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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 O\0]o!  
    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! rnX D(  
     
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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 M_BG :P5  
    function z = zernfun(n,m,r,theta,nflag) "39\@Ow  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ctk~}( 1#  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Z(h.)$yH*=  
    %   and angular frequency M, evaluated at positions (R,THETA) on the azBYh*s=5{  
    %   unit circle.  N is a vector of positive integers (including 0), and s^\ *jZ6  
    %   M is a vector with the same number of elements as N.  Each element HP,sNiw  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) C srxi'Pe  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, @yImR+^.7  
    %   and THETA is a vector of angles.  R and THETA must have the same I,8f{T!O@"  
    %   length.  The output Z is a matrix with one column for every (N,M) #];b+ T  
    %   pair, and one row for every (R,THETA) pair. od=x?uBVd  
    % MrU0Jrk4+  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 4>t'4p6{  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ovXU +8  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral d}:eLC  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, w! kWG,{C  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized nhdOo   
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 4 AWL::FU5  
    % rGDx9KR4K!  
    %   The Zernike functions are an orthogonal basis on the unit circle. w,)O*1't  
    %   They are used in disciplines such as astronomy, optics, and 3bN]2\   
    %   optometry to describe functions on a circular domain. (/ qOY  
    % ;}>g/lw  
    %   The following table lists the first 15 Zernike functions. -s6k't  
    % 7{ JIHY+  
    %       n    m    Zernike function           Normalization o)]mJb~XG-  
    %       -------------------------------------------------- `m")v0n3  
    %       0    0    1                                 1 ]I^b&N  
    %       1    1    r * cos(theta)                    2 `uh+d  
    %       1   -1    r * sin(theta)                    2 oE.59dx  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) yQz6K6p  
    %       2    0    (2*r^2 - 1)                    sqrt(3) `k;MGs)&  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) }N0$DqP  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) S*3*Q l*  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) o)2KQ$b>Q  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) EGMIw?%Y`-  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) \8<ZPqt9  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) o|cx?  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) zB68%  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) _c$F?9:  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) P P-U.  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) I<+i87=  
    %       -------------------------------------------------- Q8Fqf ;4  
    % J`[v u4  
    %   Example 1: wrhGZ=k{  
    % H.)Y*zK0.  
    %       % Display the Zernike function Z(n=5,m=1) M 8NWQ^Y  
    %       x = -1:0.01:1;  DJJd_  
    %       [X,Y] = meshgrid(x,x); 1@:BUE;jZ  
    %       [theta,r] = cart2pol(X,Y); ss0`9:z  
    %       idx = r<=1; V'^E'[Dd{  
    %       z = nan(size(X)); Liv.i;-qE  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 5< $8.a#  
    %       figure =@ d/SZ|(E  
    %       pcolor(x,x,z), shading interp ?RPVd8PUhN  
    %       axis square, colorbar w.o>G2u  
    %       title('Zernike function Z_5^1(r,\theta)') UC@Jsj~f  
    % *8Kx y@  
    %   Example 2: 7R7e3p,K  
    % ?#~km0~F)  
    %       % Display the first 10 Zernike functions 7!g"q\s  
    %       x = -1:0.01:1; H8!)zZ  
    %       [X,Y] = meshgrid(x,x); 8|) $;.  
    %       [theta,r] = cart2pol(X,Y); SpC6dkxD\  
    %       idx = r<=1; N8KH.P+  
    %       z = nan(size(X)); 6Z#$(oC  
    %       n = [0  1  1  2  2  2  3  3  3  3]; %7hf6Xo=  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; &dky_H  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; )}$]~ f4R  
    %       y = zernfun(n,m,r(idx),theta(idx)); 2|A?9aE%0  
    %       figure('Units','normalized') Qf($F,)K  
    %       for k = 1:10 x8!uI)#tS  
    %           z(idx) = y(:,k); QAzwNXE+  
    %           subplot(4,7,Nplot(k)) VOSq%hB  
    %           pcolor(x,x,z), shading interp E_ D0Nm%n  
    %           set(gca,'XTick',[],'YTick',[]) -q30tO.  
    %           axis square Q2Dh(  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) %Y-5L;MI  
    %       end 0.kC|  
    % Vji:,k=3\  
    %   See also ZERNPOL, ZERNFUN2. aQ*?L l  
    |,Kk#`lW<f  
    %   Paul Fricker 11/13/2006 5p]V/<r  
    Aa+<4 R  
    {BY(zsl  
    % Check and prepare the inputs:  r m  
    % ----------------------------- VDFs.;:s  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <Rfx`mn  
        error('zernfun:NMvectors','N and M must be vectors.') (L*<CV  
    end #.{ddY{  
    }R!t/ 8K  
    if length(n)~=length(m) ;(@' +"  
        error('zernfun:NMlength','N and M must be the same length.') H=*lj.x  
    end w0X})&,{`m  
    '{w[).c.  
    n = n(:); n s#v?D9NF  
    m = m(:); Y|6gg  
    if any(mod(n-m,2)) M#k$[w}=  
        error('zernfun:NMmultiplesof2', ... '#a;n  
              'All N and M must differ by multiples of 2 (including 0).') &NX7  
    end 39~te%;C7  
    to;^'#B  
    if any(m>n) O7oq1JI]Y  
        error('zernfun:MlessthanN', ... mwutv8?  
              'Each M must be less than or equal to its corresponding N.') UPy 4ST  
    end 7Ue&y8Yf  
    M(1cf(<+  
    if any( r>1 | r<0 ) &2nICAN[  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ! u@JH`  
    end 2^%O%Pc  
    ;` h$xB(  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 4Uhh]/  
        error('zernfun:RTHvector','R and THETA must be vectors.') C;?<WtH  
    end +4+c zfz  
    5+2qx)FZ  
    r = r(:); XAN.Plk  
    theta = theta(:); N/eus"O;  
    length_r = length(r); "E@A~<RKP  
    if length_r~=length(theta) gZBb /<  
        error('zernfun:RTHlength', ... hka%!W5  
              'The number of R- and THETA-values must be equal.') vVZ+u4y  
    end 5me#/NqLHY  
    ;ojJXH~$}  
    % Check normalization: -jzoGzC3  
    % -------------------- 9g|99Z  
    if nargin==5 && ischar(nflag) <'48mip  
        isnorm = strcmpi(nflag,'norm'); klMpiy  
        if ~isnorm XQ2 YUe]DJ  
            error('zernfun:normalization','Unrecognized normalization flag.') X]D:vuB  
        end BMtk/r/  
    else ++eT 0  
        isnorm = false; CzI s_/  
    end  @{Dfro  
    O^q~dda  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yg4#,4---b  
    % Compute the Zernike Polynomials 8|nc( $}~  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >S8 n 8U  
    %f?Zg44  
    % Determine the required powers of r: ^Rtxef  
    % ----------------------------------- h8 FV2"  
    m_abs = abs(m); hu >wcOt  
    rpowers = []; :2V|(:^ '  
    for j = 1:length(n) L F&!od9[  
        rpowers = [rpowers m_abs(j):2:n(j)]; IgRi(q^b-  
    end OdO n wY  
    rpowers = unique(rpowers); D< kf/hj  
    MEM(uBYKOb  
    % Pre-compute the values of r raised to the required powers, #xfav19{.  
    % and compile them in a matrix: ~pHuh#>  
    % ----------------------------- f\r"7j  
    if rpowers(1)==0 G.$KP  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); O0s,)8+z5D  
        rpowern = cat(2,rpowern{:}); }=JS d@`_  
        rpowern = [ones(length_r,1) rpowern]; o+L [o_er  
    else S;u.Ds&  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); B)/c]"@89  
        rpowern = cat(2,rpowern{:}); omznSL  
    end _pzYmQ  
    i'10qWz  
    % Compute the values of the polynomials: #R7hk5/8n}  
    % -------------------------------------- B%`| W@v  
    y = zeros(length_r,length(n)); ]+b?J0|P<  
    for j = 1:length(n) s8 u`v1  
        s = 0:(n(j)-m_abs(j))/2; 76] Z~^Y  
        pows = n(j):-2:m_abs(j); ,tDLpnB@;  
        for k = length(s):-1:1  ^6b5}{>  
            p = (1-2*mod(s(k),2))* ... 2 6 >9$S  
                       prod(2:(n(j)-s(k)))/              ... IwOL1\'T4  
                       prod(2:s(k))/                     ... k!G{#(++&6  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... `GlOl-  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 72/ bC  
            idx = (pows(k)==rpowers); J1 w3g,  
            y(:,j) = y(:,j) + p*rpowern(:,idx);  E(wS6  
        end s Ytn'&$\  
         Aar]eY\  
        if isnorm TU;AO%5  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 4.Fh4Y:$'  
        end 7HQL^Q  
    end <f=<r*6  
    % END: Compute the Zernike Polynomials t~)4f.F:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n*i&o;5  
    [P0c,97_ H  
    % Compute the Zernike functions: i[MBO`FF  
    % ------------------------------ ,1cpV|mAr  
    idx_pos = m>0; _\8E/4zh  
    idx_neg = m<0; -m[ tYp,q  
    kw} E0uY  
    z = y; G(wstHT;/  
    if any(idx_pos) =[[I<[BZq  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Ui-Y `  
    end 9Y2.ob!$}  
    if any(idx_neg) J`C 2}$ ~  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); s&+`>  
    end dcTZL$  
    /|#2ehE  
    % 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) ZsCwNZR  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. H|,d`@U  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated U3-MvI,Q  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive ?R4u>AHS@  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, osmCwM4O  
    %   and THETA is a vector of angles.  R and THETA must have the same __1Hx?f  
    %   length.  The output Z is a matrix with one column for every P-value, T+t7/PwC;  
    %   and one row for every (R,THETA) pair. 90,UhNz9D  
    % MtgY `p  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike :Ig9n :  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) :cIPX%S  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) H#WqO<<v  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 6 {F#_.  
    %   for all p. [3ggJcUgW>  
    % ?)-anoFyVW  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 !>wu7u-  
    %   Zernike functions (order N<=7).  In some disciplines it is EZVgTySd  
    %   traditional to label the first 36 functions using a single mode ?B)e8i<[f  
    %   number P instead of separate numbers for the order N and azimuthal ~(NFjCUY?  
    %   frequency M. ME$J?3r  
    % #6mw CA|  
    %   Example: =Lb(N61  
    % bE=[P}E  
    %       % Display the first 16 Zernike functions [#SO}'1n  
    %       x = -1:0.01:1; 0S }\ML  
    %       [X,Y] = meshgrid(x,x); SOMAs'=  
    %       [theta,r] = cart2pol(X,Y); m;IKV,  
    %       idx = r<=1; #N'9F&:V$  
    %       p = 0:15; F9(jx#J~t  
    %       z = nan(size(X)); `K[r5;QFKf  
    %       y = zernfun2(p,r(idx),theta(idx)); |mdf u=  
    %       figure('Units','normalized') |5q,%9_  
    %       for k = 1:length(p) s ya!VF]`  
    %           z(idx) = y(:,k); ^JY:$)4["  
    %           subplot(4,4,k) @Jlsx0i}}  
    %           pcolor(x,x,z), shading interp &|Rww\oJ  
    %           set(gca,'XTick',[],'YTick',[]) .m%5Esx  
    %           axis square J-G)mvkv  
    %           title(['Z_{' num2str(p(k)) '}']) G=CP17&h6  
    %       end bP|-GCKM8  
    % o/vD]Fs  
    %   See also ZERNPOL, ZERNFUN. Jvj* z6/a  
    Xi+l1xe  
    %   Paul Fricker 11/13/2006 }dqOE-"I"n  
    U+(qfa5(  
    *IF ~ab2  
    % Check and prepare the inputs: 7C_U:x  
    % ----------------------------- .lI.I  
    if min(size(p))~=1 EpCNp FQT<  
        error('zernfun2:Pvector','Input P must be vector.') hh.`Yu L  
    end bGwj` lue  
    mZ3Z8q}%P  
    if any(p)>35 5-'Z.[ImB?  
        error('zernfun2:P36', ... 8{7'w|/;.{  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Fa </  
               '(P = 0 to 35).']) JuRWR0@`  
    end dDA&\BuS  
    y.gNjc  
    % Get the order and frequency corresonding to the function number: @kba^z  
    % ---------------------------------------------------------------- 0&Iu+hv  
    p = p(:); eSW}H_3  
    n = ceil((-3+sqrt(9+8*p))/2); <K/iX%b?  
    m = 2*p - n.*(n+2); 9`@}KnvB?  
    &4M,)Q (  
    % Pass the inputs to the function ZERNFUN: qA25P<  
    % ---------------------------------------- |h((SreO  
    switch nargin >=1UhHFNI  
        case 3 l~@ -oE  
            z = zernfun(n,m,r,theta); X&@>M}  
        case 4 )sK _k U{\  
            z = zernfun(n,m,r,theta,nflag); B7%m7GM  
        otherwise [Z1,~(3  
            error('zernfun2:nargin','Incorrect number of inputs.') 9/R=_y-  
    end 0f5)]  
    9IacZ  
    % 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) WxtB:7J  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Bv6~!p  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of q~xs4?n1U  
    %   order N and frequency M, evaluated at R.  N is a vector of yoBR'$-=  
    %   positive integers (including 0), and M is a vector with the X}&Y(kOT  
    %   same number of elements as N.  Each element k of M must be a DM(c :+K-  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) }. V!|R,  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is U/\LOIs  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix cNl$ vP83z  
    %   with one column for every (N,M) pair, and one row for every SMA' VU  
    %   element in R. 1%N[DA^<\  
    % Y1{*AV6ev6  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- B+ZhQW  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is >T^BD'z@'  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to L@G~9{U>  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 D}nRH@<`  
    %   for all [n,m]. V%FWZn^  
    % "z{ rC}  
    %   The radial Zernike polynomials are the radial portion of the {9nH#yv  
    %   Zernike functions, which are an orthogonal basis on the unit A$::|2~  
    %   circle.  The series representation of the radial Zernike (Lkcx06e  
    %   polynomials is MQo/R,F }  
    % h)X"<a++N  
    %          (n-m)/2 14h0$7  
    %            __ *p^*>~i9)  
    %    m      \       s                                          n-2s 8fb<hq<  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r @dvb%A&Pur  
    %    n      s=0 /f!ze|  
    % Pbakw81!~  
    %   The following table shows the first 12 polynomials. )Tf,G[z&ge  
    % _%PEv{H0.  
    %       n    m    Zernike polynomial    Normalization wD $sKd  
    %       --------------------------------------------- bN>|4hS  
    %       0    0    1                        sqrt(2) GbBz;ZV%z,  
    %       1    1    r                           2 q_h/zPuH'  
    %       2    0    2*r^2 - 1                sqrt(6) BPypjS0?8  
    %       2    2    r^2                      sqrt(6) \7 *"M y*  
    %       3    1    3*r^3 - 2*r              sqrt(8) cGv`%  
    %       3    3    r^3                      sqrt(8) p+xjYU4^C  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) j\uPOn8k  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) g6;a2  
    %       4    4    r^4                      sqrt(10) XWf1c ~J  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) A04E <nr  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) DG 6W ^  
    %       5    5    r^5                      sqrt(12) :qK^71gz  
    %       --------------------------------------------- dZ,~yV  
    % e]3b0`E  
    %   Example: RJ$x{$r[  
    % !<4=@  
    %       % Display three example Zernike radial polynomials E:$r" oS  
    %       r = 0:0.01:1; mP -Y9*k  
    %       n = [3 2 5]; s.>;(RiJd  
    %       m = [1 2 1]; $ I|K<slV  
    %       z = zernpol(n,m,r); (L !#2Jy  
    %       figure x^6b$>1  
    %       plot(r,z) ~L=? F  
    %       grid on b%UbTb,  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') dC 8,  
    % ITBa ^P  
    %   See also ZERNFUN, ZERNFUN2. $/$ 5{<  
    C dTE~O<)  
    % A note on the algorithm. O|Y~^:ny  
    % ------------------------ T9-2"M=|<  
    % The radial Zernike polynomials are computed using the series xC-&<s  
    % representation shown in the Help section above. For many special vptBDfzz  
    % functions, direct evaluation using the series representation can 3KN})*1  
    % produce poor numerical results (floating point errors), because vQ1#Zg y  
    % the summation often involves computing small differences between sx@ %3j  
    % large successive terms in the series. (In such cases, the functions [D<"qT^*z6  
    % are often evaluated using alternative methods such as recurrence 1YvE/<6  
    % relations: see the Legendre functions, for example). For the Zernike {4HcecT  
    % polynomials, however, this problem does not arise, because the {7LNQGiJ  
    % polynomials are evaluated over the finite domain r = (0,1), and XjU/7Q  
    % because the coefficients for a given polynomial are generally all K]C@seF`  
    % of similar magnitude. "\l#q$1h  
    % oaM 3#QJ  
    % ZERNPOL has been written using a vectorized implementation: multiple L31#v$;4  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 05Y4=7,!  
    % values can be passed as inputs) for a vector of points R.  To achieve ;.&k zzvJ  
    % this vectorization most efficiently, the algorithm in ZERNPOL EOzw&M];r  
    % involves pre-determining all the powers p of R that are required to 6"u"B-cz  
    % compute the outputs, and then compiling the {R^p} into a single .dTXC'  
    % matrix.  This avoids any redundant computation of the R^p, and |,WP)  
    % minimizes the sizes of certain intermediate variables. 0E/,l``p  
    % +`'>   
    %   Paul Fricker 11/13/2006 R9)"%SO<y  
    m53~Ysq<  
    m"RSDM!  
    % Check and prepare the inputs: 9]PMti  
    % ----------------------------- Z:Y_{YAD  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 0{ !+N6MiR  
        error('zernpol:NMvectors','N and M must be vectors.') BFn4H%1  
    end G?5Vj_n  
    K)s{D ] B  
    if length(n)~=length(m) Q;y)6+VU4  
        error('zernpol:NMlength','N and M must be the same length.') cX4I+Mf  
    end 3D2i32Y@!  
    P|QM0GI  
    n = n(:); ID8u&:  
    m = m(:); /DoSU>%hK  
    length_n = length(n); O7# 8g$ZIv  
    4)NbQ[  
    if any(mod(n-m,2)) ISi^BFU  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 7L$\S[E  
    end ]lZ g }7h  
    ],l\HHQ  
    if any(m<0) =a!_H=+4  
        error('zernpol:Mpositive','All M must be positive.') fO t?2Bh  
    end !6*m<#Qm  
    H_d^Xk QZ  
    if any(m>n) 9|qzFmE#  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') B2l5}"{ `  
    end }}gtz-w  
    s&F& *5W  
    if any( r>1 | r<0 ) ^!*nhs%  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') cN%@ nW0i  
    end Y>SpV_H%  
    uG=~k O  
    if ~any(size(r)==1) ^:Fj+d  
        error('zernpol:Rvector','R must be a vector.') H_>9'(  
    end X|dlVNL8p  
    QyD(@MFxb  
    r = r(:); (DY&{vudF  
    length_r = length(r); T$*#q('1"}  
    rBZ0Fx$/[  
    if nargin==4 c)4L3W-x=  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); e&-MP;kgW9  
        if ~isnorm {wvBs87  
            error('zernpol:normalization','Unrecognized normalization flag.') JiFB<Q\  
        end ^5rB/y,  
    else mKuY=#RP  
        isnorm = false; 7[ZoUWx  
    end \Sv8c}8  
    -1}&\=8M  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% bkFO4OZd  
    % Compute the Zernike Polynomials /Csk"IfuO  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Y T'olk  
    ]]XXcQ,A  
    % Determine the required powers of r: YT}ZLx  
    % ----------------------------------- i'p6#  
    rpowers = []; xiOAj"}~  
    for j = 1:length(n) dF$&fo%  
        rpowers = [rpowers m(j):2:n(j)]; 1 RVs!;  
    end ^X ~S}MX  
    rpowers = unique(rpowers); U3~rtc*  
    QzS=oiL  
    % Pre-compute the values of r raised to the required powers, NK6 ~qWsu  
    % and compile them in a matrix: qW`DCZu  
    % ----------------------------- $g_|U:,  
    if rpowers(1)==0 \hI|I!sDWy  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); aRy" _dZ2  
        rpowern = cat(2,rpowern{:}); 1|:'jK#gE  
        rpowern = [ones(length_r,1) rpowern]; TgA>(HcO  
    else ){*9$486  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 'Lh nl3  
        rpowern = cat(2,rpowern{:}); *yYeqm  
    end Nr?CZFN#  
    h?p!uQ  
    % Compute the values of the polynomials: !GnwE  
    % -------------------------------------- @6b4YV h  
    z = zeros(length_r,length_n); jEn 9T  
    for j = 1:length_n mcTC'. 9  
        s = 0:(n(j)-m(j))/2; GD% qrK?  
        pows = n(j):-2:m(j); Q7-'5s   
        for k = length(s):-1:1 x27$h)R0v  
            p = (1-2*mod(s(k),2))* ... 2=7:6Fw  
                       prod(2:(n(j)-s(k)))/          ... pgBIYeY,  
                       prod(2:s(k))/                 ... <Vl`EfA(  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... T%4yPmY  
                       prod(2:((n(j)+m(j))/2-s(k))); 5E^P2Mlc  
            idx = (pows(k)==rpowers); O Ke 9/._  
            z(:,j) = z(:,j) + p*rpowern(:,idx); PLq]\y  
        end 8>`8p0I$+  
         gts09{"}Y  
        if isnorm Kx02 2rgDU  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); }Z)YK}_1  
        end e[6Me[b  
    end ^O<@I  
    kQ"Ax? b  
    % 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)  JY,oXA6O  
    2fNNdxdbT  
    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。 *SpE XO  
    :_`Yrx5  
    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)