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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 mi<Q3;m  
    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! o ?vGI=  
     
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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 5F~l;zT  
    function z = zernfun(n,m,r,theta,nflag) ZAgXz{!H(  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. X>o9mW  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N  rvd $4l^  
    %   and angular frequency M, evaluated at positions (R,THETA) on the E^F<"mL*  
    %   unit circle.  N is a vector of positive integers (including 0), and j %gd:-tA  
    %   M is a vector with the same number of elements as N.  Each element tn' Jkwp  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 0W*{ 1W  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, f<@!{y 2Xe  
    %   and THETA is a vector of angles.  R and THETA must have the same BM,hcT r?  
    %   length.  The output Z is a matrix with one column for every (N,M) OY`B{jV-  
    %   pair, and one row for every (R,THETA) pair. %DKFF4k  
    % 1}DA| !~  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 11yXI[  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ~#*C,4m  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral hHE~/U  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, B]"`}jn  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized R}Lk$#S#  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ( *+'k1Ea  
    % ^b+>r  
    %   The Zernike functions are an orthogonal basis on the unit circle. nL:&G'd  
    %   They are used in disciplines such as astronomy, optics, and LOx+?4|y  
    %   optometry to describe functions on a circular domain. ~U&NY7.@  
    % eTS}-  
    %   The following table lists the first 15 Zernike functions. MJ)lZ!KZ  
    % aDNB~CwZZ  
    %       n    m    Zernike function           Normalization vAUt~ X"  
    %       -------------------------------------------------- ljNwt  
    %       0    0    1                                 1 F(HfXY3  
    %       1    1    r * cos(theta)                    2 (E0   
    %       1   -1    r * sin(theta)                    2 pD$4nH4KST  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) neI7VbH4  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 9Lb96K?=>  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ~:z.Xu5m  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) !,[#,oy;  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) \#9LwC"8;  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) K?^;|m-  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) < xy@%  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 4!Js="  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .zO2g8(VR  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) l/X_CM8y~  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) AatSN@,~z  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) +NPL.b|  
    %       -------------------------------------------------- Lj1l ]OD  
    % S 5S\zTPIf  
    %   Example 1: k6Kc{kY  
    % ^Pn|Q'{/p  
    %       % Display the Zernike function Z(n=5,m=1) EMmgX*iu@  
    %       x = -1:0.01:1; *DF3juf~  
    %       [X,Y] = meshgrid(x,x); Y P2VSK2Q  
    %       [theta,r] = cart2pol(X,Y); lYx_8x2  
    %       idx = r<=1; 03 @a G  
    %       z = nan(size(X)); pr0X7 #_E5  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 7]h%?W !  
    %       figure y *i&p4Y*  
    %       pcolor(x,x,z), shading interp t}q e_c  
    %       axis square, colorbar ;28d7e}  
    %       title('Zernike function Z_5^1(r,\theta)') @k?vbq  
    % Xsq@E#@S  
    %   Example 2: ob.<j  
    % ?Z#N9Z~\  
    %       % Display the first 10 Zernike functions Y [`+7w  
    %       x = -1:0.01:1; /Y7^!3uM  
    %       [X,Y] = meshgrid(x,x); /s\ m V  
    %       [theta,r] = cart2pol(X,Y); +K 4XMf  
    %       idx = r<=1; bwsKdh  
    %       z = nan(size(X)); hw DxGiU  
    %       n = [0  1  1  2  2  2  3  3  3  3]; |`T(:ZKXZ2  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; %~LY'cfPse  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ;.>*O oe&  
    %       y = zernfun(n,m,r(idx),theta(idx)); f@OH~4FG  
    %       figure('Units','normalized') H5K Fm#  
    %       for k = 1:10 2@|`Ugjptl  
    %           z(idx) = y(:,k); uC'-: t#  
    %           subplot(4,7,Nplot(k)) gQ+]N*.  
    %           pcolor(x,x,z), shading interp F5o8@ Ib]:  
    %           set(gca,'XTick',[],'YTick',[]) ; vH2r~  
    %           axis square C(N' =-;Kl  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) V"/.An|  
    %       end `a83RX_\  
    % yZleots1  
    %   See also ZERNPOL, ZERNFUN2. |a(KVo  
    ]>n{~4a  
    %   Paul Fricker 11/13/2006 02J/=AC5  
    -$d?e%}#  
    O<m46mwM  
    % Check and prepare the inputs: 1W USp;JMl  
    % ----------------------------- jB LTEb  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) TDh)}Ms  
        error('zernfun:NMvectors','N and M must be vectors.') 7)rQf{q7  
    end ng1E'c]0@  
    ?WI v4  
    if length(n)~=length(m) q*hn5K*  
        error('zernfun:NMlength','N and M must be the same length.') W5|{A])N  
    end t~+M>Fjm?d  
    =M\yh,s!  
    n = n(:); fv;Q*; oC&  
    m = m(:); V6g*"e/8  
    if any(mod(n-m,2)) QQJGqM3a2  
        error('zernfun:NMmultiplesof2', ... AiqKf=  
              'All N and M must differ by multiples of 2 (including 0).')  ?8>a;0  
    end PR{ubM n  
    #7uH>\r  
    if any(m>n) 6{2y$'m8  
        error('zernfun:MlessthanN', ... ;z:Rj}l  
              'Each M must be less than or equal to its corresponding N.') >.?yz   
    end 1iT_mtXK$  
    /J`}o}  
    if any( r>1 | r<0 ) lu#a.41  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') CsR[@&n'  
    end )vtbA=RH?  
    -laH^<jm5  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) HSruue8  
        error('zernfun:RTHvector','R and THETA must be vectors.') {cdICWy(F3  
    end uLdHE5vr  
    l6'KIg  
    r = r(:); JsY,Q,D q  
    theta = theta(:); b_+o1Zy`  
    length_r = length(r); d6i}xnmC  
    if length_r~=length(theta) %NLd"SV  
        error('zernfun:RTHlength', ...  hb[ThQ  
              'The number of R- and THETA-values must be equal.') u(9pRr L  
    end }9OMXLbRv  
    !)M}(I}  
    % Check normalization: m(f`=+lqI`  
    % -------------------- "im5Fnu  
    if nargin==5 && ischar(nflag) H I9/  
        isnorm = strcmpi(nflag,'norm'); cW3'057  
        if ~isnorm XpAJP++  
            error('zernfun:normalization','Unrecognized normalization flag.') |!oC7!+0^  
        end l$u52e!7  
    else $QiMA,  
        isnorm = false; -jjB2xP  
    end %|jS`kj  
    a^_K@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d V%o:@Z  
    % Compute the Zernike Polynomials b:(+d"S  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~ <1s[Hu  
    -Mo4`bN  
    % Determine the required powers of r: 4~ x>]  
    % ----------------------------------- eC/{c1C  
    m_abs = abs(m); qO@vXuul,  
    rpowers = []; UP#@gxF  
    for j = 1:length(n) A!Tl  
        rpowers = [rpowers m_abs(j):2:n(j)]; BB}WfA  
    end /  Xnq0hN  
    rpowers = unique(rpowers); veDv14  
    LJrH_h8C  
    % Pre-compute the values of r raised to the required powers, 60{G 4b)  
    % and compile them in a matrix: C6ql,hR^h`  
    % ----------------------------- Z|K HF"  
    if rpowers(1)==0 W=Syo&;F8  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); gj;gl ="3  
        rpowern = cat(2,rpowern{:}); aG1Fj[,  
        rpowern = [ones(length_r,1) rpowern]; s(_z1  
    else C b'|  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); wPU5L*/*i  
        rpowern = cat(2,rpowern{:}); Rd8mn'A  
    end  W2` 3 p  
    fBX@ MedC  
    % Compute the values of the polynomials: #8jiz+1 _  
    % -------------------------------------- i, ^-9  
    y = zeros(length_r,length(n)); 14&|(M  
    for j = 1:length(n) J@_M%eN  
        s = 0:(n(j)-m_abs(j))/2; :%sG'_d  
        pows = n(j):-2:m_abs(j); g?v/ u:v>W  
        for k = length(s):-1:1 Kmx4bp4  
            p = (1-2*mod(s(k),2))* ... ;)ay uS sQ  
                       prod(2:(n(j)-s(k)))/              ... {X?Aj >l  
                       prod(2:s(k))/                     ... /Ey%aA4v  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... shB3[W{}!)  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); rk=/iD  
            idx = (pows(k)==rpowers); @o[ZJ4>*  
            y(:,j) = y(:,j) + p*rpowern(:,idx);  LcLHX  
        end 6O?zi|J[:  
         $\/i t  
        if isnorm YUSrZ9Yg  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); aVr(*s;/  
        end U/FysN_N!  
    end ,'C*?mms  
    % END: Compute the Zernike Polynomials #2|biTJ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *v#V%_o  
    *X3wf`C?  
    % Compute the Zernike functions: OGEe8Z9Jt  
    % ------------------------------ `C_qqf  
    idx_pos = m>0; Na`> pH  
    idx_neg = m<0; ~F@p}u8TV  
    L0VZ>!*o  
    z = y; q%d,E1  
    if any(idx_pos) cZ%tJ(&\7X  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ;Q3[} ]su  
    end BZLIi O  
    if any(idx_neg) I_#5gq  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); %i7U+v(d  
    end Y'1 KH}sH  
    @|h9jx|  
    % 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) M8:i]   
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ].Bx"L!B  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 5{W Aw !  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive ,ye[TQ\,M  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Slo^tqbG  
    %   and THETA is a vector of angles.  R and THETA must have the same Bi9Q8#lh  
    %   length.  The output Z is a matrix with one column for every P-value, YeT{<9p  
    %   and one row for every (R,THETA) pair. gdSqG2/&  
    % L!Tvz(_7f6  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike N,B!D~@  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 34CcZEQQ  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) vx7=I\1  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 7V@r^/`8N  
    %   for all p. P3!@}!r8  
    % o%-KO? YW  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 |d~'X%b%  
    %   Zernike functions (order N<=7).  In some disciplines it is 67/\0mV:~  
    %   traditional to label the first 36 functions using a single mode &2%|?f|  
    %   number P instead of separate numbers for the order N and azimuthal }; 7I   
    %   frequency M. feS$)H9-  
    % JXRU9`3)A  
    %   Example: k$5l kP.  
    % Hr=|xw8.  
    %       % Display the first 16 Zernike functions G*_]Lz(N  
    %       x = -1:0.01:1; =mX26l`B  
    %       [X,Y] = meshgrid(x,x); T9J&^I  
    %       [theta,r] = cart2pol(X,Y); O..{wdZy  
    %       idx = r<=1; `, ]ui*  
    %       p = 0:15; +VQD'  
    %       z = nan(size(X)); Y|wjt\M  
    %       y = zernfun2(p,r(idx),theta(idx)); z{ M2tLNb  
    %       figure('Units','normalized') 'y>Y*/  
    %       for k = 1:length(p) s5G`?/  
    %           z(idx) = y(:,k); Uu*iL< `  
    %           subplot(4,4,k) z}==6| {  
    %           pcolor(x,x,z), shading interp E_'H=QN c  
    %           set(gca,'XTick',[],'YTick',[]) f`;w@gR`=  
    %           axis square }&L%c>  
    %           title(['Z_{' num2str(p(k)) '}']) WZHw(BN{+  
    %       end SAitufS  
    % 4 7mT  
    %   See also ZERNPOL, ZERNFUN. %t6-wWM97  
    $"( 15U  
    %   Paul Fricker 11/13/2006 B#IUSHC  
    ckV\f({  
    )l! /7WKY  
    % Check and prepare the inputs: {U>N*&_`  
    % ----------------------------- nC[aEZ7  
    if min(size(p))~=1 mrsmul{  
        error('zernfun2:Pvector','Input P must be vector.') I0H]s/*C%9  
    end b{aB^a:f=L  
    y]PuY \+  
    if any(p)>35 \p.yR.  
        error('zernfun2:P36', ... < @GO]vY  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... L58#ri=  
               '(P = 0 to 35).']) /;}%E  
    end |.m)UFV  
    \6MM7x(U3  
    % Get the order and frequency corresonding to the function number: tw.GBR  
    % ---------------------------------------------------------------- a6;[Z  
    p = p(:); JF~9efWe>  
    n = ceil((-3+sqrt(9+8*p))/2); LjGZp"&{  
    m = 2*p - n.*(n+2); |By[ev"Kh%  
    ZI1]B944ni  
    % Pass the inputs to the function ZERNFUN: 7T6Zlp  
    % ---------------------------------------- cNwH Y Z'  
    switch nargin xk/-TXB 0  
        case 3 uxDM #  
            z = zernfun(n,m,r,theta); EFx>Hu/ [G  
        case 4 n6t@ e^  
            z = zernfun(n,m,r,theta,nflag); 0fvOA*UP  
        otherwise VoUo!t:(+  
            error('zernfun2:nargin','Incorrect number of inputs.') R ai 0 4  
    end H"UJBO>$  
    uJH[C>  
    % 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) ;>5`Y8s6  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. :8oJG8WH  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 1d FuoX  
    %   order N and frequency M, evaluated at R.  N is a vector of _ h#I}uJ~  
    %   positive integers (including 0), and M is a vector with the kD;pj3o&"2  
    %   same number of elements as N.  Each element k of M must be a 2yg6hR  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 7mdd}L^h Z  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 6MY<6t0a  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix F{a;=h#@Q  
    %   with one column for every (N,M) pair, and one row for every @j}%{Km]Y  
    %   element in R. X|Y(*$?D7  
    % E}S%yD[  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- hPNMp@Nm6  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ,I5SAd|dX  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to lTq"j?#E]m  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 300w\9fn&  
    %   for all [n,m]. <C(o0u&/  
    % ;XawEG7" U  
    %   The radial Zernike polynomials are the radial portion of the X)~wB7_0G  
    %   Zernike functions, which are an orthogonal basis on the unit 'n,V*9  
    %   circle.  The series representation of the radial Zernike "EMW'>&m  
    %   polynomials is RfTGTz@H  
    % 9!uiQ  
    %          (n-m)/2 CKK}Z;~:  
    %            __ ]nB|8k=J  
    %    m      \       s                                          n-2s jmk Ou5@  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r '-RacNY  
    %    n      s=0 RhHm[aN  
    % 7LVG0A2>7  
    %   The following table shows the first 12 polynomials. %Rn*oV  
    % / }$n_N\!)  
    %       n    m    Zernike polynomial    Normalization (V jU,'h  
    %       --------------------------------------------- _;;Zz&c  
    %       0    0    1                        sqrt(2) i}DS+~8v  
    %       1    1    r                           2 9ET1Er{4  
    %       2    0    2*r^2 - 1                sqrt(6) eyyME c!  
    %       2    2    r^2                      sqrt(6) 'v V7@@  
    %       3    1    3*r^3 - 2*r              sqrt(8) b@;Wh-{d  
    %       3    3    r^3                      sqrt(8) W~ET/h  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) [MFnS",7c  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) `nl n@ ;  
    %       4    4    r^4                      sqrt(10) [rT.k5_  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ^HJ?k:u  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) =zyA~}M2  
    %       5    5    r^5                      sqrt(12) |M?vFF]TN  
    %       --------------------------------------------- ;cI*"-I:F  
    % Df^F)\7!N?  
    %   Example: ~bhS$*t64  
    % *$<W"@%^J  
    %       % Display three example Zernike radial polynomials V|_ h[hXE  
    %       r = 0:0.01:1; _2!8,MX  
    %       n = [3 2 5]; ;Nj9,Va(t  
    %       m = [1 2 1]; 8 XB[CbO  
    %       z = zernpol(n,m,r); ccHf+=  
    %       figure R{H[< s+n  
    %       plot(r,z) R2Fjv@Egk  
    %       grid on 8#7qHT;cx  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 06S R74  
    % f_jhQ..g<g  
    %   See also ZERNFUN, ZERNFUN2. *i]?J  
    x)~i`$  
    % A note on the algorithm. H3D<"4Q>  
    % ------------------------ {6zNCO  
    % The radial Zernike polynomials are computed using the series DpT9"?g7  
    % representation shown in the Help section above. For many special Oo|PZ_P  
    % functions, direct evaluation using the series representation can \EySKQ=  
    % produce poor numerical results (floating point errors), because PW5]+ |#  
    % the summation often involves computing small differences between {rUg,y{v  
    % large successive terms in the series. (In such cases, the functions W[\6h Zv  
    % are often evaluated using alternative methods such as recurrence VLez<Id9(  
    % relations: see the Legendre functions, for example). For the Zernike pd|KIs%jl  
    % polynomials, however, this problem does not arise, because the At iUTA  
    % polynomials are evaluated over the finite domain r = (0,1), and >[fu&r1  
    % because the coefficients for a given polynomial are generally all OM*c7&  
    % of similar magnitude. B{nwQC b  
    % KC6Cg?y^  
    % ZERNPOL has been written using a vectorized implementation: multiple gc.Lh~  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] r=H?fTY<3E  
    % values can be passed as inputs) for a vector of points R.  To achieve 1[!v{F%]  
    % this vectorization most efficiently, the algorithm in ZERNPOL @AEH?gOX  
    % involves pre-determining all the powers p of R that are required to aOwjYl[?p  
    % compute the outputs, and then compiling the {R^p} into a single vk92j?  
    % matrix.  This avoids any redundant computation of the R^p, and 6O7s^d&K  
    % minimizes the sizes of certain intermediate variables. 5#K*75>  
    % C`[<6>&y  
    %   Paul Fricker 11/13/2006 {o}U"b<+Ra  
    $4nAb^/  
    @8|*Ndx2  
    % Check and prepare the inputs: bv[#|^/  
    % ----------------------------- s@F&N9oh  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +OE!Uqnt  
        error('zernpol:NMvectors','N and M must be vectors.') lP F326e  
    end Lx0nLJ\  
    {zwH3)|Hn  
    if length(n)~=length(m) "v8p<JfB`  
        error('zernpol:NMlength','N and M must be the same length.') ^65I,Z"  
    end vI{aF- #  
    )}ev;37<C  
    n = n(:); g#J` 7n  
    m = m(:); )+G"57p  
    length_n = length(n); +%JBr+1#\  
    s1:Wrz?4  
    if any(mod(n-m,2)) pU$k{^'UK  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') &93{>caf+  
    end *N">93:  
    @Rr=uf G  
    if any(m<0) gP2zDI   
        error('zernpol:Mpositive','All M must be positive.') M@Th^yF+8H  
    end 1BSd9Ydj  
    ~ :ASv>m  
    if any(m>n) [,o:nry'a  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') J:Cr.K`  
    end \SWTP1  
    1Bj.MQ^  
    if any( r>1 | r<0 ) 5,"c1[`-  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') RM;a]g*  
    end VOEV[?>ss  
    xfYKUOp/  
    if ~any(size(r)==1) 1'~Xn 4 f  
        error('zernpol:Rvector','R must be a vector.') *~#I5s\s!  
    end 2u3Kyn  
    {VcRur}&Y8  
    r = r(:); [o)K1>>7  
    length_r = length(r); |[SHpcq>  
    ~gDYb#p  
    if nargin==4 # T=iS(i  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); g) Lf^  
        if ~isnorm Q:-T' xk@  
            error('zernpol:normalization','Unrecognized normalization flag.') 586P~C[ic  
        end 1 G>Ud6(3<  
    else 1oQw)X  
        isnorm = false; 0AQ azhm  
    end )bUnk +_  
    ^O07GYF  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _Mw3>GNl  
    % Compute the Zernike Polynomials @{Rb]d?&F?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @8L5 UT  
    ] ZV[}7I.  
    % Determine the required powers of r: CMj =4e  
    % ----------------------------------- ;UQGi}?CD  
    rpowers = []; ? i{?Q,  
    for j = 1:length(n) W A/dt2D|  
        rpowers = [rpowers m(j):2:n(j)]; )/raTD  
    end (i~UH04r>s  
    rpowers = unique(rpowers); tOIqX0dWd  
     6}"%>9  
    % Pre-compute the values of r raised to the required powers, uo"<}>iJ  
    % and compile them in a matrix: nBy-/BU&  
    % ----------------------------- k2}DBVu1  
    if rpowers(1)==0 Od!)MQ*,  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Rl?1|$%  
        rpowern = cat(2,rpowern{:}); V]H(;+^P  
        rpowern = [ones(length_r,1) rpowern]; VGS%U8;  
    else c8uaZvfW  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); *:%&z?<Fw  
        rpowern = cat(2,rpowern{:}); S\GWMB!oF  
    end m{IlRf'  
    \s=r[0tj!  
    % Compute the values of the polynomials: odhcD;^X1  
    % -------------------------------------- =H{<}>W'  
    z = zeros(length_r,length_n); m?e/MQr  
    for j = 1:length_n K#R]of~/  
        s = 0:(n(j)-m(j))/2; LU6R"c11  
        pows = n(j):-2:m(j); 2F4<3k! &  
        for k = length(s):-1:1 5CI {&E  
            p = (1-2*mod(s(k),2))* ... XGa8tI[:X  
                       prod(2:(n(j)-s(k)))/          ... #u&fUxM:AS  
                       prod(2:s(k))/                 ... 1eI*.pt  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Rhc:szDU  
                       prod(2:((n(j)+m(j))/2-s(k))); \BHZRytQF  
            idx = (pows(k)==rpowers); pDS[ecx  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 4C;;V m4~  
        end /~,*DH$)  
         Cl0kR3Y  
        if isnorm d{fd5jv;  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 72nZ`u  
        end 9qap#A  
    end  2E*=EjGV  
    ex>7f%\  
    % 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)  Z:^<NdKe  
    EwcFxLa!F  
    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。 K"[jrvZ=  
    \X6q A-Ht  
    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)