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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 fhAK^@h  
    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! s:y=X$&M  
     
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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 ,o}!pQ  
    function z = zernfun(n,m,r,theta,nflag) SB1\SNB  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. /s>ZT8vaAs  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N qTnfiYG}  
    %   and angular frequency M, evaluated at positions (R,THETA) on the zlmb_akJ  
    %   unit circle.  N is a vector of positive integers (including 0), and 'Lft\.C  
    %   M is a vector with the same number of elements as N.  Each element AfG!(AF`  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) |*0oz=  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, `Njv#K} U  
    %   and THETA is a vector of angles.  R and THETA must have the same 1o7 pMp=  
    %   length.  The output Z is a matrix with one column for every (N,M) AAkdwo  
    %   pair, and one row for every (R,THETA) pair. zm}4=Kz}  
    % %Ysu613mz  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 2P8JLT*Tj  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), $Xw .iN]g  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral W xyQA:3s  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 7'_zJI^  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized O^I~d{M 5I  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. wxARD3%  
    % P.P3/,  
    %   The Zernike functions are an orthogonal basis on the unit circle. x"~F=jT  
    %   They are used in disciplines such as astronomy, optics, and LMWcF'l  
    %   optometry to describe functions on a circular domain. SI3ek9|XU  
    % lztPexyXZ  
    %   The following table lists the first 15 Zernike functions. HHD4#XcU  
    % _JA.~edqM  
    %       n    m    Zernike function           Normalization Zr_{Z@IpU  
    %       -------------------------------------------------- 2f>lgZ!  
    %       0    0    1                                 1 gEtD qq~y@  
    %       1    1    r * cos(theta)                    2 Xd>4n7nb$`  
    %       1   -1    r * sin(theta)                    2 p%CAicn  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) N\Bygjw|  
    %       2    0    (2*r^2 - 1)                    sqrt(3) =*qu:f\y  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 6#O n .Q  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) vbmSbZ"y  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 0 ]U ;5  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Xvm.Un< N  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Gd`qZqx#  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) A5tY4?|  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Nhn5 iN1*  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 'i_od|19~h  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /] ce?PPC  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Qv,|*bf  
    %       -------------------------------------------------- =M)>w4-  
    % +/7UM x1  
    %   Example 1: D{h1"q  
    % zTBr<:  
    %       % Display the Zernike function Z(n=5,m=1) x`w 4LF  
    %       x = -1:0.01:1; [[QrGJr  
    %       [X,Y] = meshgrid(x,x); X^#48*"a  
    %       [theta,r] = cart2pol(X,Y); *'vX:n&t  
    %       idx = r<=1; ;14[)t$  
    %       z = nan(size(X)); /s(/6~D|  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); QP)-O*+AA  
    %       figure ,IxAt&kN  
    %       pcolor(x,x,z), shading interp ~d ~$fR  
    %       axis square, colorbar 3'O+  
    %       title('Zernike function Z_5^1(r,\theta)') PkQuN;a  
    % 3k5OYUk  
    %   Example 2: eCMcr !.  
    % ]x?9lQ1&  
    %       % Display the first 10 Zernike functions zF.rsNY  
    %       x = -1:0.01:1; RS#)uC5/%  
    %       [X,Y] = meshgrid(x,x); gAC}  
    %       [theta,r] = cart2pol(X,Y); >IC.Zt@  
    %       idx = r<=1; ||cG/I&,  
    %       z = nan(size(X)); Wu<  
    %       n = [0  1  1  2  2  2  3  3  3  3]; BQmg$N,F  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; QS,IM >Nr  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; VjSb>k   
    %       y = zernfun(n,m,r(idx),theta(idx)); @3c5"  
    %       figure('Units','normalized') y'xB? >|  
    %       for k = 1:10 3 zp)!QJi  
    %           z(idx) = y(:,k); Y<X%'Wd\  
    %           subplot(4,7,Nplot(k)) li8l+5d q  
    %           pcolor(x,x,z), shading interp Am%zEt$c  
    %           set(gca,'XTick',[],'YTick',[]) EQ8jxr<p  
    %           axis square hAHl+q)w?  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ;#P@(ZVT  
    %       end ^.&uYF&  
    % _+N*4  
    %   See also ZERNPOL, ZERNFUN2. HlBw:D(z:^  
    dY68wW>d|  
    %   Paul Fricker 11/13/2006 .6+j&{WNo!  
    bdk"7N  
    9kuL1tcY  
    % Check and prepare the inputs: U")~bU  
    % ----------------------------- 7gfNe kr~W  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }k.-xaj  
        error('zernfun:NMvectors','N and M must be vectors.') )}hp[*C  
    end I1BVqIt1i  
    ez&v"J  
    if length(n)~=length(m) |8 c3%jve  
        error('zernfun:NMlength','N and M must be the same length.') vr/V_  
    end n'v[[bmu  
    a[]=*(AZI  
    n = n(:); *4Y1((1k  
    m = m(:); N\l\ M  
    if any(mod(n-m,2)) Zk"'x,]#  
        error('zernfun:NMmultiplesof2', ... 6E{HNPMb>  
              'All N and M must differ by multiples of 2 (including 0).') Uc>kCBCd  
    end SN(:\|f 2  
    ZK1d3  
    if any(m>n) EA|*|o4)  
        error('zernfun:MlessthanN', ... "n," >  
              'Each M must be less than or equal to its corresponding N.') IkFrzw p  
    end WW\u}z.QJ  
    'U.)f@L#w  
    if any( r>1 | r<0 ) n'9Wl'  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') /@X!  
    end T=(/n=  
    rS\j9@=Y4  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) "6 |j 0?Q  
        error('zernfun:RTHvector','R and THETA must be vectors.') tq H7M0Ry  
    end v{Al>v}}n  
    P{i\x#  
    r = r(:); #wK {G)J  
    theta = theta(:); vm"LPwSk>  
    length_r = length(r); c [sydl  
    if length_r~=length(theta) 5,})x]'x  
        error('zernfun:RTHlength', ... -;20|US)u  
              'The number of R- and THETA-values must be equal.') Zy|B~.@<j  
    end 9+ nB;vA  
    C$(US8:{  
    % Check normalization: }pdn-#  
    % -------------------- NQz*P.q  
    if nargin==5 && ischar(nflag) K#_&}C^-jY  
        isnorm = strcmpi(nflag,'norm'); Gole7I  
        if ~isnorm Bha#=>4FU  
            error('zernfun:normalization','Unrecognized normalization flag.') zsFzF`[k  
        end u,AP$+Qk  
    else a\>+!Vq  
        isnorm = false; Xyy;BO:  
    end H C(Vu  
    Q@?8-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C]414Ibi  
    % Compute the Zernike Polynomials < aJl i   
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0uV3J  
    g5M=$y/H  
    % Determine the required powers of r: Yz]c'M@  
    % ----------------------------------- AD K)p?  
    m_abs = abs(m); `qnp   
    rpowers = []; 7aRtw:PQn  
    for j = 1:length(n) S "'0l S   
        rpowers = [rpowers m_abs(j):2:n(j)]; qmqWMLfC  
    end 0b6jGa  
    rpowers = unique(rpowers); TwlX'iI_;  
    FlGU1%]m  
    % Pre-compute the values of r raised to the required powers, 6D|[3rXr  
    % and compile them in a matrix: 0`c|ZzY  
    % ----------------------------- SQ8xfD*  
    if rpowers(1)==0 vz5x{W  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5{Q5?M]  
        rpowern = cat(2,rpowern{:}); })W9=xO~  
        rpowern = [ones(length_r,1) rpowern]; V5:ad  
    else 2 j.6  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8C]K36q  
        rpowern = cat(2,rpowern{:}); h ` qlI1]  
    end */c4b:s  
    >*s_)IH2  
    % Compute the values of the polynomials: k%uR!cL  
    % -------------------------------------- WX .Ax$fT  
    y = zeros(length_r,length(n)); %"-bG'Yc  
    for j = 1:length(n) "| Oj!&0  
        s = 0:(n(j)-m_abs(j))/2; m}A|W[p<  
        pows = n(j):-2:m_abs(j); A12EUr5$  
        for k = length(s):-1:1 A,67)li3  
            p = (1-2*mod(s(k),2))* ... 9gq+,g>E_  
                       prod(2:(n(j)-s(k)))/              ... 2[|52+zhc  
                       prod(2:s(k))/                     ... `#HtVI  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... V=^B7a.;>  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); F!7dGa$  
            idx = (pows(k)==rpowers); ezimQ  
            y(:,j) = y(:,j) + p*rpowern(:,idx); (P!r^87  
        end r$[`A_  
         '41'Gn  
        if isnorm aeZ$Wu>]W  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); YI+ clh;%9  
        end "&Hr)yyWG  
    end (4o<U%3kGq  
    % END: Compute the Zernike Polynomials 88Nx/:#Y*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8\WV.+  
    #[ f]-c(!  
    % Compute the Zernike functions: Z(j"\d!y  
    % ------------------------------ Hg&.U;n  
    idx_pos = m>0; ^'9.VVyz  
    idx_neg = m<0; /RVwhA+c  
    PRJ  
    z = y; ~c,CngeL0  
    if any(idx_pos) 8Q%g<jX*  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); >|X )  
    end vB74r]'F  
    if any(idx_neg) |I[/Fl:  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); yPrF2@#XZ/  
    end 6VUs:iO1j5  
    \?v?%}x  
    % 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) li hIPMU  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. y2W|,=Vd  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated /#WvC;B  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive @(bg#  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, aFaioE#h(  
    %   and THETA is a vector of angles.  R and THETA must have the same _9g-D9  
    %   length.  The output Z is a matrix with one column for every P-value, hkb&]XWi[  
    %   and one row for every (R,THETA) pair. -MRX@a^1  
    % 9X?RJ."J  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ,ZghV1z  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 6hMKAk  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 2E8G 5?qe)  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 f8 BZkh  
    %   for all p. v,C~5J3h)  
    % Sn S$5o  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 6P3h955c  
    %   Zernike functions (order N<=7).  In some disciplines it is ZIKSHC9  
    %   traditional to label the first 36 functions using a single mode jDb"|l  
    %   number P instead of separate numbers for the order N and azimuthal T|8:_4/l  
    %   frequency M. ;L,i">_%u[  
    % zYrJ Hn#vB  
    %   Example: o$eo\X?J?  
    % )=#e*1!b  
    %       % Display the first 16 Zernike functions []v$QR&u#v  
    %       x = -1:0.01:1; hq&|   
    %       [X,Y] = meshgrid(x,x); ue^HhZ9  
    %       [theta,r] = cart2pol(X,Y); h%U}Y5Ps~  
    %       idx = r<=1; [GPCd@  
    %       p = 0:15; Y)@Y$_  
    %       z = nan(size(X)); s7afj t  
    %       y = zernfun2(p,r(idx),theta(idx)); ^2H;  
    %       figure('Units','normalized') |h }4J  
    %       for k = 1:length(p) ZNne 8  
    %           z(idx) = y(:,k); n$`+03a  
    %           subplot(4,4,k) -#v1/L/=  
    %           pcolor(x,x,z), shading interp 99.F'Gz  
    %           set(gca,'XTick',[],'YTick',[]) ~o#mX?'7  
    %           axis square -%5#0Ogh M  
    %           title(['Z_{' num2str(p(k)) '}']) .%y'q!?  
    %       end pHuR_U5*?  
    % }K8e(i6z  
    %   See also ZERNPOL, ZERNFUN. HCsd$M;Hbv  
    AT)b/ycC  
    %   Paul Fricker 11/13/2006 jz`3xFy *]  
    I?S t}Tl  
    k_{?{:X;y  
    % Check and prepare the inputs: }tw+8YWkz  
    % ----------------------------- *L9v(Kc  
    if min(size(p))~=1 F)KR8 (  
        error('zernfun2:Pvector','Input P must be vector.') 0PqI^|!  
    end 'da 'WZG  
    6_<~]W&  
    if any(p)>35  od{\z  
        error('zernfun2:P36', ... iMt3h8  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... e@qH!.g)  
               '(P = 0 to 35).']) O^3kPVr  
    end 4uzMO<  
    D HT^.UM28  
    % Get the order and frequency corresonding to the function number: `<yQ`Y_X  
    % ---------------------------------------------------------------- gs;^SRE I  
    p = p(:); O >pv/Ns  
    n = ceil((-3+sqrt(9+8*p))/2); ^P^%Q)QXl  
    m = 2*p - n.*(n+2); @J&korU  
    C+uW]]~I)  
    % Pass the inputs to the function ZERNFUN: t))MZw&@  
    % ---------------------------------------- m0 As t<u  
    switch nargin EwX&Cj".  
        case 3 w8>h6x "  
            z = zernfun(n,m,r,theta); 5e$1KN`  
        case 4 \7i_2|w  
            z = zernfun(n,m,r,theta,nflag); u1L^INo/  
        otherwise Jn^b}bk t  
            error('zernfun2:nargin','Incorrect number of inputs.') QOo'Iv+EL  
    end Vn4wk>b}$2  
    &:g:7l]g  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) KF00=HE|]  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. xy[#LX)RW  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of /3,Lp-kp  
    %   order N and frequency M, evaluated at R.  N is a vector of NDP" @  
    %   positive integers (including 0), and M is a vector with the :${tts2g  
    %   same number of elements as N.  Each element k of M must be a Q0Ft.b  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) VwE4:/7YN  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is >< $LV&  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix d(o=)!p  
    %   with one column for every (N,M) pair, and one row for every ![^pAEgx  
    %   element in R. uy'seJ  
    % Zt!A!Afu  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- lb. Q^TghU  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 4}h}`KZZ  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to }I&.xzJ  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 e4YP$}_L  
    %   for all [n,m]. Ctz#9[|  
    % qK a}O*  
    %   The radial Zernike polynomials are the radial portion of the :.,9}\LK  
    %   Zernike functions, which are an orthogonal basis on the unit o=3hWbe  
    %   circle.  The series representation of the radial Zernike O`9c!_lis  
    %   polynomials is `SFeln{1B  
    % cdt9hH`Cd  
    %          (n-m)/2 V_gl#e#  
    %            __ Bzrnmz5S  
    %    m      \       s                                          n-2s cK+TE8ao  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r (QiA5!wg  
    %    n      s=0 g0tnt)]  
    % !k)6r6  
    %   The following table shows the first 12 polynomials. +:.Jl:fx4  
    % aDK b78 1d  
    %       n    m    Zernike polynomial    Normalization 8|i'~BFHs  
    %       --------------------------------------------- +-^>B%/&Z  
    %       0    0    1                        sqrt(2) 1IA1;  
    %       1    1    r                           2 ^m w]u"5\  
    %       2    0    2*r^2 - 1                sqrt(6) dT|f<E/P  
    %       2    2    r^2                      sqrt(6) /h0bBP  
    %       3    1    3*r^3 - 2*r              sqrt(8) ZwS:Te9-  
    %       3    3    r^3                      sqrt(8) tk%f_"}  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) iYStl  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) }tW-l*\U  
    %       4    4    r^4                      sqrt(10) eBrNhE-[G]  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ,x8;| o5  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 7y'":1  
    %       5    5    r^5                      sqrt(12) w(Z?j%b  
    %       --------------------------------------------- JXK\mah  
    % y&zFS4"x  
    %   Example: dH^6K0J  
    % *y*tI}  
    %       % Display three example Zernike radial polynomials "tz6O0D  
    %       r = 0:0.01:1; Y<xqws  
    %       n = [3 2 5]; N'v3 |g  
    %       m = [1 2 1]; U>E: Ub0r  
    %       z = zernpol(n,m,r); 1MLL  
    %       figure 2tq2   
    %       plot(r,z) m^D'p  
    %       grid on tK%ie\  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') %":3xj'EEI  
    % ?G,4N<]Nu  
    %   See also ZERNFUN, ZERNFUN2. !DZ=`a?y  
    egaX[ j r  
    % A note on the algorithm. jSY[Y:6md  
    % ------------------------ 1>J.kQR^  
    % The radial Zernike polynomials are computed using the series p R'J4~  
    % representation shown in the Help section above. For many special ,n/]ALz>~  
    % functions, direct evaluation using the series representation can f^$,;  
    % produce poor numerical results (floating point errors), because Qg*\aa94  
    % the summation often involves computing small differences between SyvoN, ;Q  
    % large successive terms in the series. (In such cases, the functions J/je/PC  
    % are often evaluated using alternative methods such as recurrence M~:_^B  
    % relations: see the Legendre functions, for example). For the Zernike m TE(J Zt  
    % polynomials, however, this problem does not arise, because the 9F/I",EA  
    % polynomials are evaluated over the finite domain r = (0,1), and "\b>JV5  
    % because the coefficients for a given polynomial are generally all %Rk|B`ST  
    % of similar magnitude. BsQ;`2  
    % GE/!$3  
    % ZERNPOL has been written using a vectorized implementation: multiple Pd91<L  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] +U o NJ   
    % values can be passed as inputs) for a vector of points R.  To achieve 4\;zz8 5E  
    % this vectorization most efficiently, the algorithm in ZERNPOL 9{u8fDm!  
    % involves pre-determining all the powers p of R that are required to 2)f_L|o,m  
    % compute the outputs, and then compiling the {R^p} into a single Y Zj-%5  
    % matrix.  This avoids any redundant computation of the R^p, and nGF +a[Z  
    % minimizes the sizes of certain intermediate variables. 1sqE/-v1_^  
    % TA[%eMvA  
    %   Paul Fricker 11/13/2006 ?xj8a3F  
    u H[WlZ4  
    Rt8[P6e"q  
    % Check and prepare the inputs: ?:;;0kSk  
    % ----------------------------- V\L;EHtc$  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) tu -a`h_NJ  
        error('zernpol:NMvectors','N and M must be vectors.') ,h*gd^i  
    end n7!T{+ge  
    A,~3oQV  
    if length(n)~=length(m) S#/BWNz|  
        error('zernpol:NMlength','N and M must be the same length.') 8M5)fDu*?  
    end $BwWQ?lp  
    % N8I'*u  
    n = n(:); P#O" {+`  
    m = m(:); <o(;~  
    length_n = length(n); hG1$YE  
    WyO*8b_ D  
    if any(mod(n-m,2)) v vErzUxN  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') pv]@}+<Dt  
    end ziM{2Fs>  
    T)! }Wvv  
    if any(m<0) ;8]HCC@:  
        error('zernpol:Mpositive','All M must be positive.') PL:(Se%  
    end gT)(RS`_)  
    B"43o7C  
    if any(m>n) `c Gks  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') DG TLlBkT  
    end mA(kq   
    TNu% _ 34  
    if any( r>1 | r<0 ) ?0Q3F  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') l#0zHBc  
    end eb_.@.a  
    ('z=/"(l  
    if ~any(size(r)==1) Z518J46o  
        error('zernpol:Rvector','R must be a vector.') QV[&2&&^<<  
    end FWW4n_74  
    ufL,K q4  
    r = r(:); ?_]Y8f  
    length_r = length(r); s\*p|vc  
    qCI&H7u@  
    if nargin==4 RZz?_1'  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); !@z9n\Yj  
        if ~isnorm 01n!T2;yW}  
            error('zernpol:normalization','Unrecognized normalization flag.') !.R-|<2|6  
        end sUF$eVAT  
    else eu(Fhs   
        isnorm = false; DwBe_h.  
    end O@$>'Z  
    =]@Bc 7@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `q}D#0  
    % Compute the Zernike Polynomials r9f- C  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TXB!Y!RG#  
    (u?s@/e:`/  
    % Determine the required powers of r: m-{DhJV  
    % ----------------------------------- \KV.lG!  
    rpowers = []; kHK<~srB  
    for j = 1:length(n) I(6%'s2  
        rpowers = [rpowers m(j):2:n(j)]; +C=vuR  
    end }-[l)<F:  
    rpowers = unique(rpowers); g!0 j1  
    wU%uO/sU9  
    % Pre-compute the values of r raised to the required powers, oypLE=H  
    % and compile them in a matrix: >Iij,J5i  
    % ----------------------------- (CQ! &Z8  
    if rpowers(1)==0 <(E)M@2  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); q;SD+%tI  
        rpowern = cat(2,rpowern{:}); "|6(.S+o  
        rpowern = [ones(length_r,1) rpowern]; 9^Xndo]y  
    else mpYBMSLM  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); #o&T$D5  
        rpowern = cat(2,rpowern{:}); <@7j37,R7V  
    end C5=^cH8  
    1XS~b-St  
    % Compute the values of the polynomials: ^ iu)vED  
    % -------------------------------------- |mhKD#:  
    z = zeros(length_r,length_n); XzAXcxC6G  
    for j = 1:length_n yc0 1\o  
        s = 0:(n(j)-m(j))/2; #mH28UT  
        pows = n(j):-2:m(j); ejg!1*H@n  
        for k = length(s):-1:1 |(~IfSE2  
            p = (1-2*mod(s(k),2))* ... 7:~3B-Tb  
                       prod(2:(n(j)-s(k)))/          ... `y'%dY}$n  
                       prod(2:s(k))/                 ... _~Lu%   
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ,$]m1|t@z  
                       prod(2:((n(j)+m(j))/2-s(k))); ;$eY#ypx  
            idx = (pows(k)==rpowers); T #E{d  
            z(:,j) = z(:,j) + p*rpowern(:,idx); e ,XT(KY  
        end &'\-M6GW  
         K%9!1'  
        if isnorm ?r;F'%N=  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); %~eu&\os  
        end Xk:x=4u&  
    end SP0ueAa}  
    6@Q; LV+  
    % 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)  r`CsR0[  
    ||kUi=5  
    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。 ER$qL"H U  
    GZqy.AE,  
    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)