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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 mmsPLv6  
    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! l0] EX>"E  
     
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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 W`&hp6Jq  
    function z = zernfun(n,m,r,theta,nflag) m3ff;,  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~v83pu1!2s  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N d1kJRJ   
    %   and angular frequency M, evaluated at positions (R,THETA) on the fX)# =c|5  
    %   unit circle.  N is a vector of positive integers (including 0), and smLQS+UE  
    %   M is a vector with the same number of elements as N.  Each element /@Zrq#o zx  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) &[SC|=U'M  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, QM]YJr3r E  
    %   and THETA is a vector of angles.  R and THETA must have the same `lPfb[b  
    %   length.  The output Z is a matrix with one column for every (N,M) 'RRE|L,  
    %   pair, and one row for every (R,THETA) pair. JLi|Td "1%  
    % 'QIqBU'~  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike o]:9')5^  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), G9 :l'\  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 7)k\{&+P  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, )ANmIwmC#  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized BUR*n;V`  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ]q-Y }1di8  
    % `@ FYkH  
    %   The Zernike functions are an orthogonal basis on the unit circle. "vslZ`RU  
    %   They are used in disciplines such as astronomy, optics, and : c[L3rJl  
    %   optometry to describe functions on a circular domain. U?=Dg1  
    % rD>f|kA?L  
    %   The following table lists the first 15 Zernike functions. hzRYec(  
    % 7= DdrG<  
    %       n    m    Zernike function           Normalization IMfqiH)  
    %       -------------------------------------------------- m_l[MG\  
    %       0    0    1                                 1 5D l/aHb  
    %       1    1    r * cos(theta)                    2 ;'Nd~:-]  
    %       1   -1    r * sin(theta)                    2 H4JTGt1"  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) pD74+/DD  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 7!$^r$t   
    %       2    2    r^2 * sin(2*theta)             sqrt(6) @]#1(9P  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) t_suF$  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) e!r-+.i(  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) @<Yy{ ~L|  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) I9Fr5p-%O  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 2>H24F  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) : \}(& >  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 9$m|'$p3sG  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~WN:DXn  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 3Le{\}-$.  
    %       -------------------------------------------------- orvp*F{7[H  
    % FkRo _?  
    %   Example 1: f4Rf?w*  
    % nJLFfXWx  
    %       % Display the Zernike function Z(n=5,m=1) fg{n(TE"8  
    %       x = -1:0.01:1; 4NIRmDEd  
    %       [X,Y] = meshgrid(x,x); (@}!0[[^  
    %       [theta,r] = cart2pol(X,Y); Ip]KPrw p  
    %       idx = r<=1; &yol_%C  
    %       z = nan(size(X)); v6Vcjm  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); H$KTo/  
    %       figure S/I/-Bp~  
    %       pcolor(x,x,z), shading interp LYg- .~<I  
    %       axis square, colorbar 3<zp  
    %       title('Zernike function Z_5^1(r,\theta)') ~| 6[j<ziL  
    % C{XmVc.  
    %   Example 2: L z1ME(  
    % EUgs6[w 4  
    %       % Display the first 10 Zernike functions 6B ?twh)  
    %       x = -1:0.01:1; 9RI-Lq`  
    %       [X,Y] = meshgrid(x,x); o7LuKRl   
    %       [theta,r] = cart2pol(X,Y); d&s9t;@=  
    %       idx = r<=1; u=_mvN  
    %       z = nan(size(X)); :$9tF >  
    %       n = [0  1  1  2  2  2  3  3  3  3]; P_#bow  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; qWKAM@  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; y<bDTeoo  
    %       y = zernfun(n,m,r(idx),theta(idx)); SG4%}wn%  
    %       figure('Units','normalized') M[112%[+4  
    %       for k = 1:10 dmN&+t  
    %           z(idx) = y(:,k); ~<OSYb  
    %           subplot(4,7,Nplot(k)) Ezv Y"T@  
    %           pcolor(x,x,z), shading interp Q&| \r  
    %           set(gca,'XTick',[],'YTick',[]) :TC@tM~Oy  
    %           axis square x7x\Y(@  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) LAe6`foW/  
    %       end H? y,ie#u  
    % az|N-?u  
    %   See also ZERNPOL, ZERNFUN2. !GEJIefx_  
    -{vKus  
    %   Paul Fricker 11/13/2006  y%b F&  
    \A6B,|@  
    f! .<$ih  
    % Check and prepare the inputs: r!a3\ep  
    % ----------------------------- B i<Q=x'Z;  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) r s?R:+  
        error('zernfun:NMvectors','N and M must be vectors.') y[_Q-   
    end '1)$'   
     \qK&q  
    if length(n)~=length(m) yw3$2EW  
        error('zernfun:NMlength','N and M must be the same length.') -n<pPau2  
    end g]yBA7/S"  
    A;|D:;x3G  
    n = n(:); qXtC^n@x  
    m = m(:); x6ARzH\  
    if any(mod(n-m,2)) cX OK)g#  
        error('zernfun:NMmultiplesof2', ... "E?2xf|.  
              'All N and M must differ by multiples of 2 (including 0).') tlp@?(u  
    end @w!PaP  
    "?I y(*^  
    if any(m>n) ce3YCflt  
        error('zernfun:MlessthanN', ... t; {F%9j{  
              'Each M must be less than or equal to its corresponding N.') Ev(>z-{F  
    end "s_lP&nq  
    zb<6 Ov  
    if any( r>1 | r<0 ) 2eol gXp  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') BC<^a )D=  
    end .oUTqki  
    z}ddqZ27G$  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) J 9iy  
        error('zernfun:RTHvector','R and THETA must be vectors.') K_ ~"}  
    end k<{{*  
    O[)kboY  
    r = r(:);  J@Q7p}  
    theta = theta(:); 1sdLDw_)p  
    length_r = length(r); 28J^DMOW  
    if length_r~=length(theta) Y@ksQ_u  
        error('zernfun:RTHlength', ... 0C6-GKbZ  
              'The number of R- and THETA-values must be equal.') > eIP.,9  
    end 6WJ)by  
    Z>Wg*sZy)  
    % Check normalization: * 8_wYYH  
    % -------------------- Uu(SR/R}  
    if nargin==5 && ischar(nflag) 9g"2^^wD  
        isnorm = strcmpi(nflag,'norm');  Qq;Foa  
        if ~isnorm M={V|H0  
            error('zernfun:normalization','Unrecognized normalization flag.') ],a5)kV  
        end 1@1U/ss1  
    else Rt!FPoN,y  
        isnorm = false; (/j/>9iro  
    end 4k_vdz  
    C$D -Pt"+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !F1N~6f  
    % Compute the Zernike Polynomials ,+xB$e  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #[~pD:qqM  
    9"A`sGZ  
    % Determine the required powers of r: CtAwBQO  
    % ----------------------------------- h+&OQ%e=8  
    m_abs = abs(m); j=aI9p  
    rpowers = []; 'JfdV%M  
    for j = 1:length(n) 8UyMVY  
        rpowers = [rpowers m_abs(j):2:n(j)]; IrhA+)pdse  
    end "4+ WZR]  
    rpowers = unique(rpowers); ( _)jkI \  
    $5< #n@  
    % Pre-compute the values of r raised to the required powers, ]d0tE?9  
    % and compile them in a matrix: kDN:ep{/  
    % ----------------------------- cm[&?  
    if rpowers(1)==0 _EMwm&!  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); W!&'pg  
        rpowern = cat(2,rpowern{:}); e`TH91@  
        rpowern = [ones(length_r,1) rpowern]; C:C}5<fk x  
    else )V6Hl@v  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); U4-g^S[  
        rpowern = cat(2,rpowern{:}); *HO}~A%Lx  
    end ps%q9}J  
    X+S9{X#Cm  
    % Compute the values of the polynomials: `-l6S  
    % -------------------------------------- DV-;4AxxRq  
    y = zeros(length_r,length(n)); lfz2~Si5A  
    for j = 1:length(n) -[!P!d=  
        s = 0:(n(j)-m_abs(j))/2; O 8u j`G 9  
        pows = n(j):-2:m_abs(j); I@%t.%O Jp  
        for k = length(s):-1:1 L>%o[tS  
            p = (1-2*mod(s(k),2))* ... r{ef.^&:  
                       prod(2:(n(j)-s(k)))/              ... %_L\z*+  
                       prod(2:s(k))/                     ... % !>I*H  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... "a"]o  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); pDcjwlA%  
            idx = (pows(k)==rpowers); 9Hu/u=vB<  
            y(:,j) = y(:,j) + p*rpowern(:,idx); * %M3PTY\  
        end i2(1ki/|O  
         ;YX4:OBqr  
        if isnorm ); dT_  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); %S nd\  
        end mkF"   
    end \":m!K;Z  
    % END: Compute the Zernike Polynomials f[~L?B;_L  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,7NZu0  
    V8-oYwOR  
    % Compute the Zernike functions: U1RpLkibQ  
    % ------------------------------  !@'6)/  
    idx_pos = m>0; T{Uc:Z  
    idx_neg = m<0; &PK\|\\2  
    7`8Ik`lY  
    z = y; dJ""XaHqf  
    if any(idx_pos) rT5Ycm@  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); %V{7DA&C  
    end Qj6/[mUr~  
    if any(idx_neg) $8[r9L!  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); e9[|!/./5  
    end y2vUthRwo  
    4NG?_D5&  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) e-*.Ca  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. i?:_:"^x  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated YH_7=0EJ  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 9n5<]Q (  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, `zt_7MD  
    %   and THETA is a vector of angles.  R and THETA must have the same z,:a8LB#[  
    %   length.  The output Z is a matrix with one column for every P-value, `o?Ph&p}  
    %   and one row for every (R,THETA) pair. T%n2$  
    % BQ2wnGc  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike { e5/+W  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) j"@93D~  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 4bEf  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 dVo.Czyd  
    %   for all p. R$Tp8G>j  
    % ;/*6U  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ^F"iP7   
    %   Zernike functions (order N<=7).  In some disciplines it is (%:>T Q(  
    %   traditional to label the first 36 functions using a single mode T,OwM\`.X{  
    %   number P instead of separate numbers for the order N and azimuthal 4r0b)Y &I  
    %   frequency M. :4T("a5aM  
    % $<|l E/_]  
    %   Example: j]m|7]  
    % 6q6FB  
    %       % Display the first 16 Zernike functions 3 Lsj}p  
    %       x = -1:0.01:1; \yGsr Bl  
    %       [X,Y] = meshgrid(x,x); okFvn;  
    %       [theta,r] = cart2pol(X,Y); NAzX". g  
    %       idx = r<=1; |s)?cpb  
    %       p = 0:15; a9?y`{%L  
    %       z = nan(size(X)); hw~a:kD  
    %       y = zernfun2(p,r(idx),theta(idx)); lM[XS4/TRa  
    %       figure('Units','normalized') HH>:g(bu  
    %       for k = 1:length(p) *cg( ?yg  
    %           z(idx) = y(:,k); ?[MsQQd~  
    %           subplot(4,4,k) iIGbHn,/  
    %           pcolor(x,x,z), shading interp v^7LctcVm  
    %           set(gca,'XTick',[],'YTick',[]) e~T@~(fft  
    %           axis square q0bHB_|wL  
    %           title(['Z_{' num2str(p(k)) '}']) Y05P'Q  
    %       end o(Cey7  
    % N8`4veVBx'  
    %   See also ZERNPOL, ZERNFUN. 5I@w~z  
    A[YpcG'9  
    %   Paul Fricker 11/13/2006 ACK1@eF  
    [|3>MZ2/  
    45H!;Q sk  
    % Check and prepare the inputs: irZFV  
    % ----------------------------- Px>va01n  
    if min(size(p))~=1 pBC<u  
        error('zernfun2:Pvector','Input P must be vector.') h`}3h< 8  
    end LN_OD5gZ  
    2w$t wW-  
    if any(p)>35 U`x bPQ  
        error('zernfun2:P36', ... {3Vk p5%l  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... *AIEl"29  
               '(P = 0 to 35).']) =)OC|?9 C\  
    end l#wdpD a{  
    D8# on!  
    % Get the order and frequency corresonding to the function number: 1SV^){5I  
    % ---------------------------------------------------------------- ag4`n:1  
    p = p(:); l~Lb!;,dN  
    n = ceil((-3+sqrt(9+8*p))/2); ib0g3p-Lc  
    m = 2*p - n.*(n+2); T/P7F\R  
    Ab1/.~^  
    % Pass the inputs to the function ZERNFUN: @l UlY2  
    % ---------------------------------------- Q^Bt1C  
    switch nargin i NWC6y  
        case 3 v1.q$ f^(  
            z = zernfun(n,m,r,theta); www`=)A;  
        case 4 |k{-l!HI  
            z = zernfun(n,m,r,theta,nflag); (HN4g;{  
        otherwise s2v(=  
            error('zernfun2:nargin','Incorrect number of inputs.') *V;3~x!  
    end Q:|w%L*E  
    p&K\]l}  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) -0,4eg j3  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. mPVE?jnR^0  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of D(r:}pyU  
    %   order N and frequency M, evaluated at R.  N is a vector of "6I[4U"@  
    %   positive integers (including 0), and M is a vector with the s=EiH  
    %   same number of elements as N.  Each element k of M must be a hE!7RM+Y  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) GF--riyfB  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is iG[? ]]  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix <uF [,  
    %   with one column for every (N,M) pair, and one row for every >v0:qN7|  
    %   element in R. (buw^ ,NwZ  
    % ;WI]vn  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- mPmB6q%)]  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is )45_]tk >  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Qm);6X   
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 IG / $!* E  
    %   for all [n,m]. vg5NY =O  
    % XS0V:<+,  
    %   The radial Zernike polynomials are the radial portion of the 9)yG.9d1  
    %   Zernike functions, which are an orthogonal basis on the unit i@$-0%,  
    %   circle.  The series representation of the radial Zernike qiNliJ>40E  
    %   polynomials is c d%hW  
    % Gm&2R4)EP  
    %          (n-m)/2 xtJAMo>g  
    %            __ 0MpS4tW0=  
    %    m      \       s                                          n-2s w6EI{  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r X7e/:._SAH  
    %    n      s=0 hmGdjw t$  
    % v'nHFC+p  
    %   The following table shows the first 12 polynomials. )bYez  
    % org*z!;.   
    %       n    m    Zernike polynomial    Normalization PqhlXqX9  
    %       --------------------------------------------- aii'}c  
    %       0    0    1                        sqrt(2) g[!Cj,  
    %       1    1    r                           2 Jf+7"![|  
    %       2    0    2*r^2 - 1                sqrt(6) DM2Q1Dh3  
    %       2    2    r^2                      sqrt(6) %\yK5V5  
    %       3    1    3*r^3 - 2*r              sqrt(8) "3t\em!  
    %       3    3    r^3                      sqrt(8) Gj`f--2GE  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) h3h8lt_ |  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 3zb)"\(R  
    %       4    4    r^4                      sqrt(10) /;+,mp4  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) d8.ajeN]o  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) P|N?OocE  
    %       5    5    r^5                      sqrt(12) ZW* fOaj  
    %       --------------------------------------------- WLy7'3@  
    % p#M!S2&z  
    %   Example: fNEz  
    % fm6]CU1^  
    %       % Display three example Zernike radial polynomials :bw6k  
    %       r = 0:0.01:1; M,L@k  
    %       n = [3 2 5]; HWR& C  
    %       m = [1 2 1]; O<a3DyUa;  
    %       z = zernpol(n,m,r); kGj]i@(PA4  
    %       figure 2B'^`>+8S  
    %       plot(r,z) Vw?P.4  
    %       grid on 2;R/.xI6v  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') P6E1^$e  
    % J=L`]XE  
    %   See also ZERNFUN, ZERNFUN2. \I xzdFF#  
    h *waRD  
    % A note on the algorithm. 1Q_ ``.M  
    % ------------------------ 165WO}(;/  
    % The radial Zernike polynomials are computed using the series T Xl\hL\+  
    % representation shown in the Help section above. For many special dAwS<5!  
    % functions, direct evaluation using the series representation can 9!S^^;PN&  
    % produce poor numerical results (floating point errors), because ;.r2$/E  
    % the summation often involves computing small differences between 1G_xP^H!  
    % large successive terms in the series. (In such cases, the functions oP,RlR  
    % are often evaluated using alternative methods such as recurrence 9H8=eJd  
    % relations: see the Legendre functions, for example). For the Zernike *e,CDV  
    % polynomials, however, this problem does not arise, because the i/M+t~   
    % polynomials are evaluated over the finite domain r = (0,1), and ,{TQ ~LP  
    % because the coefficients for a given polynomial are generally all 9 G((wiE  
    % of similar magnitude. p1uN ]T7>  
    % 4Q/r[x/&C  
    % ZERNPOL has been written using a vectorized implementation: multiple 5#BF,-Jv  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 7ozYq_ $  
    % values can be passed as inputs) for a vector of points R.  To achieve 2:n|x5\H  
    % this vectorization most efficiently, the algorithm in ZERNPOL ;HT0w_,  
    % involves pre-determining all the powers p of R that are required to w=gQ3j#s  
    % compute the outputs, and then compiling the {R^p} into a single ],$6&Cm  
    % matrix.  This avoids any redundant computation of the R^p, and x:vrK#8D>  
    % minimizes the sizes of certain intermediate variables. (S3jZ  
    % mf#fA2[  
    %   Paul Fricker 11/13/2006 #VQ36pCd  
    4KZSL: A  
    w8U2y/:>  
    % Check and prepare the inputs: tc5M$b3^2  
    % ----------------------------- 7ia "u+Y  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) #[C< J#;  
        error('zernpol:NMvectors','N and M must be vectors.') e=-YP8l  
    end t0+t9w/fTP  
    4'_L W?DS  
    if length(n)~=length(m) %pd5w~VP  
        error('zernpol:NMlength','N and M must be the same length.') jf2y0W>6s  
    end D@2Ya/c  
    YlG; A\]k  
    n = n(:); "C?:T'dW  
    m = m(:); MyK^i2eD  
    length_n = length(n); z{@= _5;  
    IBzHR[#,^  
    if any(mod(n-m,2)) R<_mK33hd  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') uFMs ^^#  
    end f*UBigk  
    D1"1MUSod  
    if any(m<0) -%saeX Wo  
        error('zernpol:Mpositive','All M must be positive.') 1g+LF[*-~  
    end %+/f'6kR  
    u2f `|+1^y  
    if any(m>n) e1:u1(".  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') !2/l9SUi  
    end sTJJE3TBI  
    yl[2et  
    if any( r>1 | r<0 ) Y#GT*V  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') t- !h X/  
    end ojiM2QT}m  
    @+[Y0_  
    if ~any(size(r)==1) e OO!jrT:  
        error('zernpol:Rvector','R must be a vector.') ux)<&p.  
    end IJ+O),'  
    5R$=^gE  
    r = r(:); (D:KqGqoT  
    length_r = length(r); ]Fb8.q5(Y  
    39'X$!  
    if nargin==4 sxf}Mmsk  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); *'kC8 ZR5  
        if ~isnorm dO Y lI`4  
            error('zernpol:normalization','Unrecognized normalization flag.') {LjK_J'  
        end /5Gnb.zN)  
    else l#mqV@?A~  
        isnorm = false; J(H??9(s  
    end _:oMyK'  
    $IZ *|>(  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X20<r?^,,  
    % Compute the Zernike Polynomials $Ui]hA-:?y  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]W89.><%14  
    \"<GL;  
    % Determine the required powers of r: 7Y|Wy Oq  
    % ----------------------------------- C@l +\M(  
    rpowers = []; H620vlC}V  
    for j = 1:length(n) Fj[ dO&  
        rpowers = [rpowers m(j):2:n(j)]; S(q4OQ B{  
    end ~hxeD" w  
    rpowers = unique(rpowers); iPRJA{$b_  
    VQZT.^  
    % Pre-compute the values of r raised to the required powers, W$x K^}  
    % and compile them in a matrix: krnvFZRTQ  
    % ----------------------------- gDUoc*+h  
    if rpowers(1)==0 { &6l\|  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); #d7)$ub  
        rpowern = cat(2,rpowern{:}); Dg?Ho2ih  
        rpowern = [ones(length_r,1) rpowern]; _R>s5|_  
    else )wyu+_:  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); {Tjtj@-  
        rpowern = cat(2,rpowern{:}); {|t?   
    end 7z0;FW3>9  
    X"]mR7k  
    % Compute the values of the polynomials: 72B zvY.  
    % -------------------------------------- 4cv|ok8P  
    z = zeros(length_r,length_n); M[&.kH  
    for j = 1:length_n RQ_#rYmT  
        s = 0:(n(j)-m(j))/2; A `H]q5d  
        pows = n(j):-2:m(j); 1{Sx V  
        for k = length(s):-1:1 \Ho#[k=y*/  
            p = (1-2*mod(s(k),2))* ... SO8|]Fk  
                       prod(2:(n(j)-s(k)))/          ... d- _93  
                       prod(2:s(k))/                 ... KJ05Zx~uma  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ,sy / r V  
                       prod(2:((n(j)+m(j))/2-s(k))); 'h+4zvI"8  
            idx = (pows(k)==rpowers); `rRg(fCN!M  
            z(:,j) = z(:,j) + p*rpowern(:,idx); a*e|>pDO  
        end .5$V7t.t$\  
         iIC9rso"Q1  
        if isnorm :;#c:RKi:  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); o 2$<>1^  
        end h;mQ%9 Yd  
    end _^,[wD  
    r.W"@vc>  
    % 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)  ] lTfi0}g_  
    s o s&  
    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。 ddxv.kIj.  
    9|DC<Zn&B#  
    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)