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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 mV0.9pxS  
    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! }-:s9Lt  
     
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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 _VFxzM9f  
    function z = zernfun(n,m,r,theta,nflag) %Y"@VcN  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. I^pD=1Y]  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N J+3PUfg>@R  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ] Ma2*E !p  
    %   unit circle.  N is a vector of positive integers (including 0), and  hfpSxL  
    %   M is a vector with the same number of elements as N.  Each element ITa8*Myj  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) K8{Ub  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, FpjpsD~ Qu  
    %   and THETA is a vector of angles.  R and THETA must have the same A+Nf]([  
    %   length.  The output Z is a matrix with one column for every (N,M) zK`z*\  
    %   pair, and one row for every (R,THETA) pair. }v[*V   
    % ~U+SK4SK:o  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike eJ+V!K'H2  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), u%FG% j?C  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral n22k<@y  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, {umdW x.*  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized )J&1uMp{  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. F0O"rN{  
    % R=jIVw'  
    %   The Zernike functions are an orthogonal basis on the unit circle. >r] bfN,  
    %   They are used in disciplines such as astronomy, optics, and f S50  
    %   optometry to describe functions on a circular domain. m&xyw9a  
    % U$R+&@;  
    %   The following table lists the first 15 Zernike functions. kYwk'\s  
    % %xE\IRlR  
    %       n    m    Zernike function           Normalization Ur`Ri?  
    %       -------------------------------------------------- 5I ,5da  
    %       0    0    1                                 1 R9X* R3nB  
    %       1    1    r * cos(theta)                    2 iX 0s4  
    %       1   -1    r * sin(theta)                    2 P!q U8AJkt  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) <X}@afS  
    %       2    0    (2*r^2 - 1)                    sqrt(3) N>?R,XM V  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) T&6W>VQ|[>  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) (P {o9  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) iGmBG1a\  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) TY[{)aH{S  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) E5.3wOE  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 8YJ8_$Z  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) UTw f!  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) $Yh7N5XH,  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,6Ulj+l  
    %       4    4    r^4 * sin(4*theta)             sqrt(10)  PDaD:}9  
    %       -------------------------------------------------- Wu]D pe  
    % /P bN!r<1  
    %   Example 1: Z)cGe1?q  
    % @RW=(&<1  
    %       % Display the Zernike function Z(n=5,m=1) Gj]*_"T  
    %       x = -1:0.01:1; FBpf_=(_1  
    %       [X,Y] = meshgrid(x,x); Ie(vTP1Cj  
    %       [theta,r] = cart2pol(X,Y); NLHF3h=?1p  
    %       idx = r<=1; @l~zn%!X  
    %       z = nan(size(X)); xh[De}@  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); `~'yy q  
    %       figure 5\Sm^t|Tx  
    %       pcolor(x,x,z), shading interp J%c4-'l  
    %       axis square, colorbar t(FI Bf3  
    %       title('Zernike function Z_5^1(r,\theta)') |T:' G  
    % t><AaYij_  
    %   Example 2: X_Vj&{  
    % / $7E  
    %       % Display the first 10 Zernike functions r=YprVX  
    %       x = -1:0.01:1; ;`IZ&m$  
    %       [X,Y] = meshgrid(x,x); Y#Pl)sRr  
    %       [theta,r] = cart2pol(X,Y); QEIu}e6b  
    %       idx = r<=1; .c~`{j}  
    %       z = nan(size(X)); {R b|";  
    %       n = [0  1  1  2  2  2  3  3  3  3]; QGE)Xn#_bN  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; >D'Kt?L<]m  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; U  JO  
    %       y = zernfun(n,m,r(idx),theta(idx)); $y~!ePKh  
    %       figure('Units','normalized') >(Mu9ie*`  
    %       for k = 1:10 )*_4=-8H  
    %           z(idx) = y(:,k); ).HYW _Yih  
    %           subplot(4,7,Nplot(k)) dZ'hTzw~  
    %           pcolor(x,x,z), shading interp HhkubG)\  
    %           set(gca,'XTick',[],'YTick',[]) zb/w^~J_i  
    %           axis square ^ s< p5V  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _l}"gUtiw  
    %       end L7i^?40  
    % ?0HPd5=<v  
    %   See also ZERNPOL, ZERNFUN2. sr(f9Vl  
    C"w>U   
    %   Paul Fricker 11/13/2006 ,<]X0;~oB  
    |ho|Kl `=  
    ao>`[-  
    % Check and prepare the inputs: K1c@]]y)  
    % ----------------------------- <a_Q1 l  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) f(Jz*el S  
        error('zernfun:NMvectors','N and M must be vectors.') Y/Yp+W6n  
    end z Qtg]@S  
    -' 7I|r  
    if length(n)~=length(m) > D:( HWL  
        error('zernfun:NMlength','N and M must be the same length.') J6}J/  
    end S0+nQM%  
    j_2-  
    n = n(:); Dk&@AjJga  
    m = m(:); 8jyg1NN D  
    if any(mod(n-m,2)) qF!oP  
        error('zernfun:NMmultiplesof2', ... *G|w#-\.c  
              'All N and M must differ by multiples of 2 (including 0).') JGjqBuz#A*  
    end kI5`[\  
     h"<-^=b  
    if any(m>n) &sJZSrk|  
        error('zernfun:MlessthanN', ... !9+xKr99  
              'Each M must be less than or equal to its corresponding N.') 6`$HBX%.K  
    end 8t3,}}TJ  
    [43:E*\$  
    if any( r>1 | r<0 ) >q{E9.~b  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Q)}_S@v|%  
    end 9Yg=4>#$  
    <4!SQgL  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) e)I-|Q4^%  
        error('zernfun:RTHvector','R and THETA must be vectors.') ]mEY/)~7  
    end Vo*38c2  
    V> K sbPqR  
    r = r(:); We]mm3M3  
    theta = theta(:); MH;5gC@ `  
    length_r = length(r); \%fl`+`  
    if length_r~=length(theta) ,[6N64fy  
        error('zernfun:RTHlength', ... 7VWq8FH`  
              'The number of R- and THETA-values must be equal.') |y+<|fb,a  
    end $6~ J#;  
     XI+m  
    % Check normalization: A1{ 7g<k6  
    % -------------------- Ji<^s@8Zc  
    if nargin==5 && ischar(nflag) 8 /3`rEW  
        isnorm = strcmpi(nflag,'norm'); e RiPC  
        if ~isnorm Qs(WyP#  
            error('zernfun:normalization','Unrecognized normalization flag.') y8/ 7@qw  
        end saMv.;s 1^  
    else  [o]^\a y  
        isnorm = false; =b+W*vUAw  
    end r=8(n<;Co  
    IBYRuaEB  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?2D1gjr  
    % Compute the Zernike Polynomials C(( 7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Er; @nOyD  
    tBSHMz  
    % Determine the required powers of r: y_bb//IAG  
    % ----------------------------------- i|zs Li/  
    m_abs = abs(m); |TCHPKN  
    rpowers = []; QH:PClW![  
    for j = 1:length(n) -*;-T9  
        rpowers = [rpowers m_abs(j):2:n(j)]; Rlvb@aXgy  
    end }cDw9;~D  
    rpowers = unique(rpowers); m:EO}ws=  
    yQ5F'.m9e  
    % Pre-compute the values of r raised to the required powers, * !4r}h`  
    % and compile them in a matrix: <w@ziUr  
    % ----------------------------- j*uc$hC"  
    if rpowers(1)==0 FbxrBM  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); p$r=jF&  
        rpowern = cat(2,rpowern{:}); /b3b0VfF  
        rpowern = [ones(length_r,1) rpowern]; QIZ }7  
    else k+8K[ ?K-  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); u,]?_bK)  
        rpowern = cat(2,rpowern{:}); qY^OO~[  
    end Kh{C$b  
    ,Jqi J?,4C  
    % Compute the values of the polynomials: _M.7%k/U8  
    % -------------------------------------- KMFvi_8  
    y = zeros(length_r,length(n)); N%8O9Dp8;  
    for j = 1:length(n) ~j}7Fre  
        s = 0:(n(j)-m_abs(j))/2;  U/v }4b  
        pows = n(j):-2:m_abs(j); 5[^pU$Y  
        for k = length(s):-1:1 `~${fs{-`/  
            p = (1-2*mod(s(k),2))* ... C'4gve 7!  
                       prod(2:(n(j)-s(k)))/              ... Y", :u@R  
                       prod(2:s(k))/                     ... ["N{6d&Q  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ,Mt/*^|  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 5i 56J1EC  
            idx = (pows(k)==rpowers); !U}dYB:O  
            y(:,j) = y(:,j) + p*rpowern(:,idx); NkWU5E!  
        end rnB-e?>  
         :el]IH  
        if isnorm 3ya_47D  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); .nXOv]  
        end eUa2"=M  
    end @.JhL[f  
    % END: Compute the Zernike Polynomials njO5 YYOu  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nJEm&"AI  
    ,yZvT7  
    % Compute the Zernike functions: KW&5&~)2  
    % ------------------------------ XJ\ j0  
    idx_pos = m>0; \EP<r  
    idx_neg = m<0; lO?dI=}]  
    r!DUsE  
    z = y; 2(5HPRQ  
    if any(idx_pos) ;xp^F KP  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); xp+Z%0D  
    end Q?e]N I^  
    if any(idx_neg) N{6 - rR  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); M oIq)5/  
    end D;~c`G "f  
    $kc*~V~   
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) "mBX$t'gb  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. *)+K+J  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated U9uy (KOW  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive v61'fQ1Qg!  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, az5 $.  
    %   and THETA is a vector of angles.  R and THETA must have the same +W{ELdup%q  
    %   length.  The output Z is a matrix with one column for every P-value, DOk(5gR  
    %   and one row for every (R,THETA) pair. znNJ?  
    % s<}d)L(  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike t%xD epFQ  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) rD7L==Ld  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ynWF Y<VX  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 H<hFA(M  
    %   for all p. x *p>l !  
    % ]VarO'  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 rKK{*%n  
    %   Zernike functions (order N<=7).  In some disciplines it is KF'DOXBw>  
    %   traditional to label the first 36 functions using a single mode `pB]_"b  
    %   number P instead of separate numbers for the order N and azimuthal Rds_Cd C  
    %   frequency M. 0N" VOEvG  
    % m2j&v$  
    %   Example: "4 Lt:o4x  
    % sBsf{%I[{  
    %       % Display the first 16 Zernike functions SCXH{8SS  
    %       x = -1:0.01:1; G;^},%<  
    %       [X,Y] = meshgrid(x,x); f{m,?[1C,  
    %       [theta,r] = cart2pol(X,Y); WAzYnl'p  
    %       idx = r<=1; ]\fXy?2  
    %       p = 0:15; C`p)S`d  
    %       z = nan(size(X)); '+@q  
    %       y = zernfun2(p,r(idx),theta(idx)); @W{VT7w  
    %       figure('Units','normalized') L<GF1I)  
    %       for k = 1:length(p) .V4w+:i  
    %           z(idx) = y(:,k); 0`{3|g  
    %           subplot(4,4,k) #u<^  
    %           pcolor(x,x,z), shading interp fR:BF47  
    %           set(gca,'XTick',[],'YTick',[]) eM3-S=R?<g  
    %           axis square `|{6U"n  
    %           title(['Z_{' num2str(p(k)) '}']) X {["4  
    %       end UJ}Xa&*H\  
    % 7K*\F}2)q  
    %   See also ZERNPOL, ZERNFUN. s8/sH];  
    f{} zqCK  
    %   Paul Fricker 11/13/2006 {iz,iv/U  
    u]D>O$_ s  
    \R m2c8Z2  
    % Check and prepare the inputs: v#HaZT]u  
    % ----------------------------- J ejDF*Q  
    if min(size(p))~=1 ]bPj%sb*@  
        error('zernfun2:Pvector','Input P must be vector.') 3)? v  
    end 5BztOYn,  
    mnZS](>  
    if any(p)>35 UG]]Vk1d]  
        error('zernfun2:P36', ... yB>5p]$P  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Lq-Di|6q  
               '(P = 0 to 35).']) c h_1 -  
    end QG|KZ8uO  
    13:yaRo  
    % Get the order and frequency corresonding to the function number: 0%K/gd#S<  
    % ---------------------------------------------------------------- aMUy^>  
    p = p(:); /Ir|& <yB  
    n = ceil((-3+sqrt(9+8*p))/2); 't3@dz_dG  
    m = 2*p - n.*(n+2); =nq9)4o  
    Oq7R^t`b  
    % Pass the inputs to the function ZERNFUN: N33{vx  
    % ---------------------------------------- y .+d3  
    switch nargin 9PU9BYBG  
        case 3 t Q0vX@I<v  
            z = zernfun(n,m,r,theta); 17itC9U  
        case 4 500qg({2]  
            z = zernfun(n,m,r,theta,nflag); R5y+bMZ  
        otherwise dzK]F/L]  
            error('zernfun2:nargin','Incorrect number of inputs.') mt0ZD}E  
    end .U66Uet>RX  
    ?|&plf |  
    % 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) %(EUZu2  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. V4('}Q!  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of H%K,2/Nj  
    %   order N and frequency M, evaluated at R.  N is a vector of Kn#3^>D  
    %   positive integers (including 0), and M is a vector with the 7c:5 Ey  
    %   same number of elements as N.  Each element k of M must be a L5"|RI}  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) #4WA2EW  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 6l{=[\.Xa  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix @.4e^Km  
    %   with one column for every (N,M) pair, and one row for every \F|L y >g  
    %   element in R. Jkc1ih`^  
    % ,| \62B`  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- v7"Hvp3w  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is QQd%V#M?  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to [n53 eC  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 aS7[s6  
    %   for all [n,m]. Jb~nu  
    % )u. ut8![T  
    %   The radial Zernike polynomials are the radial portion of the `7;I*|  
    %   Zernike functions, which are an orthogonal basis on the unit *-!&5~o/U  
    %   circle.  The series representation of the radial Zernike \2rCT~x  
    %   polynomials is 7[V6@K!Al[  
    % .kBZ(`K  
    %          (n-m)/2 &DqeO8?Q  
    %            __ 8:BPXdiK  
    %    m      \       s                                          n-2s 5UFR^\e  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r I+) Acy;  
    %    n      s=0 (Ea)`'/  
    % QU&b5!;&  
    %   The following table shows the first 12 polynomials. Jy,Dcl  
    % Wcgy:4K3  
    %       n    m    Zernike polynomial    Normalization H:~41f[  
    %       --------------------------------------------- )xJo/{?  
    %       0    0    1                        sqrt(2) uW.)(l  
    %       1    1    r                           2 ^,Sl^ 9K  
    %       2    0    2*r^2 - 1                sqrt(6)  c`'2  
    %       2    2    r^2                      sqrt(6) ;2m<#~@0  
    %       3    1    3*r^3 - 2*r              sqrt(8) S?Y,sl+A:  
    %       3    3    r^3                      sqrt(8) }y-b<J ?H  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) jLULf+ 8&  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Q k`yK|(0=  
    %       4    4    r^4                      sqrt(10) cVzOW|NVx  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) G3j&8[  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) [qHLo>HaL  
    %       5    5    r^5                      sqrt(12) /<(d.6T[}:  
    %       --------------------------------------------- %8?s3^ o  
    % erG;M!9\  
    %   Example: zh hH A9  
    % \78w1Rkl  
    %       % Display three example Zernike radial polynomials |eEcEu?/b  
    %       r = 0:0.01:1; t~xp&LQiY  
    %       n = [3 2 5]; [!@&t:A  
    %       m = [1 2 1]; I_Q*uH.Y5  
    %       z = zernpol(n,m,r); M PMa  
    %       figure bK)gB!  
    %       plot(r,z) oGzZ.K3 A  
    %       grid on FFTh}>>  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ]jUxL=]r  
    % |[k/%  
    %   See also ZERNFUN, ZERNFUN2. 8:j8>K*6  
    cLN(yL  
    % A note on the algorithm. r^uo7?gZ^  
    % ------------------------ *mQOW]x%  
    % The radial Zernike polynomials are computed using the series LB<,(dyh  
    % representation shown in the Help section above. For many special ^cYm.EHI  
    % functions, direct evaluation using the series representation can giy4<  
    % produce poor numerical results (floating point errors), because x,Im%!h  
    % the summation often involves computing small differences between =K<8X!xUW  
    % large successive terms in the series. (In such cases, the functions 8ODrW!o  
    % are often evaluated using alternative methods such as recurrence G)A5;u\P9  
    % relations: see the Legendre functions, for example). For the Zernike P^-tGo!  
    % polynomials, however, this problem does not arise, because the } k[gR I]  
    % polynomials are evaluated over the finite domain r = (0,1), and zOR  
    % because the coefficients for a given polynomial are generally all r)jj]$0  
    % of similar magnitude. Bc {#ia  
    % 90%alG 1>y  
    % ZERNPOL has been written using a vectorized implementation: multiple )'{:4MX  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] MB,;HeP!  
    % values can be passed as inputs) for a vector of points R.  To achieve 8$fiq}a  
    % this vectorization most efficiently, the algorithm in ZERNPOL kRBPl9 9  
    % involves pre-determining all the powers p of R that are required to C7ZU)MEUd/  
    % compute the outputs, and then compiling the {R^p} into a single ,B>Rc#  
    % matrix.  This avoids any redundant computation of the R^p, and +tz^ &(  
    % minimizes the sizes of certain intermediate variables. dP(*IOO.  
    % h9)QQPP  
    %   Paul Fricker 11/13/2006 a7#J af  
    ~F`t[p  
    rC>')`uk  
    % Check and prepare the inputs: 9}`A_KzFx  
    % ----------------------------- L]o 5=K  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ;;E "+.  
        error('zernpol:NMvectors','N and M must be vectors.') Lh~Ym<CeN  
    end z5CZ!"&v  
    ;C8'7  
    if length(n)~=length(m) )-FQ_K%  
        error('zernpol:NMlength','N and M must be the same length.') Z:|2PQ4  
    end 7d0E9t;W  
    Z6<vLc  
    n = n(:); f Q.ea#xh^  
    m = m(:); !$'s?rnh  
    length_n = length(n); [c@14]e  
    *N"bn'>3  
    if any(mod(n-m,2)) 0:^L>MO  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') YShtoaCx>  
    end GI}h )T  
    :%Dw3IrOM  
    if any(m<0) t 7D2k2x9  
        error('zernpol:Mpositive','All M must be positive.') EQ&E C  
    end v_*E:E  
    :U`8s#  
    if any(m>n) @b=b>V[d6  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') {q2H_H  
    end ^a@Vn\V1  
    W! FmC$Kc  
    if any( r>1 | r<0 ) 9k_3=KS3N  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') /IM5#M5~  
    end `, ?T;JRc  
    p1p4t40<l  
    if ~any(size(r)==1) I(r^q"  
        error('zernpol:Rvector','R must be a vector.') K;2tY+I  
    end RaFk/mSw  
    ': Gk~   
    r = r(:); =4 &/Pr  
    length_r = length(r); _s./^B_w!  
    |2!/<%Yr`  
    if nargin==4 8o~<\eF%  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); )J+vmY~&  
        if ~isnorm )2mi6[qs0l  
            error('zernpol:normalization','Unrecognized normalization flag.') *7hr3x  
        end 4NxtU/5-sU  
    else '\dau>  
        isnorm = false; *ms?UFV[r  
    end Dqu1!f  
    LQSno)OZ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >S5:zz\  
    % Compute the Zernike Polynomials )+[IR  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dX0A(6  
    [#H$@g|CT  
    % Determine the required powers of r: :0pxacD"!  
    % ----------------------------------- D,+I)-k<  
    rpowers = []; c$ Kn.<a  
    for j = 1:length(n) "V:B-q  
        rpowers = [rpowers m(j):2:n(j)]; ]* -9zo0  
    end ulsr)Ik  
    rpowers = unique(rpowers); GE=#8-@g~p  
    Owalt4}C  
    % Pre-compute the values of r raised to the required powers, W&y%fd\&3  
    % and compile them in a matrix: @AL,@P/9=  
    % ----------------------------- VF=$'Bl|  
    if rpowers(1)==0 bh:;ovH  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); =vBxwa^  
        rpowern = cat(2,rpowern{:}); r'fNQJ >  
        rpowern = [ones(length_r,1) rpowern]; ( 3IM7  
    else pU)wxv[~  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); mfLS< /A  
        rpowern = cat(2,rpowern{:}); Y;S+2])R2  
    end >L?)f3_a  
    \}t(g}7T  
    % Compute the values of the polynomials: nK Rx_D$d  
    % -------------------------------------- iUqL /  
    z = zeros(length_r,length_n); waXA%u50  
    for j = 1:length_n (`gqLPx[  
        s = 0:(n(j)-m(j))/2; S'vi +_  
        pows = n(j):-2:m(j); YD$fN"}-  
        for k = length(s):-1:1 xtN%v0ZZ  
            p = (1-2*mod(s(k),2))* ... @Y*ONnl  
                       prod(2:(n(j)-s(k)))/          ... ws4a(1  
                       prod(2:s(k))/                 ... ?f[#O&#  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... VN|P(S6  
                       prod(2:((n(j)+m(j))/2-s(k))); 3]0ETcT  
            idx = (pows(k)==rpowers); R@t?!`f!+  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 2 5h.u>6@{  
        end $I!vQbi  
         u*Eb4  
        if isnorm k2N[B(&4J  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 71nXROB  
        end S/ ]2Qt#T  
    end 9f;\fe  
    hf>JW[>Xo  
    % 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)  dJxdrs  
    MIma:N_c  
    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。 @i2"+_}*  
    ;]c@%LX  
    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)