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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 8tC+ lc  
    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! =_pmy>_z  
     
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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 \;'_|bu3.  
    function z = zernfun(n,m,r,theta,nflag) VoWA tNU  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. <tGI]@Nwk  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N #R5we3&p  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 4|I7:~  
    %   unit circle.  N is a vector of positive integers (including 0), and C8! 8u?k  
    %   M is a vector with the same number of elements as N.  Each element b "`ru~]  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 5+J 64_  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 0@JilGk1u  
    %   and THETA is a vector of angles.  R and THETA must have the same jM{(8aUG  
    %   length.  The output Z is a matrix with one column for every (N,M) rwasH,+  
    %   pair, and one row for every (R,THETA) pair. G*8+h  
    % BYkVg2D(  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 1y_fQ+\2A  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), TB ;3`  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ce 7Yr*ZB  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, t$ACQ*O  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized f%`*ba" v  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ^uKnP>*l  
    % b EoB;]  
    %   The Zernike functions are an orthogonal basis on the unit circle. {d&X/tT  
    %   They are used in disciplines such as astronomy, optics, and ocb%&m ;i  
    %   optometry to describe functions on a circular domain. A73V6"  
    % +9Xu"OFm  
    %   The following table lists the first 15 Zernike functions. Kx(76_XD  
    % V=G b>_d  
    %       n    m    Zernike function           Normalization fho=<|-  
    %       -------------------------------------------------- 4r68`<mn[  
    %       0    0    1                                 1 y|&.v <  
    %       1    1    r * cos(theta)                    2 YlZYS'_  
    %       1   -1    r * sin(theta)                    2 U)O?| VN^o  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) yEMX`  
    %       2    0    (2*r^2 - 1)                    sqrt(3) !$%/ rQ9  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) JL}hOBqfI  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) *u:;:W&5y  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) J3]qg.B%z  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) .( TQ5/ ~  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) fxLE]VJQ  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) l044c,AW(  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0A #9C09  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ~u O:tL  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) U.zRIhA ]  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 4?P%M"\Iv  
    %       -------------------------------------------------- 7eaA]y~H  
    % ~#HH;q_7m  
    %   Example 1: kxr6sO~  
    % XwHu:v'=  
    %       % Display the Zernike function Z(n=5,m=1) Z`SWZ<  
    %       x = -1:0.01:1; .!7Fe)(x  
    %       [X,Y] = meshgrid(x,x); 9^#zxmH)  
    %       [theta,r] = cart2pol(X,Y); &;r'{$  
    %       idx = r<=1; ft~|  
    %       z = nan(size(X)); 5WtQwN~  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); i/C -{+}U  
    %       figure l`~a}y"n  
    %       pcolor(x,x,z), shading interp I>YtWY|ed  
    %       axis square, colorbar ?34EJ !  
    %       title('Zernike function Z_5^1(r,\theta)') p[af[!  
    % >Rl0%!  
    %   Example 2: CA~em_dC  
    % v;N1'  
    %       % Display the first 10 Zernike functions O&rD4#  
    %       x = -1:0.01:1; zezofW]a  
    %       [X,Y] = meshgrid(x,x); !R] CmK  
    %       [theta,r] = cart2pol(X,Y); BCa90  
    %       idx = r<=1; 34+)-\xt:  
    %       z = nan(size(X)); m-Z'K_oQ  
    %       n = [0  1  1  2  2  2  3  3  3  3]; WcZo+r  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; +[ZMrTW!0C  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Z,N7nMJf  
    %       y = zernfun(n,m,r(idx),theta(idx)); I9Edw]  
    %       figure('Units','normalized') >~^mIu_BH  
    %       for k = 1:10 3;t@KuQ66  
    %           z(idx) = y(:,k); (:j+[3Ht  
    %           subplot(4,7,Nplot(k)) U l7pxzj  
    %           pcolor(x,x,z), shading interp r+V(1<`2X  
    %           set(gca,'XTick',[],'YTick',[]) iaaH9X %  
    %           axis square eK=m02  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Mi %1+  
    %       end NXWIE4T>*^  
    % YQB]t=Ha  
    %   See also ZERNPOL, ZERNFUN2. w ufKb.4`  
    Chb 4VoE  
    %   Paul Fricker 11/13/2006 1=/MT#d^?  
    9m#H24{V'  
    69<rsp(p  
    % Check and prepare the inputs: +*:x#$phx  
    % ----------------------------- F-reb5pt.=  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) k DceBs s  
        error('zernfun:NMvectors','N and M must be vectors.') T`RQUJO  
    end =?I1V#.  
    J7 *G/F  
    if length(n)~=length(m) 1Hk<_no5  
        error('zernfun:NMlength','N and M must be the same length.') 3' :[i2[  
    end :+gCO!9Y  
    0=(-8vwd  
    n = n(:); eqUn8<<s  
    m = m(:); D\_*,Fc  
    if any(mod(n-m,2)) O+8ApicjTc  
        error('zernfun:NMmultiplesof2', ... EDa08+Y  
              'All N and M must differ by multiples of 2 (including 0).') K9z_=c+  
    end Ie`SWg*WL  
    %;B(_ht<-w  
    if any(m>n) Lct+cKKU  
        error('zernfun:MlessthanN', ... >{LJ#Dc6  
              'Each M must be less than or equal to its corresponding N.') QF.wtMGF&  
    end 9>$%F;JP44  
    ^v'g~+@o  
    if any( r>1 | r<0 ) ezq q@t9  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') )l!&i?h%  
    end ^) b7m  
    U0|j^.)  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) y 4,T  
        error('zernfun:RTHvector','R and THETA must be vectors.') b09#+CH?  
    end <x%my4M  
    EJ &ZZg  
    r = r(:); as!|8JE`  
    theta = theta(:); $Bwvw)(%  
    length_r = length(r); yn ?U7`V  
    if length_r~=length(theta) ~E:/oV:4 >  
        error('zernfun:RTHlength', ... ['N#aDh.?  
              'The number of R- and THETA-values must be equal.') 5-QvQ&eH.  
    end 3 z/O`z  
    <&m  
    % Check normalization: Z5^,!6  
    % -------------------- C6T 9  
    if nargin==5 && ischar(nflag) )mo|.L0  
        isnorm = strcmpi(nflag,'norm'); MT#[ - M\  
        if ~isnorm s)&R W#:X  
            error('zernfun:normalization','Unrecognized normalization flag.') NYV0<z@M2M  
        end G}hkr  
    else >8mW-p  
        isnorm = false; ])ZJ1QL1  
    end ^&w'`-ra  
    GPhwq n{  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ea[a)Z7#  
    % Compute the Zernike Polynomials z )}wo3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G?/8&%8  
    I8pxo7(-  
    % Determine the required powers of r: RV@(&eM  
    % ----------------------------------- +WEO]q?K  
    m_abs = abs(m); 8#JyK+NU  
    rpowers = []; RkXLE"G '  
    for j = 1:length(n) Z(`K6`KM  
        rpowers = [rpowers m_abs(j):2:n(j)]; P9HPr2  
    end P*^UU\x'4I  
    rpowers = unique(rpowers); H(ftOd.y  
    ?B31 t9  
    % Pre-compute the values of r raised to the required powers, K]azUK7  
    % and compile them in a matrix: +wgUs*(W  
    % ----------------------------- o>k-~v7  
    if rpowers(1)==0 [z t&8g  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Y?SJQhN6W  
        rpowern = cat(2,rpowern{:}); `fE:5y  
        rpowern = [ones(length_r,1) rpowern]; HQ#L |LN  
    else ;0}"2aGY  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); .;sPG  
        rpowern = cat(2,rpowern{:}); Tf]VcEF  
    end -8J@r2\  
    gG z_t,=  
    % Compute the values of the polynomials: RPqn#B  
    % -------------------------------------- o+23?A~+  
    y = zeros(length_r,length(n)); ~CTRPH   
    for j = 1:length(n) vP;tgW9Qk  
        s = 0:(n(j)-m_abs(j))/2; k5\ zGsol  
        pows = n(j):-2:m_abs(j); s5|)4Z ac  
        for k = length(s):-1:1 9Yne=R/]  
            p = (1-2*mod(s(k),2))* ... 7.'j~hJL  
                       prod(2:(n(j)-s(k)))/              ... ) W7H{#  
                       prod(2:s(k))/                     ... BHBR_7  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... o~N-x*   
                       prod(2:((n(j)+m_abs(j))/2-s(k))); X~VZ61vNu  
            idx = (pows(k)==rpowers); |&*rSp2iH  
            y(:,j) = y(:,j) + p*rpowern(:,idx); #Yb9w3N  
        end ep1Ajz.l  
          ^*>no=A  
        if isnorm E*]L]vR  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Tfs9< k>G#  
        end X7!A(q+h  
    end #3jZ7RqzQ  
    % END: Compute the Zernike Polynomials d`*vJ#$> 2  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zT40,rk  
    ,tc]E45  
    % Compute the Zernike functions: ol>=tk 8}  
    % ------------------------------ 4p g(QeR  
    idx_pos = m>0; h.%Qn vL  
    idx_neg = m<0; lw lW.C  
    nr%^:u  
    z = y; PU\q.y0R  
    if any(idx_pos) )CU(~s|s  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); A}}t86T  
    end nsn,8a38  
    if any(idx_neg) {i?K~| h  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); J)~=b_'<  
    end SaScP  
    :~(^b;yhZ  
    % 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) 'S ;vv]}Gs  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ,4UJ| D=J  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated -4a&R=%p  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive  iSax-Mc  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, L~zet-3UNf  
    %   and THETA is a vector of angles.  R and THETA must have the same vDL/PXNC  
    %   length.  The output Z is a matrix with one column for every P-value, 247>+:7z  
    %   and one row for every (R,THETA) pair. (\_d'Js(;  
    % qdZYaS ~  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike  EZFWxR/  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) hWJc A.A  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) p5hP}Z4r  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 8t"DQ Y-R  
    %   for all p. h Nwb.[  
    % &ICO{#v5  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 F3'G9Xf8Q=  
    %   Zernike functions (order N<=7).  In some disciplines it is S^.=j oI  
    %   traditional to label the first 36 functions using a single mode x]M1UBnMN  
    %   number P instead of separate numbers for the order N and azimuthal >skS`/6  
    %   frequency M. w9BH>56/"  
    % ; U4X U  
    %   Example: 5sUnEHN  
    % &!Vp'l\9  
    %       % Display the first 16 Zernike functions t$2{U  
    %       x = -1:0.01:1; 1x)%9u}  
    %       [X,Y] = meshgrid(x,x); |^&e\8>.  
    %       [theta,r] = cart2pol(X,Y); lX7^LB  
    %       idx = r<=1; ^m9cEl^:nQ  
    %       p = 0:15; >oNs_{  
    %       z = nan(size(X)); ZvJx01F{  
    %       y = zernfun2(p,r(idx),theta(idx)); D%btlw ?{  
    %       figure('Units','normalized') SfR_#"Uu  
    %       for k = 1:length(p) pcG q  
    %           z(idx) = y(:,k); b/`' ?| C  
    %           subplot(4,4,k) :|TBsd|/x  
    %           pcolor(x,x,z), shading interp )*B.y|b #  
    %           set(gca,'XTick',[],'YTick',[]) |d8o<Q  
    %           axis square ~]Jfg$'  
    %           title(['Z_{' num2str(p(k)) '}']) 3i1>EjML  
    %       end &~EOM  
    % Zuf&maa S  
    %   See also ZERNPOL, ZERNFUN. %qhaVM$]  
    4{=Em5`HbO  
    %   Paul Fricker 11/13/2006 *np|PyLP:  
    }ls>~uN  
    i|e-N?l  
    % Check and prepare the inputs: jw)t"S/E  
    % ----------------------------- Q]C1m<x  
    if min(size(p))~=1 Bdi~ B")  
        error('zernfun2:Pvector','Input P must be vector.') XR(kR{yo  
    end  R !HL+  
    5Z'pMkn3  
    if any(p)>35 #<:khs6  
        error('zernfun2:P36', ... li?Gb1  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... bV ym  
               '(P = 0 to 35).']) <Z Ls+|1  
    end Bb6_['y  
    `B~%TEvMh  
    % Get the order and frequency corresonding to the function number: .W\Fa2}%av  
    % ---------------------------------------------------------------- }N NyUwFa  
    p = p(:); I`7[0jA~  
    n = ceil((-3+sqrt(9+8*p))/2); P_}$|zj7  
    m = 2*p - n.*(n+2); <v?-$3YT  
    d?JVB  
    % Pass the inputs to the function ZERNFUN: .+.Pc_fv  
    % ---------------------------------------- sE% n=Ww  
    switch nargin &f$jpIyVX  
        case 3 w$_ooQ(_;Q  
            z = zernfun(n,m,r,theta); /@K?W=w4  
        case 4 ugz1R+f_4{  
            z = zernfun(n,m,r,theta,nflag); gg=z.`}  
        otherwise G 8@%)$A  
            error('zernfun2:nargin','Incorrect number of inputs.') TX;|g1K  
    end >i '3\  
    (Xx @_  
    % 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) /%~`B[4F  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 3.soCyxmc  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of LGK}oL'  
    %   order N and frequency M, evaluated at R.  N is a vector of R6ywc "xE  
    %   positive integers (including 0), and M is a vector with the 'Z';$N ]  
    %   same number of elements as N.  Each element k of M must be a ;kdJxxUox  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) :wMZ&xERDZ  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is r3|vu"Uei  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix MYu-[Hg  
    %   with one column for every (N,M) pair, and one row for every p[cC%3  
    %   element in R.  7p{lDQ  
    % O\x Uv  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- u4%-e )$X  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is |>m@]s7Z  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to w(VH>t  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 - iU7'  
    %   for all [n,m]. )V3G~p=0  
    % 9+pnpaZB0  
    %   The radial Zernike polynomials are the radial portion of the |3s-BKbN4  
    %   Zernike functions, which are an orthogonal basis on the unit 4!D!.t~r  
    %   circle.  The series representation of the radial Zernike z`{x1*w_  
    %   polynomials is ?=aQG0  
    % pDW .Pav  
    %          (n-m)/2 VPK)HzPG,  
    %            __ \SyfEcSf2v  
    %    m      \       s                                          n-2s %h^; "|Z  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r /|Zk$q.\  
    %    n      s=0 pj'Yv  
    % ofsua?lSe  
    %   The following table shows the first 12 polynomials. I6\ l 6o  
    % " @.hz@>  
    %       n    m    Zernike polynomial    Normalization }6`#u :OZ  
    %       --------------------------------------------- IU&n!5d$)|  
    %       0    0    1                        sqrt(2) 1_ %3cN.  
    %       1    1    r                           2 RxcX\:  
    %       2    0    2*r^2 - 1                sqrt(6) l{6fR(d ?  
    %       2    2    r^2                      sqrt(6) ZjbMk 3Y  
    %       3    1    3*r^3 - 2*r              sqrt(8) TEv3;Z*N  
    %       3    3    r^3                      sqrt(8) [i\K#O +f  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) x]w%?BlS  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) [Qr#JJ  
    %       4    4    r^4                      sqrt(10) pLNv\M+  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) {o AJL  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) z;D[7tT  
    %       5    5    r^5                      sqrt(12) 8H;yrNL  
    %       --------------------------------------------- j&u{a[Y/}  
    % Ea 0 j}  
    %   Example: 3 (jI  
    % kk& ([ xqU  
    %       % Display three example Zernike radial polynomials _>0 I9.[5  
    %       r = 0:0.01:1; =56O-l7T*w  
    %       n = [3 2 5]; R FWJ ZN"  
    %       m = [1 2 1]; Xbe=_9l&p  
    %       z = zernpol(n,m,r); '8PZmS8X9  
    %       figure ~Cm_=[  
    %       plot(r,z) U%_BgLwy%  
    %       grid on PIl:z?q({  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') [s"xOP9R  
    % lh{U@,/  
    %   See also ZERNFUN, ZERNFUN2. 9n%vz@X  
    Igrr"NuDZ  
    % A note on the algorithm. CN+[|Mz*p  
    % ------------------------  YH@p\#Y  
    % The radial Zernike polynomials are computed using the series Bz!SZpW(M  
    % representation shown in the Help section above. For many special M*$#j|  
    % functions, direct evaluation using the series representation can V\vt!wBcB  
    % produce poor numerical results (floating point errors), because &C6Z-bS"  
    % the summation often involves computing small differences between nF`_3U8e  
    % large successive terms in the series. (In such cases, the functions ,Y  ./9F  
    % are often evaluated using alternative methods such as recurrence E.45 s? r  
    % relations: see the Legendre functions, for example). For the Zernike c>mTd{Abi  
    % polynomials, however, this problem does not arise, because the k> SPtiAs  
    % polynomials are evaluated over the finite domain r = (0,1), and jI[Y< (F ;  
    % because the coefficients for a given polynomial are generally all "V'<dn  
    % of similar magnitude. QbrR=[8b  
    % epXvk &  
    % ZERNPOL has been written using a vectorized implementation: multiple :"{("!x   
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] &P'd&B1   
    % values can be passed as inputs) for a vector of points R.  To achieve 7wKT:~~oS3  
    % this vectorization most efficiently, the algorithm in ZERNPOL 3qggdi  
    % involves pre-determining all the powers p of R that are required to ';'gKX!9V  
    % compute the outputs, and then compiling the {R^p} into a single Qa>t$`o`  
    % matrix.  This avoids any redundant computation of the R^p, and V(wm?Cc]  
    % minimizes the sizes of certain intermediate variables. 4z6kFQgu  
    % M-7^\wXTA  
    %   Paul Fricker 11/13/2006 w]1Ltq*g/  
    xU9T8Lw  
    ;iq H:wO  
    % Check and prepare the inputs: Bc&Y[u-n  
    % ----------------------------- #|-i*2@oR  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) sYa;vg4[  
        error('zernpol:NMvectors','N and M must be vectors.') &1 t84p:^=  
    end o@2Y98~Q}  
    #7v=#Jco  
    if length(n)~=length(m) h\-3Y U  
        error('zernpol:NMlength','N and M must be the same length.') zpgRK4p,I"  
    end efN5(9*9R  
    y8%QS*  
    n = n(:); Yc^;?n`x  
    m = m(:); 6 H{G$[2  
    length_n = length(n); 0H!J  
    erlg\-H   
    if any(mod(n-m,2)) o3C GG  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Tji*\<?  
    end wNvq['P  
    Q{= DLm`  
    if any(m<0) _D"V^4^yqu  
        error('zernpol:Mpositive','All M must be positive.') 9w!PA-) L  
    end )kIZm Q|f1  
    Fzmc#?  
    if any(m>n) ,76Q*p  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') FQ_a= v  
    end 5J~@jPU  
    %_ (Xn  
    if any( r>1 | r<0 ) /JjSx/  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') F e1^9ja  
    end .C|dGE?,  
    T deHs{|  
    if ~any(size(r)==1) Lrjp  
        error('zernpol:Rvector','R must be a vector.') l3N '@GO  
    end <IBWA0A=8a  
    -[}AhNYK  
    r = r(:); HC!5AJ&+}v  
    length_r = length(r); @Ta0v:Y  
    g|Xjw Ti8$  
    if nargin==4 IE:;`e:\D  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); Ve\.7s  
        if ~isnorm lk4U/:  
            error('zernpol:normalization','Unrecognized normalization flag.') dnzZ\t>U  
        end b Bkg/p]  
    else m^$KDrkD  
        isnorm = false; Owz>g4l r  
    end NqC}}N\,  
    @rE+H 5  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O:j=L{,d^  
    % Compute the Zernike Polynomials 7*eIs2aY  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ("s!t?!&YS  
    V~7Oa2'#B  
    % Determine the required powers of r: L"L a|  
    % ----------------------------------- ?$@E}t8g\  
    rpowers = []; ,l#f6H7p  
    for j = 1:length(n) R^6]v`j;  
        rpowers = [rpowers m(j):2:n(j)]; W~& QcSWqD  
    end %Vb~}sT:  
    rpowers = unique(rpowers); Y;=GM:*H  
    J\?d+}hynX  
    % Pre-compute the values of r raised to the required powers, H2{&da@D5  
    % and compile them in a matrix: uQl=?0 85  
    % ----------------------------- \Icd>>)*  
    if rpowers(1)==0 o4H'  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); g$U7bCHG  
        rpowern = cat(2,rpowern{:}); v*&WqVg  
        rpowern = [ones(length_r,1) rpowern]; xJ^pqb  
    else TOsHb+Uv  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); YR"IPyj  
        rpowern = cat(2,rpowern{:}); W]5sqtF;6  
    end mr+8[0  
    )U+&XjK  
    % Compute the values of the polynomials: 7Ga'FT.F  
    % -------------------------------------- ~}Z'/ zCZf  
    z = zeros(length_r,length_n); \|7Y"WEQ  
    for j = 1:length_n oz'^.+uvE  
        s = 0:(n(j)-m(j))/2; ^o(C\\>{&  
        pows = n(j):-2:m(j);  z:d+RMA  
        for k = length(s):-1:1 5N[H@%>QO  
            p = (1-2*mod(s(k),2))* ... ^,#MfF6  
                       prod(2:(n(j)-s(k)))/          ... 6oLZH6fG  
                       prod(2:s(k))/                 ... yHmNO*(  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... @rlL'|&X*  
                       prod(2:((n(j)+m(j))/2-s(k))); -^,wQW:o)  
            idx = (pows(k)==rpowers);  WYW@%t  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Fv3:J~Yf  
        end +ooQ-Gh  
         kOV6O?h  
        if isnorm `l-R?C?*!  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 71G00@&w9D  
        end y*j8OA.S  
    end 2(>=@q.1H  
    *h~(LH"tN  
    % 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)  g~@0p7]Y  
    RfB""b8]=  
    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。 3Ecm Nwr  
    SJ-g2aAT  
    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)