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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 ;wj|@cF  
    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! ux)Wh.5  
     
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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 6U,:J'5gP  
    function z = zernfun(n,m,r,theta,nflag) s !I I}'Je  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. -CALU X  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 0*j\i@  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 2o 7o~r  
    %   unit circle.  N is a vector of positive integers (including 0), and "$q"Kilj%  
    %   M is a vector with the same number of elements as N.  Each element Z/;hbbG  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) g@ ]1H41  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, n.,ZgLx["  
    %   and THETA is a vector of angles.  R and THETA must have the same ^c"\%!w"O  
    %   length.  The output Z is a matrix with one column for every (N,M) N9vNSmm  
    %   pair, and one row for every (R,THETA) pair. .5tXwxad"  
    % ssmJ?sl  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike (e9hp2m  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), K3&k+~$  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral slLTZ]  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Djf~8q V!  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized a;(,$q3M  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. mn(MgJKQ\  
    % QRF:6bAxsL  
    %   The Zernike functions are an orthogonal basis on the unit circle. 9QkssI  
    %   They are used in disciplines such as astronomy, optics, and aw7pr464  
    %   optometry to describe functions on a circular domain. 3Q,p,  
    % NkQain9  
    %   The following table lists the first 15 Zernike functions. uL^X$8K;(  
    % lxBcO/  
    %       n    m    Zernike function           Normalization !_?HSDAj"n  
    %       -------------------------------------------------- \P*%u  
    %       0    0    1                                 1 YL[y3&K  
    %       1    1    r * cos(theta)                    2 (D+%*ax  
    %       1   -1    r * sin(theta)                    2 9~ifST \  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) FH;)5GGnv  
    %       2    0    (2*r^2 - 1)                    sqrt(3) bf[l4$3k  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) - @KT#  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) y;hco  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) (unJwh{7Q  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) qLB(Th\&'  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) %F<3_#Y  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) NNRKYdp,  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) PG'I7)Bv  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) =g=Vv"B_  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #QW% ;^  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) r?`7i'  
    %       -------------------------------------------------- 5jTA6s9zA  
    % d"+ _`d=`  
    %   Example 1: 3W3d $  
    % J^Wqa$<;"  
    %       % Display the Zernike function Z(n=5,m=1) 5zt5]zl'  
    %       x = -1:0.01:1; 6|1#Prj  
    %       [X,Y] = meshgrid(x,x); be.Kx< I  
    %       [theta,r] = cart2pol(X,Y); =I+5sCF{g  
    %       idx = r<=1; CS"p3$7,  
    %       z = nan(size(X)); 1 EHNg<J(  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); <"S/M]9  
    %       figure <a2Kc '  
    %       pcolor(x,x,z), shading interp a0  w  
    %       axis square, colorbar KpO%)M!/Z#  
    %       title('Zernike function Z_5^1(r,\theta)') EtcXzq>w  
    % XP65  
    %   Example 2: U9RpHh`  
    % C}]rx{xC  
    %       % Display the first 10 Zernike functions  TJb&f<  
    %       x = -1:0.01:1; iEMIzaR  
    %       [X,Y] = meshgrid(x,x); td2bL4  
    %       [theta,r] = cart2pol(X,Y); 2V*<J:;wb  
    %       idx = r<=1; l" H/PB<.  
    %       z = nan(size(X)); 79U 7<]-!  
    %       n = [0  1  1  2  2  2  3  3  3  3]; m RtE~~p  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 23`pog{n  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 0y#TGM|0D  
    %       y = zernfun(n,m,r(idx),theta(idx)); j<i: rk|  
    %       figure('Units','normalized') 1;+(HB  
    %       for k = 1:10 {>#4{D00  
    %           z(idx) = y(:,k); ;[-y>qU0  
    %           subplot(4,7,Nplot(k)) Q__1QUu  
    %           pcolor(x,x,z), shading interp =/HTe&  
    %           set(gca,'XTick',[],'YTick',[]) 65pC#$F<x  
    %           axis square p5=VGKp  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ;#?+i`9'q  
    %       end 79MB_Is]s  
    % z^9df(  
    %   See also ZERNPOL, ZERNFUN2. t`A5wqm  
    Gt?ckMB  
    %   Paul Fricker 11/13/2006 I*8_5?)g<  
    c::Vh  
    Hd=!  
    % Check and prepare the inputs: !rgdOlTR^  
    % ----------------------------- *:V"C\`^n  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) lD)QB!*v  
        error('zernfun:NMvectors','N and M must be vectors.') '=m ?l  
    end "8`f x  
    XZ&q5]PJI  
    if length(n)~=length(m) `LCxxpHi|  
        error('zernfun:NMlength','N and M must be the same length.') NU|T`gP  
    end F!yejn [  
    \9dC z;  
    n = n(:); ?QCHkhU  
    m = m(:); :. a}pgh  
    if any(mod(n-m,2)) :ug j+  
        error('zernfun:NMmultiplesof2', ... K)t+lJ  
              'All N and M must differ by multiples of 2 (including 0).') B (dq$+4  
    end p[-bu B]  
    rgg3{bU/  
    if any(m>n) F>A&L8  
        error('zernfun:MlessthanN', ... d/:zO4v3  
              'Each M must be less than or equal to its corresponding N.') @~<M_63  
    end Y>[u(q&09O  
    bi[gyl#  
    if any( r>1 | r<0 ) hSD uByoi  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') n,NKJt  
    end iw^(3FcP@C  
    G@igxnm}  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) skP2IMa75  
        error('zernfun:RTHvector','R and THETA must be vectors.') O486:tF  
    end mam2]St"  
    -kd_gbnr3  
    r = r(:); `$D2w|  
    theta = theta(:); p V^hZ.  
    length_r = length(r); r$~ f[cA  
    if length_r~=length(theta) v-@xO&<  
        error('zernfun:RTHlength', ... ,-*oc>  
              'The number of R- and THETA-values must be equal.') Jm8#M z  
    end C.$`HGv  
    Y8 a![  
    % Check normalization: niV=Ijt{5  
    % -------------------- +kKfx!  
    if nargin==5 && ischar(nflag) g^DPb pWxu  
        isnorm = strcmpi(nflag,'norm'); P=V=\T<4_  
        if ~isnorm D=nuK25  
            error('zernfun:normalization','Unrecognized normalization flag.') vxzOG?Xc:  
        end QNH5Cq;Y  
    else w%[ `'_[  
        isnorm = false; 7.PG*q  
    end =?f\o*J)  
     w|>O!]K]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,#42ebGHR  
    % Compute the Zernike Polynomials c91rc>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9+\3E4K  
    ;Qc_Tf=,  
    % Determine the required powers of r: 'i|z>si[*  
    % ----------------------------------- YRYAQj/7  
    m_abs = abs(m); wV;qc3  
    rpowers = []; Y|=/*?o}  
    for j = 1:length(n) H}QOoXWkg  
        rpowers = [rpowers m_abs(j):2:n(j)]; L;0ZB=3n  
    end FXPw 5  
    rpowers = unique(rpowers); Ncu\;K\N  
    Ii,Lj1Q  
    % Pre-compute the values of r raised to the required powers, b:nHcxDU<  
    % and compile them in a matrix: ?2;r#)  
    % ----------------------------- f`?Y+nu}  
    if rpowers(1)==0 lk6*?EJ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); HUtuUX  
        rpowern = cat(2,rpowern{:}); }F1|& A  
        rpowern = [ones(length_r,1) rpowern]; ]3C&l+m$ot  
    else ~/6m|k  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); k 4+F  
        rpowern = cat(2,rpowern{:}); 6U k[_)1  
    end W,i SN}  
    ?+S&`%?  
    % Compute the values of the polynomials: L "L@4 B  
    % -------------------------------------- 0SXWt? }  
    y = zeros(length_r,length(n)); :mU,g|~55  
    for j = 1:length(n) ;Bo{.916  
        s = 0:(n(j)-m_abs(j))/2; t>h<XPJi  
        pows = n(j):-2:m_abs(j); 95,y@~ *]  
        for k = length(s):-1:1 !+4}x;!8  
            p = (1-2*mod(s(k),2))* ... 6<+R55  
                       prod(2:(n(j)-s(k)))/              ... :cmfy6h]  
                       prod(2:s(k))/                     ... gg(^:`+  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... mfO:#]K  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); x[3kCa|4A  
            idx = (pows(k)==rpowers); _^'fp  
            y(:,j) = y(:,j) + p*rpowern(:,idx); xQC.ap  
        end u2^ oXl  
         (u-i{<   
        if isnorm e*e}X&|(g  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); MPMJkL$F^  
        end &E$jAqc  
    end 9)Y]05us  
    % END: Compute the Zernike Polynomials rp.S4;=Q9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C:g2E[#  
    '2a}1?  
    % Compute the Zernike functions: 4w^B&e%  
    % ------------------------------ P8e1J0A  
    idx_pos = m>0; K3&v6 #]  
    idx_neg = m<0;  gM20n^  
    C_?L$3 U0  
    z = y; @c{=:kg5  
    if any(idx_pos) *TA${$K  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); NjVuwIm+  
    end %O;"Z`I  
    if any(idx_neg) Zgo^M,g  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); dRyK'Xr  
    end 9 kzytx  
    !SIGzj  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) @"-<m|lM  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. /s(/6~D|  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated QP)-O*+AA  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Ot(U_rJCi  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, C',D"  
    %   and THETA is a vector of angles.  R and THETA must have the same \&]'GsfF  
    %   length.  The output Z is a matrix with one column for every P-value, n[CESo%[  
    %   and one row for every (R,THETA) pair. DIH.c7o  
    % Gk*Mx6|N  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike D|,d_W  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ($' rV!}  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) @P6K`'.0  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 eHi|_3A&*  
    %   for all p. b4!(~"b.  
    % f GE+DjeA  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 MFaK=1  
    %   Zernike functions (order N<=7).  In some disciplines it is .t4IR =Z  
    %   traditional to label the first 36 functions using a single mode JSt%L|}Y  
    %   number P instead of separate numbers for the order N and azimuthal #tKc!]m  
    %   frequency M. 7qV_QZ!.  
    % rv1kIc5Za<  
    %   Example: "wmQ,=  
    % Z_D8}$!  
    %       % Display the first 16 Zernike functions b-U LoV  
    %       x = -1:0.01:1; o8BbSZVu  
    %       [X,Y] = meshgrid(x,x); !v<r=u  
    %       [theta,r] = cart2pol(X,Y); Zaf].R  
    %       idx = r<=1; K_bF)6"  
    %       p = 0:15; LpeQx\  
    %       z = nan(size(X)); ^IOf%  
    %       y = zernfun2(p,r(idx),theta(idx)); NZ/gp"D?  
    %       figure('Units','normalized') Kjc"K36{L  
    %       for k = 1:length(p) ]8*g%  
    %           z(idx) = y(:,k); $`.7XD}  
    %           subplot(4,4,k) oh-Y  
    %           pcolor(x,x,z), shading interp Cy4@\X%W  
    %           set(gca,'XTick',[],'YTick',[]) f}0(qN/G  
    %           axis square 2B3H -`  
    %           title(['Z_{' num2str(p(k)) '}']) ;RB]awE  
    %       end {*ATY+  
    % SN(:\|f 2  
    %   See also ZERNPOL, ZERNFUN. ZK1d3  
    EA|*|o4)  
    %   Paul Fricker 11/13/2006 "n," >  
    m{x[q  
    7f#e#_sM;  
    % Check and prepare the inputs: y!u)q3J0&  
    % ----------------------------- C$SuFL(pb  
    if min(size(p))~=1 'U.)f@L#w  
        error('zernfun2:Pvector','Input P must be vector.') n'9Wl'  
    end )~v`dwKj;  
    |)* K#%j  
    if any(p)>35 B^P)(Nu+  
        error('zernfun2:P36', ... H)}>&Z4  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 7/a[;`i*!  
               '(P = 0 to 35).']) U748$%}]  
    end F$ShhZgi  
    M' e<\wqm  
    % Get the order and frequency corresonding to the function number: vm"LPwSk>  
    % ---------------------------------------------------------------- c [sydl  
    p = p(:); B\r2M`N5  
    n = ceil((-3+sqrt(9+8*p))/2); ~m8".Z"  
    m = 2*p - n.*(n+2); Zy|B~.@<j  
    u n\!K  
    % Pass the inputs to the function ZERNFUN: g(B&A P_e  
    % ---------------------------------------- LQ`s>q  
    switch nargin X0Y1I}gD  
        case 3 R8I%Cyc  
            z = zernfun(n,m,r,theta); &l"/G%W  
        case 4 '#!nK O2<  
            z = zernfun(n,m,r,theta,nflag); Oq@+/UWX  
        otherwise 7DDd 1"jE  
            error('zernfun2:nargin','Incorrect number of inputs.') }(A`aB_  
    end ukpbx;O:hc  
    "3.v(GVr  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) /4/'&tY  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. G%^jgr)  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of  9uR+  
    %   order N and frequency M, evaluated at R.  N is a vector of V[ju7\>$Z  
    %   positive integers (including 0), and M is a vector with the d-c<dS+R  
    %   same number of elements as N.  Each element k of M must be a Q,R>dkS  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ?lq  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is B|pO2d e  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix #(swVo:+E  
    %   with one column for every (N,M) pair, and one row for every ze2%#<  
    %   element in R. 0t*e#,y  
    % Lh%z2 5t  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- EP,j+^RVf  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is xfoQx_]$Im  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 9$[6\jMh  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 c?5?TJpm  
    %   for all [n,m]. uN>JX/-  
    % cq]JD6937  
    %   The radial Zernike polynomials are the radial portion of the p3r("\Za,  
    %   Zernike functions, which are an orthogonal basis on the unit aItQ(+y  
    %   circle.  The series representation of the radial Zernike ' ` _TFTO  
    %   polynomials is GWFF.Mo^  
    % `_aX>fw  
    %          (n-m)/2 Drm#z05i[g  
    %            __ /2^"c+/'p  
    %    m      \       s                                          n-2s !LI6_Oq  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r r$[`A_  
    %    n      s=0 '41'Gn  
    % aeZ$Wu>]W  
    %   The following table shows the first 12 polynomials. YI+ clh;%9  
    % "&Hr)yyWG  
    %       n    m    Zernike polynomial    Normalization (4o<U%3kGq  
    %       --------------------------------------------- 88Nx/:#Y*  
    %       0    0    1                        sqrt(2) 8\WV.+  
    %       1    1    r                           2 W(pq_H'  
    %       2    0    2*r^2 - 1                sqrt(6) yFoPCA86y  
    %       2    2    r^2                      sqrt(6) |RjAp.pm  
    %       3    1    3*r^3 - 2*r              sqrt(8) }1fi#  
    %       3    3    r^3                      sqrt(8) Q,nJz*AJ  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) wZ4w`|'  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Gj_7wP$  
    %       4    4    r^4                      sqrt(10) 'oKen!?A  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) D:] QBA)C  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) d| #&j. "  
    %       5    5    r^5                      sqrt(12) )ifjK6*  
    %       --------------------------------------------- Qb%o%z?hee  
    % p{ Xde   
    %   Example: }7non  
    % dx@|M{jz'  
    %       % Display three example Zernike radial polynomials fj|b;8_}l  
    %       r = 0:0.01:1; M*!WXQlud  
    %       n = [3 2 5]; @An}  
    %       m = [1 2 1]; za%gD  
    %       z = zernpol(n,m,r); <%o9*)F  
    %       figure Y~ xo=v(  
    %       plot(r,z) &(7=NAQsE  
    %       grid on Gv[s86AP,  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') pMHF u/|Pr  
    % _aeIK  
    %   See also ZERNFUN, ZERNFUN2. y'aK92pF:  
    }u9#S  
    % A note on the algorithm. _01wRsm%2  
    % ------------------------ =oBlUE  
    % The radial Zernike polynomials are computed using the series HYg! <y  
    % representation shown in the Help section above. For many special T;G<62`.h  
    % functions, direct evaluation using the series representation can Q,K$)bM  
    % produce poor numerical results (floating point errors), because l`uI K.  
    % the summation often involves computing small differences between lD^c_b  
    % large successive terms in the series. (In such cases, the functions ;FQNO:NP  
    % are often evaluated using alternative methods such as recurrence i;rcg d  
    % relations: see the Legendre functions, for example). For the Zernike 7u{V1_ n1  
    % polynomials, however, this problem does not arise, because the C}#$wge  
    % polynomials are evaluated over the finite domain r = (0,1), and wn^#`s!]U  
    % because the coefficients for a given polynomial are generally all T>f6V 5  
    % of similar magnitude. G6QD`ED  
    % Eug RC  
    % ZERNPOL has been written using a vectorized implementation: multiple 7Db}bDU1 |  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ,<N{Y[n]e  
    % values can be passed as inputs) for a vector of points R.  To achieve uNkJe  
    % this vectorization most efficiently, the algorithm in ZERNPOL iC3C~?,7  
    % involves pre-determining all the powers p of R that are required to JK`$/l|7  
    % compute the outputs, and then compiling the {R^p} into a single uu9IUqEq2  
    % matrix.  This avoids any redundant computation of the R^p, and l?QA;9_R'  
    % minimizes the sizes of certain intermediate variables. j]FK.G'  
    % l\F71pwSI  
    %   Paul Fricker 11/13/2006 35Fxzj $  
    pd B\D  
    ;yajt\a  
    % Check and prepare the inputs: `Xz!apA  
    % ----------------------------- [UXN= 76N  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) w3ATsIw  
        error('zernpol:NMvectors','N and M must be vectors.') U?an\rv  
    end GeDI\-  
    C{,^4Eh3r  
    if length(n)~=length(m) m}u)C&2>  
        error('zernpol:NMlength','N and M must be the same length.') k3H0$1  
    end Jz=|-F(Sy  
    F^$;hMh%  
    n = n(:); IIT UM)  
    m = m(:); a2Nxpxho  
    length_n = length(n); =[8K#PZ$w  
    U(0FL6sPC  
    if any(mod(n-m,2)) $|xSM2  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 7Q]c=i cg  
    end 5D.Sg;\  
    JO`r)_  
    if any(m<0) ~+j2a3rv-{  
        error('zernpol:Mpositive','All M must be positive.') #CPLvg#  
    end >s 6ye  
    &e6UEG  
    if any(m>n) UOsK(mB  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') DI8<0.L  
    end e@qH!.g)  
    'B{FRK  
    if any( r>1 | r<0 ) i^e8.zgywF  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ~uH_y-  
    end 2cUT bRm  
    F$(ak;v}  
    if ~any(size(r)==1) l 9 wO x  
        error('zernpol:Rvector','R must be a vector.') #GLW3}  
    end hVmnXT 3Z  
    mE`qA*=?  
    r = r(:); M,S'4Sz uk  
    length_r = length(r); 'prHXzi(h  
    :^PksR  
    if nargin==4 HO['o{>BL  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); xI1{Wo*2C}  
        if ~isnorm ,5"(m?[m  
            error('zernpol:normalization','Unrecognized normalization flag.') JC%&d1  
        end ;<N:!$p  
    else }rI:pp^KS  
        isnorm = false; Hc =QSP  
    end *Q^ z4UY  
    :u./"[G  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (z>t4(%\  
    % Compute the Zernike Polynomials -1Dq_!i  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Oo@o$\+v  
    1Vdi5;dn  
    % Determine the required powers of r: 8k95IJR1  
    % ----------------------------------- -z~!%4 a  
    rpowers = []; sa4w.9O1GS  
    for j = 1:length(n) jOVF+9M  
        rpowers = [rpowers m(j):2:n(j)]; R__:~ uv,  
    end Mn(iAsg  
    rpowers = unique(rpowers); '"fJA/O  
    V-}}?c1 F  
    % Pre-compute the values of r raised to the required powers, IO)#O<  
    % and compile them in a matrix: @]vY[O!&;  
    % ----------------------------- -1,0hmn=+  
    if rpowers(1)==0 1f}(=Hv{  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 4_kN';a4Q  
        rpowern = cat(2,rpowern{:}); ,JE_aje7  
        rpowern = [ones(length_r,1) rpowern]; `,-mXxTNT  
    else A vq+s.h  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); !Fp %2gt|  
        rpowern = cat(2,rpowern{:}); d]] z )  
    end *u1q7JFQk  
    X n$ZA-  
    % Compute the values of the polynomials: U_(>eVi7F  
    % -------------------------------------- A,9JbX  
    z = zeros(length_r,length_n); x{SlJ%V  
    for j = 1:length_n 2Qp}f^  
        s = 0:(n(j)-m(j))/2; h9)fXW  
        pows = n(j):-2:m(j);  ~2"hh$  
        for k = length(s):-1:1 hr}f5Z)^v  
            p = (1-2*mod(s(k),2))* ... Q2VF+g,  
                       prod(2:(n(j)-s(k)))/          ... 1j$\ 48Z  
                       prod(2:s(k))/                 ... \~l_w ,Poo  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... &)mZ~cPU3  
                       prod(2:((n(j)+m(j))/2-s(k))); 9pqsr~  
            idx = (pows(k)==rpowers); ZpVkgX4  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ZOqS"3j! j  
        end :J`@@H  
         mfr aw2H  
        if isnorm ._}}@V_/  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); b'z $S+  
        end :aNjh  
    end {T4_Xn-I  
    pv]@}+<Dt  
    % 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)  5P*jGOg.  
    `~axOp9N  
    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。 ]_ LAy  
    JNU9RxR  
    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)