z = zeros(length_r,length_n); w~{| S7/  
    for j = 1:length_n s@ z{dmL  
        s = 0:(n(j)-m(j))/2; YJc%h@_=]  
        pows = n(j):-2:m(j); v\'r Xy  
        for k = length(s):-1:1 N GSS:  
            p = (1-2*mod(s(k),2))* ... W CoF{ *  
                       prod(2:(n(j)-s(k)))/          ... W[GQ[h  
                       prod(2:s(k))/                 ... u&tFb]1@)  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ~BtKd*~*  
                       prod(2:((n(j)+m(j))/2-s(k))); Hy;901( %  
            idx = (pows(k)==rpowers); g#Mv&tU  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 5=m3J !?  
        end DH/L`$  
         }z?xGW/k  
        if isnorm `>\4"`I  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); %awVVt{aG  
        end 363cuRP  
    end 6I5o2i  
    /_HwifRQ  
    % 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)  x?%rx}h  
    AeNyZ[40T  
    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。 )$ ofl%+  
    Xy[4f=X}z  
    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)