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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 D= LLm$y  
    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! bG[)r  
     
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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 ic_q<Y}  
    function z = zernfun(n,m,r,theta,nflag) Y^~Dr|5%  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. eURj'8o),  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ;"77? )  
    %   and angular frequency M, evaluated at positions (R,THETA) on the D[ -Gzqh  
    %   unit circle.  N is a vector of positive integers (including 0), and [Q5>4WY  
    %   M is a vector with the same number of elements as N.  Each element p%+uv\Ix  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 2,,t+8"`  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 4)XZ'~|  
    %   and THETA is a vector of angles.  R and THETA must have the same c0%.GcF0{  
    %   length.  The output Z is a matrix with one column for every (N,M) <+wbnnK  
    %   pair, and one row for every (R,THETA) pair. +7]]=e<[E  
    % wZ_k]{J  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike -U"h3Ye^  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), o2C{V1nB  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral U94Tp A6  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ..g?po  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ^t{2k[@  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ]a}K%D)H  
    % hkhk,bhI  
    %   The Zernike functions are an orthogonal basis on the unit circle. 2MapB*  
    %   They are used in disciplines such as astronomy, optics, and `X06JTqf:  
    %   optometry to describe functions on a circular domain. mrgieb%  
    % '4""Gz  
    %   The following table lists the first 15 Zernike functions. KiDL]2  
    % 2# y!(D8  
    %       n    m    Zernike function           Normalization +hJ@w-u,G  
    %       -------------------------------------------------- iVg3=R)[1  
    %       0    0    1                                 1 M@=eWZ<  
    %       1    1    r * cos(theta)                    2  gh{Z=_  
    %       1   -1    r * sin(theta)                    2 `(rnD  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) @FBlF$vG  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ?!4xtOA  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 0A}'@N@G)  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) YFF\m{#  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) o'8`>rb  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 6!eI=h2P  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) EqV]/0-\  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) wInJ!1  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) xElHYh(\  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) t[ Zoe+&  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) m1mA:R\zM  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) I}&`IUP  
    %       -------------------------------------------------- f`dQ $Kh  
    % {O4y Y=G  
    %   Example 1: rk$$gXg9/  
    % ZT \=:X*e  
    %       % Display the Zernike function Z(n=5,m=1) aOj(=s  
    %       x = -1:0.01:1; W.fsW<{4j  
    %       [X,Y] = meshgrid(x,x); 8hK\Ya:mP  
    %       [theta,r] = cart2pol(X,Y); y$f{P:!"{3  
    %       idx = r<=1; ^` 96L  
    %       z = nan(size(X)); jgfl|;I?pg  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); a=m7pe ^  
    %       figure d'4^c,d  
    %       pcolor(x,x,z), shading interp 'k?%39  
    %       axis square, colorbar \,b@^W6e>  
    %       title('Zernike function Z_5^1(r,\theta)') COF_a%  
    % iL%Q@!ka  
    %   Example 2: ?G48GxJ  
    % Xlw8> .\  
    %       % Display the first 10 Zernike functions zO)>(E?  
    %       x = -1:0.01:1; ] X9e|  
    %       [X,Y] = meshgrid(x,x); uEK9  
    %       [theta,r] = cart2pol(X,Y); s C/5N  
    %       idx = r<=1; w _*|u  
    %       z = nan(size(X)); XpFo SW#K  
    %       n = [0  1  1  2  2  2  3  3  3  3]; -27uh  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; X/5\L.g2  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; |m^qA](M  
    %       y = zernfun(n,m,r(idx),theta(idx)); WxN@&g(  
    %       figure('Units','normalized') AS} FRNIVx  
    %       for k = 1:10 ^sWsP`DV  
    %           z(idx) = y(:,k); ?\ qfuA9.  
    %           subplot(4,7,Nplot(k)) ugZ-*e7  
    %           pcolor(x,x,z), shading interp DQ<{FN  
    %           set(gca,'XTick',[],'YTick',[]) `YZK$ -,  
    %           axis square Eagl7'x  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Ux<2!vh  
    %       end _o 2pyV&  
    % 8f^QO:  
    %   See also ZERNPOL, ZERNFUN2. :f~[tox  
    Slk__eC  
    %   Paul Fricker 11/13/2006 Mn- f  
    Lq&;`)BJ  
    U_-9rkUa  
    % Check and prepare the inputs: 0X`sQNx  
    % ----------------------------- xHA6  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) * 5H  
        error('zernfun:NMvectors','N and M must be vectors.') \Bg;^6U  
    end -|?I'~[#(  
    / _N*6a~  
    if length(n)~=length(m) @E(_H$|E  
        error('zernfun:NMlength','N and M must be the same length.') 7rc6  
    end EXdx$I=X  
    OZ/P@`kN.f  
    n = n(:); A,tmy',d"  
    m = m(:); cGevFlnh  
    if any(mod(n-m,2)) QbF!V%+a's  
        error('zernfun:NMmultiplesof2', ... i|z=q  
              'All N and M must differ by multiples of 2 (including 0).') N W/RQ(  
    end  h :[8$]  
    %s+H& vfQs  
    if any(m>n) [hLSK-K 9  
        error('zernfun:MlessthanN', ... Ur`jmB  
              'Each M must be less than or equal to its corresponding N.') ,qx;kJJ  
    end Fq]ht*  
    'nK(cKDIG  
    if any( r>1 | r<0 ) ICJp-  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') X3z$f(lF%)  
    end y>:-6)pv  
    ;1S~'B&1Q  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) )  H %Cb  
        error('zernfun:RTHvector','R and THETA must be vectors.') $ BEIG@qG  
    end X1 A~#w>  
    ,rvw E  
    r = r(:); Dr;-2$Kt/&  
    theta = theta(:); E>/kNl  
    length_r = length(r); b(hnouS  
    if length_r~=length(theta) H5L~[\ 5t  
        error('zernfun:RTHlength', ... QKj-"y[  
              'The number of R- and THETA-values must be equal.') IZ<d~ [y  
    end Ig9gGI,  
    C*;g!~{  
    % Check normalization: Ueq*R(9>  
    % -------------------- DNARe!pK  
    if nargin==5 && ischar(nflag) HFJna2B`  
        isnorm = strcmpi(nflag,'norm'); uQ^r1 $#  
        if ~isnorm "pb$[*_@$  
            error('zernfun:normalization','Unrecognized normalization flag.') Q(P'4XCm  
        end `Qf$]Eoft  
    else uXs.7+f  
        isnorm = false; }y<p_dZI  
    end C%s+o0b  
    T gpf0(  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *z2G(Uac  
    % Compute the Zernike Polynomials bB|UQaCl  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~! *xi  
    =")}wl=s  
    % Determine the required powers of r: 2>l =oXq  
    % ----------------------------------- * u_ nu>  
    m_abs = abs(m); \q2#ef@2  
    rpowers = []; hJqLH ?Ri  
    for j = 1:length(n) GpjyF_L  
        rpowers = [rpowers m_abs(j):2:n(j)]; BPp`r_m8w}  
    end `rt  
    rpowers = unique(rpowers); ()< E?D=  
    P@ gVzx)M  
    % Pre-compute the values of r raised to the required powers, ^DL}J>F9G  
    % and compile them in a matrix: w"s;R8  
    % ----------------------------- )7U^&I,  
    if rpowers(1)==0 OnNWci|7  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); -WDU~VSU  
        rpowern = cat(2,rpowern{:}); _ >)+ u  
        rpowern = [ones(length_r,1) rpowern]; (=v :@\r  
    else ^/|agQ7D2  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); =0te.io)3O  
        rpowern = cat(2,rpowern{:}); QXXB>gOY5  
    end {1RI!#[\  
    vwVK ^B  
    % Compute the values of the polynomials: +T*=JHOD  
    % -------------------------------------- Xb0$BAP  
    y = zeros(length_r,length(n)); Z`5jX;Z!  
    for j = 1:length(n) 2V6=F[T  
        s = 0:(n(j)-m_abs(j))/2; {H]xA3[]  
        pows = n(j):-2:m_abs(j); r-M:YB  
        for k = length(s):-1:1 8@Zg@>,  
            p = (1-2*mod(s(k),2))* ... .\r=1HZ3  
                       prod(2:(n(j)-s(k)))/              ... a;=)`  
                       prod(2:s(k))/                     ... "N"$B~W*  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... #fq%903=  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); >s 4"2X  
            idx = (pows(k)==rpowers); ?LJDBN  
            y(:,j) = y(:,j) + p*rpowern(:,idx); %4F Q~  
        end ET]PF,`  
         j]-0m4QF  
        if isnorm 8>T#sO?+  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 3 [R<JrO  
        end }2WscxL  
    end qJjXN+/D  
    % END: Compute the Zernike Polynomials iFJ2dFA  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~}uv4;0l]  
    N_dHPa  
    % Compute the Zernike functions: iD*%' #u  
    % ------------------------------ DtXQLL*fl(  
    idx_pos = m>0; #BB,6E   
    idx_neg = m<0; "Di27Rq  
    C$"N)6%q  
    z = y; sK)fEx  
    if any(idx_pos) ~UrKyA  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 1u?h4w C  
    end ;kSRv=S  
    if any(idx_neg) eWKFs)C]  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)');  {{hp;&x  
    end  HaJs)j  
    [}xVz"8V  
    % 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) j6!C/UgQ  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. -cgLEl1J  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated i-.]onR  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive TLcev*  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, \,UpFuU\  
    %   and THETA is a vector of angles.  R and THETA must have the same <z'Pj7c[  
    %   length.  The output Z is a matrix with one column for every P-value, ;(&$Iw9X  
    %   and one row for every (R,THETA) pair. ;jgk53lo  
    % 4>x$I9^Y!  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike A-n@:` n~  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 1c5+X Cr  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) NO)Hi)$X6Y  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ;mT|0&o>#  
    %   for all p. \d'>Ky;GD  
    % \ltbiDP2  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 'CQ~ZV5  
    %   Zernike functions (order N<=7).  In some disciplines it is 7 XNZEi9o  
    %   traditional to label the first 36 functions using a single mode L3/SIoqd  
    %   number P instead of separate numbers for the order N and azimuthal Zz,j,w0 Z  
    %   frequency M. u%t/W0xi  
    % AvmI<U  
    %   Example: O{vVW9Q  
    % ojc m%yd  
    %       % Display the first 16 Zernike functions f|v5i tO2  
    %       x = -1:0.01:1; W u C2 LM  
    %       [X,Y] = meshgrid(x,x); VyMFALSe]h  
    %       [theta,r] = cart2pol(X,Y); >(?}'pS8  
    %       idx = r<=1; Sg&0a$  
    %       p = 0:15; Y)O88C  
    %       z = nan(size(X)); 00 9[`Z  
    %       y = zernfun2(p,r(idx),theta(idx)); Ub,5~I+`  
    %       figure('Units','normalized') dguN<yS- E  
    %       for k = 1:length(p) 0?$jC-@k:  
    %           z(idx) = y(:,k); e2"<3  
    %           subplot(4,4,k) La2f]+sV  
    %           pcolor(x,x,z), shading interp &Z3u(Eb  
    %           set(gca,'XTick',[],'YTick',[]) +^6a$ N  
    %           axis square +vr|J:  
    %           title(['Z_{' num2str(p(k)) '}']) 3>T2k }  
    %       end 3wYhDxY1  
    % [g/ &%n0^  
    %   See also ZERNPOL, ZERNFUN. h5o6G1ur  
    H2jF=U"=  
    %   Paul Fricker 11/13/2006 `o4%UkBpM  
    Hhzi(<e^  
    /;X+<Wj  
    % Check and prepare the inputs: c]/X >8;  
    % ----------------------------- ?wi^R:2|j  
    if min(size(p))~=1 5y~ Srb?2  
        error('zernfun2:Pvector','Input P must be vector.') &cpqn2Z  
    end CcJ%; .V,T  
    , 3&D A  
    if any(p)>35 p 2>\  
        error('zernfun2:P36', ... TWeup6k  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... CJYpgSr  
               '(P = 0 to 35).']) 9 df GV!Z  
    end y$K[ArqX  
    =h>jo&=Wad  
    % Get the order and frequency corresonding to the function number: (|2:^T+  
    % ---------------------------------------------------------------- }~#Tsv  
    p = p(:); YlC$L$%Zd.  
    n = ceil((-3+sqrt(9+8*p))/2); o.g)[$M8cF  
    m = 2*p - n.*(n+2); LOEiV  
    h( DmSW  
    % Pass the inputs to the function ZERNFUN: Jm`{MzqL  
    % ---------------------------------------- Z]SUr`Z  
    switch nargin `'E(L&  
        case 3 iu iVr$E  
            z = zernfun(n,m,r,theta); e9;5.m  
        case 4 zq'KX/o  
            z = zernfun(n,m,r,theta,nflag); 8b:GyC5L  
        otherwise WB.w3w [f  
            error('zernfun2:nargin','Incorrect number of inputs.') ._[uSBR'  
    end Ew )1O9f  
    $/kZKoF{f  
    % 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) #2dH2k\F  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. lNo]]a+_  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of K*P:FCz  
    %   order N and frequency M, evaluated at R.  N is a vector of =<X4LO)C  
    %   positive integers (including 0), and M is a vector with the f2 ?01PM,Q  
    %   same number of elements as N.  Each element k of M must be a !8I80 :e_~  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) N (0%C?  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is W.c>("gC  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix !}hG|Y6s  
    %   with one column for every (N,M) pair, and one row for every ODxCD%L  
    %   element in R. @5h(bLEP  
    % ,0@QBr5P  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 1b<[/g9  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is y`Zn{mQ@[  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to mq+x=  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 l^2m7 7)  
    %   for all [n,m]. Eih6?Lpu  
    % g*;z V i  
    %   The radial Zernike polynomials are the radial portion of the bAEwjZ  
    %   Zernike functions, which are an orthogonal basis on the unit P@0J!  
    %   circle.  The series representation of the radial Zernike ZKJhmk  
    %   polynomials is nP0rg  
    % ~{ucr#]C  
    %          (n-m)/2 @!*I mNMI  
    %            __ Z3f}'vr  
    %    m      \       s                                          n-2s !8|?0>3)  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r xk7VuS *  
    %    n      s=0 L=gG23U&  
    % jt0f*e YE8  
    %   The following table shows the first 12 polynomials. )  D5JA`  
    % V6&6I  
    %       n    m    Zernike polynomial    Normalization U U3o (Yq  
    %       --------------------------------------------- _q}^#-  
    %       0    0    1                        sqrt(2) JvF0s}#4  
    %       1    1    r                           2 w&*oWI$i  
    %       2    0    2*r^2 - 1                sqrt(6) A&{eC C  
    %       2    2    r^2                      sqrt(6) pp+z5  
    %       3    1    3*r^3 - 2*r              sqrt(8) ^;[^L=}8$  
    %       3    3    r^3                      sqrt(8) U lPhW~F)  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) _FCg5F2U  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) C[CNJ66  
    %       4    4    r^4                      sqrt(10) )O8w'4P5  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ,MUgww!.  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) SP;1XXlL  
    %       5    5    r^5                      sqrt(12) p: u@? k  
    %       --------------------------------------------- Oo/@A_JO@  
    % [*g'Y;W  
    %   Example: }[y_Fr0  
    % AG|:mQO  
    %       % Display three example Zernike radial polynomials v?l*jr1-2  
    %       r = 0:0.01:1; ,WE2MAjhT  
    %       n = [3 2 5]; 5Vr#>W  
    %       m = [1 2 1]; esd9N'.Q*  
    %       z = zernpol(n,m,r); 4!A(7 s4t  
    %       figure ^*F'[!. p  
    %       plot(r,z) 6M[OEI5  
    %       grid on or(P?Ro  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') v!#koqd1y.  
    % SmV}Wf  
    %   See also ZERNFUN, ZERNFUN2. (8(P12l  
    |P-kyY34  
    % A note on the algorithm. .SDE6nvbW  
    % ------------------------ &X,6v  
    % The radial Zernike polynomials are computed using the series {d%hkbN+{  
    % representation shown in the Help section above. For many special @&GY5<&b  
    % functions, direct evaluation using the series representation can U%)*I~9  
    % produce poor numerical results (floating point errors), because dvLL~VP  
    % the summation often involves computing small differences between gR wRhA/  
    % large successive terms in the series. (In such cases, the functions ]Aj5 K  
    % are often evaluated using alternative methods such as recurrence ]'<"qY  
    % relations: see the Legendre functions, for example). For the Zernike u6 4{w,  
    % polynomials, however, this problem does not arise, because the EJ(z]M`f  
    % polynomials are evaluated over the finite domain r = (0,1), and #<vzQ\~Y  
    % because the coefficients for a given polynomial are generally all IO"q4(&;P4  
    % of similar magnitude. V]/ $ dJ  
    % :M.]-+(  
    % ZERNPOL has been written using a vectorized implementation: multiple @Py?.H   
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] G4%dah 5  
    % values can be passed as inputs) for a vector of points R.  To achieve %1 rN6A!%  
    % this vectorization most efficiently, the algorithm in ZERNPOL ?>mpUH  
    % involves pre-determining all the powers p of R that are required to 4{%-r[C9k  
    % compute the outputs, and then compiling the {R^p} into a single W{t- UK   
    % matrix.  This avoids any redundant computation of the R^p, and ue -a/a  
    % minimizes the sizes of certain intermediate variables. {*X|)nr  
    % @DlN;r ?Cv  
    %   Paul Fricker 11/13/2006 54Vb[;`Kkb  
    ?+L7Bd(EF%  
    N`LY$U+N|  
    % Check and prepare the inputs: c LfPSA  
    % ----------------------------- ]:Pkh./  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 8< "lEL|  
        error('zernpol:NMvectors','N and M must be vectors.') K*5Ij]j&  
    end 7e Hj"_;  
    <o@__l.  
    if length(n)~=length(m) Wv30;7~  
        error('zernpol:NMlength','N and M must be the same length.') pEY zB;  
    end Q7_#k66gb7  
    r|3<UR%  
    n = n(:); twp~#s:\z  
    m = m(:); uExYgI`<%&  
    length_n = length(n); 5yf`3vV|3@  
    rGzGbI=  
    if any(mod(n-m,2)) ht*;,[ea  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') @"afEMd  
    end :!fU+2$`^(  
    IW=%2n(<1  
    if any(m<0) ,PX7}//X^  
        error('zernpol:Mpositive','All M must be positive.') l?KP /0`  
    end vH@b  
    X`7O%HiX/`  
    if any(m>n) AZnFOS  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') &zHY0fxX  
    end ,9W!cD+0  
    Q.SqOHeJ  
    if any( r>1 | r<0 ) 1t7T\~ +F  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') :~~\{fm  
    end DK<}q1xi  
    Obc wmL  
    if ~any(size(r)==1) N._^\FRyn  
        error('zernpol:Rvector','R must be a vector.') qcfg 55]'c  
    end }LX.gm  
    cLIeo{H  
    r = r(:); !lI1jb"  
    length_r = length(r); !uhh_3RH  
    :(i=> ~O  
    if nargin==4 Zc=#Y  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ^F-AZP /5F  
        if ~isnorm V""3#Tw   
            error('zernpol:normalization','Unrecognized normalization flag.') ORt)sn&~d  
        end tA-p!#V<k1  
    else K?=g IC:  
        isnorm = false; ^Fwdi#g  
    end MwWN;_#EO)  
    ]GzfU'fOn|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% VB~Do?]*k%  
    % Compute the Zernike Polynomials 2&:nHZ)  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _+qtH< F/  
    2~@Cj@P]  
    % Determine the required powers of r: f1I/aRV:+  
    % -----------------------------------  bRx}ih  
    rpowers = []; |L6 +e *  
    for j = 1:length(n) vH{JLN2  
        rpowers = [rpowers m(j):2:n(j)]; m!:sDQn{3  
    end k'K 1zUBj  
    rpowers = unique(rpowers); J_&G\b.9/  
    !H`uN  
    % Pre-compute the values of r raised to the required powers, |@dY[VK>  
    % and compile them in a matrix: }1YQ?:@  
    % ----------------------------- @&2# kO~=  
    if rpowers(1)==0 l8d }g  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ]Waa7)}DM  
        rpowern = cat(2,rpowern{:}); zC!Pb{IaH  
        rpowern = [ones(length_r,1) rpowern]; }?Tz=hP  
    else zmU>  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7@ mP;K0  
        rpowern = cat(2,rpowern{:}); m"~),QwF9  
    end Y NGS"3F  
    86~q pN  
    % Compute the values of the polynomials: 't3nh  
    % -------------------------------------- Bmi9U   
    z = zeros(length_r,length_n); k; vhQ=  
    for j = 1:length_n \PK}4<x}  
        s = 0:(n(j)-m(j))/2; g;!,2,De}  
        pows = n(j):-2:m(j); &=l aZxe  
        for k = length(s):-1:1 vFsl]|<;8  
            p = (1-2*mod(s(k),2))* ... Ai5D[ykX  
                       prod(2:(n(j)-s(k)))/          ... Q}]RB$ZS  
                       prod(2:s(k))/                 ... ]]|vQA^  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... {(^%2dk83C  
                       prod(2:((n(j)+m(j))/2-s(k))); ?yAjxoE~?  
            idx = (pows(k)==rpowers); <*DP G\6Ma  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 6g'+1%O  
        end G":u::hR  
         O+o_{t\R  
        if isnorm C8 "FTH'  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); _E xd:  
        end pAc "Wo(Q  
    end RU,!F99'1  
    ]s~%1bd  
    % 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)  OGqsQ  
    fK{[=xMr@  
    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。 <UMT:`h1MZ  
    b&1-tYV  
    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)