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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 | a i#rU  
    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! e?07o!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 D0S^Msk9L  
    function z = zernfun(n,m,r,theta,nflag) :AuKQ`c  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 3w[uc~f  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 3qNuv];2  
    %   and angular frequency M, evaluated at positions (R,THETA) on the UaQW<6+  
    %   unit circle.  N is a vector of positive integers (including 0), and ]PL\;[b>  
    %   M is a vector with the same number of elements as N.  Each element $SFreyI;Uf  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) xZV|QVY;  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, I7'v;*  
    %   and THETA is a vector of angles.  R and THETA must have the same =bvLMpa  
    %   length.  The output Z is a matrix with one column for every (N,M) *(/b{!~  
    %   pair, and one row for every (R,THETA) pair. _XrlCLp: d  
    % 0s}gg[lj  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike _wW"Tn]  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ?G&J_L=@Y  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral PqyR,Bcx0  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ~WB-WI\  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized +>a(9r|:  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. d=\\ik8  
    % k4:=y9`R}$  
    %   The Zernike functions are an orthogonal basis on the unit circle. '?{L gj^R  
    %   They are used in disciplines such as astronomy, optics, and Q4N0j' QA  
    %   optometry to describe functions on a circular domain. %t:13eM  
    % kqC7^x  
    %   The following table lists the first 15 Zernike functions. OH 88d:  
    % >w\3.6A  
    %       n    m    Zernike function           Normalization 0.(7R,-  
    %       -------------------------------------------------- P{2ED1T\  
    %       0    0    1                                 1 w5Ucj*A\  
    %       1    1    r * cos(theta)                    2 XwU1CejP0  
    %       1   -1    r * sin(theta)                    2 {K/xI  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) O=!EqaExW  
    %       2    0    (2*r^2 - 1)                    sqrt(3) >7W8_6sC<  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) /B{c L`<  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) $O\]cQD`u  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) nnd-d+$  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) /" &Jf}r  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ah!RQ2hDrV  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) HXqG;Fds(  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) O G7U+d6  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) H}1XK|K3#H  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) N{!@M_C^%R  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) x.(Sv]+[  
    %       -------------------------------------------------- cI <T/~P  
    % c^,8eb7c  
    %   Example 1: 0{Zwg0&  
    % _]+ \ B  
    %       % Display the Zernike function Z(n=5,m=1) D;DI8.4`N  
    %       x = -1:0.01:1; #CB`7 }jq  
    %       [X,Y] = meshgrid(x,x); 09Z\F^*$F  
    %       [theta,r] = cart2pol(X,Y); {E1^Wn1M  
    %       idx = r<=1; 5@i(pVWZ  
    %       z = nan(size(X)); ~llw_ w  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx));  JU=4v!0  
    %       figure >?$qKu  
    %       pcolor(x,x,z), shading interp U,~Z2L  
    %       axis square, colorbar If@%^'^ON=  
    %       title('Zernike function Z_5^1(r,\theta)') >>h0(G|  
    % j5 W)9HW:  
    %   Example 2: $\nAGmp@  
    % l9NET  
    %       % Display the first 10 Zernike functions <gY.2#6C\%  
    %       x = -1:0.01:1; rPJbbV",+^  
    %       [X,Y] = meshgrid(x,x); O-<nL B!Wf  
    %       [theta,r] = cart2pol(X,Y); Aq&H-g]s  
    %       idx = r<=1; MrS~u  
    %       z = nan(size(X)); 6 &MATMR  
    %       n = [0  1  1  2  2  2  3  3  3  3]; <\\,L@  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; .+`Z:{:BC&  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; B%Z,Xjq  
    %       y = zernfun(n,m,r(idx),theta(idx)); QPz3IK%   
    %       figure('Units','normalized')  'v&f  
    %       for k = 1:10 XSo$;q\  
    %           z(idx) = y(:,k); G:|=d0  
    %           subplot(4,7,Nplot(k)) )^Md ^\?  
    %           pcolor(x,x,z), shading interp *W1:AGpz  
    %           set(gca,'XTick',[],'YTick',[]) Hl*/s  
    %           axis square zT _[pa)O`  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) roWg~U(S  
    %       end um mkAeWb  
    % ! d" i  
    %   See also ZERNPOL, ZERNFUN2. ,Je9]XT  
    ADlLodG  
    %   Paul Fricker 11/13/2006 EY.Z.gMZI(  
    ?C|b>wM/  
    +"SYG  
    % Check and prepare the inputs: vsCy?  
    % ----------------------------- VaFv%%w  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <a$'tw-8  
        error('zernfun:NMvectors','N and M must be vectors.')  *4{GI D  
    end P }$DCD<$U  
    t3FfPV!P"  
    if length(n)~=length(m) . ^JsnP  
        error('zernfun:NMlength','N and M must be the same length.') ^CQVqa${]  
    end ^/v!hq_#%&  
    CXhE+oS5z'  
    n = n(:); H83/X,"!w  
    m = m(:); + |d[q?  
    if any(mod(n-m,2)) c=\H&x3X  
        error('zernfun:NMmultiplesof2', ... $+Vp>  
              'All N and M must differ by multiples of 2 (including 0).') ugMf pT)  
    end c27\S?\ Jd  
    hG%J:}  
    if any(m>n) M}b[;/~  
        error('zernfun:MlessthanN', ... jMB&(r  
              'Each M must be less than or equal to its corresponding N.') VT`C<'   
    end N13wVx  
    dQH9NsV7g  
    if any( r>1 | r<0 ) b'5L|1d  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') j?cE0 hz  
    end v6_fF5N/  
    > z1q\cz  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) YU24wTe;k  
        error('zernfun:RTHvector','R and THETA must be vectors.') |dQ-l !  
    end p$OkWSi~  
    )M(-EDL>Qk  
    r = r(:); B&k"B?9mL  
    theta = theta(:); 1me16 5y<B  
    length_r = length(r); O &De!Gx  
    if length_r~=length(theta) $bT<8:g  
        error('zernfun:RTHlength', ... gls %<A{C  
              'The number of R- and THETA-values must be equal.') nq"U`z@R  
    end A5LTgGzaW  
    R#i{eE*WF  
    % Check normalization: W|aFEY  
    % -------------------- n%Gk {h5  
    if nargin==5 && ischar(nflag) Y< drRK!  
        isnorm = strcmpi(nflag,'norm'); l^*'W(%  
        if ~isnorm [N4#R  
            error('zernfun:normalization','Unrecognized normalization flag.') Y$ To)qo  
        end UL   
    else 8KrqJN0\  
        isnorm = false; \9GJa"xA`  
    end Gh#$[5&`  
    F 7~T=X)1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?$&iVN^UA  
    % Compute the Zernike Polynomials r.T!R6v}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8KU5x#  
    pAd 8-a  
    % Determine the required powers of r: "TboIABp:H  
    % ----------------------------------- LmQS;/:  
    m_abs = abs(m); ^dF?MQA<@  
    rpowers = []; 0j )D[K  
    for j = 1:length(n) C0$KpUB  
        rpowers = [rpowers m_abs(j):2:n(j)]; OLS.0UEc  
    end 9e*v&A2Y'  
    rpowers = unique(rpowers); G uLU7a  
    FV->226o%  
    % Pre-compute the values of r raised to the required powers, i`}nv,  
    % and compile them in a matrix: N-O"y3W}  
    % ----------------------------- &n)=OConge  
    if rpowers(1)==0 L)`SNN\ipR  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 8qY\T0  
        rpowern = cat(2,rpowern{:}); Z*Fxr;)d  
        rpowern = [ones(length_r,1) rpowern];  A/zZ%h  
    else / .ddx<  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 4.}{B_)LK  
        rpowern = cat(2,rpowern{:}); e0ea2 2  
    end DiLZ5^`]  
    ^t'mfG|DV  
    % Compute the values of the polynomials: O4mSr{HCp  
    % -------------------------------------- x8]5> G8(r  
    y = zeros(length_r,length(n)); E0Y>2HOuL  
    for j = 1:length(n) lSu\VCG  
        s = 0:(n(j)-m_abs(j))/2; quPNwNy  
        pows = n(j):-2:m_abs(j); &2EimP  
        for k = length(s):-1:1 /d\#|[S  
            p = (1-2*mod(s(k),2))* ... l6wN&JHTh  
                       prod(2:(n(j)-s(k)))/              ... n\ yDMY  
                       prod(2:s(k))/                     ... )_\ZUem  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Hmi]qK[F  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); >*A"tk#oR  
            idx = (pows(k)==rpowers); K~ 6[zJ4  
            y(:,j) = y(:,j) + p*rpowern(:,idx); TC%ENxDR  
        end &u@<0 1=  
         CE'd`_;HLn  
        if isnorm BmP!/i_  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); X?'v FC  
        end P'dH*}H  
    end |H LU5=Y  
    % END: Compute the Zernike Polynomials PSM~10l,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (")IU{>c6  
    >*hY1@N1  
    % Compute the Zernike functions: GjmPpKIu\  
    % ------------------------------ Y30e7d* qr  
    idx_pos = m>0; cM= ? {W7~  
    idx_neg = m<0; j~IX  
    Z?7XuELKV  
    z = y; p%8v+9+h2  
    if any(idx_pos) = %O@%v  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); + ~6Nq(kV  
    end |V 3AA   
    if any(idx_neg) V@QWJZ"  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Y] ZNAR  
    end :slVja$e  
    O$2= Z  
    % EOF zernfun
    离线niuhelen
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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-M| 6fv  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. &_L FV@/  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated C1/<t)^  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive ((2 g  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 1qR[& =/  
    %   and THETA is a vector of angles.  R and THETA must have the same <rO0t9OH  
    %   length.  The output Z is a matrix with one column for every P-value, 7nzNBtk  
    %   and one row for every (R,THETA) pair. 'pJ46"D@m  
    % TTJFF\$?  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike "I)*W8wTn  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) jK[~d Y  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) kiW|h)w_,v  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 (d L;A0L  
    %   for all p. ]w3-No  
    % <`B4+:;w6  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 *bl*R';  
    %   Zernike functions (order N<=7).  In some disciplines it is Z/|oCwR  
    %   traditional to label the first 36 functions using a single mode YPjjSi:#  
    %   number P instead of separate numbers for the order N and azimuthal xHA6  
    %   frequency M. * 5H  
    % n^svRM]eQ  
    %   Example: syEWc(5  
    % Kc6p||<  
    %       % Display the first 16 Zernike functions 'w'P rM,:  
    %       x = -1:0.01:1; JAjXhk<=  
    %       [X,Y] = meshgrid(x,x); y?ps+ce93  
    %       [theta,r] = cart2pol(X,Y); F~NmLm  
    %       idx = r<=1; }`O_  
    %       p = 0:15; \m>mE/N  
    %       z = nan(size(X)); ^!={=No]  
    %       y = zernfun2(p,r(idx),theta(idx)); B1EI'<S  
    %       figure('Units','normalized') &0E>&1`7  
    %       for k = 1:length(p) kl0!*j  
    %           z(idx) = y(:,k); ,?OV39h  
    %           subplot(4,4,k) CaSoR |  
    %           pcolor(x,x,z), shading interp MH"{N "|  
    %           set(gca,'XTick',[],'YTick',[]) AgOw{bJ%  
    %           axis square VSK!Pc.G}  
    %           title(['Z_{' num2str(p(k)) '}']) : MOr?"  
    %       end (QO8_  
    % '7+e!>"  
    %   See also ZERNPOL, ZERNFUN. H`js1b1n  
    h";0i:  
    %   Paul Fricker 11/13/2006 cJ6n@\  
    e?Pzhh a  
    >S3,_@C  
    % Check and prepare the inputs: GI&XL'K&  
    % ----------------------------- !>~W5c^  
    if min(size(p))~=1 .{cka]9WJz  
        error('zernfun2:Pvector','Input P must be vector.') N36<EHq  
    end 5h"moh9tG  
    :YL`GSl  
    if any(p)>35 Ig9gGI,  
        error('zernfun2:P36', ... RXMzwk  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... o%!8t_1mR  
               '(P = 0 to 35).']) `/zx2Tkk  
    end lJ+05\pE  
    :UjF<V  
    % Get the order and frequency corresonding to the function number: j@JhxCe1+R  
    % ---------------------------------------------------------------- (+@ Lnz\  
    p = p(:); X[Lwx.Ly8  
    n = ceil((-3+sqrt(9+8*p))/2); \#(3r1(  
    m = 2*p - n.*(n+2); >~;MQDU5*Y  
    d?j_L`?+  
    % Pass the inputs to the function ZERNFUN: 8Ol#-2>k$  
    % ---------------------------------------- C%s+o0b  
    switch nargin T gpf0(  
        case 3 *z2G(Uac  
            z = zernfun(n,m,r,theta); E0;KTcZi  
        case 4 o gcEv>0  
            z = zernfun(n,m,r,theta,nflag); `$IuN *  
        otherwise M;BDo(1  
            error('zernfun2:nargin','Incorrect number of inputs.') &WSxg&YG)\  
    end (dOC ^i  
    #WBlEVx;Z  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) H4s^&--  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. OH`| c  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of o:H^ L,<Tl  
    %   order N and frequency M, evaluated at R.  N is a vector of cC{eu[ XW  
    %   positive integers (including 0), and M is a vector with the 08J[9a0[  
    %   same number of elements as N.  Each element k of M must be a pwg$% lv  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) nz72w_  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is X$o$8s  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix \)\uAI-  
    %   with one column for every (N,M) pair, and one row for every 3 ;M7^DM  
    %   element in R. _ZM$&6EC  
    % >]6f!;Rt  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 9FB[`}  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 2nSX90@:  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 9"KO!w  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 \;al@yC=T  
    %   for all [n,m]. bSOxM /N  
    % :}QBrd  
    %   The radial Zernike polynomials are the radial portion of the Tr}z&efY  
    %   Zernike functions, which are an orthogonal basis on the unit `Gct_6  
    %   circle.  The series representation of the radial Zernike v'R{lXE  
    %   polynomials is ?xftr(  
    % A 1b</2  
    %          (n-m)/2 DuESLMhz  
    %            __ \rXmWzl{  
    %    m      \       s                                          n-2s BMubN   
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r zf@gAvJ  
    %    n      s=0 do {E39  
    % 6f"jl  
    %   The following table shows the first 12 polynomials. $]V,H"  
    % 4 &r5M  
    %       n    m    Zernike polynomial    Normalization bve_*7CEM  
    %       --------------------------------------------- 1J`<'{*  
    %       0    0    1                        sqrt(2) T:v.]0l~  
    %       1    1    r                           2 ;kSRv=S  
    %       2    0    2*r^2 - 1                sqrt(6) Wo&WO e  
    %       2    2    r^2                      sqrt(6) J1i{n7f=@  
    %       3    1    3*r^3 - 2*r              sqrt(8) rF9|xgFK  
    %       3    3    r^3                      sqrt(8) MQs!+Z"m>  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) w %4SNR  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) $8/=@E{51  
    %       4    4    r^4                      sqrt(10) nWfzwXP>_  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) K|i:tHF]@  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) n$2Ia E;v  
    %       5    5    r^5                      sqrt(12) /_WA F90R?  
    %       --------------------------------------------- w}8 ,ICL  
    % ,;3bPjey  
    %   Example: _?]0b7X  
    % 0P{^aSxTP  
    %       % Display three example Zernike radial polynomials k#eH Q!  
    %       r = 0:0.01:1; u|;?FQ$M  
    %       n = [3 2 5]; vbt0G-%Z  
    %       m = [1 2 1]; <WXGDCj  
    %       z = zernpol(n,m,r); JD`IPQb~E  
    %       figure myq@X(K  
    %       plot(r,z) #'DrgZ)W  
    %       grid on {Ad4H[]|]  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') +8<|P&fH  
    % X8}m %  
    %   See also ZERNFUN, ZERNFUN2. s ;3k#-w  
    lN(|EI  
    % A note on the algorithm. M =/+q  
    % ------------------------ Tu!2lHK;  
    % The radial Zernike polynomials are computed using the series jHPkfwfAF  
    % representation shown in the Help section above. For many special BlLK6"gJT  
    % functions, direct evaluation using the series representation can ,9A1p06  
    % produce poor numerical results (floating point errors), because 'CQ~ZV5  
    % the summation often involves computing small differences between B$=oU   
    % large successive terms in the series. (In such cases, the functions DOaTp f  
    % are often evaluated using alternative methods such as recurrence  :EGvI  
    % relations: see the Legendre functions, for example). For the Zernike @AB}r1E2  
    % polynomials, however, this problem does not arise, because the }3825  
    % polynomials are evaluated over the finite domain r = (0,1), and FT F`-}Hz  
    % because the coefficients for a given polynomial are generally all :VkuK@Th`  
    % of similar magnitude. OLH[F  
    % T!f+H?6  
    % ZERNPOL has been written using a vectorized implementation: multiple _p^?_  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] #QUQC2P(~  
    % values can be passed as inputs) for a vector of points R.  To achieve u=6LPwiI  
    % this vectorization most efficiently, the algorithm in ZERNPOL Cs!z3QU  
    % involves pre-determining all the powers p of R that are required to 7 @W}>gnf  
    % compute the outputs, and then compiling the {R^p} into a single 2_/H,  
    % matrix.  This avoids any redundant computation of the R^p, and +YJpVxYmZ  
    % minimizes the sizes of certain intermediate variables. ~=P#7l\o1  
    % < `Xt?K  
    %   Paul Fricker 11/13/2006  JT,[;  
    @u>:(9bp  
    =x xN3Ay  
    % Check and prepare the inputs: h tuYctu`  
    % ----------------------------- 7Dt* ++:  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) op2<~v0?  
        error('zernpol:NMvectors','N and M must be vectors.') .5^7Jwh  
    end kC_Kb&Q0  
    H2jF=U"=  
    if length(n)~=length(m) `o4%UkBpM  
        error('zernpol:NMlength','N and M must be the same length.') Hhzi(<e^  
    end v:IpZ;^  
    qo*%S  
    n = n(:); eqY8;/  
    m = m(:); .)g7s? K  
    length_n = length(n); NiSybyR$  
    @$7'{*  
    if any(mod(n-m,2)) d)WGI RUx  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') WvoJ^{\4N*  
    end !hugn6  
    H3xMoSs  
    if any(m<0) 3j6Am{9  
        error('zernpol:Mpositive','All M must be positive.') "$I8EW/1  
    end ,%T sfB  
    <~bvf A=  
    if any(m>n) 6no&2a|D  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') l9Av@|  
    end 01 <Ti"  
    >^~W'etX|  
    if any( r>1 | r<0 ) PJ))p6 9  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') grxlGS~Q  
    end D &Bdl5g  
    8U)*kmq  
    if ~any(size(r)==1) x+bC\,q  
        error('zernpol:Rvector','R must be a vector.') 8zO;=R A7%  
    end NX&Z=ObHu}  
    {+^&7JX  
    r = r(:); `]I p`_{  
    length_r = length(r); 7P bwCRg  
    STL+tLJ  
    if nargin==4 fyF8RTm{  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); veAdk9  
        if ~isnorm s^"*]9B"  
            error('zernpol:normalization','Unrecognized normalization flag.') NtG^t}V  
        end P r2WF~NuO  
    else 1wy?<B.f  
        isnorm = false; or`D-x)+@  
    end $}t;c62  
    pS~=T}o  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?s@=DDB\u  
    % Compute the Zernike Polynomials ^@ Xzh:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% > ofWHl[-  
    ROJ=ZYof  
    % Determine the required powers of r: G.~Ffk  
    % ----------------------------------- >ra)4huZ  
    rpowers = []; HP,{/ $i:  
    for j = 1:length(n) QT4&Ix,4T1  
        rpowers = [rpowers m(j):2:n(j)]; {#,?K  
    end J%IKdxa  
    rpowers = unique(rpowers); KjK-#F,@  
    $#-O^0D  
    % Pre-compute the values of r raised to the required powers, y[';@t7CC  
    % and compile them in a matrix: (H;,E-  
    % ----------------------------- {XH3zMk[  
    if rpowers(1)==0 UmLBoy&*  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 07G'"=  
        rpowern = cat(2,rpowern{:}); pwVaSnre`  
        rpowern = [ones(length_r,1) rpowern]; 7;a  
    else =J`M}BBx  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); PU-L,]K  
        rpowern = cat(2,rpowern{:}); s]pNT1,  
    end [JEf P/n|.  
    m>f8RBp]'  
    % Compute the values of the polynomials: t]hfq~Ft  
    % -------------------------------------- +t8#rT ^B  
    z = zeros(length_r,length_n); ;+*/YTkC+P  
    for j = 1:length_n _ZE&W  
        s = 0:(n(j)-m(j))/2; s;#,c(   
        pows = n(j):-2:m(j); K?Jo"oy7  
        for k = length(s):-1:1 \;1nEjIA  
            p = (1-2*mod(s(k),2))* ... 0py29>"t  
                       prod(2:(n(j)-s(k)))/          ... j/F:j5O*  
                       prod(2:s(k))/                 ... HHL7z,%f  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... T%yGSk  
                       prod(2:((n(j)+m(j))/2-s(k))); fW$1f5g"  
            idx = (pows(k)==rpowers); U^kk0OT^  
            z(:,j) = z(:,j) + p*rpowern(:,idx); RBpv40n0  
        end 3F$N@K~s  
         i)(-Ad_  
        if isnorm /H$:Q|T}  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); jd.w7.8  
        end Zd]ua_)I%[  
    end MaZVGrcC  
    Ap> H-/C  
    % 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
    在线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)  oSH]TL2@Cd  
    Pk:b:(4  
    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。 :Y4G^i  
    V# JuNJ  
    07年就写过这方面的计算程序了。
    让光学不再神秘,让光学变得容易,快速实现客户关于光学的设想与愿望。