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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 %5RY Ea  
    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! jNB|98NN  
     
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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 qyGVyi3  
    function z = zernfun(n,m,r,theta,nflag) dQ@ e+u5  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. >/nS<y>  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N QgO@oV*S  
    %   and angular frequency M, evaluated at positions (R,THETA) on the YOwo\'|=  
    %   unit circle.  N is a vector of positive integers (including 0), and "12.Bi.O"[  
    %   M is a vector with the same number of elements as N.  Each element S*Un$ngAh  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) qPuxYU  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ,,S5 8\x  
    %   and THETA is a vector of angles.  R and THETA must have the same K2>(C$Z  
    %   length.  The output Z is a matrix with one column for every (N,M) B5*{85p(u  
    %   pair, and one row for every (R,THETA) pair. `YAqR?Xj_<  
    % 2-j+-B|i  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike J!O5`k*.C  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), HiCNs;t  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral GiJ|5"  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Xg?hh 0s  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Y*;Z(W.V#  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. BRYhL|d~.  
    % u*Z>&]W_  
    %   The Zernike functions are an orthogonal basis on the unit circle. j0^~="p%C  
    %   They are used in disciplines such as astronomy, optics, and } *|_P  
    %   optometry to describe functions on a circular domain. 'A .c*<_  
    % %sP C3L  
    %   The following table lists the first 15 Zernike functions. st P~/}  
    % ]WR+>)ERb  
    %       n    m    Zernike function           Normalization b>=MG8  
    %       -------------------------------------------------- p#hs8xz  
    %       0    0    1                                 1 8<t6_* f  
    %       1    1    r * cos(theta)                    2 gN1b?_g  
    %       1   -1    r * sin(theta)                    2 L0ig%  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) DvHcT] l>5  
    %       2    0    (2*r^2 - 1)                    sqrt(3) F7gipCc1We  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 7S LJLn3d  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) K,bv\j;f  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Re-~C[zwT  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) *Uie{^p?  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) I!&|L0Qq  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) /R^Moj<  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;E>5<[aa  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 8o#*0d|  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) sufidi  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) e  p~3e5  
    %       -------------------------------------------------- -v .\CtpHv  
    % w'z ?1M(*  
    %   Example 1: $'*@g1v Y  
    % Gf\Dc   
    %       % Display the Zernike function Z(n=5,m=1) cP%mkh_ri  
    %       x = -1:0.01:1; A9\m .3jo  
    %       [X,Y] = meshgrid(x,x); vJVL%,7  
    %       [theta,r] = cart2pol(X,Y); BM!\U 6  
    %       idx = r<=1; z OD5a=[1  
    %       z = nan(size(X)); A|1 TE$  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); S%<RV6{aiM  
    %       figure CwZ+P n0  
    %       pcolor(x,x,z), shading interp /KjRB_5~q}  
    %       axis square, colorbar U1bhd}MoR  
    %       title('Zernike function Z_5^1(r,\theta)') azR<Y_tw  
    % P1)f-:;  
    %   Example 2: [~9rp]<  
    % {i y[8eLg  
    %       % Display the first 10 Zernike functions pV{MW#e  
    %       x = -1:0.01:1; ,0%P3  
    %       [X,Y] = meshgrid(x,x); l?v`kAMR  
    %       [theta,r] = cart2pol(X,Y); :L#t?~  
    %       idx = r<=1; (G $nN*rlu  
    %       z = nan(size(X)); {Ak{ ct\t  
    %       n = [0  1  1  2  2  2  3  3  3  3];  {I+   
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; l{F^"_U  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; rq?:I:0  
    %       y = zernfun(n,m,r(idx),theta(idx)); }VxbO8\b(  
    %       figure('Units','normalized') J/S 47J~  
    %       for k = 1:10 xO)vn\uJ  
    %           z(idx) = y(:,k); jjbBv~vs  
    %           subplot(4,7,Nplot(k)) /Y@^B,6 \  
    %           pcolor(x,x,z), shading interp u}Vc2a,WV  
    %           set(gca,'XTick',[],'YTick',[]) UOHU 1.3$T  
    %           axis square +6 t<FH  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _yY(&(]#  
    %       end D,%R[F? 5O  
    % "@U9'rKx  
    %   See also ZERNPOL, ZERNFUN2. =KqcWN3k  
    x'kwk  
    %   Paul Fricker 11/13/2006 @r4ZN6Wn  
    7sKN`  
    Kk+IUs  
    % Check and prepare the inputs: Sp$~)f'  
    % ----------------------------- Z*S 9pkWcF  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) | n5F_RL  
        error('zernfun:NMvectors','N and M must be vectors.') m<)0 XE6w  
    end l<5O\?Vo]  
    N|hNh$J[  
    if length(n)~=length(m) v(D{_  
        error('zernfun:NMlength','N and M must be the same length.') Qb}7lm{r  
    end OrP-+eg  
    n ^P=a'+  
    n = n(:); BE. v+'c"  
    m = m(:); )R$+dPu>  
    if any(mod(n-m,2)) 9z7^0Ruw  
        error('zernfun:NMmultiplesof2', ... C{>@b:]p  
              'All N and M must differ by multiples of 2 (including 0).') ModwJ w  
    end <![tn#_  
    YVt#( jl  
    if any(m>n) 6*,'A|t?y  
        error('zernfun:MlessthanN', ... -5,QrMM<  
              'Each M must be less than or equal to its corresponding N.') 9n{tbabJ  
    end 02E-|p;  
    jv7-i'I@  
    if any( r>1 | r<0 ) ^|y6oj  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 2?YN8 n9n  
    end 3qOq:ZkQ  
    (pM5B8U  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) N %N %  
        error('zernfun:RTHvector','R and THETA must be vectors.') UwOZBF<  
    end ?8[,0l:|  
    DpjiE/*  
    r = r(:); jn vJ`7zFP  
    theta = theta(:); v#*9rNEj0  
    length_r = length(r); NIufL }6\  
    if length_r~=length(theta) &ywAzGV{s  
        error('zernfun:RTHlength', ... P5s'cPX  
              'The number of R- and THETA-values must be equal.') z =1 J{]  
    end %T@3-V_  
    hJY= )  
    % Check normalization: -1).'aJ^  
    % -------------------- y<mmv~=  
    if nargin==5 && ischar(nflag) }~pT saw  
        isnorm = strcmpi(nflag,'norm'); q<(yNqMKP  
        if ~isnorm `tA~"J$32l  
            error('zernfun:normalization','Unrecognized normalization flag.') OAPR wOQ^=  
        end :0 G "EM4  
    else %!%G\nv  
        isnorm = false; HqNM31)  
    end >qh8em  
    SA_5..  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -w nlJi1f  
    % Compute the Zernike Polynomials S^nshQI  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A41*4!L=  
    OZ 4uk.)  
    % Determine the required powers of r: ?U'c;*O-  
    % ----------------------------------- l/9V59Fv9  
    m_abs = abs(m); 2)}ic2]pn  
    rpowers = []; iu!j#VO  
    for j = 1:length(n) !f5I.r~  
        rpowers = [rpowers m_abs(j):2:n(j)]; !K a!f1  
    end # \9sCnb  
    rpowers = unique(rpowers); ,b;eU[!]  
    w@&g9e6E  
    % Pre-compute the values of r raised to the required powers, 5dc24GB>_  
    % and compile them in a matrix: :m*r( i3  
    % ----------------------------- USF&;M3  
    if rpowers(1)==0 J6pQ){;6  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); [ySO  
        rpowern = cat(2,rpowern{:}); 1_JtD|Jy  
        rpowern = [ones(length_r,1) rpowern]; Pd-0u> k  
    else EfA*w/y  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); m(Ghe2T:  
        rpowern = cat(2,rpowern{:}); dIk/vg  
    end <aps)vF  
    L3[r7 b  
    % Compute the values of the polynomials: Q/[|/uNw?  
    % -------------------------------------- HPl'u'.Hg  
    y = zeros(length_r,length(n)); E__^>=  
    for j = 1:length(n) On%21L;JG  
        s = 0:(n(j)-m_abs(j))/2; Fw ,'a  
        pows = n(j):-2:m_abs(j);  c(Liwuj  
        for k = length(s):-1:1 y9W6e "  
            p = (1-2*mod(s(k),2))* ... b0W~*s [4  
                       prod(2:(n(j)-s(k)))/              ... +$Q.N{LV  
                       prod(2:s(k))/                     ... xXG-yh  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... S!!i  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ap|7./yg  
            idx = (pows(k)==rpowers); Y r3h=XY  
            y(:,j) = y(:,j) + p*rpowern(:,idx); W vh3Y,|3  
        end Gvg)@VNr  
         ,\*PpcU  
        if isnorm 3I0=^ >A  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); A gKG>%0  
        end nNuv 0  
    end b[+G+V   
    % END: Compute the Zernike Polynomials e}|UVoeH  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {#>>dILPr  
    @C[]o.r  
    % Compute the Zernike functions: Y:|_M3&'o  
    % ------------------------------ sg@)IEg</v  
    idx_pos = m>0; aLr\Uq,83  
    idx_neg = m<0; jP*5(*[&y  
    5Fh?YS=  
    z = y; 5I#L|+  
    if any(idx_pos) RmXC ^VQ  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Y{c_5YYf  
    end Z}#, E ;  
    if any(idx_neg) J:s^F n  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 0*?/s\>PS;  
    end n _G< /8  
    &?~OV:r9  
    % 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) C$MaJHkiF  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ~G0\57;h  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ?m&?BsW$)  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive J*)Vpk  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, @^g/`{j>J  
    %   and THETA is a vector of angles.  R and THETA must have the same :',.I  
    %   length.  The output Z is a matrix with one column for every P-value, XXuIWIhm  
    %   and one row for every (R,THETA) pair. {XC1B  
    % y*ae 5=6(  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ~udi=J |  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) $umh&z/  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) c^^[~YW j  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 yKJKQ9  
    %   for all p. j$%KKl8j  
    % sGx"j a +  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Uj!3H]d  
    %   Zernike functions (order N<=7).  In some disciplines it is oj.f uJD  
    %   traditional to label the first 36 functions using a single mode U1pL `P1  
    %   number P instead of separate numbers for the order N and azimuthal .+1.??8:+  
    %   frequency M. //C3tW  
    % R"Q=U}?$  
    %   Example: SrMg=a  
    % HWe?vz$4"  
    %       % Display the first 16 Zernike functions ka_]s:>+  
    %       x = -1:0.01:1; <gU^#gsGra  
    %       [X,Y] = meshgrid(x,x); O 0Fw!IQk  
    %       [theta,r] = cart2pol(X,Y); -phwzR\(t  
    %       idx = r<=1; "#uXpCuw  
    %       p = 0:15; HCP' V  
    %       z = nan(size(X)); xE/r:D#  
    %       y = zernfun2(p,r(idx),theta(idx)); b&k !DeE  
    %       figure('Units','normalized') H8[ L:VeNT  
    %       for k = 1:length(p) ia.95H;  
    %           z(idx) = y(:,k); 6?-vj2,  
    %           subplot(4,4,k) ?yKW^,q+  
    %           pcolor(x,x,z), shading interp w_-v!s2  
    %           set(gca,'XTick',[],'YTick',[]) 5mNd5IM  
    %           axis square CRy;>UI  
    %           title(['Z_{' num2str(p(k)) '}']) (rfU=E  
    %       end [}snKogp  
    % X}?`G?'  
    %   See also ZERNPOL, ZERNFUN. ^8S'=Bk  
    ,DrE4")4  
    %   Paul Fricker 11/13/2006 l4c9.'6  
    zUg-M  
    CB(Qy9C%h[  
    % Check and prepare the inputs: Y{P0?`  
    % ----------------------------- C[h"w'A2  
    if min(size(p))~=1 gC-3ghmgS  
        error('zernfun2:Pvector','Input P must be vector.') zcCX;N  
    end \(9hg.E  
    B4k ~~;|  
    if any(p)>35 >"qnuv G  
        error('zernfun2:P36', ... Ps3wg=ni[  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 2CX'J8Sy  
               '(P = 0 to 35).']) =n_r\z  
    end >5Vv6_CI0?  
    BLzl XhHn  
    % Get the order and frequency corresonding to the function number: `]L&2RS  
    % ---------------------------------------------------------------- Y#3<w  
    p = p(:); |/,XdTSy  
    n = ceil((-3+sqrt(9+8*p))/2); 'UGkL;  
    m = 2*p - n.*(n+2); D/1{v  
    *g =ey?1S  
    % Pass the inputs to the function ZERNFUN: @ToY,@]e  
    % ---------------------------------------- AS[yNCsjC  
    switch nargin -LAYj:4  
        case 3 )H&rr(  
            z = zernfun(n,m,r,theta); ?1\rf$l8  
        case 4 &)?ECj0`  
            z = zernfun(n,m,r,theta,nflag); EHByo[  
        otherwise C*A!`Q?1Y  
            error('zernfun2:nargin','Incorrect number of inputs.') FsI51@V72Q  
    end d{?X:*F  
    H$2<N@'4z  
    % 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) 49vKb(bz{  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. neN #Mo'A  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of jQDXl  
    %   order N and frequency M, evaluated at R.  N is a vector of d\% |!ix  
    %   positive integers (including 0), and M is a vector with the X?PcEAi;w  
    %   same number of elements as N.  Each element k of M must be a Li[ :L  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) `cee tr=  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is |Tn+Aq7  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix /DxaKZ ;b  
    %   with one column for every (N,M) pair, and one row for every m0*bz5  
    %   element in R. 7f!"vhCXM;  
    % 8|FHr,  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- /G}TPXA  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is \]bAXa{ p  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to #^|"dIZ_M  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 >NL4&MV:  
    %   for all [n,m]. VJp; XM  
    % ;onhc*{lv  
    %   The radial Zernike polynomials are the radial portion of the 6x?3%0Km  
    %   Zernike functions, which are an orthogonal basis on the unit E?,O>bCJ5  
    %   circle.  The series representation of the radial Zernike l_Ee us  
    %   polynomials is 0O,Q]P 82f  
    % &-l(nr]h]  
    %          (n-m)/2 1DzI@c~X  
    %            __ i@Vi.oc4[  
    %    m      \       s                                          n-2s "n,? )  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r :PJ 5~7C  
    %    n      s=0 F^La\cZ*'  
    % QQ1|]/)  
    %   The following table shows the first 12 polynomials. hq.XO=0"k  
    % M`1pze_A  
    %       n    m    Zernike polynomial    Normalization PcxCal4  
    %       --------------------------------------------- zh{:zT)(1  
    %       0    0    1                        sqrt(2) HN7C+e4U~  
    %       1    1    r                           2 /j}"4_. 8  
    %       2    0    2*r^2 - 1                sqrt(6) CFo>D\*J  
    %       2    2    r^2                      sqrt(6) ~57.0?IK  
    %       3    1    3*r^3 - 2*r              sqrt(8) uH"W07  
    %       3    3    r^3                      sqrt(8) 25bLU?x5B  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 9^D5Sl$g  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 5 5$J% ;&  
    %       4    4    r^4                      sqrt(10) Dht,!LVb;  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) S.i CkX  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) w-2?|XvDmf  
    %       5    5    r^5                      sqrt(12) y5oC|v7  
    %       --------------------------------------------- 57nSyd] PR  
    % 3W <_J_[  
    %   Example: I=vGS  
    % 7Pb: z4j  
    %       % Display three example Zernike radial polynomials  9hbn<Y  
    %       r = 0:0.01:1; OE{PP9 eh  
    %       n = [3 2 5]; at(oepq  
    %       m = [1 2 1]; 'f'zV@)  
    %       z = zernpol(n,m,r); zQPQP`  
    %       figure ;";#{B:  
    %       plot(r,z) iS/faXe5  
    %       grid on .|Ee,Un  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') `XmT)C  
    % klUW_d-  
    %   See also ZERNFUN, ZERNFUN2. L("zS%qr  
    sTmY'5ry  
    % A note on the algorithm. pb`!_GmB  
    % ------------------------ $N@EH;{_0  
    % The radial Zernike polynomials are computed using the series [*{G,=tF`Y  
    % representation shown in the Help section above. For many special N51g<K  
    % functions, direct evaluation using the series representation can VH>?%aL  
    % produce poor numerical results (floating point errors), because PF6w'T 5  
    % the summation often involves computing small differences between ]&oQ6  
    % large successive terms in the series. (In such cases, the functions +?&|p0  
    % are often evaluated using alternative methods such as recurrence n"Gow/-;  
    % relations: see the Legendre functions, for example). For the Zernike =x QLf4>  
    % polynomials, however, this problem does not arise, because the nKR=/5a4Y  
    % polynomials are evaluated over the finite domain r = (0,1), and v&#=1Zb  
    % because the coefficients for a given polynomial are generally all QOfqW@g  
    % of similar magnitude. /a'cP  
    % et-<ib<lY  
    % ZERNPOL has been written using a vectorized implementation: multiple H0 %;t  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] c;M&;'#x  
    % values can be passed as inputs) for a vector of points R.  To achieve GM{J3O=  
    % this vectorization most efficiently, the algorithm in ZERNPOL IegZ)&_n  
    % involves pre-determining all the powers p of R that are required to Z` ;.62S  
    % compute the outputs, and then compiling the {R^p} into a single zfexaf!  
    % matrix.  This avoids any redundant computation of the R^p, and `8D)j>Yh~  
    % minimizes the sizes of certain intermediate variables. L2N O_N  
    % ea\b7a*  
    %   Paul Fricker 11/13/2006 fc,^H&  
    ]TTQ;F  
    _7^4sR8=  
    % Check and prepare the inputs: (*XSr Q  
    % ----------------------------- 5VLJ:I?0O  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) KcW]"K>p!  
        error('zernpol:NMvectors','N and M must be vectors.') Uiz#QGt  
    end  n}f*>Mn  
    p%?VW  
    if length(n)~=length(m) eHt |O~  
        error('zernpol:NMlength','N and M must be the same length.') 3%J7_e'  
    end Gv$}>YJ  
    ~?BN4ptc  
    n = n(:); F~C9,`#Wf@  
    m = m(:); Mu~DB:Y9e  
    length_n = length(n); W/?\8AE  
    (:TZ~"VY  
    if any(mod(n-m,2)) q|r/%[[!o  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') L{i,.aE/nO  
    end +OTNn@!9  
    mv0JD(  
    if any(m<0) 'u)zQAaw.  
        error('zernpol:Mpositive','All M must be positive.') w4"4(SR.  
    end LYV\|a{Y  
    SFaG`T=  
    if any(m>n) a2vZ'  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.')  > h>  
    end QL{^  
    u\uYq  
    if any( r>1 | r<0 ) /2&:sHWW  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') XoO#{7a  
    end Pv>W`/*_,s  
    [!Jd.zm  
    if ~any(size(r)==1) qa!3lb_'M  
        error('zernpol:Rvector','R must be a vector.') "j<l=l!  
    end $v#\bqY  
    ": G\  
    r = r(:); )j^~=Sio.  
    length_r = length(r); ar#Xe;T!  
    Alh"ZT^*  
    if nargin==4 ! ,*4d $  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); F79!B  
        if ~isnorm i!U,qV1  
            error('zernpol:normalization','Unrecognized normalization flag.') #*"5F*  
        end lls-Nir%  
    else ;hcOD4or  
        isnorm = false; : K#z~#n  
    end @ 7WWoy  
    UmC_C[/n?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6y4&nTq[  
    % Compute the Zernike Polynomials zt23on2  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yN4K^#  
    ;YYo^9Lh}  
    % Determine the required powers of r: ohod)8  
    % ----------------------------------- 9|}u"jJB%E  
    rpowers = []; FU~xKNr  
    for j = 1:length(n) $^ wqoW%t  
        rpowers = [rpowers m(j):2:n(j)]; @+,J^[ y  
    end K:osfd  
    rpowers = unique(rpowers); Xc!0'P0T  
    aJmSagr69C  
    % Pre-compute the values of r raised to the required powers, $XOs(>~"r  
    % and compile them in a matrix: ?df*Y5I2  
    % ----------------------------- v_7?Zik8E  
    if rpowers(1)==0 .0[ zZ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5R?[My  
        rpowern = cat(2,rpowern{:}); ?)# qBE ]  
        rpowern = [ones(length_r,1) rpowern]; Vh1R!>XY  
    else #KOr-Yg|U  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); +li<y`aw0  
        rpowern = cat(2,rpowern{:}); n&4 4Acs[  
    end xlW`4\ Pa  
    V2&O]bR  
    % Compute the values of the polynomials: {[<o)k.A  
    % -------------------------------------- 6~t;&)6J  
    z = zeros(length_r,length_n); XlPi)3m4/S  
    for j = 1:length_n >3v j<v}m  
        s = 0:(n(j)-m(j))/2; [nx OGa2  
        pows = n(j):-2:m(j); [lmghI!  
        for k = length(s):-1:1 Uzx,aYo X  
            p = (1-2*mod(s(k),2))* ... 'DDlX3W-  
                       prod(2:(n(j)-s(k)))/          ... Tf|?j=f  
                       prod(2:s(k))/                 ... N3Yf3rK  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 2EU((Q`>=(  
                       prod(2:((n(j)+m(j))/2-s(k))); D;@nrj`.  
            idx = (pows(k)==rpowers); 42Qfv%*c  
            z(:,j) = z(:,j) + p*rpowern(:,idx); nNN~Z'bG  
        end YEg(QOn3Q  
         A$H;2T5N  
        if isnorm mn>$K"_k  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); #%=6DHsK  
        end D<DSK~  
    end ++HHUM  
    =tS1|_  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  ,?3)L   
    }[$qn|  
    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。 }#b[@3/T  
    VbwB<nQl  
    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)