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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 VYo;[ue([  
    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! .`N` M9  
     
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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 L$c%u  
    function z = zernfun(n,m,r,theta,nflag) +{i "G,3  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. P3ev 4DL  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N _|wY[YJ[  
    %   and angular frequency M, evaluated at positions (R,THETA) on the  >E ;o"  
    %   unit circle.  N is a vector of positive integers (including 0), and )6 0f  
    %   M is a vector with the same number of elements as N.  Each element bK$D lBZ  
    %   k of M must be a positive integer, with possible values M(k) = -N(k)  / !  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, U{uWk3I_b  
    %   and THETA is a vector of angles.  R and THETA must have the same G:C6`uiy`  
    %   length.  The output Z is a matrix with one column for every (N,M) }6,bq`MN  
    %   pair, and one row for every (R,THETA) pair. ';|>`<  
    % !vVjZ  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike as |c`4r\O  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), =)1YYJTe9  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ^ O Xr: P  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ^npS==Y]!.  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Iki+5  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 4\SBf\ c  
    % gXLZ)>+A+  
    %   The Zernike functions are an orthogonal basis on the unit circle. $F`<&o  
    %   They are used in disciplines such as astronomy, optics, and ~EWfEHf*BJ  
    %   optometry to describe functions on a circular domain. <bXWkj  
    % qb&N S4#  
    %   The following table lists the first 15 Zernike functions. 1o~U+s_r  
    % YEPG[W<kg  
    %       n    m    Zernike function           Normalization mc=! X  
    %       -------------------------------------------------- $N+ {r=  
    %       0    0    1                                 1 HZ<f(  
    %       1    1    r * cos(theta)                    2 Nw>T $RzS  
    %       1   -1    r * sin(theta)                    2 d7tD|[(J  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) R ms01m>Y  
    %       2    0    (2*r^2 - 1)                    sqrt(3) W*rU,F|9  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) R{xyme@"^  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) &J/4J  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ctUF/[_w;  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) w H_n$w  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) .Lr)~  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) *[1u[H9Cv  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) CVSsB:H6e  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) s3_e7D ^H  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) _?]BVw  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 'UvS3]bSYW  
    %       -------------------------------------------------- ^Im%D(MY  
    % Rp`_Grcd  
    %   Example 1: JfP\7  
    % :OQ:@Yk  
    %       % Display the Zernike function Z(n=5,m=1) 2hwXWTSu  
    %       x = -1:0.01:1; ^'u;e(AaE  
    %       [X,Y] = meshgrid(x,x);  kulQR>u  
    %       [theta,r] = cart2pol(X,Y); U_}A{bFG  
    %       idx = r<=1; \abAPo  
    %       z = nan(size(X)); o&XMgY~  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); |N=@E,33  
    %       figure r0g/:lJi  
    %       pcolor(x,x,z), shading interp bDFCZH-:'O  
    %       axis square, colorbar 4j/iG\  
    %       title('Zernike function Z_5^1(r,\theta)') d7_g u  
    % m]=oaj@9  
    %   Example 2: u_6BHsU  
    % !,6v=n[Nz  
    %       % Display the first 10 Zernike functions v<7Gln  
    %       x = -1:0.01:1; B/sBYVU  
    %       [X,Y] = meshgrid(x,x); 5e/qgI)M5  
    %       [theta,r] = cart2pol(X,Y); |DFvZ6}  
    %       idx = r<=1; Hr<C2p^a  
    %       z = nan(size(X)); u $% D9Z^  
    %       n = [0  1  1  2  2  2  3  3  3  3]; %7(kP}y*  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; :B*vkwT  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Bd[L6J)  
    %       y = zernfun(n,m,r(idx),theta(idx)); Nr]8P/[~  
    %       figure('Units','normalized') 1t\b a1x  
    %       for k = 1:10 3u?`q%Y-e  
    %           z(idx) = y(:,k); {n'qKur xY  
    %           subplot(4,7,Nplot(k)) "Ql}Y1  
    %           pcolor(x,x,z), shading interp "'F;lzq  
    %           set(gca,'XTick',[],'YTick',[]) gP %|:"  
    %           axis square L*UV  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) U7]<U-.&  
    %       end S[L#M;n  
    % I NPYJ#%  
    %   See also ZERNPOL, ZERNFUN2. 2GiUPtO&Gj  
    ,XeyE;||  
    %   Paul Fricker 11/13/2006 yWv<A^C &  
    MS st  
    |ilv|UV  
    % Check and prepare the inputs: U BhciZ  
    % ----------------------------- ?y>Y$-v/C  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) uOG-IHuF  
        error('zernfun:NMvectors','N and M must be vectors.') dcl.wD0~V  
    end $*AC>i\  
    && DD  
    if length(n)~=length(m) o9m  
        error('zernfun:NMlength','N and M must be the same length.') ^zaKO'KcV  
    end y^mWG1"O  
    N>A{)_k3  
    n = n(:); aJ5H3X}Y  
    m = m(:); X!7VyE+n  
    if any(mod(n-m,2)) 2/v35| ?  
        error('zernfun:NMmultiplesof2', ... RHwaJ;:)#  
              'All N and M must differ by multiples of 2 (including 0).') *3_f &Y  
    end `%t$s,TiP  
    I #M%%5e  
    if any(m>n) VG<Hw{ c3r  
        error('zernfun:MlessthanN', ... tjZ\h=  
              'Each M must be less than or equal to its corresponding N.') HDF!`  
    end i\=z'  
    SUH mBo"}  
    if any( r>1 | r<0 ) OuOk=  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') *<*0".#  
    end HYjMNj0  
    ;dqk@@O"(  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 5q|+p?C  
        error('zernfun:RTHvector','R and THETA must be vectors.') ioC@n8_[G  
    end <i_> y~v`  
    u\{qH!?t  
    r = r(:); y4xT:G/M  
    theta = theta(:); gohAp  
    length_r = length(r); May&@x/oMS  
    if length_r~=length(theta) \4h>2y  
        error('zernfun:RTHlength', ... 87QZun%  
              'The number of R- and THETA-values must be equal.') ds*m6#1b  
    end ,c4c@|Bh?  
    *:=];1 O  
    % Check normalization: I86e&"40  
    % -------------------- xn(+G$m  
    if nargin==5 && ischar(nflag) D9qX->p  
        isnorm = strcmpi(nflag,'norm'); nE/=:{~Ws  
        if ~isnorm 5/& 1Oxo  
            error('zernfun:normalization','Unrecognized normalization flag.') ss? ]  
        end 5cD XWF  
    else Xzl KP;r0  
        isnorm = false; R<f#r03@|  
    end 9o-!ecx}  
    ]>tq|R78  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %mY|  
    % Compute the Zernike Polynomials z^4KU\/JK  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9<xTu>7J  
    M[ x_#m|  
    % Determine the required powers of r: F\>oxttS1  
    % ----------------------------------- `kv1@aQPL  
    m_abs = abs(m); 'oleB_B  
    rpowers = []; ]e^R@w  
    for j = 1:length(n) w[ Axs8N'  
        rpowers = [rpowers m_abs(j):2:n(j)]; PC*m% ?+  
    end ~O \}/I28  
    rpowers = unique(rpowers); (# JMB)  
    h^}_YaT\  
    % Pre-compute the values of r raised to the required powers, }<vvxi  
    % and compile them in a matrix: mO#I nTO  
    % ----------------------------- N<9w{zIK(  
    if rpowers(1)==0 Rr%tbt.sE  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ./$ <J6-J  
        rpowern = cat(2,rpowern{:}); b.QpHrnhtK  
        rpowern = [ones(length_r,1) rpowern]; x+4v s s  
    else >G]?  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); e#tIk;9Xz  
        rpowern = cat(2,rpowern{:}); m7JPH7P@BM  
    end *5 e<\{!  
    f%c06Un=  
    % Compute the values of the polynomials: 3 h#s([uL  
    % -------------------------------------- hQXxG/yFm  
    y = zeros(length_r,length(n)); Q~phGD3!~  
    for j = 1:length(n) Q/p(#/y#b  
        s = 0:(n(j)-m_abs(j))/2; yL.^ =  
        pows = n(j):-2:m_abs(j); l$F_"o?&S@  
        for k = length(s):-1:1 My. dD'C  
            p = (1-2*mod(s(k),2))* ... P*0f~eu  
                       prod(2:(n(j)-s(k)))/              ... JfMJF[Mb  
                       prod(2:s(k))/                     ... h-7A9:  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ~L~]QN\3  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 29%=:*R$  
            idx = (pows(k)==rpowers); b7bSTFZxC  
            y(:,j) = y(:,j) + p*rpowern(:,idx); >;,gGH  
        end pDGT@qJ  
         j~epbl)pC  
        if isnorm F#su5<d  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); V"O 9n[|  
        end {(;B5rs  
    end {gsW(T>)  
    % END: Compute the Zernike Polynomials VUp. j  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "=qv#mZ#9  
    o5P&JBX<  
    % Compute the Zernike functions: q-YL]PgV  
    % ------------------------------ I:F <vE  
    idx_pos = m>0; .:8[wI_f  
    idx_neg = m<0; \7yJ\I  
    q3+I<qsAz  
    z = y; EY~7oNfc`R  
    if any(idx_pos) 6+iK!&+=  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Hq?&Qo  
    end w,Q)@]_  
    if any(idx_neg) ~ 7}]  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); QWw"K$l  
    end IP04l;p/  
    hfg O  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) dQ<(lzS~  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 0R|K0XH#$  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated zw]3Vg{T  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Y%g "Y  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, cz#_<8'N  
    %   and THETA is a vector of angles.  R and THETA must have the same Q2"WV  
    %   length.  The output Z is a matrix with one column for every P-value, n' ?4.tb  
    %   and one row for every (R,THETA) pair. j;iL&eo>  
    % f>niFPW"  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike hO6RQ0Iv@  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) n+?-�  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) nW GR5*e:  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 @Dj:4  
    %   for all p. ufP Cx|x~  
    % @+&'%1  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 {2:d` fqD  
    %   Zernike functions (order N<=7).  In some disciplines it is BSKEh"f  
    %   traditional to label the first 36 functions using a single mode 4%7s259%  
    %   number P instead of separate numbers for the order N and azimuthal YBvd q1  
    %   frequency M. G#0,CLGN^  
    % pds*2p)2  
    %   Example:  eu9w|g  
    % N@()F&e  
    %       % Display the first 16 Zernike functions -NzTqLBn  
    %       x = -1:0.01:1; Pbe7SRdr^  
    %       [X,Y] = meshgrid(x,x); ?E7=:h(@t  
    %       [theta,r] = cart2pol(X,Y); 9|=nV|R'6  
    %       idx = r<=1; {y6C0A*  
    %       p = 0:15; U:n*<l-k}  
    %       z = nan(size(X)); :B.G)M\  
    %       y = zernfun2(p,r(idx),theta(idx)); A"4@L*QV  
    %       figure('Units','normalized') S?4KC^Y5  
    %       for k = 1:length(p) dIJGB==  
    %           z(idx) = y(:,k); y(v_-6b  
    %           subplot(4,4,k) L\L"mc|O  
    %           pcolor(x,x,z), shading interp tOH0IE c  
    %           set(gca,'XTick',[],'YTick',[]) Pm^lr!3p  
    %           axis square 0`n 5x0R  
    %           title(['Z_{' num2str(p(k)) '}']) nU17L6'$  
    %       end }g{_AiP rv  
    % "PzP; Br  
    %   See also ZERNPOL, ZERNFUN. iBoEZEHjw  
    g 1@wf  
    %   Paul Fricker 11/13/2006 *1bzg/T<  
    ug#<LO-.Rd  
    $.:3$et@/  
    % Check and prepare the inputs: tD-gc ''H  
    % ----------------------------- "8HE^Po/pn  
    if min(size(p))~=1 y|sma;D  
        error('zernfun2:Pvector','Input P must be vector.') eYEc^nC,c)  
    end C:GvP>  
    ^ ulps**e  
    if any(p)>35 85z;Zt0{  
        error('zernfun2:P36', ... zvr\36  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Q2c*.Y  
               '(P = 0 to 35).']) Qra>}e%*  
    end 4j3oT)+8  
    ?"j@;/=  
    % Get the order and frequency corresonding to the function number: -H\,2FO  
    % ---------------------------------------------------------------- ]l[2hy= cV  
    p = p(:); +'XhC#:  
    n = ceil((-3+sqrt(9+8*p))/2); hYb9`0G"2  
    m = 2*p - n.*(n+2); IN^_BKQt  
    yvxl_*Ds8  
    % Pass the inputs to the function ZERNFUN: dJl^ADX[@  
    % ---------------------------------------- gs`> C(  
    switch nargin *]x_,:R6Ow  
        case 3 }q'WC4.  
            z = zernfun(n,m,r,theta); f&yQhe6q  
        case 4 2-v\3voN  
            z = zernfun(n,m,r,theta,nflag); TpP8=8_Lh  
        otherwise io4aYB\  
            error('zernfun2:nargin','Incorrect number of inputs.') 0=;YnsY  
    end kG^dqqn6  
    xvDI 4x&  
    % 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) s;{K!L@  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. %S#WPD'Y  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of >5Lexj  
    %   order N and frequency M, evaluated at R.  N is a vector of FFe) e>bH  
    %   positive integers (including 0), and M is a vector with the <4mQ*6  
    %   same number of elements as N.  Each element k of M must be a qI2'u%  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) Fe$/t(  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is T=\!2gt  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix .Z%G@X*  
    %   with one column for every (N,M) pair, and one row for every *^h_z;{,  
    %   element in R. HomN/wKh  
    % ~O4|KY  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 5I,$EGG  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is /cY^]VLe  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to _e'Y3:  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ^l!L)iw  
    %   for all [n,m]. \0AiCMX[  
    % P(h5=0`*PR  
    %   The radial Zernike polynomials are the radial portion of the /F~X,lm*~  
    %   Zernike functions, which are an orthogonal basis on the unit ;nB2o-%  
    %   circle.  The series representation of the radial Zernike 0*7*RX  
    %   polynomials is & _K*kI:  
    % 8D~x\!(p\  
    %          (n-m)/2 '6zd;l9Z  
    %            __ zWIeHIt  
    %    m      \       s                                          n-2s }<[Db}?9  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ek)Xrp:2  
    %    n      s=0 97!5Q~I  
    % 40K2uT{cq  
    %   The following table shows the first 12 polynomials. slvq9,  
    % gyus8#sT  
    %       n    m    Zernike polynomial    Normalization  c8DZJSO  
    %       --------------------------------------------- L11L23:  
    %       0    0    1                        sqrt(2) p|VcMxT9-  
    %       1    1    r                           2 n33kb/q*  
    %       2    0    2*r^2 - 1                sqrt(6) Alz~-hqQ  
    %       2    2    r^2                      sqrt(6) =! m JG  
    %       3    1    3*r^3 - 2*r              sqrt(8) S,vu]?-8  
    %       3    3    r^3                      sqrt(8) 'G#SLqZy  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) Lk.h.ST  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) /iC_!nu  
    %       4    4    r^4                      sqrt(10) ;^-:b(E  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) J22r v(  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) c8ZCs?   
    %       5    5    r^5                      sqrt(12) ,w`~K:b.  
    %       --------------------------------------------- G5c7:iGm/c  
    % b5 Q NEi  
    %   Example: nj2gs,k  
    % K $-;;pUl  
    %       % Display three example Zernike radial polynomials |.w;r   
    %       r = 0:0.01:1; V}9;eJRvw  
    %       n = [3 2 5]; SrZ50Se  
    %       m = [1 2 1]; xzk}[3P{  
    %       z = zernpol(n,m,r); Tf-CEHWD  
    %       figure +qkMQETV6  
    %       plot(r,z) 5"=:#zN  
    %       grid on l{%a&/  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') t[2i$%NVM  
    % wkikD  
    %   See also ZERNFUN, ZERNFUN2. 6k9LxC:M  
    ]!Zty[  
    % A note on the algorithm. }gCHQ;U7`  
    % ------------------------ A3s57.Z]|  
    % The radial Zernike polynomials are computed using the series i6>R qP!69  
    % representation shown in the Help section above. For many special y8?t-Pp]1  
    % functions, direct evaluation using the series representation can -e*BqH2t  
    % produce poor numerical results (floating point errors), because Yx1 D)  
    % the summation often involves computing small differences between g42T#p8^  
    % large successive terms in the series. (In such cases, the functions _4R,Ej}  
    % are often evaluated using alternative methods such as recurrence 6%?A>  
    % relations: see the Legendre functions, for example). For the Zernike og-]tEWA1  
    % polynomials, however, this problem does not arise, because the Y6VQ:glDT-  
    % polynomials are evaluated over the finite domain r = (0,1), and qG9qN.|dC  
    % because the coefficients for a given polynomial are generally all g rbTcLSF  
    % of similar magnitude. (~#G'Hd  
    % cU+>|'f &  
    % ZERNPOL has been written using a vectorized implementation: multiple s*JE)  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] c{>|o  
    % values can be passed as inputs) for a vector of points R.  To achieve {A3 m+_8  
    % this vectorization most efficiently, the algorithm in ZERNPOL sXB+s  
    % involves pre-determining all the powers p of R that are required to I:t^S.,  
    % compute the outputs, and then compiling the {R^p} into a single ]kS7n @8  
    % matrix.  This avoids any redundant computation of the R^p, and pULsGb  
    % minimizes the sizes of certain intermediate variables. u(hC^T1  
    % a:u}d7T3e  
    %   Paul Fricker 11/13/2006 l0`'5>  
    }ywi"k4>  
    ;3 UvkN  
    % Check and prepare the inputs: 0j$OE  
    % ----------------------------- jo0Pd_W8&  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Z#7U "G-A  
        error('zernpol:NMvectors','N and M must be vectors.') h{/ve`F>@  
    end b1=pO]3u  
    XGIpUz  
    if length(n)~=length(m) 1+^n!$  
        error('zernpol:NMlength','N and M must be the same length.') J px'W  
    end ^s*\Qw{Ii  
    1Z:R,\+L  
    n = n(:); 4$Pr|gx  
    m = m(:); KjYDFrR4  
    length_n = length(n); uF<S  
    (C\r&N  
    if any(mod(n-m,2)) R 3G@ G  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') (1 yGg==W.  
    end C1 {ZW~"YI  
    XnY"oDg^>  
    if any(m<0) )E`+BH  
        error('zernpol:Mpositive','All M must be positive.') ][t 6VA  
    end BD'NuI  
    d*%Mv[X:<  
    if any(m>n) mJ$Htyr  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') BWEv1' v  
    end 8.2`~'V  
    g)cY\`&W8  
    if any( r>1 | r<0 ) Omb.53+  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') %uuH^A  
    end ^0Q'./A{&  
    ,'_( DJX  
    if ~any(size(r)==1) K;C_Z/<%  
        error('zernpol:Rvector','R must be a vector.') G>0)I  
    end k<O y%+C  
    R&!]Rl9hf  
    r = r(:); UoRDeYQ`E  
    length_r = length(r); v= 8VvT 8  
    raHVkE{<  
    if nargin==4 y.ql#eQ,  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ;^ov~PPl  
        if ~isnorm _H2tZ%RM  
            error('zernpol:normalization','Unrecognized normalization flag.') DZP*x  
        end * gHCy4u{  
    else l/F!Bq[*g  
        isnorm = false; QQ~23TlA  
    end qmID-t"  
    xFX&9^Uk  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d<v~=  
    % Compute the Zernike Polynomials cIZ[[(Db  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% K @C4*?P  
    q_pmwJ:UL  
    % Determine the required powers of r: Y"oDFo,  
    % ----------------------------------- g8v[)o(qd  
    rpowers = []; f?5A"-NS  
    for j = 1:length(n) e&ts\0  
        rpowers = [rpowers m(j):2:n(j)]; %-r?=L  
    end NQ{-&#@/v  
    rpowers = unique(rpowers); %r P !  
    _'lmCj8L  
    % Pre-compute the values of r raised to the required powers, m:"+J  
    % and compile them in a matrix: 9.]kOs_  
    % ----------------------------- KcnjF^k  
    if rpowers(1)==0 8? F 2jv  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); nj"m^PmWo3  
        rpowern = cat(2,rpowern{:}); d.I%k1`(  
        rpowern = [ones(length_r,1) rpowern]; S[sr 'ZW  
    else ]Y=S  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); aPt{C3<  
        rpowern = cat(2,rpowern{:}); qzHU)Ns(_  
    end ,@479ZvvR3  
    u ]SZ{[ e  
    % Compute the values of the polynomials: n5\}KZh  
    % -------------------------------------- u`+ 'lBE,  
    z = zeros(length_r,length_n); d^y86pq.  
    for j = 1:length_n _1\poAy  
        s = 0:(n(j)-m(j))/2; k|5k8CRX  
        pows = n(j):-2:m(j); @Rf^P(  
        for k = length(s):-1:1 SlT7L||Ww  
            p = (1-2*mod(s(k),2))* ... cPSti  
                       prod(2:(n(j)-s(k)))/          ... "G@E6{/  
                       prod(2:s(k))/                 ... df yrn%^Ia  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... St_S l:m$  
                       prod(2:((n(j)+m(j))/2-s(k))); hu&n=6  
            idx = (pows(k)==rpowers); ~9bv Wd1D  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Z) t{JHm:  
        end <a fO 6?`  
         eZ A6D\  
        if isnorm cXMa\#P  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); yduuFK  
        end h*l cEzG?A  
    end lZBv\JE  
    1Lc8fP$  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  Z!q2F%02FO  
    ]>]#zu$=c  
    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。 ujI 3tsl  
    !=8L.^5c  
    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)