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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 P^'TI[\L9  
    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! W*2d!/;7>  
     
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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 :%-w/QwTR  
    function z = zernfun(n,m,r,theta,nflag) F|a'^:Qs  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. m'zve%G  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N \ )WS^KR%  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ({3Ap{Q}  
    %   unit circle.  N is a vector of positive integers (including 0), and nIr:a|}[  
    %   M is a vector with the same number of elements as N.  Each element KCIya[$*  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Xf#+^cQ  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, =PF2p'.o  
    %   and THETA is a vector of angles.  R and THETA must have the same ]Z nASlc)  
    %   length.  The output Z is a matrix with one column for every (N,M) YK\pV'&+  
    %   pair, and one row for every (R,THETA) pair. >PzZt8e  
    % c)3.AgT  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike }K^v Ujl  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), xa'^:H $X  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral &\=Tm~  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, #;[0:jU0  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized .?vHoNvo  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. JZdRAL2#v  
    % !r8_'K5R(  
    %   The Zernike functions are an orthogonal basis on the unit circle. [vY#9W"!  
    %   They are used in disciplines such as astronomy, optics, and bcq&yL'D  
    %   optometry to describe functions on a circular domain. OqWm5(u&S  
    % : *XAQb0  
    %   The following table lists the first 15 Zernike functions. g< xE}[gF  
    % -2[#1S*  
    %       n    m    Zernike function           Normalization <+-=j  
    %       -------------------------------------------------- + ZK U2N*  
    %       0    0    1                                 1 ;F|#m,2Q-  
    %       1    1    r * cos(theta)                    2 :R`e<g~4  
    %       1   -1    r * sin(theta)                    2 zO2=o5nF.  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) @j!(at4B  
    %       2    0    (2*r^2 - 1)                    sqrt(3) HSWki';G  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) XzPOqZ`Nv  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ]>Ym   
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ;\v&4+3S  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) xL*J9&~iG  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) {P_i5V?  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) H| _@9V  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }N} Js*  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) twu,yC!  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) x`c 7*q%  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) nU' qE  
    %       -------------------------------------------------- c`/VYgcTqB  
    % R7"7 Rx   
    %   Example 1: Y0Tad?iC  
    % D w/vXyZ  
    %       % Display the Zernike function Z(n=5,m=1) b*Q3j}cZ  
    %       x = -1:0.01:1; z#Fel/L`O  
    %       [X,Y] = meshgrid(x,x); P z~jW):E  
    %       [theta,r] = cart2pol(X,Y); }K={HW1>  
    %       idx = r<=1; 7H09\g&  
    %       z = nan(size(X)); $E&T6=Wn  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); =IW!ZN_  
    %       figure |gWA'O0S  
    %       pcolor(x,x,z), shading interp ">G*hS  
    %       axis square, colorbar =tbfBK+  
    %       title('Zernike function Z_5^1(r,\theta)') @dk-+YxG  
    % 0 @!huk  
    %   Example 2: Ka6u*:/  
    % $#-rOi /  
    %       % Display the first 10 Zernike functions ImG8v[Q E  
    %       x = -1:0.01:1; Q=8YAiCu  
    %       [X,Y] = meshgrid(x,x); Xy%||\P{)  
    %       [theta,r] = cart2pol(X,Y); IIih9I`IR  
    %       idx = r<=1;  =   
    %       z = nan(size(X)); 2GORGS%  
    %       n = [0  1  1  2  2  2  3  3  3  3]; yuy\T(7BN  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ]\KVA)\  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; h]h"-3  
    %       y = zernfun(n,m,r(idx),theta(idx)); q01 L{~>bz  
    %       figure('Units','normalized') m5iCvOP  
    %       for k = 1:10 U#c Gd\b  
    %           z(idx) = y(:,k); JRi:MWR<r  
    %           subplot(4,7,Nplot(k)) "T_9_6tH  
    %           pcolor(x,x,z), shading interp .Sn{a }XP4  
    %           set(gca,'XTick',[],'YTick',[]) Zj!,3{jX^  
    %           axis square V]; i$  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) t VO}{[U}  
    %       end 4~3 n =T*  
    % G"` }"T0}  
    %   See also ZERNPOL, ZERNFUN2. u.|%@  
    SGSyO0O  
    %   Paul Fricker 11/13/2006 \?]U*)B.r  
     {ibu 0  
    h$kz3r;b,"  
    % Check and prepare the inputs: lHtywZ@%3  
    % ----------------------------- *d jLf.I@  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) , + G  
        error('zernfun:NMvectors','N and M must be vectors.') >zqaV@T  
    end 4P[MkMoC  
    WM`3QJb  
    if length(n)~=length(m) zwZvKV/g  
        error('zernfun:NMlength','N and M must be the same length.') +HBizJ9K  
    end Et!J*{s  
    jQ;/=9  
    n = n(:); cN0 *<  
    m = m(:); :Bmn<2[Y;  
    if any(mod(n-m,2)) ttUK~%wSx  
        error('zernfun:NMmultiplesof2', ... \894 Jqh  
              'All N and M must differ by multiples of 2 (including 0).') {iX#  
    end F$)l8}  
    ~w3u(X$m"  
    if any(m>n) beBG40  
        error('zernfun:MlessthanN', ... E+i*u   
              'Each M must be less than or equal to its corresponding N.') o *J*} y  
    end &Gh0f"?  
    LWv<mtuYf  
    if any( r>1 | r<0 ) 4?1Qe\A^  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') R1X'}#mU  
    end RbL?(  
    e?.j8 Q ~  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ^T!Zz"/:  
        error('zernfun:RTHvector','R and THETA must be vectors.') V*b/N  
    end oh< -&3Jn  
    P !i_?M  
    r = r(:); (O{OQk;CF  
    theta = theta(:); 0TmEa59P  
    length_r = length(r); n#g_)\  
    if length_r~=length(theta) Q"dq_8\`U  
        error('zernfun:RTHlength', ... &Gjpc>d  
              'The number of R- and THETA-values must be equal.') (p{%]M  
    end gLX<> |)*  
    w\acgQ^%e  
    % Check normalization: uK@d?u!`  
    % -------------------- 9$\s v5  
    if nargin==5 && ischar(nflag) p[JIH~nb  
        isnorm = strcmpi(nflag,'norm'); 4>=M"D hB  
        if ~isnorm M5h r0 R{  
            error('zernfun:normalization','Unrecognized normalization flag.') u9"yU:1keb  
        end ?YW~7zG  
    else `f;w  
        isnorm = false; ;[::&qf  
    end ?Z 2,?G  
    QFx3N%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =$J(]KPv!?  
    % Compute the Zernike Polynomials zbxW U]<S?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :|s8v2am  
    D6Ad "|Z  
    % Determine the required powers of r: vW=-RTRH  
    % ----------------------------------- nZbI}kcm  
    m_abs = abs(m); wnokP  
    rpowers = []; 8X7??f1;Y  
    for j = 1:length(n) ~pRgTXbz  
        rpowers = [rpowers m_abs(j):2:n(j)]; |T6K?:U7  
    end Y/5M)AyJt  
    rpowers = unique(rpowers); A0Mjk  
    @3?>[R  
    % Pre-compute the values of r raised to the required powers, =:&xdphZ+  
    % and compile them in a matrix: ,,{;G'R|  
    % ----------------------------- /xk7Z q  
    if rpowers(1)==0 P~trxp=k  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); DEZww9T2Qs  
        rpowern = cat(2,rpowern{:}); =IC.FT}  
        rpowern = [ones(length_r,1) rpowern]; S[F06.(1  
    else ~(V\.hq  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); L~6%Fi&n4  
        rpowern = cat(2,rpowern{:}); j9NF|  
    end 2%pED xui  
    r !Aj5  
    % Compute the values of the polynomials: cX-M9Cz  
    % -------------------------------------- 6j(/uF4!#  
    y = zeros(length_r,length(n)); W'@ |ob  
    for j = 1:length(n) (L/>LZn|  
        s = 0:(n(j)-m_abs(j))/2; ^Gk`n  
        pows = n(j):-2:m_abs(j); R])Eg&  
        for k = length(s):-1:1 V\Cl""`XN  
            p = (1-2*mod(s(k),2))* ... ({!!b"B2  
                       prod(2:(n(j)-s(k)))/              ... XR+ SjCA  
                       prod(2:s(k))/                     ... $.jG O!  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... =K`.$R  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 3NpB1lgh&:  
            idx = (pows(k)==rpowers); efQ8jO  
            y(:,j) = y(:,j) + p*rpowern(:,idx); |q w0:c=7!  
        end <T_3s\  
         e#C v*i_<  
        if isnorm ZQfxlzj+X  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 4EqThvI{  
        end }:#WjH^  
    end wm`<+K  
    % END: Compute the Zernike Polynomials Nj>6TD81u  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :lB*kmg  
    P-\f-FS  
    % Compute the Zernike functions: &42 ]#B"*  
    % ------------------------------ _@ao$)q{J  
    idx_pos = m>0; &ys>z<Z  
    idx_neg = m<0; /L^g. ~  
    '3l TI  
    z = y; ,clbD4  
    if any(idx_pos) zq};{~u(  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Q VTL}AT2:  
    end yzS^8,  
    if any(idx_neg) ETHcZ  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); N!K%aH~O  
    end Pm/<^z%  
    _KH91$iW8m  
    % 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) ;73S;IPR  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. !PA:#]J  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated gnp.!-  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive !K-1tp$  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Xw4Eti._D  
    %   and THETA is a vector of angles.  R and THETA must have the same 2w.FC  
    %   length.  The output Z is a matrix with one column for every P-value, u n v:sV#b  
    %   and one row for every (R,THETA) pair. R (f:UC  
    %  ew1L+  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike #<0Hvde  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) &ivU4rEG  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) | 1B0  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 Bq8#'K2i,  
    %   for all p. tYD8Y  
    % NljpkeX'  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Dmh$@Uu#F  
    %   Zernike functions (order N<=7).  In some disciplines it is if'=W6W  
    %   traditional to label the first 36 functions using a single mode S F)$b  
    %   number P instead of separate numbers for the order N and azimuthal r)t[QoD1  
    %   frequency M. >fC&bab  
    % iV{_?f1jo  
    %   Example: "uBnK!  
    % \g34YY^L3  
    %       % Display the first 16 Zernike functions I1 ]YT  
    %       x = -1:0.01:1; >|SIqB<%:  
    %       [X,Y] = meshgrid(x,x); 31G:[;g  
    %       [theta,r] = cart2pol(X,Y); $wM..ee  
    %       idx = r<=1; B /;(#{U;  
    %       p = 0:15; g}+|0FTV  
    %       z = nan(size(X)); XC3)#D#HGh  
    %       y = zernfun2(p,r(idx),theta(idx)); L0![SE>  
    %       figure('Units','normalized') q-z1ElrN7u  
    %       for k = 1:length(p) V>Jr4z  
    %           z(idx) = y(:,k); IUOf/mM5  
    %           subplot(4,4,k) Mq91HmC(@  
    %           pcolor(x,x,z), shading interp 2O kID WcM  
    %           set(gca,'XTick',[],'YTick',[]) =CQfs6np:N  
    %           axis square 2c Xae  
    %           title(['Z_{' num2str(p(k)) '}']) gvc@q`_]  
    %       end P`Zon  
    % 8#QT[H 4F  
    %   See also ZERNPOL, ZERNFUN. ':4ny]F  
    2VV>?s  
    %   Paul Fricker 11/13/2006 E]#;K-j  
    ] G["TX,  
    v/ry" W  
    % Check and prepare the inputs: K\-N'M!Z  
    % ----------------------------- ]>~.U ~  
    if min(size(p))~=1 "==c  
        error('zernfun2:Pvector','Input P must be vector.') f,ro1Nke  
    end 1:eWZ]B5"  
    j}Tv/O,f  
    if any(p)>35  m}yu4  
        error('zernfun2:P36', ... Oem1=QpaC  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... l4RqQ+[KA;  
               '(P = 0 to 35).']) @JSWqi>  
    end !8p>4|VM  
    UA.Tp[u  
    % Get the order and frequency corresonding to the function number: />xEpR3_A  
    % ---------------------------------------------------------------- e,OXngC  
    p = p(:); :Ou~?q%X  
    n = ceil((-3+sqrt(9+8*p))/2); $@VJ@JAe  
    m = 2*p - n.*(n+2); fS}Eu4Xe  
    Uv59 XF$  
    % Pass the inputs to the function ZERNFUN: $l ,U)  
    % ---------------------------------------- q;AD#A|\  
    switch nargin %ZRv+}z  
        case 3 }e7/F[c.U  
            z = zernfun(n,m,r,theta); <x`yoVPiZg  
        case 4 3JD62wtx  
            z = zernfun(n,m,r,theta,nflag); /,I?"&FWc  
        otherwise VY/r2o#  
            error('zernfun2:nargin','Incorrect number of inputs.') ,q*|R O  
    end (U5XB [r_P  
    5^uX!_ r`  
    % 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) T72Li"00  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. C^C'!  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of bIT[\Q  
    %   order N and frequency M, evaluated at R.  N is a vector of k&yBB%g  
    %   positive integers (including 0), and M is a vector with the djf8FNnn  
    %   same number of elements as N.  Each element k of M must be a Wr Wz+5M8  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) h9S f  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is qw4wg9w5p  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix o ^w^dgJ  
    %   with one column for every (N,M) pair, and one row for every L^^f.w#m  
    %   element in R. (q"S0{  
    % -X EK[  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- J{Ij  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is e>Q:j_?.e  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to b0f6?s  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 6jr}l  
    %   for all [n,m]. >Dv=lgPF  
    % Gyu =}  
    %   The radial Zernike polynomials are the radial portion of the M|{KQ3q:9  
    %   Zernike functions, which are an orthogonal basis on the unit L%7WHtU*#  
    %   circle.  The series representation of the radial Zernike #\~m}O,  
    %   polynomials is ;|rFP  
    % Uwiy@ T Z  
    %          (n-m)/2 %Y`)ZKh  
    %            __ ,vi6<C\  
    %    m      \       s                                          n-2s :@~3wD[y  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r @ 6jKjI  
    %    n      s=0 a6T!)g  
    % C 1HNcfa7  
    %   The following table shows the first 12 polynomials. ~O;?;@  
    % !H^R_GC  
    %       n    m    Zernike polynomial    Normalization yaj1nq! *"  
    %       --------------------------------------------- w4y ???90)  
    %       0    0    1                        sqrt(2) Z _<Wr7D  
    %       1    1    r                           2 H_JT"~_2  
    %       2    0    2*r^2 - 1                sqrt(6) j~2t^Qz  
    %       2    2    r^2                      sqrt(6) a;7gy419<p  
    %       3    1    3*r^3 - 2*r              sqrt(8) qB3& F pgW  
    %       3    3    r^3                      sqrt(8) KG5B6Om5'  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) YcaLc_pUx  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) :fG9p`  
    %       4    4    r^4                      sqrt(10) wu0J XB%&^  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) "^pF2JI  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) NLWj5K)1P  
    %       5    5    r^5                      sqrt(12) h 7\EN  
    %       --------------------------------------------- imS&N.*3m  
    % ]=^NTm,  
    %   Example: )N ^g0 L  
    % AQBr{^inH|  
    %       % Display three example Zernike radial polynomials p t{/|P  
    %       r = 0:0.01:1; \]/ 6>yT  
    %       n = [3 2 5]; YF");itH  
    %       m = [1 2 1]; ~i@Z4t j7  
    %       z = zernpol(n,m,r); j"+R*H(#  
    %       figure 2L2)``*   
    %       plot(r,z) f#vVk  
    %       grid on 47/YD y%  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') FCr>$  
    % [k +fkr]  
    %   See also ZERNFUN, ZERNFUN2. hzk]kM/OC  
    -/qu."9(B  
    % A note on the algorithm. ! +Hc(i  
    % ------------------------ My >{;n=}  
    % The radial Zernike polynomials are computed using the series [=xO>  
    % representation shown in the Help section above. For many special DCtrTX  
    % functions, direct evaluation using the series representation can dJg72?"ka  
    % produce poor numerical results (floating point errors), because ZYW=#df R  
    % the summation often involves computing small differences between eYjr/`>O  
    % large successive terms in the series. (In such cases, the functions _q\w9gN  
    % are often evaluated using alternative methods such as recurrence {wf e!f  
    % relations: see the Legendre functions, for example). For the Zernike r`'n3#O*  
    % polynomials, however, this problem does not arise, because the i%_nH"h  
    % polynomials are evaluated over the finite domain r = (0,1), and 4THGHS^  
    % because the coefficients for a given polynomial are generally all mm<rdo(`  
    % of similar magnitude. \.dvRI'  
    % \xaK?_hv  
    % ZERNPOL has been written using a vectorized implementation: multiple  t"'aQr  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] &mvC<_1n  
    % values can be passed as inputs) for a vector of points R.  To achieve of:xj$dQ_  
    % this vectorization most efficiently, the algorithm in ZERNPOL {#1}YGpiVM  
    % involves pre-determining all the powers p of R that are required to j1,ir  
    % compute the outputs, and then compiling the {R^p} into a single <yrl_vl{  
    % matrix.  This avoids any redundant computation of the R^p, and PM%Gsy]q  
    % minimizes the sizes of certain intermediate variables. >'lte&  
    % !n/"39KT  
    %   Paul Fricker 11/13/2006 X}3o  
    d1>Nn!m  
    /e}NZo{)g  
    % Check and prepare the inputs: o;@T6-VH  
    % ----------------------------- @ (A[H^E  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) `=3:*.T*  
        error('zernpol:NMvectors','N and M must be vectors.') ',p`B-dw  
    end A|d(5{:N  
    ON=6w_  
    if length(n)~=length(m) FCEFg)c5=  
        error('zernpol:NMlength','N and M must be the same length.') =sW(2Im  
    end wGf SVA-q\  
    vN%SN>=L<  
    n = n(:); mMvt#+O  
    m = m(:); 5)GO  
    length_n = length(n); anTS8b   
    u>.qhtm[  
    if any(mod(n-m,2)) 5}4r'P$m:  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Ie~#k[X  
    end AA um1xl  
    bEE'50 D  
    if any(m<0) 2 -uL  
        error('zernpol:Mpositive','All M must be positive.') ,$96bF "#  
    end <x),HTJ  
    +mN]VO*y  
    if any(m>n) 0ZXG{Gp9S  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') IOsitMOX:  
    end =5jX#Dc5.+  
    >8nRP%r[5,  
    if any( r>1 | r<0 ) bi bjFg   
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') t .&YD x  
    end Q!:J.J  
    gI qYIt  
    if ~any(size(r)==1) 39QAj&  
        error('zernpol:Rvector','R must be a vector.') [g:ZIl4p\P  
    end w"O^CR)  
    [ENm(e$sI  
    r = r(:); Ii /#cdgF  
    length_r = length(r); fKMbOqU_  
    Lh6G"f(n  
    if nargin==4 spV/+jy{  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); /,z4tf  
        if ~isnorm "3A.x1uQ  
            error('zernpol:normalization','Unrecognized normalization flag.') !K#Q[Ee  
        end Ax4nx!W,   
    else V&E)4KBOs  
        isnorm = false; 0S0 ?\r  
    end bBBW7',[a  
    'dp3>4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Lc!% 3,#.  
    % Compute the Zernike Polynomials HgP9evz,0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7c9-MP)  
    S$Tc\ /{  
    % Determine the required powers of r:  h+Dp<b  
    % ----------------------------------- ++Qg5FukR  
    rpowers = []; -gl7mO*  
    for j = 1:length(n) W~J@v@..4  
        rpowers = [rpowers m(j):2:n(j)]; 4 e1=b,  
    end WhsTKy&E  
    rpowers = unique(rpowers); Lf|5miO  
    z!quA7s<]  
    % Pre-compute the values of r raised to the required powers, `w1|(Sk$h  
    % and compile them in a matrix: %l!Gt"\xm  
    % ----------------------------- ML!Z m[I9  
    if rpowers(1)==0 0!YB.=\{_q  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); pt&(c[  
        rpowern = cat(2,rpowern{:}); GpV"KVJJ/  
        rpowern = [ones(length_r,1) rpowern];  q[#2`  
    else #8G (r9  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ~{hcJ:bI  
        rpowern = cat(2,rpowern{:}); /pZ]:.A  
    end lV/-jkR  
    K'EGm #I  
    % Compute the values of the polynomials: s_A<bW566F  
    % -------------------------------------- |'xVU8  
    z = zeros(length_r,length_n); z{w!yMp"  
    for j = 1:length_n *P,dR]-m  
        s = 0:(n(j)-m(j))/2; ]42bd  
        pows = n(j):-2:m(j); !N--  
        for k = length(s):-1:1 a,3} o:f  
            p = (1-2*mod(s(k),2))* ... D/C)Rrq"a  
                       prod(2:(n(j)-s(k)))/          ... fGDR<t3yiQ  
                       prod(2:s(k))/                 ... l(Dkmt>^  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 6vySOVMj  
                       prod(2:((n(j)+m(j))/2-s(k))); (a0q*iC%  
            idx = (pows(k)==rpowers); 3VZeUOxY\W  
            z(:,j) = z(:,j) + p*rpowern(:,idx); '`$z!rA  
        end X(nbfh?n  
         7yGc@kJ?  
        if isnorm ~wmc5L/!?  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); b13XHR)0  
        end kZXsL  
    end #gzY _)E  
    O`_!G`E  
    % 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)  l@}BWSx&ms  
    :sT\-MpQvn  
    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。 %,9iY&;U"  
    [oOA@  
    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)