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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 4@ PA+(kvS  
    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! ;77K&#1  
     
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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 7L!JP:v   
    function z = zernfun(n,m,r,theta,nflag) Tc WCr  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. EeuYRyK  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N H1bR+2s  
    %   and angular frequency M, evaluated at positions (R,THETA) on the xRh 22z  
    %   unit circle.  N is a vector of positive integers (including 0), and =X$ieXq|  
    %   M is a vector with the same number of elements as N.  Each element >US*7m }  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) H [=\_X1o(  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, yXJhOCa  
    %   and THETA is a vector of angles.  R and THETA must have the same fkV@3sj  
    %   length.  The output Z is a matrix with one column for every (N,M) 7Uenr9)M  
    %   pair, and one row for every (R,THETA) pair. ~7]V^tG  
    % jI-a+LnEm  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike \x<8   
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^6s im2  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Ew8@{X y  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, eADCT  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Uj!3MF  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. <:S qMf  
    % fQ<sq0' e\  
    %   The Zernike functions are an orthogonal basis on the unit circle. m^A2 8X7  
    %   They are used in disciplines such as astronomy, optics, and 'a~@q~!  
    %   optometry to describe functions on a circular domain. pj>R9zpn_  
    % /3b *dsYsl  
    %   The following table lists the first 15 Zernike functions. SI7rTJ]/  
    % 1NZ"\9=U  
    %       n    m    Zernike function           Normalization q}{E![ZTu  
    %       -------------------------------------------------- 8D*7{Q  
    %       0    0    1                                 1 l]*RiK2AC  
    %       1    1    r * cos(theta)                    2 VvhfD2*T  
    %       1   -1    r * sin(theta)                    2 ;blL\|ch;  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) vW+6_41ZM  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Z\!,f.>g  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) g3^s_*A  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) }[p{%:tP  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) cx\"r  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) il0K ^i  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) DX_ mrG  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) e" v%m 'G  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) bZu'5+(@  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) YI0 wr1N  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) X=)V<2WO  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) R5HT EB  
    %       -------------------------------------------------- bvox7V>  
    % %>|FJ  
    %   Example 1: (J:+'u  
    % \:vF FK4a  
    %       % Display the Zernike function Z(n=5,m=1) [txOh!sxD  
    %       x = -1:0.01:1; BA;r%?MRL  
    %       [X,Y] = meshgrid(x,x); ,KY;NbL-Jp  
    %       [theta,r] = cart2pol(X,Y); T7[@ lMa?  
    %       idx = r<=1; J Enjc/  
    %       z = nan(size(X)); omG2p  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); u0C:q`;z  
    %       figure d4Uw+3ikW  
    %       pcolor(x,x,z), shading interp g6M>S1oOO  
    %       axis square, colorbar Liqo)m  
    %       title('Zernike function Z_5^1(r,\theta)') !=9x=  
    % +R'8$  
    %   Example 2: c`O~I<(Pm  
    % I%T+H[,  
    %       % Display the first 10 Zernike functions nrEI0E9  
    %       x = -1:0.01:1; /!6'K  
    %       [X,Y] = meshgrid(x,x); }x'*3zI  
    %       [theta,r] = cart2pol(X,Y); GS;GJsAs  
    %       idx = r<=1; j<AOC?  
    %       z = nan(size(X)); *D]:{#C*  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 7oZ :/6_>  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; {yn,u)@r9S  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; :jiEn y  
    %       y = zernfun(n,m,r(idx),theta(idx)); `z!AjAT-G  
    %       figure('Units','normalized') FXCBX:LnvU  
    %       for k = 1:10 u8f\)m  
    %           z(idx) = y(:,k); *>m[ZJd%=  
    %           subplot(4,7,Nplot(k)) J;4x$BI  
    %           pcolor(x,x,z), shading interp WjV Bz   
    %           set(gca,'XTick',[],'YTick',[]) Qz(D1>5I?  
    %           axis square $QJ3~mG2  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @-@Coy 4Tt  
    %       end z{XB_j6\=  
    % Mc,79Ix"  
    %   See also ZERNPOL, ZERNFUN2. -H?c4? 5  
    /|EdpHx0  
    %   Paul Fricker 11/13/2006 ]\yIHdcDi  
    d`sZ"8}j  
    }7.A~h  
    % Check and prepare the inputs: 5U 84 *RY  
    % ----------------------------- NaR} 0  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) j0l{Mc5  
        error('zernfun:NMvectors','N and M must be vectors.') jcCAXk055  
    end EX)&|2w  
    L>Y+}]~  
    if length(n)~=length(m) 2Zu9? L ,I  
        error('zernfun:NMlength','N and M must be the same length.') l*ltS(?  
    end 1RAkqw<E  
    ]d*9@+Iu  
    n = n(:); Zkp~qx  
    m = m(:); !W}sOK7#  
    if any(mod(n-m,2))  AG(6.  
        error('zernfun:NMmultiplesof2', ... {B$CqsvJ  
              'All N and M must differ by multiples of 2 (including 0).') hFV,FBsAO  
    end [6VB&   
    TXImmkC  
    if any(m>n) c>bns/f  
        error('zernfun:MlessthanN', ... @eYpARF  
              'Each M must be less than or equal to its corresponding N.') <) * U/r  
    end X,Ql6uO  
    "uH>S+%|b  
    if any( r>1 | r<0 ) (cj9xROx  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') oidZWy  
    end  +n1!xv]  
    >LBA0ynh {  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) *7Vb([x4;  
        error('zernfun:RTHvector','R and THETA must be vectors.') J v}  
    end [8QK @5[  
    hjL;B 'IL  
    r = r(:); VMah3T!  
    theta = theta(:); N[Z`tk?-  
    length_r = length(r); s^u  Y   
    if length_r~=length(theta) 66val"^W  
        error('zernfun:RTHlength', ... N,Y)'s<  
              'The number of R- and THETA-values must be equal.') z:Am1B  
    end \%7*@&  
    e!VtDJDS  
    % Check normalization: [CQR  
    % -------------------- ysnW3q!@  
    if nargin==5 && ischar(nflag) P<pv@ l9)  
        isnorm = strcmpi(nflag,'norm'); .SC *!,  
        if ~isnorm FJvY`zqB  
            error('zernfun:normalization','Unrecognized normalization flag.') yTZev|ej@  
        end t}+/GSwT  
    else *'((_ NZ>  
        isnorm = false; xQsxc  
    end |k.'w<6mb9  
    "L3mW=!*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5dj" UxH  
    % Compute the Zernike Polynomials *PF<J/Pr  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Dg&6@c|  
    V 20h\(\\  
    % Determine the required powers of r: U[wx){[|  
    % ----------------------------------- o[>d"Kp  
    m_abs = abs(m); wR%Ta-  
    rpowers = []; um,f!ho-U  
    for j = 1:length(n) cC~RW71  
        rpowers = [rpowers m_abs(j):2:n(j)]; B4.: 9Od3  
    end 4aO/^Hl  
    rpowers = unique(rpowers); +byOThuE  
    7d;|?R-8D  
    % Pre-compute the values of r raised to the required powers, SAP/jD$5]>  
    % and compile them in a matrix: gPd K%"B@  
    % ----------------------------- AE rPd)yk0  
    if rpowers(1)==0 6n]+(=  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Gxw1P@<F:  
        rpowern = cat(2,rpowern{:}); 6ll!7U(9(  
        rpowern = [ones(length_r,1) rpowern];  9q"kM  
    else 5cP yi/  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); }n^Rcz6HeO  
        rpowern = cat(2,rpowern{:}); 01A{\O1$j  
    end A.>mk598  
    3E*|^*  
    % Compute the values of the polynomials: x;~:p;]J2F  
    % -------------------------------------- 4>,X.|9{  
    y = zeros(length_r,length(n)); A3eCI  
    for j = 1:length(n) >~o- 6g  
        s = 0:(n(j)-m_abs(j))/2; "D'e  
        pows = n(j):-2:m_abs(j); ?X@!jB,Pv  
        for k = length(s):-1:1 S f?;j{?G  
            p = (1-2*mod(s(k),2))* ... /2p*uv }IP  
                       prod(2:(n(j)-s(k)))/              ... !Gmnck&+  
                       prod(2:s(k))/                     ... 2>o[  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... |N/d }  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); >V6t L;+  
            idx = (pows(k)==rpowers); &J 3QO%  
            y(:,j) = y(:,j) + p*rpowern(:,idx); V_h&9]RL  
        end ,ua1sTgQ  
         D, ")n75  
        if isnorm n\+ c3  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 5f*_K6,v  
        end R /=rNUe  
    end 4aHogheg  
    % END: Compute the Zernike Polynomials iVFOOsJ@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >ai,6!  
    {;{U@Z  
    % Compute the Zernike functions: VM$n|[C~  
    % ------------------------------ t'U=K>7  
    idx_pos = m>0; kyHli~Nr"  
    idx_neg = m<0; ji?Hw  
    qHk{5O3  
    z = y; <Z^by;d|z  
    if any(idx_pos) PK+sGV  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Uj5-x%~  
    end ^.A*mMQ  
    if any(idx_neg) .lcp5D[(  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); @} Ig*@  
    end :-RB< Lj  
    pA!-spgX  
    % 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) %M96 m   
    %ZERNFUN2 Single-index Zernike functions on the unit circle. t=Oq<r  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated "}71z  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Ll|-CY $  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, NO~G4PUM0C  
    %   and THETA is a vector of angles.  R and THETA must have the same v]cw})l  
    %   length.  The output Z is a matrix with one column for every P-value, =NY;#Jjn  
    %   and one row for every (R,THETA) pair. n.@HT"  
    % )^(gwE  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike wh(_<VZ  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 4 {M   
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) /NQrE#pb  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 'pt(  
    %   for all p. @ZISv'F  
    % J_7w _T/  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 5qM$ahN3wH  
    %   Zernike functions (order N<=7).  In some disciplines it is +<Gp >c  
    %   traditional to label the first 36 functions using a single mode ejd_ 85$  
    %   number P instead of separate numbers for the order N and azimuthal ,8384'  
    %   frequency M. cq[9#@ 4=  
    % z\eQB%aM  
    %   Example: ^;F/^ _  
    % & R_?6*n  
    %       % Display the first 16 Zernike functions ^t3>Z|DiB^  
    %       x = -1:0.01:1; ~R;/u")@e  
    %       [X,Y] = meshgrid(x,x); 0PkX-.  
    %       [theta,r] = cart2pol(X,Y); E7O3$B8  
    %       idx = r<=1; S]@;`_?m{  
    %       p = 0:15; k[8F: T-  
    %       z = nan(size(X)); 6'r;6T *  
    %       y = zernfun2(p,r(idx),theta(idx)); 5$ik|e^:y  
    %       figure('Units','normalized') $ 5"  
    %       for k = 1:length(p) !\#Wq{p>W*  
    %           z(idx) = y(:,k); GFT@Pqq  
    %           subplot(4,4,k) O[ z0+Q?6Z  
    %           pcolor(x,x,z), shading interp I3>8B  
    %           set(gca,'XTick',[],'YTick',[]) *Hx*s_F  
    %           axis square +[ ?!@)  
    %           title(['Z_{' num2str(p(k)) '}']) -. *E<%  
    %       end , >WH)+a  
    % y`j_]qvt  
    %   See also ZERNPOL, ZERNFUN. ~ g!!#ad  
    s={>{,E  
    %   Paul Fricker 11/13/2006 n>)aw4  
    = 9Yf o,F  
    }36AeJ7L  
    % Check and prepare the inputs: (RI)<zaK ;  
    % ----------------------------- 3u j|jwL  
    if min(size(p))~=1 #U-y<[ 3  
        error('zernfun2:Pvector','Input P must be vector.') 80Y% C-Y:  
    end Y-})/zFc  
    4AL,=C3  
    if any(p)>35 ~2>Adp  
        error('zernfun2:P36', ... J3y _JoS  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... oOprzxf"+Z  
               '(P = 0 to 35).']) `]65&hWZL  
    end '|gsmO  
    N/F_,>E  
    % Get the order and frequency corresonding to the function number: fK:4jl-r  
    % ---------------------------------------------------------------- V06*qQ[  
    p = p(:); ]0'cdC  
    n = ceil((-3+sqrt(9+8*p))/2); l1]N&jN{  
    m = 2*p - n.*(n+2); 6)sKg{H  
    _b8KK4UR  
    % Pass the inputs to the function ZERNFUN: Vl+UC1M}B>  
    % ---------------------------------------- ?8$`GyjS  
    switch nargin 1M`>;fjYa  
        case 3 ;uAh)|;S#  
            z = zernfun(n,m,r,theta); 0m1V@ 3]7>  
        case 4 dm/3{\ 4  
            z = zernfun(n,m,r,theta,nflag); ~Q=;L>Qd  
        otherwise {*bx8*y1  
            error('zernfun2:nargin','Incorrect number of inputs.') 274j7Y'  
    end a)W|gx6Y  
    8/p ]'BLf  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) Uy=yA  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. i5(qJ/u  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of CSjd&G *ZB  
    %   order N and frequency M, evaluated at R.  N is a vector of >#n"r1  
    %   positive integers (including 0), and M is a vector with the An>ai N]  
    %   same number of elements as N.  Each element k of M must be a t0cS.hi  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) i'CK/l.H  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Zl%)#=kO  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix hwk] ;6[  
    %   with one column for every (N,M) pair, and one row for every _fgsHx>l7  
    %   element in R. jSBz),.XU}  
    % s8A"x`5(  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- Vxrj(knck,  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is w8 S pt  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to V*JqC  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 tMdSdJ8  
    %   for all [n,m]. y)L X?d  
    % #/I+[|=[O  
    %   The radial Zernike polynomials are the radial portion of the D4"<suU|.  
    %   Zernike functions, which are an orthogonal basis on the unit Msvs98LvW  
    %   circle.  The series representation of the radial Zernike N (\n$bpTt  
    %   polynomials is ox<6qW  
    % nGTGX  
    %          (n-m)/2 CUTjRWQ  
    %            __ 3;b)pQ~6CJ  
    %    m      \       s                                          n-2s _3u3b/%J?  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r dVq9'{[3  
    %    n      s=0 yS K81`  
    % ?.ObHV*k  
    %   The following table shows the first 12 polynomials. `B&E?x  
    % P$Y w'3v/  
    %       n    m    Zernike polynomial    Normalization > mCH!ey  
    %       --------------------------------------------- })8D3kzX)  
    %       0    0    1                        sqrt(2) oFyB-vpYQV  
    %       1    1    r                           2 vp&.  
    %       2    0    2*r^2 - 1                sqrt(6) hG`@#9|f  
    %       2    2    r^2                      sqrt(6) f@9XSZ<.71  
    %       3    1    3*r^3 - 2*r              sqrt(8) 5mVO9Q j  
    %       3    3    r^3                      sqrt(8) >8{{H"$;(  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) LIJ#nb  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) G]xYQ]  
    %       4    4    r^4                      sqrt(10) Tw%1m  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) o=7e8l  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Dg~m}La  
    %       5    5    r^5                      sqrt(12) 6ym$8^  
    %       --------------------------------------------- hX,RuI  
    % #v$wjqK5  
    %   Example: 3XUVUd~  
    % ?t}s3P!Q3w  
    %       % Display three example Zernike radial polynomials Qa2p34Z/  
    %       r = 0:0.01:1; B(FM~TVZ  
    %       n = [3 2 5]; |gk4X%o6  
    %       m = [1 2 1]; Y$, ++wx  
    %       z = zernpol(n,m,r); d/+s-g p  
    %       figure `o9:6X?RA  
    %       plot(r,z) T6?03cSE  
    %       grid on E>#@ H  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ZLVgK@l  
    % 1H%p|'FKA  
    %   See also ZERNFUN, ZERNFUN2. S+Ia2O)BA  
    ' 8R5 Tl  
    % A note on the algorithm. c9@3=6S/  
    % ------------------------ WAuT`^"u  
    % The radial Zernike polynomials are computed using the series 2ER_?y  
    % representation shown in the Help section above. For many special rT-.'aQ2t  
    % functions, direct evaluation using the series representation can 9 M?UPE  
    % produce poor numerical results (floating point errors), because ~[aV\r?  
    % the summation often involves computing small differences between x~m$(LT  
    % large successive terms in the series. (In such cases, the functions eC 2~&:$L  
    % are often evaluated using alternative methods such as recurrence 3DCR n :  
    % relations: see the Legendre functions, for example). For the Zernike 8.%wnH  
    % polynomials, however, this problem does not arise, because the 7On.y*  
    % polynomials are evaluated over the finite domain r = (0,1), and 4S+E% b|)  
    % because the coefficients for a given polynomial are generally all | "b|Q  
    % of similar magnitude. g@t..xJ,  
    %  MkdC*|  
    % ZERNPOL has been written using a vectorized implementation: multiple B1I{@\z0G  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] PxWH)4  
    % values can be passed as inputs) for a vector of points R.  To achieve l  ~xXy<  
    % this vectorization most efficiently, the algorithm in ZERNPOL 2A*,9S|Y  
    % involves pre-determining all the powers p of R that are required to G&Yo2aADR  
    % compute the outputs, and then compiling the {R^p} into a single '%+LQ"Bp  
    % matrix.  This avoids any redundant computation of the R^p, and #;1RStb:zj  
    % minimizes the sizes of certain intermediate variables. ~rq:I<5  
    % BTO l`U  
    %   Paul Fricker 11/13/2006 b9wC:NgQx  
    _TJk Yz$  
    uHZjpMoM  
    % Check and prepare the inputs: "-5FUKI-  
    % ----------------------------- c/F!cW{z^  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Qi qRx  
        error('zernpol:NMvectors','N and M must be vectors.') P uQ  
    end {65Y Tt%  
    S,'ekWVD  
    if length(n)~=length(m) yk^2<?z>2  
        error('zernpol:NMlength','N and M must be the same length.') A;^{%S  
    end o5+7Lt]  
    %Zfh6Bl\X  
    n = n(:); t82*rC IB{  
    m = m(:); u ^2/:L  
    length_n = length(n); jCx*{TO  
    R +U*]5~R  
    if any(mod(n-m,2)) ojJu a c4  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') a>ye  
    end %rl<%%T#.M  
    J!Rqm!)q  
    if any(m<0) d;3f80Kd*  
        error('zernpol:Mpositive','All M must be positive.') V.+a}J=Cw  
    end gId :IR  
    $fPiR  
    if any(m>n) c*(=Glzn  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') D51O/.:U2  
    end Pc+,iK>  
    hsK(09:J  
    if any( r>1 | r<0 ) pJo4&Ff  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') @P=n{-pIW  
    end h9nh9a(2  
    A~s6~  
    if ~any(size(r)==1) ZY6%%7?1  
        error('zernpol:Rvector','R must be a vector.') i4SWFa``  
    end F"tM?V.|  
    <BdC#t:*L  
    r = r(:); i1HO>X:ea  
    length_r = length(r); ~PN[ #e]  
    x;4m@)Mu  
    if nargin==4 XNBzA3W  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); {-|El}.M  
        if ~isnorm 5["n] i  
            error('zernpol:normalization','Unrecognized normalization flag.') 20Rm|CNH?  
        end [2 Rz8e^  
    else [eDRghK  
        isnorm = false; B\dhw@hM  
    end E /ycPqD  
    1aUu:#c  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (tg+C\ S.  
    % Compute the Zernike Polynomials nF 'U*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ml_!)b  
    S! .N3ezn  
    % Determine the required powers of r: YIl,8! z~  
    % ----------------------------------- 1-G-p:|  
    rpowers = []; G<~P||Lu^  
    for j = 1:length(n) 2T"[$iH!7  
        rpowers = [rpowers m(j):2:n(j)]; /DSy/p0%  
    end 7l'1  
    rpowers = unique(rpowers); kPnuU!  
    Z~"8C Kz  
    % Pre-compute the values of r raised to the required powers, gg Hl{cl)  
    % and compile them in a matrix: 1fh6A`c  
    % ----------------------------- <9Ytv|t@0  
    if rpowers(1)==0 tkHUX!Ow;  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 9H%ixBnM  
        rpowern = cat(2,rpowern{:}); T zHR  
        rpowern = [ones(length_r,1) rpowern]; Lo9 \[4FP  
    else ?&b"/sRS  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ^;EwZwH[  
        rpowern = cat(2,rpowern{:}); tkdyR1-  
    end YgkQF0+  
    G;AV~1i:~  
    % Compute the values of the polynomials: >>>MTV f  
    % -------------------------------------- / DS T|2  
    z = zeros(length_r,length_n); c{/KkmI  
    for j = 1:length_n MIc(B_q  
        s = 0:(n(j)-m(j))/2; O$Dj_R#  
        pows = n(j):-2:m(j); qh wl  
        for k = length(s):-1:1 j<vU[J+gx~  
            p = (1-2*mod(s(k),2))* ... 7"{CBbT  
                       prod(2:(n(j)-s(k)))/          ... wV{VV?h}  
                       prod(2:s(k))/                 ... Q,\S3>1n  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... p"NuR4   
                       prod(2:((n(j)+m(j))/2-s(k))); uU8L93  
            idx = (pows(k)==rpowers); gL wNHS  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Mwd.S  
        end M3q7{w*bM  
         oeG?2!Zh  
        if isnorm !p 70g0+  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 8j@ADfZ9  
        end S%w67sGl4n  
    end 9OM&&Ue<E  
    zXeBUbVi  
    % 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)  Yl6\}_h`  
    :-kXZe  
    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。 tcxs%yWO1  
    lhA<wV1-9G  
    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)