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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 Q1hHK'3w  
    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! [9N>*dKB  
     
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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 NP#6'eH\  
    function z = zernfun(n,m,r,theta,nflag) #OMFv.  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. /BN_K8nb`  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Ez)hArxns  
    %   and angular frequency M, evaluated at positions (R,THETA) on the XK+" x!   
    %   unit circle.  N is a vector of positive integers (including 0), and _A/q bm  
    %   M is a vector with the same number of elements as N.  Each element VY1&YR}Y  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) on^m2pQ *p  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, *r90IS}A$2  
    %   and THETA is a vector of angles.  R and THETA must have the same <6rc 8jYz  
    %   length.  The output Z is a matrix with one column for every (N,M) :MPfCiAv  
    %   pair, and one row for every (R,THETA) pair. .91@T.  
    % rGDx9KR4K!  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike L ^E#"f  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), rWMG6+Scb  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 5Q$.q &,  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 1fOH$33  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized zBjtPtiiI8  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. iVSN>APe  
    % je#OV,uHM  
    %   The Zernike functions are an orthogonal basis on the unit circle. Kg;u.4.-M  
    %   They are used in disciplines such as astronomy, optics, and WeiDg,]e$b  
    %   optometry to describe functions on a circular domain. &02I-lD4+  
    % b0| ;v-v  
    %   The following table lists the first 15 Zernike functions. ^0tO2$  
    % 6"djX47j  
    %       n    m    Zernike function           Normalization Abc%VRsT  
    %       -------------------------------------------------- @,^c?v  
    %       0    0    1                                 1 (~IoRhp^  
    %       1    1    r * cos(theta)                    2 3 BQZ[%0@  
    %       1   -1    r * sin(theta)                    2 V] 0T P#  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) oniVC',  
    %       2    0    (2*r^2 - 1)                    sqrt(3) VFI\2n`  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) k}&7!G@T  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) )45#lE3TH  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) $a#-d;  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) X/BcS[a  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) t9eEcq Mg  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) /$'|`jKsB  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 259R5X<V  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 2 r';)8:  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) oAprM Z 7Y  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Y a/+|mv  
    %       -------------------------------------------------- 0&$,?CL?  
    % vrq5 +K&||  
    %   Example 1: IQ\5!e  
    % i9+qU  
    %       % Display the Zernike function Z(n=5,m=1) csjCXT=Ve  
    %       x = -1:0.01:1; 3j7Na#<tL3  
    %       [X,Y] = meshgrid(x,x); Z{}+7P  
    %       [theta,r] = cart2pol(X,Y); 5q,ZH6\ {  
    %       idx = r<=1; $)#?4v<  
    %       z = nan(size(X)); %'w?fqk  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); H8!)zZ  
    %       figure ?1.W F}X'  
    %       pcolor(x,x,z), shading interp nKnQ%R  
    %       axis square, colorbar 5V*R  Dh  
    %       title('Zernike function Z_5^1(r,\theta)') q|ZzGEj:OV  
    % ( yK@(euG  
    %   Example 2: U ATF}x   
    % ~J![Nx/  
    %       % Display the first 10 Zernike functions 83!{?EPE  
    %       x = -1:0.01:1; ('z:XW96  
    %       [X,Y] = meshgrid(x,x); f=hT o!i  
    %       [theta,r] = cart2pol(X,Y); 7e:eL5f>~  
    %       idx = r<=1;  rrP_7D  
    %       z = nan(size(X)); F*-+5nJ&@  
    %       n = [0  1  1  2  2  2  3  3  3  3]; {YK7';_E*  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 25Uw\rKeO  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; j8)rz  
    %       y = zernfun(n,m,r(idx),theta(idx)); G{74o8  
    %       figure('Units','normalized') {,B. OM)J  
    %       for k = 1:10 B:96E&  
    %           z(idx) = y(:,k); kB9@ &t +  
    %           subplot(4,7,Nplot(k)) `-w,6  
    %           pcolor(x,x,z), shading interp t{-*@8Ke  
    %           set(gca,'XTick',[],'YTick',[]) 0uu)0:  
    %           axis square 1*f*}M  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @TJ2 |_s6]  
    %       end \SN>Yy  
    % Q9Vj8JO"{  
    %   See also ZERNPOL, ZERNFUN2. s`en8%  
    H=*lj.x  
    %   Paul Fricker 11/13/2006 $It3}?>C'  
    HX{K5+  
    l5nm.i<M  
    % Check and prepare the inputs: ?c<uN~fC=  
    % ----------------------------- xW|8-q  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) .*B@1q  
        error('zernfun:NMvectors','N and M must be vectors.') w>e+UW25Y  
    end LP'~7FG  
    O7oq1JI]Y  
    if length(n)~=length(m) mwutv8?  
        error('zernfun:NMlength','N and M must be the same length.') vNHvuw K  
    end hmB`+?,z*  
    NJCSo(O  
    n = n(:); v7/k0D .  
    m = m(:); uO>pl37@  
    if any(mod(n-m,2)) 7+;.Q  
        error('zernfun:NMmultiplesof2', ... qpjiQ,\:b  
              'All N and M must differ by multiples of 2 (including 0).') Y;"jsK{$  
    end 2UG>(R:  
    d;nk>6<|  
    if any(m>n) 3^iVDbAW{  
        error('zernfun:MlessthanN', ... ?pWda<&  
              'Each M must be less than or equal to its corresponding N.') 6_&S ?yA  
    end p fR~?jYzm  
    `! xI!Y\  
    if any( r>1 | r<0 ) yeam-8  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') L}7 TM:%  
    end L2c\i  
    ^{YK'60  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ;9<?~S  
        error('zernfun:RTHvector','R and THETA must be vectors.') {55f{5y3 c  
    end m%nRHT0KAf  
    6~l+wu<$  
    r = r(:); TR%8O;  
    theta = theta(:); gnYo/q=K  
    length_r = length(r); @; tM R|p  
    if length_r~=length(theta) N8DouDq  
        error('zernfun:RTHlength', ... +6x}yc:yd  
              'The number of R- and THETA-values must be equal.') G#~U\QlG-  
    end $b[Ha{9(v  
    ] &SmeTe  
    % Check normalization: A-, hm=?  
    % -------------------- hj\A-Yf  
    if nargin==5 && ischar(nflag) 4aKppj  
        isnorm = strcmpi(nflag,'norm'); X3] [C  
        if ~isnorm +-T|ov<  
            error('zernfun:normalization','Unrecognized normalization flag.') 4];>O  
        end p(cnSvg  
    else I%b5a`7  
        isnorm = false; 2.^CIJc  
    end 96S$Y~G# &  
    WM%w_,Z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Dk&(QajL  
    % Compute the Zernike Polynomials l;$FR4}d  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #guK&?Fye  
     F&lH5  
    % Determine the required powers of r: Uw| -d[!  
    % ----------------------------------- h(^c5#.  
    m_abs = abs(m); ArScJ\/Nwv  
    rpowers = []; ^Nu j/  
    for j = 1:length(n) T`,G57-5  
        rpowers = [rpowers m_abs(j):2:n(j)]; RR|X4h0.  
    end Z|fi$2k0!  
    rpowers = unique(rpowers); Hy -)yR  
    "Pu917_P  
    % Pre-compute the values of r raised to the required powers, 4`Zo Ar-5|  
    % and compile them in a matrix: n]7rHV}G  
    % ----------------------------- 76] Z~^Y  
    if rpowers(1)==0 !/ dH"h  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); l5]R*mR  
        rpowern = cat(2,rpowern{:}); hL&7D @  
        rpowern = [ones(length_r,1) rpowern]; S(^YTb7  
    else /q8B | (U  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); stMxlG"d  
        rpowern = cat(2,rpowern{:}); R+!oPWfb  
    end 'n^?DPvD  
    {NcJL< ;tS  
    % Compute the values of the polynomials: Aar]eY\  
    % -------------------------------------- TU;AO%5  
    y = zeros(length_r,length(n)); #DARZhU)  
    for j = 1:length(n) !T2{xmHKv$  
        s = 0:(n(j)-m_abs(j))/2; }x& X vI  
        pows = n(j):-2:m_abs(j); lHPnAaue@  
        for k = length(s):-1:1 rP,|  
            p = (1-2*mod(s(k),2))* ...  @' %XdH  
                       prod(2:(n(j)-s(k)))/              ... K4H27SH  
                       prod(2:s(k))/                     ... BG)zkn$  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 2Nx:Y+[  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); a8v\H8@X  
            idx = (pows(k)==rpowers); X-Ev>3H  
            y(:,j) = y(:,j) + p*rpowern(:,idx); +t&+f7  
        end ,izp^,`  
         ^uphpABpD  
        if isnorm >o%X;U 3  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1r*yYm'  
        end (kyRx+gA  
    end K>5 bb  
    % END: Compute the Zernike Polynomials Yakrsi/jV}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S6\E  I5S  
    X\w["! B  
    % Compute the Zernike functions: u~ VXe  
    % ------------------------------ *3OlWnZ?  
    idx_pos = m>0; vl6|i)D  
    idx_neg = m<0; eu8a<  
    j^v<rCzc (  
    z = y; $FDGHFM  
    if any(idx_pos) `:R9M+ OX  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); pAq PHD=  
    end Nf2lw]-G4  
    if any(idx_neg) 2yD ?f8P4  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Z-pZyDz  
    end N})vrB;1  
    N)a5~<fBG  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) aqImW  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. m*h O@M  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated LaZ @4/z!  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive |Q@(<'8=  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Im};wJ&  
    %   and THETA is a vector of angles.  R and THETA must have the same G(o6/  
    %   length.  The output Z is a matrix with one column for every P-value, BT^=p  
    %   and one row for every (R,THETA) pair. n=0^8QQ  
    % beT[7uVj_  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike D8xE"6T>  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) Q,tjODc6n  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) u[4h|*'"|  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 NXz/1ut%  
    %   for all p. uINEq{yo  
    % q$s)(D  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 m]'+Eye ]r  
    %   Zernike functions (order N<=7).  In some disciplines it is Lm.N {NV'  
    %   traditional to label the first 36 functions using a single mode ti]8_vP}*  
    %   number P instead of separate numbers for the order N and azimuthal 2#CN:b]+  
    %   frequency M. 7TU77  
    % q1 BpE8  
    %   Example: m(5LXH Jnv  
    % Q&@<?K9  
    %       % Display the first 16 Zernike functions P]2 /}\f  
    %       x = -1:0.01:1; muBl~6_mb2  
    %       [X,Y] = meshgrid(x,x); 1Mx2%  
    %       [theta,r] = cart2pol(X,Y); a^X% (@Sg  
    %       idx = r<=1; "]=XB0)  
    %       p = 0:15; *+2BZ ZwT  
    %       z = nan(size(X)); '!4\H"t  
    %       y = zernfun2(p,r(idx),theta(idx)); !+YSc&R_fW  
    %       figure('Units','normalized') xk,1 D  
    %       for k = 1:length(p) CSwB+yN  
    %           z(idx) = y(:,k); ' ~z`kah  
    %           subplot(4,4,k) 5nmE*(  
    %           pcolor(x,x,z), shading interp ,?%o ~  
    %           set(gca,'XTick',[],'YTick',[]) :; La V  
    %           axis square .#K\u![@N  
    %           title(['Z_{' num2str(p(k)) '}']) 4 'vjU6gW  
    %       end &t'P>6)  
    % ;7JyL|2  
    %   See also ZERNPOL, ZERNFUN. bIk4?S  
    7E?60^Tve  
    %   Paul Fricker 11/13/2006 V4W(> g  
    S3QX{5t\  
    QYAt)Ik9q  
    % Check and prepare the inputs: - s{&_]A~  
    % ----------------------------- u)/i$N  
    if min(size(p))~=1 Q(Pc  
        error('zernfun2:Pvector','Input P must be vector.') >{rD3X"d  
    end Tv% Z|%*  
    JiXN"s^mcb  
    if any(p)>35 [Z1,~(3  
        error('zernfun2:P36', ... 9/R=_y-  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... M{Vi4ehOq  
               '(P = 0 to 35).']) =}e{U&CX  
    end 6}\J-A/  
    1vq c8lC  
    % Get the order and frequency corresonding to the function number: =;?Maexp3$  
    % ---------------------------------------------------------------- 6HpiG`  
    p = p(:); cz$*6P<9J  
    n = ceil((-3+sqrt(9+8*p))/2); q _:7uQ  
    m = 2*p - n.*(n+2); _gCi@uXS3  
    e4.G9(  
    % Pass the inputs to the function ZERNFUN: >bO}sx1?  
    % ---------------------------------------- M=EV^Tw-=  
    switch nargin )Oj{x0{\Q  
        case 3 'm/`= QX  
            z = zernfun(n,m,r,theta); #g1,U7vv8  
        case 4 RTL@WI  
            z = zernfun(n,m,r,theta,nflag); HLq2a vs\  
        otherwise E1qf N>0Z  
            error('zernfun2:nargin','Incorrect number of inputs.') S;nlC  
    end NnY+=#j7L  
    \YsLVOv%:d  
    % 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) h$$i@IO0  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. FyllVrK  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 1?".R]<{2T  
    %   order N and frequency M, evaluated at R.  N is a vector of 4ZT0~37(  
    %   positive integers (including 0), and M is a vector with the oUN;u*  
    %   same number of elements as N.  Each element k of M must be a G"*ch$:  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) @$o^(my  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is g+KuK`\N%  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix .:SY:v r  
    %   with one column for every (N,M) pair, and one row for every p8E6_%Rw  
    %   element in R. tE:6  
    % "J%dI9tM{  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- aByd,uSe)_  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is }h9f(ZyJn  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to nSbcq>3  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 U)qG]RI  
    %   for all [n,m]. ;:w0%>X^  
    % KhNO xMZ  
    %   The radial Zernike polynomials are the radial portion of the e>b|13X  
    %   Zernike functions, which are an orthogonal basis on the unit >s>{+6e  
    %   circle.  The series representation of the radial Zernike 2U'Vq  
    %   polynomials is 9Cq"Szs  
    % j55OG~)  
    %          (n-m)/2 HP[M"u  
    %            __ V7,;N@FL  
    %    m      \       s                                          n-2s p-2PC{% t|  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r !<4=@  
    %    n      s=0 H>|*D~RdT  
    % l1" *  
    %   The following table shows the first 12 polynomials. b|u0a6  
    % @-aMj  
    %       n    m    Zernike polynomial    Normalization e!1am%aE  
    %       --------------------------------------------- f^@D uI  
    %       0    0    1                        sqrt(2) !mu1e=bY>  
    %       1    1    r                           2 w72\'  
    %       2    0    2*r^2 - 1                sqrt(6) ^:^8M4:  
    %       2    2    r^2                      sqrt(6) crr#tad.  
    %       3    1    3*r^3 - 2*r              sqrt(8) 8'0I$Qa4  
    %       3    3    r^3                      sqrt(8) q#P@,|nc:  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) })5I/   
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Jm]P,jaLc  
    %       4    4    r^4                      sqrt(10) Og9:MFI  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) oNIt<T  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) o@DlK`  
    %       5    5    r^5                      sqrt(12) y"Jma`Vjq  
    %       --------------------------------------------- g I@I.=y  
    % p JM&R<i:  
    %   Example: A%% Vyz  
    % &Q[|FO;[  
    %       % Display three example Zernike radial polynomials :Wd@Qy?;  
    %       r = 0:0.01:1; ^,6c9Dxy  
    %       n = [3 2 5]; B1(T-pr  
    %       m = [1 2 1]; P] qL&_  
    %       z = zernpol(n,m,r); \EQCR[7qu7  
    %       figure =4:]V\o):'  
    %       plot(r,z) ,O 1/|Y  
    %       grid on 2#xz,RM.  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') gEkH5|*Y  
    % Ae_:Kc6  
    %   See also ZERNFUN, ZERNFUN2. ]seOc],4  
    \jHIjFwQ  
    % A note on the algorithm. !A&>Eeai  
    % ------------------------ 9?4:},FRmE  
    % The radial Zernike polynomials are computed using the series m-MfFEZ  
    % representation shown in the Help section above. For many special rtZEK:.#  
    % functions, direct evaluation using the series representation can I|vfxf  
    % produce poor numerical results (floating point errors), because }BJR/r  
    % the summation often involves computing small differences between )^LiAL h  
    % large successive terms in the series. (In such cases, the functions ,]_<8@R  
    % are often evaluated using alternative methods such as recurrence Q;y)6+VU4  
    % relations: see the Legendre functions, for example). For the Zernike y.Y;<UGu  
    % polynomials, however, this problem does not arise, because the 0c$ ')`! m  
    % polynomials are evaluated over the finite domain r = (0,1), and cOvdC4  
    % because the coefficients for a given polynomial are generally all aP/Ff%5T  
    % of similar magnitude. U\x $@J  
    % 9 1ndr@*|  
    % ZERNPOL has been written using a vectorized implementation: multiple I7Xm~w!{qk  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] S$ Z?T  
    % values can be passed as inputs) for a vector of points R.  To achieve c, FZ{O@  
    % this vectorization most efficiently, the algorithm in ZERNPOL Ktn:6=,  
    % involves pre-determining all the powers p of R that are required to EdC/]  
    % compute the outputs, and then compiling the {R^p} into a single pRGag~h|E  
    % matrix.  This avoids any redundant computation of the R^p, and {Xv0=P  
    % minimizes the sizes of certain intermediate variables. 5LJ0V  
    % -X_dY>>s  
    %   Paul Fricker 11/13/2006 <7Ry"z6g;  
    ZXC_kmBN/  
    D&!c7_^  
    % Check and prepare the inputs: Vi'zSR28Z  
    % ----------------------------- S$NJmXhx5  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) OZ6:u^OS]  
        error('zernpol:NMvectors','N and M must be vectors.') ~+CEek  
    end H_>9'(  
    X|dlVNL8p  
    if length(n)~=length(m) h8hyQd$!  
        error('zernpol:NMlength','N and M must be the same length.') k\KI#.>  
    end hkl9 EVO)  
    }0AoV&75  
    n = n(:); 6d/1PGB  
    m = m(:); ?(Ytc)   
    length_n = length(n); ) m(!lDz3  
    }j;G`mV2  
    if any(mod(n-m,2)) j] J-#J  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') WYd9p;k  
    end 7[ZoUWx  
    \Sv8c}8  
    if any(m<0) -1}&\=8M  
        error('zernpol:Mpositive','All M must be positive.') )LTX.Kg  
    end K8RV=3MBLD  
    i$lp8Y2ih  
    if any(m>n) p9![8VU  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 3@wio[  
    end !nL>Ly  
    1'f&  
    if any( r>1 | r<0 ) ;L[N.ZY!  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') TGHyBPJb  
    end W<>R;~)  
    2B b,ZC*  
    if ~any(size(r)==1) A$70!5*  
        error('zernpol:Rvector','R must be a vector.') jbWgL$  
    end ~- eB  
    %\T#Ik~3  
    r = r(:); F+)g!NQZ  
    length_r = length(r); ?D;7ut$~  
    {$Z S 2 7  
    if nargin==4 DdqE6qE  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); rqz48~\lJ  
        if ~isnorm ^~^=$fz  
            error('zernpol:normalization','Unrecognized normalization flag.') Y2[ik<  
        end m>djoe  
    else wizLA0W  
        isnorm = false; X}g"_wN,g>  
    end B*:W`}G]_c  
    lUd4`r"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8Y [4JXUK  
    % Compute the Zernike Polynomials l~mj>$  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CbFO9q  
    i?861Hu  
    % Determine the required powers of r: m-]F]c=)w<  
    % ----------------------------------- u Au'2M,_  
    rpowers = []; -ufaV#  
    for j = 1:length(n) $}B&u)  
        rpowers = [rpowers m(j):2:n(j)]; <[vsGUbc  
    end M[P1hFuna  
    rpowers = unique(rpowers); 2=,d.1E3d  
    E Q]>^VE2B  
    % Pre-compute the values of r raised to the required powers, wRg[Mu,Q5  
    % and compile them in a matrix: Z-3("%_$/  
    % ----------------------------- kQ"Ax? b  
    if rpowers(1)==0 ki|OowP  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); rJ(AO'=  
        rpowern = cat(2,rpowern{:}); z9w]{Zd_,d  
        rpowern = [ones(length_r,1) rpowern]; cZ3A~dTOR  
    else Tnas$=J  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); mQ3gp&d3W  
        rpowern = cat(2,rpowern{:}); |Xmzq X%  
    end b^A7R{G7  
    Q)\[wYMt  
    % Compute the values of the polynomials: -$I$zo  
    % -------------------------------------- !LCy:>i!d  
    z = zeros(length_r,length_n); @ 6*eS+t\  
    for j = 1:length_n =.l>Uw!  
        s = 0:(n(j)-m(j))/2; bnN&E?{hF1  
        pows = n(j):-2:m(j); B<ZCuVWH:  
        for k = length(s):-1:1 lo-VfKvy  
            p = (1-2*mod(s(k),2))* ... P MI?PC[;  
                       prod(2:(n(j)-s(k)))/          ... i!eY"|o  
                       prod(2:s(k))/                 ... )5fly%-r)  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... "n{JH9sA:  
                       prod(2:((n(j)+m(j))/2-s(k))); 4#W*f3d[@:  
            idx = (pows(k)==rpowers); %Vfr#j$=  
            z(:,j) = z(:,j) + p*rpowern(:,idx); [LrO"9q(  
        end Gn4XVzB`O  
         `Om W#\  
        if isnorm (yoF  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); GK1P7Qy?V  
        end k"E|E";B  
    end Wu/:ES)C  
    !wC( ]Y  
    % 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)   =WEDQ\ c  
    0%,?z`UY  
    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。 IGab~`c-[  
    l)'*jZ  
    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)