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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 {gkwOMW  
    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! j]> uZalr  
     
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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 AF[>fMI  
    function z = zernfun(n,m,r,theta,nflag) x^2 W?<  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. %c0z)R~  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N { y/-:=S)A  
    %   and angular frequency M, evaluated at positions (R,THETA) on the hT=f;6$  
    %   unit circle.  N is a vector of positive integers (including 0), and (w2(qT&O  
    %   M is a vector with the same number of elements as N.  Each element j];G*-iv{  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 51/sTx<Z}  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, J{H?xc o  
    %   and THETA is a vector of angles.  R and THETA must have the same *.dKR  
    %   length.  The output Z is a matrix with one column for every (N,M) r /yHmEk&  
    %   pair, and one row for every (R,THETA) pair. `r.N  
    %  7kM4Ei  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike R9E6uz.j  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), {kG;."S+K  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral \)GR\~z0h  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, )8]3kQffJ=  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized UC#"=Xd 4  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. #XL`S  
    % 8q*";>*  
    %   The Zernike functions are an orthogonal basis on the unit circle. dk4D+*R  
    %   They are used in disciplines such as astronomy, optics, and =VCQ*  
    %   optometry to describe functions on a circular domain. w=$'Lt!  
    % '{W3j^m7  
    %   The following table lists the first 15 Zernike functions. \d$Rd")w  
    % UhA_1A'B  
    %       n    m    Zernike function           Normalization , #Ln/;  
    %       -------------------------------------------------- |P~q/Wff  
    %       0    0    1                                 1 L B<UC?e  
    %       1    1    r * cos(theta)                    2 @|]G0&gn&?  
    %       1   -1    r * sin(theta)                    2 Xiw@  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) G)4SWu0<t  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ytob/tc  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) F b2p(.  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ip674'bq7R  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) s%bUgO%&  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) :OX$LCi  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) lkN'uZ  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) [DL|Ht>  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `M6YblnJZ  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Ba<#1p7_  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) n8Q* _?Z/  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) m/KjJ"s,  
    %       -------------------------------------------------- _Z0\`kba+  
    % ' me:Zd  
    %   Example 1: {[N?+ZJD*L  
    % |thad!?  
    %       % Display the Zernike function Z(n=5,m=1) a6P!Wzb  
    %       x = -1:0.01:1; " C&x ,Ic  
    %       [X,Y] = meshgrid(x,x); $oc9 |Q 7  
    %       [theta,r] = cart2pol(X,Y); BZ}`4W'  
    %       idx = r<=1; tz3]le|ml  
    %       z = nan(size(X)); ;i}i5yv2  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); )L|C'dJ<k`  
    %       figure G6<HO7\  
    %       pcolor(x,x,z), shading interp Qz# 3p3N?  
    %       axis square, colorbar 8Y7 @D$=w  
    %       title('Zernike function Z_5^1(r,\theta)') #*\Ry/9Q  
    % l-Fmn/V  
    %   Example 2: cJ2y)`  
    % y3Y2 QC(  
    %       % Display the first 10 Zernike functions # UjEY9"M  
    %       x = -1:0.01:1; \y@ eBW  
    %       [X,Y] = meshgrid(x,x); {GAsFnZk  
    %       [theta,r] = cart2pol(X,Y); ?${V{=)*X'  
    %       idx = r<=1; 4YBf ~Pp  
    %       z = nan(size(X)); iq,ah"L  
    %       n = [0  1  1  2  2  2  3  3  3  3]; aQxe)  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 3V"dG1?  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; f8R+7Ykx  
    %       y = zernfun(n,m,r(idx),theta(idx)); eS* *L 3  
    %       figure('Units','normalized') ktU9LW~  
    %       for k = 1:10 ?-4OfGN  
    %           z(idx) = y(:,k); 9x4wk*z  
    %           subplot(4,7,Nplot(k)) 8 H,_vf  
    %           pcolor(x,x,z), shading interp vi^z5n  
    %           set(gca,'XTick',[],'YTick',[]) [2=^C=52  
    %           axis square Pu1GCr(  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) )_X;9%L7  
    %       end 4$ ..r4@  
    % >\Z lZ  
    %   See also ZERNPOL, ZERNFUN2. G,+xT}@wu  
    -6(h@F%E  
    %   Paul Fricker 11/13/2006 bb*c+XN0  
    }{P&idkv  
    nR(#F9  
    % Check and prepare the inputs: i?lX,9%  
    % ----------------------------- G[ ,,L  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ="/R5fp  
        error('zernfun:NMvectors','N and M must be vectors.') jM{qRfOrg  
    end \o0z@Ntq  
    11PLH0  
    if length(n)~=length(m) ;|Y2r^c  
        error('zernfun:NMlength','N and M must be the same length.') Ar\IZ_Q  
    end I|GV :D  
    pHq{S;R2G  
    n = n(:); s4^[3|Zrr0  
    m = m(:); f<Va<TL6-  
    if any(mod(n-m,2)) !a.3OpQ  
        error('zernfun:NMmultiplesof2', ... hz&^_ G6`  
              'All N and M must differ by multiples of 2 (including 0).') ZJ;wRd@  
    end n%7A;l!{  
    ,| $|kO/  
    if any(m>n) %Y#[% ~|(  
        error('zernfun:MlessthanN', ... BnY\FQ)K  
              'Each M must be less than or equal to its corresponding N.') MBnK&GS  
    end |:!E HFr  
    JrY"J]/  
    if any( r>1 | r<0 ) 5JJg"yuY"  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') !~6'@UYo  
    end ~ nLkn#Z  
    2<`gs(oxXe  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) KfJ c  
        error('zernfun:RTHvector','R and THETA must be vectors.') pZni,< Q  
    end ^(E"3 c  
    I^rZgp<'i  
    r = r(:); }TXp<E"\  
    theta = theta(:); Enq6K1@%G  
    length_r = length(r); FCS5@l,'<  
    if length_r~=length(theta) `?Y_0Nh>  
        error('zernfun:RTHlength', ... oyi7YRvwd  
              'The number of R- and THETA-values must be equal.') NgDZ4&L  
    end f(w#LuW<  
    4GmSG,]  
    % Check normalization: -f-O2G=  
    % -------------------- ')Dp%"\?  
    if nargin==5 && ischar(nflag) p*(U*8Q  
        isnorm = strcmpi(nflag,'norm'); 6KBzlj0T+  
        if ~isnorm GN~[xXJU  
            error('zernfun:normalization','Unrecognized normalization flag.') x"zjN'|  
        end S'v V"  
    else RE(=! 8lGR  
        isnorm = false; B.CH9M  
    end ,?7xb]h  
    y~4SKv $  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q9g[+*9]$  
    % Compute the Zernike Polynomials 4EaS g#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @8 oDy$j  
    [~Z'xY y  
    % Determine the required powers of r: ,YAPCj  
    % ----------------------------------- 5kRwSOG%'  
    m_abs = abs(m); O,V6hU/ *  
    rpowers = []; GDNh?R  
    for j = 1:length(n) a V+o\fId  
        rpowers = [rpowers m_abs(j):2:n(j)]; S1x.pLHj8  
    end B~ 'VDOG$Z  
    rpowers = unique(rpowers); buxI-wv  
    <?=mLOo =  
    % Pre-compute the values of r raised to the required powers, ^R8U-V8:  
    % and compile them in a matrix: O[5_ 9W 4  
    % ----------------------------- pJ)+}vascR  
    if rpowers(1)==0 {YO%JTQ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 6S&=OK^  
        rpowern = cat(2,rpowern{:}); \h'E5LO  
        rpowern = [ones(length_r,1) rpowern]; GWA!Ab'<U  
    else >TQBRA;'  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8R??J>h5\  
        rpowern = cat(2,rpowern{:}); Ndug9j\2  
    end [iO$ c]!H  
    XYxm8ee"j  
    % Compute the values of the polynomials: N8MlT \+r  
    % -------------------------------------- 3Q!J9t5dc  
    y = zeros(length_r,length(n)); n'&`9M['%d  
    for j = 1:length(n) +{=_|3(  
        s = 0:(n(j)-m_abs(j))/2; 3A}nNHpN  
        pows = n(j):-2:m_abs(j); 44fq1<.K  
        for k = length(s):-1:1 Jv4D^>yj[  
            p = (1-2*mod(s(k),2))* ... C^\*|=*\  
                       prod(2:(n(j)-s(k)))/              ... r PRuSk-f  
                       prod(2:s(k))/                     ... 9,EaN{GM  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 5qtmb4R~  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); @7[.> I(  
            idx = (pows(k)==rpowers); ek;&<Z_ ]  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ah!O&ECh  
        end 5[j!\d}U  
         0Z) ;.l^  
        if isnorm %&=(,;d  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); mZ0oa-Iy  
        end ;MRC~F=  
    end !$KhL.4P  
    % END: Compute the Zernike Polynomials @BHS5^|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% QSs$   
    ?od}~G4s#  
    % Compute the Zernike functions: 1f pS"_}  
    % ------------------------------ mP$G9R  
    idx_pos = m>0; x 1xj\O  
    idx_neg = m<0; 3}#XA+Z  
    @;n$caw  
    z = y; |n6 Q  
    if any(idx_pos) kj3o1Y  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); }MavI'  
    end ^tKOxW# a  
    if any(idx_neg) 1-NX>E5  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); L..X)-D2 n  
    end wq_oh*"  
    ssJDaf79  
    % 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) o(>-:l i0  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ~q T1<k  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated L|1zHDxQ  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive h. (;GJO  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, F3 l^^ Mc  
    %   and THETA is a vector of angles.  R and THETA must have the same O"^a.`27  
    %   length.  The output Z is a matrix with one column for every P-value, PUZXmnB  
    %   and one row for every (R,THETA) pair. L,A-G"z0Z  
    % Is6']bYh  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike aq,)6P`  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) u r.T YKF  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) n `T[eb~  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 =O'%)Y&  
    %   for all p. rWfurB5f  
    % )>M@hIV5>  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Ce'2lo  
    %   Zernike functions (order N<=7).  In some disciplines it is m3xj5]#^$  
    %   traditional to label the first 36 functions using a single mode gL}Y5U+s  
    %   number P instead of separate numbers for the order N and azimuthal 8(/f!~  
    %   frequency M. OUk5c$M(  
    % i[\u-TF  
    %   Example: S1= JdN  
    % :+^$?[6]  
    %       % Display the first 16 Zernike functions zu*G4?]~h  
    %       x = -1:0.01:1; ApJf4D<V  
    %       [X,Y] = meshgrid(x,x); 6ym)F!t8l  
    %       [theta,r] = cart2pol(X,Y); d<'Yt|zt  
    %       idx = r<=1; MirBJL  
    %       p = 0:15; 8U:dgXz  
    %       z = nan(size(X)); tMBy ^@p  
    %       y = zernfun2(p,r(idx),theta(idx)); g7LW?Ewr  
    %       figure('Units','normalized') .d!*<`S|  
    %       for k = 1:length(p) Cl.T'A$  
    %           z(idx) = y(:,k); j"sO<Q{6%  
    %           subplot(4,4,k) u&_U CJCf  
    %           pcolor(x,x,z), shading interp [gdPHXs  
    %           set(gca,'XTick',[],'YTick',[]) })SdaZ  
    %           axis square L.:QI<n  
    %           title(['Z_{' num2str(p(k)) '}']) \ J:T]  
    %       end gI5nWEM0{  
    % N&h!14]{ Z  
    %   See also ZERNPOL, ZERNFUN. UYrzsUjg&  
    'I>#0VRr  
    %   Paul Fricker 11/13/2006 4bzn^  
    OwIy(ukTI  
    Jo$Dxa z  
    % Check and prepare the inputs: []3}(8yxGb  
    % -----------------------------  de47O  
    if min(size(p))~=1 *>$)#?t  
        error('zernfun2:Pvector','Input P must be vector.') 4^ 6L])y  
    end fToI,FA  
    _1c_TMh}9  
    if any(p)>35 6jo&i  
        error('zernfun2:P36', ... 6MNA.{Jdd  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... *9(1:N;#  
               '(P = 0 to 35).']) 9ufs6 z  
    end Z2jb>%  
    [gp:nxyfQm  
    % Get the order and frequency corresonding to the function number: TPFmSDq  
    % ---------------------------------------------------------------- /(pChY>  
    p = p(:); BIf].RY  
    n = ceil((-3+sqrt(9+8*p))/2); slfVQ809  
    m = 2*p - n.*(n+2); \o)4m[oF  
    u`@FA?+E1  
    % Pass the inputs to the function ZERNFUN: 2vQ^519  
    % ---------------------------------------- (+ anTA=  
    switch nargin a`iAA1HJ  
        case 3 I'b]s~u  
            z = zernfun(n,m,r,theta); .{Oq)^!ot  
        case 4 >! .9g  
            z = zernfun(n,m,r,theta,nflag); #de^~  
        otherwise DJ0T5VE W3  
            error('zernfun2:nargin','Incorrect number of inputs.') }c5`~ LLK  
    end 8mLU ~P |  
    E2kRt'~N  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) Asu"#sd  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 'FFc"lqj  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 3[Iw%% q  
    %   order N and frequency M, evaluated at R.  N is a vector of (SA*9%  
    %   positive integers (including 0), and M is a vector with the 3y,?>-  
    %   same number of elements as N.  Each element k of M must be a Ps\^OJR  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) :q1r2&ne  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is N&`ay{&`:  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ]t;5kj/  
    %   with one column for every (N,M) pair, and one row for every %WN2 xCSf  
    %   element in R. hz<J8'U  
    % ]!:Y]VYN)\  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- We?:DM [  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ZE` {J =,  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to >K%x44|  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 &[5az/Hj*  
    %   for all [n,m]. a"aV&t  
    % w,9F riW  
    %   The radial Zernike polynomials are the radial portion of the j3&*wU_  
    %   Zernike functions, which are an orthogonal basis on the unit musxX58%  
    %   circle.  The series representation of the radial Zernike 5K{h)* *5  
    %   polynomials is e*H$c?7NL  
    % 0{F.DDiNT  
    %          (n-m)/2 nVzo=+Yp  
    %            __ PM7/fv*,  
    %    m      \       s                                          n-2s CV"Y40  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 55p=veq \  
    %    n      s=0 `0:@`)&g1  
    % e,8-P-h~T  
    %   The following table shows the first 12 polynomials. Q,`kfxA`O  
    % _@2G]JD  
    %       n    m    Zernike polynomial    Normalization y9)",G!  
    %       --------------------------------------------- O]lfs >>x  
    %       0    0    1                        sqrt(2) {eUfwPAa3  
    %       1    1    r                           2 +)S X  
    %       2    0    2*r^2 - 1                sqrt(6) }}_l@5  
    %       2    2    r^2                      sqrt(6) T`sM4 VWqU  
    %       3    1    3*r^3 - 2*r              sqrt(8) rI/KrBM  
    %       3    3    r^3                      sqrt(8) ]U%Tm>s.  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) zhE7+``g  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) MzD0F#Y  
    %       4    4    r^4                      sqrt(10) ?f..N,s  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) f6 nltZ  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ^ZG1  
    %       5    5    r^5                      sqrt(12) HrGX-6`  
    %       --------------------------------------------- LKcrr;  
    % 9OUhV [D  
    %   Example: 4NV1v&"  
    % YP l{5 =  
    %       % Display three example Zernike radial polynomials gp=0;#4 4  
    %       r = 0:0.01:1; ~55>uw<  
    %       n = [3 2 5]; &&O=v]6,V  
    %       m = [1 2 1]; O5 SX"A  
    %       z = zernpol(n,m,r); ZV;yXLx|  
    %       figure x5ia<V>=d  
    %       plot(r,z) UlrY  
    %       grid on l<0V0R(  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ]mSVjF3l  
    % iQF93:#  
    %   See also ZERNFUN, ZERNFUN2. X!Q"p$D4(  
    7Y/_/t~Y  
    % A note on the algorithm. f$|v  
    % ------------------------ >nX'RE|F  
    % The radial Zernike polynomials are computed using the series V EzIWNV  
    % representation shown in the Help section above. For many special h*LIS@&9C5  
    % functions, direct evaluation using the series representation can EX_& wep@1  
    % produce poor numerical results (floating point errors), because WlUE&=|Oz2  
    % the summation often involves computing small differences between |UG)*t/  
    % large successive terms in the series. (In such cases, the functions yrw!b\  
    % are often evaluated using alternative methods such as recurrence Jp- hFD  
    % relations: see the Legendre functions, for example). For the Zernike Vs >1%$If  
    % polynomials, however, this problem does not arise, because the &3<]FK  
    % polynomials are evaluated over the finite domain r = (0,1), and !?{5ET,gtN  
    % because the coefficients for a given polynomial are generally all GfDA5v[  
    % of similar magnitude. 8J} J;Ga  
    % 1Q<a+ l  
    % ZERNPOL has been written using a vectorized implementation: multiple L6T_&AiL$  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] * 7CI q  
    % values can be passed as inputs) for a vector of points R.  To achieve $3>|R lxYA  
    % this vectorization most efficiently, the algorithm in ZERNPOL *d(Dk*(  
    % involves pre-determining all the powers p of R that are required to vJ!t.Vou  
    % compute the outputs, and then compiling the {R^p} into a single g:HIiGN0Ic  
    % matrix.  This avoids any redundant computation of the R^p, and rlD@O~P4  
    % minimizes the sizes of certain intermediate variables. "2mVW_k  
    % y}A-o_u@cD  
    %   Paul Fricker 11/13/2006 \ C Yu;  
    3I]5DW %-  
    5gGr|d|(  
    % Check and prepare the inputs: gIeo7>u  
    % ----------------------------- "LYob}_z  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) fw<'ygd  
        error('zernpol:NMvectors','N and M must be vectors.') {PZe!EQ  
    end t1kD5^  
    lG2){){j  
    if length(n)~=length(m) Ks4TBi&J   
        error('zernpol:NMlength','N and M must be the same length.') [30e>bSf`  
    end p~t$ll0s  
    @ B+];lr/-  
    n = n(:); - 0zo>[c/p  
    m = m(:); .fgoEB,(  
    length_n = length(n); Js'|N%pi  
    :H~r _>E  
    if any(mod(n-m,2)) 6`'^$wKs  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') bkb}M)C  
    end rS=6d6@  
    dpy,;nqzeN  
    if any(m<0) s:%>H|-  
        error('zernpol:Mpositive','All M must be positive.') _v-sb(* J  
    end *{uu_O  
    l! GPOmf9`  
    if any(m>n) s;bqUY?LD  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') jk~< si  
    end GE>&fG  
    k vb"n}  
    if any( r>1 | r<0 ) {2!.3<#  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') nv|&|6?`oK  
    end Ba"Z^(:  
    h-<+Pjc  
    if ~any(size(r)==1) kM.zX|_  
        error('zernpol:Rvector','R must be a vector.') ;lGjj9we>  
    end dme_Ivt  
    E5B:79BGO  
    r = r(:); Zvc{o8^z  
    length_r = length(r); ZW2U9  
    wuPx6hCl  
    if nargin==4 T7[ItLZ  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); C@xh$(y  
        if ~isnorm nfc&.(6x<  
            error('zernpol:normalization','Unrecognized normalization flag.') Rt+s\MC^r  
        end <MoWS9s!yb  
    else xand%XNv  
        isnorm = false; Y #KgaZ7N  
    end a4c~ThbI  
    }psJ'aiG*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nM@S`"  
    % Compute the Zernike Polynomials Uc.K6%iI  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F;kNc:X`)  
    QHK$2xtq|  
    % Determine the required powers of r: NI3_wV  
    % ----------------------------------- -e30!A  
    rpowers = []; jfk`%C Ek=  
    for j = 1:length(n) z`lDD  
        rpowers = [rpowers m(j):2:n(j)]; 8dP^zjPj  
    end WUKYwA/t  
    rpowers = unique(rpowers); TeQpmhN  
    7Y:1ji0l  
    % Pre-compute the values of r raised to the required powers, ~h -0rE  
    % and compile them in a matrix: op;OPf,  
    % ----------------------------- I U/gYFT  
    if rpowers(1)==0 l9\ *G;  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); q* +}wP  
        rpowern = cat(2,rpowern{:}); 'RXh E  
        rpowern = [ones(length_r,1) rpowern]; PC/Oo~Gx  
    else >osY?9  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); sT|8a  
        rpowern = cat(2,rpowern{:}); OT+LQ TE  
    end u[})|x*N  
    c5pF?kFaD  
    % Compute the values of the polynomials: }Dm-Ibdg(  
    % -------------------------------------- _dj_+<Y?  
    z = zeros(length_r,length_n); K%O%#Kk  
    for j = 1:length_n z.--"cF  
        s = 0:(n(j)-m(j))/2; 4Z,MqG>  
        pows = n(j):-2:m(j); .hXxh)F  
        for k = length(s):-1:1 k68\ _NUL  
            p = (1-2*mod(s(k),2))* ... }/Pz1,/  
                       prod(2:(n(j)-s(k)))/          ... "1t%J7c_  
                       prod(2:s(k))/                 ... wUv Zc  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ng"R[/)In  
                       prod(2:((n(j)+m(j))/2-s(k))); > T=($:n  
            idx = (pows(k)==rpowers); CtfI&rb[  
            z(:,j) = z(:,j) + p*rpowern(:,idx); |#>\GU=!  
        end g?qm >X  
         !ffdeWHR  
        if isnorm 7E 6gXf.  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); (w}iEm\b  
        end :2vk vLM  
    end "k[-eFz/@M  
    r>+\9q1  
    % 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)  ,V`zW<8  
    JkfVsmc<{h  
    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。 Tr\6 AN?o  
    (7zdbJX  
    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)