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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 {m~)~/z?  
    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! L{&2 P  
     
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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 Mi|13[p{  
    function z = zernfun(n,m,r,theta,nflag) yXDjM2oR/2  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. eo4z!@pRN  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N %?].( Lc  
    %   and angular frequency M, evaluated at positions (R,THETA) on the W7uX  
    %   unit circle.  N is a vector of positive integers (including 0), and 'pIrwA^6N  
    %   M is a vector with the same number of elements as N.  Each element pu/5#[MC)^  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) +&VY6(Zj+*  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 6Y ]P7j  
    %   and THETA is a vector of angles.  R and THETA must have the same o[_,r]%+D  
    %   length.  The output Z is a matrix with one column for every (N,M) J?m/u6  
    %   pair, and one row for every (R,THETA) pair. vi^YtA  
    % GIE QD$vy  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Ds"%=  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), K1J |\!o  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral p P@q `  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, bLG7{qp  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized k-n`R)p:  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. >v@3]a i  
    % kEC^_sO"  
    %   The Zernike functions are an orthogonal basis on the unit circle. pp(09y`]  
    %   They are used in disciplines such as astronomy, optics, and p1d%&e  
    %   optometry to describe functions on a circular domain. Cscu   
    % E~WbV+,3  
    %   The following table lists the first 15 Zernike functions. #6|ve?`I  
    % 8Snv, Lb`^  
    %       n    m    Zernike function           Normalization td%J.&K_*'  
    %       -------------------------------------------------- k;cX,*DIn  
    %       0    0    1                                 1 TPBQfp%HU  
    %       1    1    r * cos(theta)                    2 WZ6{9/%:  
    %       1   -1    r * sin(theta)                    2 na $MR3@e  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 02[m{a-  
    %       2    0    (2*r^2 - 1)                    sqrt(3) "1Hn?4nz5  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) H*k\C  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) "t^RZ45  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) B/a`5&G]  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) wg0_J<y]  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) pJ8F+`*  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) "Y:>^F;  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !c)F;  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) )tJaw#Mih  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) C)i8XX  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) >e/>@ J*  
    %       -------------------------------------------------- aE)1LP  
    % SPlt=*C#_  
    %   Example 1: v=G*K11@  
    % ``g  
    %       % Display the Zernike function Z(n=5,m=1) .yfp-n4H  
    %       x = -1:0.01:1; Brs6RkRf  
    %       [X,Y] = meshgrid(x,x); rWJ5C\R  
    %       [theta,r] = cart2pol(X,Y); =\2gnk~  
    %       idx = r<=1; 9O&gR46.  
    %       z = nan(size(X)); 0/DO"pnL@  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); w?u3e+  
    %       figure s'N<  
    %       pcolor(x,x,z), shading interp REU&8J@k&?  
    %       axis square, colorbar ;\A_-a_(#  
    %       title('Zernike function Z_5^1(r,\theta)') OHAU@*[lM  
    % C;:=r:bth  
    %   Example 2: e?;c9]XO,o  
    % } x r0m+/  
    %       % Display the first 10 Zernike functions +p>h` fc  
    %       x = -1:0.01:1; L9e<hRZ$  
    %       [X,Y] = meshgrid(x,x); /PSXuVtu5  
    %       [theta,r] = cart2pol(X,Y); -?#iPvk6  
    %       idx = r<=1; |)>+& xk  
    %       z = nan(size(X)); 36co 'a4,  
    %       n = [0  1  1  2  2  2  3  3  3  3]; qZ>_{b0f  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; EZiLXQd_  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; \Cq4r4'  
    %       y = zernfun(n,m,r(idx),theta(idx)); T&/ n.-@nk  
    %       figure('Units','normalized') a Tm R~k  
    %       for k = 1:10 0sw;h.VY  
    %           z(idx) = y(:,k); khR[8j..  
    %           subplot(4,7,Nplot(k)) b 4^O=  
    %           pcolor(x,x,z), shading interp 4Dzg r,V  
    %           set(gca,'XTick',[],'YTick',[]) bnL!PsG$K,  
    %           axis square c ZYvP  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]pB5cq7o  
    %       end w3 vZ}1|  
    % e%ro7~  
    %   See also ZERNPOL, ZERNFUN2. AfO.D ?4x  
    Jjj;v2uSK  
    %   Paul Fricker 11/13/2006 |9 5K  
    p9G+la~;VM  
    a.UYBRP/l  
    % Check and prepare the inputs: -a|b.p  
    % ----------------------------- F(/<ADx  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) tR9iFv_  
        error('zernfun:NMvectors','N and M must be vectors.') ;TDvk ]:  
    end R*cef  
    E83$(6z  
    if length(n)~=length(m) O\cc=7  
        error('zernfun:NMlength','N and M must be the same length.') uAnL`  
    end JP"#9f  
    F> Ika=z,  
    n = n(:); /#{~aCOi)  
    m = m(:); Q~f]?a`  
    if any(mod(n-m,2)) )O*h79t^Q  
        error('zernfun:NMmultiplesof2', ... ,if~%'9j  
              'All N and M must differ by multiples of 2 (including 0).') _&gO>G,uy  
    end @kDY c8 t9  
    .EWjeVq  
    if any(m>n) #+Bz$CO  
        error('zernfun:MlessthanN', ... DU,B  
              'Each M must be less than or equal to its corresponding N.') c^H#[<6p  
    end 7Cz=;  
    xa_ IdkV  
    if any( r>1 | r<0 ) XD6Kp[s  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Z3wdk6%:}  
    end :0%[u(  
    2 7dS.6  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) IY!.j5q8  
        error('zernfun:RTHvector','R and THETA must be vectors.') {%('|(57  
    end >_]Ov:5  
     )D+eWo  
    r = r(:); %kkDitmI{  
    theta = theta(:); Sa)L=5Nr  
    length_r = length(r); hB>FJZQ_  
    if length_r~=length(theta) sng6U;Z  
        error('zernfun:RTHlength', ... _(=g[=Mer  
              'The number of R- and THETA-values must be equal.') O['[_1n_u]  
    end gL| 9hvHr[  
    B&KIM{j\  
    % Check normalization: )Mflt0fp  
    % -------------------- d5 ]-{+V+  
    if nargin==5 && ischar(nflag) n]w%bKc-9  
        isnorm = strcmpi(nflag,'norm'); 32j#kJW  
        if ~isnorm AGwdM-$iT  
            error('zernfun:normalization','Unrecognized normalization flag.') DN*M-o9  
        end ebL0cK?  
    else w D6QN  
        isnorm = false;  0RCp  
    end i 28TH Jh  
    4Rp[>}L  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d"3x11|  
    % Compute the Zernike Polynomials =b)!l9TX  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :SMf (E 5  
    %F-yF N"  
    % Determine the required powers of r: ?a, `{1m0\  
    % ----------------------------------- J1M9) ,  
    m_abs = abs(m); P()&?C  
    rpowers = []; \q!TI x  
    for j = 1:length(n) "f3mi[  
        rpowers = [rpowers m_abs(j):2:n(j)]; / a}N6KUi  
    end D&N3LH  
    rpowers = unique(rpowers); 2=7[r-*E  
    z+0#H39&  
    % Pre-compute the values of r raised to the required powers, & R<K>i  
    % and compile them in a matrix: "K|':3n|  
    % ----------------------------- HmsXV_B8[Y  
    if rpowers(1)==0 N/2WUp  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); .[:WMCc\  
        rpowern = cat(2,rpowern{:}); Qe9}%k6@E  
        rpowern = [ones(length_r,1) rpowern]; WwKpZ67$R  
    else u1z!OofN>  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); .",BLuce  
        rpowern = cat(2,rpowern{:}); BT -Y9j  
    end xo-}t5w6t  
    %f&Bt,xEo  
    % Compute the values of the polynomials: m60hTJ?N)  
    % -------------------------------------- h,fahbH -  
    y = zeros(length_r,length(n)); B.b sU  
    for j = 1:length(n) 3c`  
        s = 0:(n(j)-m_abs(j))/2; op&j4R  
        pows = n(j):-2:m_abs(j); I.2>d_^<  
        for k = length(s):-1:1 \D%n8O  
            p = (1-2*mod(s(k),2))* ... >k}Kf1I  
                       prod(2:(n(j)-s(k)))/              ... ^d9o \  
                       prod(2:s(k))/                     ... 5!6iAS+I  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... dleLX%P  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); d(Yuz#Qcrh  
            idx = (pows(k)==rpowers); sv\=/F@n  
            y(:,j) = y(:,j) + p*rpowern(:,idx); QNcl    
        end `+Mva  
         0V2~  
        if isnorm 85FzIX-F%  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); PDh!B _+  
        end [#:yOZt  
    end KWw?W1H  
    % END: Compute the Zernike Polynomials FT gt$I  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% D_w<igu!3  
    ]7|qhAh<L  
    % Compute the Zernike functions: eQ#"-i  
    % ------------------------------ PXDJ[Oj7(0  
    idx_pos = m>0; 3/su1M[  
    idx_neg = m<0; XlwyD  
    T(kG"dz   
    z = y; Ojp|/yd^YL  
    if any(idx_pos) 1Zp^X:(  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Ao *{#z   
    end URyY^+s  
    if any(idx_neg) * ^\u%Ir"  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); }OgZZ8-_M  
    end B@vup {Kg  
    &e4EZ  
    % 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) xC$CRzAe5p  
    %ZERNFUN2 Single-index Zernike functions on the unit circle.  Y}Nd2  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated pH.&OW%  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive h{VGh kU9f  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 1,sD'iNb  
    %   and THETA is a vector of angles.  R and THETA must have the same ARid   
    %   length.  The output Z is a matrix with one column for every P-value, ]~m2#g%  
    %   and one row for every (R,THETA) pair. ^Pc&`1Ap  
    % 0^ $6U  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike #xxs^Kbqa#  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) J|o )c~  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) }{)>aJ  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 K1fnHpK  
    %   for all p. ;c>IM]  
    % &28%~&L  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 nx`I9j\  
    %   Zernike functions (order N<=7).  In some disciplines it is Yg3emn|a  
    %   traditional to label the first 36 functions using a single mode kT{d pGU9  
    %   number P instead of separate numbers for the order N and azimuthal ;kF+V*  
    %   frequency M. !W45X}/o  
    % C%kIxa)  
    %   Example: K(p6P3Z  
    % JXF@b-c  
    %       % Display the first 16 Zernike functions +# tmsv]2  
    %       x = -1:0.01:1; Q2!vO4!<N  
    %       [X,Y] = meshgrid(x,x); LD)P. f  
    %       [theta,r] = cart2pol(X,Y); AU^5N3%j  
    %       idx = r<=1; Ba]^0Y u  
    %       p = 0:15; dht*1i3v  
    %       z = nan(size(X)); 6 VuMx7W1  
    %       y = zernfun2(p,r(idx),theta(idx)); c{K[bppJ*  
    %       figure('Units','normalized') r4Jc9Tv d  
    %       for k = 1:length(p) 7 a_99? J  
    %           z(idx) = y(:,k); i@#fyU)[G  
    %           subplot(4,4,k) A<s9c=d6  
    %           pcolor(x,x,z), shading interp =LMM]'no,  
    %           set(gca,'XTick',[],'YTick',[]) :/'oh]T|  
    %           axis square la[>C:8IG  
    %           title(['Z_{' num2str(p(k)) '}']) VTvNn  
    %       end 6.g k6  
    % <ULydBom  
    %   See also ZERNPOL, ZERNFUN. \ POQeZ  
    O0 ,=@nw8.  
    %   Paul Fricker 11/13/2006  q<Zza  
    nf9NJ_8}4H  
    GbN|!,X1m  
    % Check and prepare the inputs: +yo1&b R/  
    % ----------------------------- use` y^c  
    if min(size(p))~=1 eww/tGa  
        error('zernfun2:Pvector','Input P must be vector.') _mn2bc9M  
    end Z`Sbq{Kx  
    X[KHI1@w  
    if any(p)>35 w [7vxQ!-  
        error('zernfun2:P36', ... &i?>mt  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... x 2Cp{+}  
               '(P = 0 to 35).']) %T'<vw0  
    end eJwHeG  
    DDwm;,eZ  
    % Get the order and frequency corresonding to the function number: :927y  
    % ---------------------------------------------------------------- V+&C_PyC  
    p = p(:); #J09Eka;J  
    n = ceil((-3+sqrt(9+8*p))/2); wmnh7'|0u  
    m = 2*p - n.*(n+2); %uy5la  
    Vmf !0-  
    % Pass the inputs to the function ZERNFUN: 6@; P  
    % ---------------------------------------- #1oyRD-  
    switch nargin M"Q{lR  
        case 3 DZE@C^ 0%  
            z = zernfun(n,m,r,theta); -oR P ZtW  
        case 4 5isqBu  
            z = zernfun(n,m,r,theta,nflag); T.?}iz=ZEq  
        otherwise Ty;P`Uv]r  
            error('zernfun2:nargin','Incorrect number of inputs.') %{HeXe  
    end Ek%mX"  
    w=feXA3-S  
    % 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) YgL{*XYAt  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. $1}Y4>3  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of &ku.Q3xGs  
    %   order N and frequency M, evaluated at R.  N is a vector of f 6 k=ew  
    %   positive integers (including 0), and M is a vector with the '4"c#kCKL  
    %   same number of elements as N.  Each element k of M must be a }NpN<C+  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) $QB/n63  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is B3g # )  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ^ZPynduR  
    %   with one column for every (N,M) pair, and one row for every 5/YGu=,  
    %   element in R. {u)>W@Lr  
    % yB2}[1  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- kr>4%Ndm7  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is hnBX enT6  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to JpE7"Z"~MS  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 i# bcjH  
    %   for all [n,m]. b>]k=zd  
    % \zLKSJ]  
    %   The radial Zernike polynomials are the radial portion of the "el}9OitC  
    %   Zernike functions, which are an orthogonal basis on the unit ~`X$b F  
    %   circle.  The series representation of the radial Zernike )0?u_Z]w9  
    %   polynomials is _?v&\j  
    % .oH)eD  
    %          (n-m)/2 g1v=a  
    %            __ ,s`4k?y  
    %    m      \       s                                          n-2s 8h,=yAn5  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Dgc}T8R  
    %    n      s=0  !U=o<)I  
    % e?_uJh"  
    %   The following table shows the first 12 polynomials. sT'j36Nc<,  
    % pS+hE4D  
    %       n    m    Zernike polynomial    Normalization +$$5Cv5#<&  
    %       --------------------------------------------- TpcJ1*t  
    %       0    0    1                        sqrt(2) N$N 7aE$  
    %       1    1    r                           2 Qv6-,6<  
    %       2    0    2*r^2 - 1                sqrt(6) N"8'=wB  
    %       2    2    r^2                      sqrt(6) _E2W%N  
    %       3    1    3*r^3 - 2*r              sqrt(8) xSrjN  
    %       3    3    r^3                      sqrt(8) `Z^\<{z  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) @%BsQm  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) B&m6N,  
    %       4    4    r^4                      sqrt(10) ~s*kuj'%+  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) )F+wk"`+6  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) r;_*.|AH  
    %       5    5    r^5                      sqrt(12) w@WPp0mny  
    %       --------------------------------------------- \+j:d9?  
    % mO2u9?N  
    %   Example: <w3_EO  
    % 4s 6,`-  
    %       % Display three example Zernike radial polynomials y({lE3P  
    %       r = 0:0.01:1;  kMZo7 y  
    %       n = [3 2 5]; 5,J.$Sax  
    %       m = [1 2 1]; '| p"HbJ  
    %       z = zernpol(n,m,r); a66Ns7Rb  
    %       figure fd$nAE  
    %       plot(r,z) $8}'h  
    %       grid on OlP1Zd/l  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') p z\8Bp}yo  
    % i0F6eqe=J  
    %   See also ZERNFUN, ZERNFUN2. p<+Y;,+  
    Ca -.&$f  
    % A note on the algorithm. m- bu{  
    % ------------------------ o)n= n!A  
    % The radial Zernike polynomials are computed using the series ZCuoYE$g  
    % representation shown in the Help section above. For many special kD(#LM<9s  
    % functions, direct evaluation using the series representation can 3wg1wl|  
    % produce poor numerical results (floating point errors), because + w'q5/`  
    % the summation often involves computing small differences between \5}*;O@  
    % large successive terms in the series. (In such cases, the functions *7w!~mn[m  
    % are often evaluated using alternative methods such as recurrence 9_O6Sl  
    % relations: see the Legendre functions, for example). For the Zernike KL./  
    % polynomials, however, this problem does not arise, because the .FN 6/N\  
    % polynomials are evaluated over the finite domain r = (0,1), and <}T7;knO  
    % because the coefficients for a given polynomial are generally all +8Y|kC{9"  
    % of similar magnitude. .03Rp5+v  
    % &?}A/(#  
    % ZERNPOL has been written using a vectorized implementation: multiple izzX$O[=:  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Y]7 6y>|e  
    % values can be passed as inputs) for a vector of points R.  To achieve nok-![  
    % this vectorization most efficiently, the algorithm in ZERNPOL @}2EEo#  
    % involves pre-determining all the powers p of R that are required to >pp#>{}  
    % compute the outputs, and then compiling the {R^p} into a single -@ra~li,yQ  
    % matrix.  This avoids any redundant computation of the R^p, and OZA^L;#>  
    % minimizes the sizes of certain intermediate variables. XRHngW_A  
    % "L!U7|9J  
    %   Paul Fricker 11/13/2006 &8I }q]'k  
    `Tei  
    9Y@ eXP  
    % Check and prepare the inputs: =WHI/|&  
    % ----------------------------- D8{ ,}@  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) \_YDSmjy  
        error('zernpol:NMvectors','N and M must be vectors.') ^$X|Lq  
    end B*t1Y<>x  
    P&Uj?et"  
    if length(n)~=length(m) "dT"6,  
        error('zernpol:NMlength','N and M must be the same length.') V(8,94vm  
    end FmFjRYA W  
    GaV}@Q  
    n = n(:); M| Nh(kvH  
    m = m(:); m41%?uC/  
    length_n = length(n); 7dv!  
    2j#Dwa(lZQ  
    if any(mod(n-m,2)) [%O f  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') o/Q|R+yXV  
    end 4H NaE{O4  
    mi.,Z`]o  
    if any(m<0) ?^2nrh,n+  
        error('zernpol:Mpositive','All M must be positive.') !8D>Bczq)  
    end w!z* ?k=Da  
    BMqr YW  
    if any(m>n) ) iZU\2L  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') | KY-kRN7  
    end R>]7l!3^1  
    KMK8jJ  
    if any( r>1 | r<0 ) .6C6ZUB;  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') T "hjL  
    end {9X mFa  
    BzS\p3&  
    if ~any(size(r)==1) Xk/iyp/  
        error('zernpol:Rvector','R must be a vector.') ;9~ WB X"  
    end )EQz9  
    )2#&l  
    r = r(:); *X{7m]5  
    length_r = length(r); \.}ZvM$  
    u! &T}i:  
    if nargin==4 i]J.WFu  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 4HR36=E6  
        if ~isnorm CBF<53TshR  
            error('zernpol:normalization','Unrecognized normalization flag.') *8uS,s6g  
        end a+\<2NXYD  
    else '2hy%  
        isnorm = false; <QO1Yg7}  
    end e&*b{>1*  
    5!cp^[rGL  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >3pT).wH|M  
    % Compute the Zernike Polynomials M@P%k`6C  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% K~2sX>l  
    &3;"$P  
    % Determine the required powers of r: 1KbZ6Msy  
    % ----------------------------------- ZNM9@;7  
    rpowers = []; D[?;+g/  
    for j = 1:length(n) *W 2)!C|  
        rpowers = [rpowers m(j):2:n(j)]; MnZljB  
    end "(vK.-T  
    rpowers = unique(rpowers); ~\i(bFd)  
    7(uz*~Z?`0  
    % Pre-compute the values of r raised to the required powers, rsLkH&aM  
    % and compile them in a matrix: 9P)!v.,T/  
    % ----------------------------- +RJKJ:W  
    if rpowers(1)==0 X 6tJ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); dQZdL4  
        rpowern = cat(2,rpowern{:}); ~*"ZF-c,  
        rpowern = [ones(length_r,1) rpowern]; ('Qq"cn#  
    else $5.52  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); t72u%M6  
        rpowern = cat(2,rpowern{:}); 0nL #-`S  
    end y`L.#5T  
    c"-X: m"  
    % Compute the values of the polynomials: 2O*At%CzW  
    % -------------------------------------- 6i~|<vcSP  
    z = zeros(length_r,length_n); dNNXMQ0"  
    for j = 1:length_n Du65>O  
        s = 0:(n(j)-m(j))/2; 24k]X`/n  
        pows = n(j):-2:m(j); A%?c1`ZxF  
        for k = length(s):-1:1 r5ldK?=k+*  
            p = (1-2*mod(s(k),2))* ... 7<*0fy5nn  
                       prod(2:(n(j)-s(k)))/          ... t&Eiz H$  
                       prod(2:s(k))/                 ... {:*G/*1[.  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... CHZ/@gc  
                       prod(2:((n(j)+m(j))/2-s(k))); TWGn: mi  
            idx = (pows(k)==rpowers); Hg<aU*o;  
            z(:,j) = z(:,j) + p*rpowern(:,idx); IN<nZ?D#  
        end yj#FO'UY  
         T4Vp0i  
        if isnorm o$l8"Uv  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ^|p D(v  
        end - _ 8-i1?  
    end UPr& `kaJ  
    =}Zl E  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  s?m_zJh  
    qonStIP  
    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。  24 [cU  
    fA<os+*9i  
    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)