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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 '#6e Ub  
    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! |k.'w<6mb9  
     
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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 9nR\7!_  
    function z = zernfun(n,m,r,theta,nflag) ;wwc;wQ'  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. /p !A:8  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N PYCN3s#Gi  
    %   and angular frequency M, evaluated at positions (R,THETA) on the #8Bh5L!SJ1  
    %   unit circle.  N is a vector of positive integers (including 0), and 2>o[  
    %   M is a vector with the same number of elements as N.  Each element |N/d }  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) >V6t L;+  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, OjGI !  
    %   and THETA is a vector of angles.  R and THETA must have the same -Q20af-  
    %   length.  The output Z is a matrix with one column for every (N,M) G^.N$wcv  
    %   pair, and one row for every (R,THETA) pair. D0Q9A]bD;  
    % ^8 VW$}  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike jrp>Y:  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), u''Ce`N  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral =v:?rY}  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, T]tP!a;K  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Cx TAd[az  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *L^W[o  
    % rI>x'0Go*  
    %   The Zernike functions are an orthogonal basis on the unit circle. $yx\2   
    %   They are used in disciplines such as astronomy, optics, and eIvZhi  
    %   optometry to describe functions on a circular domain. ` @QZK0Ox  
    % :;_ khno  
    %   The following table lists the first 15 Zernike functions. zM0}(5$m  
    % i(.e=  
    %       n    m    Zernike function           Normalization ei5YxV6I  
    %       -------------------------------------------------- F{x+1hct0  
    %       0    0    1                                 1 8 W  
    %       1    1    r * cos(theta)                    2 qPPe)IM'Sc  
    %       1   -1    r * sin(theta)                    2 Wk[a|>  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) FI^Wh7J  
    %       2    0    (2*r^2 - 1)                    sqrt(3) cKVFykwM  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) M/ 64`lcb  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) VlV X  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) -DkD*64wu  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) y-hTTd"{  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 'C5id7O&  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ':n`0+Eh  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |S!R Q-CF  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) o898pg  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) j:%,lcF  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ~uhW~bT  
    %       -------------------------------------------------- `jeATxWv  
    % xeF>"6\  
    %   Example 1: YYT;a$GTo  
    % 'APx  
    %       % Display the Zernike function Z(n=5,m=1) Pxl,"  
    %       x = -1:0.01:1; Z:{| ?4  
    %       [X,Y] = meshgrid(x,x); `Abd=1nH  
    %       [theta,r] = cart2pol(X,Y); ,SIS3A>s  
    %       idx = r<=1; "}3sL#|z  
    %       z = nan(size(X)); pN^g.  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); m.+h@  
    %       figure hNXZL>6  
    %       pcolor(x,x,z), shading interp JRB6T_U  
    %       axis square, colorbar sTd@/>S?p  
    %       title('Zernike function Z_5^1(r,\theta)') L-Q8iFW'  
    % ?-j/X6(\(  
    %   Example 2: tl_3 %$s  
    % DzR,ou  
    %       % Display the first 10 Zernike functions e(s0mbJE  
    %       x = -1:0.01:1; $z_yx `5  
    %       [X,Y] = meshgrid(x,x); atZe`0  
    %       [theta,r] = cart2pol(X,Y); 5M=U*BI  
    %       idx = r<=1;  N}5  
    %       z = nan(size(X)); JL,Y9G*]s  
    %       n = [0  1  1  2  2  2  3  3  3  3]; S})f`X9_}  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 6)1PDlB  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; }F]Z1('  
    %       y = zernfun(n,m,r(idx),theta(idx)); U$5x#{AFp  
    %       figure('Units','normalized') fnX[R2KZ  
    %       for k = 1:10 8oE`>Y  
    %           z(idx) = y(:,k); {H/%2  
    %           subplot(4,7,Nplot(k)) {|oWU8.l  
    %           pcolor(x,x,z), shading interp u4hn9**a1  
    %           set(gca,'XTick',[],'YTick',[]) suQTi'K1  
    %           axis square DCp8rvUI  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _S) K+C|@  
    %       end N5K(yY_T  
    % brTNwRze  
    %   See also ZERNPOL, ZERNFUN2. a]Pi2:S  
    6c!F%xU}  
    %   Paul Fricker 11/13/2006 }aOqoi7w  
    F`4W5~`  
    eZ:iW#YF  
    % Check and prepare the inputs: )<HvIr(xr  
    % ----------------------------- `!cdxKLR  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) d*|RFU  
        error('zernfun:NMvectors','N and M must be vectors.') y CHOg  
    end 4Wgzp51Aq!  
    qeMDC#N  
    if length(n)~=length(m) [.>=> KJ_  
        error('zernfun:NMlength','N and M must be the same length.') 80Y% C-Y:  
    end =+_nVO*  
    .iV=ybMT  
    n = n(:); P DY :?/  
    m = m(:); fYuSfB+<  
    if any(mod(n-m,2)) Do(G;D`h+_  
        error('zernfun:NMmultiplesof2', ... !%$[p'  
              'All N and M must differ by multiples of 2 (including 0).') Y*@7/2,  
    end sq=EL+=j  
    CE M4E  
    if any(m>n) A o* IshVh  
        error('zernfun:MlessthanN', ... [NE!  
              'Each M must be less than or equal to its corresponding N.') d_(>:|o h  
    end c9c]1XJ  
    @Nb/n  
    if any( r>1 | r<0 ) hRXnig{;3  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') J t.<Z&  
    end I._ A  
    / xv5we~  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) egsP\ '  
        error('zernfun:RTHvector','R and THETA must be vectors.') / ^)3V}  
    end (P? |Bk [  
    :sw5@JdJ  
    r = r(:); *i*\ dl  
    theta = theta(:); *JImP9SE  
    length_r = length(r); 3]1 ! g6  
    if length_r~=length(theta) +E9G"Z65iP  
        error('zernfun:RTHlength', ... V^tD@N  
              'The number of R- and THETA-values must be equal.') |};d:LwX  
    end .pZYPKMaE  
    $MvKwQ/  
    % Check normalization: W\j'8^kI9  
    % -------------------- Q^<amM!  
    if nargin==5 && ischar(nflag) f'ld6jt|%  
        isnorm = strcmpi(nflag,'norm'); VEa"^{,w  
        if ~isnorm &(<>} r  
            error('zernfun:normalization','Unrecognized normalization flag.') +h-% {  
        end t$qIJt$  
    else \roJf&O }  
        isnorm = false; jh z*Y}MX  
    end VS4Glx73  
    Ib{#dhV  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3_G0eIE"u  
    % Compute the Zernike Polynomials $-^& AKc  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +D @B eQu  
    sh,4n{+  
    % Determine the required powers of r: enxb pq#  
    % ----------------------------------- V %[t'uh  
    m_abs = abs(m); >4bw4 Z1  
    rpowers = []; \a0{9Xx F  
    for j = 1:length(n) q8ZxeMqx%  
        rpowers = [rpowers m_abs(j):2:n(j)]; |5>A^a  
    end J|jvqt9C  
    rpowers = unique(rpowers); tHaHBx1P  
    +EA ")T<l  
    % Pre-compute the values of r raised to the required powers, 8=bn TJf  
    % and compile them in a matrix: ?$)a[UnqX  
    % ----------------------------- cb'Y a_  
    if rpowers(1)==0 6VQQI9  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); F+VNrt-  
        rpowern = cat(2,rpowern{:}); 1 39T*0C  
        rpowern = [ones(length_r,1) rpowern]; x xzUey  
    else QNE/SSL  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ;*K;)C  
        rpowern = cat(2,rpowern{:}); **-rPonM[  
    end = ZoNkj/^,  
    'H`:c+KDG`  
    % Compute the values of the polynomials: 5WHqD!7u  
    % -------------------------------------- KiMlbF.~V  
    y = zeros(length_r,length(n)); vS ( Y_6  
    for j = 1:length(n) +(`D'5EB(  
        s = 0:(n(j)-m_abs(j))/2; G \a`F'Oo  
        pows = n(j):-2:m_abs(j); HQF@@  
        for k = length(s):-1:1 B.?F^m@zS  
            p = (1-2*mod(s(k),2))* ... %qJgtu"8  
                       prod(2:(n(j)-s(k)))/              ... KBi(Ns#+  
                       prod(2:s(k))/                     ... {B#w9>'b  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... N:'GNMu  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); DB|w&tygq  
            idx = (pows(k)==rpowers); LdOqV'&r  
            y(:,j) = y(:,j) + p*rpowern(:,idx); *Q2 oc:6  
        end Tw%1m  
         o=7e8l  
        if isnorm Dg~m}La  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 6ym$8^  
        end hX,RuI  
    end #v$wjqK5  
    % END: Compute the Zernike Polynomials RI<s mt.Ng  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _8SB+s*  
    Qa2p34Z/  
    % Compute the Zernike functions: C_ ;nlG6  
    % ------------------------------ Y1AZ%{^0a  
    idx_pos = m>0; uZf 6W<a  
    idx_neg = m<0; m' j1  
    OP=oSfa  
    z = y; V"gKk$j7  
    if any(idx_pos) $M,Q"QL  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); n}9<7e~/  
    end ZJFF4($qN  
    if any(idx_neg) 8)s0$64Ra  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)');  $AZ=;iP-  
    end }"RVUYU  
    DIP%*b#l$\  
    % 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)  D5Jg(-  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. @yQ1F> t  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 7N5M=f.DS(  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive a3:45[SO4e  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 4QPHT#eqX  
    %   and THETA is a vector of angles.  R and THETA must have the same HsRoiqo  
    %   length.  The output Z is a matrix with one column for every P-value, Cnc=GTR i  
    %   and one row for every (R,THETA) pair. [EVyCIcY,h  
    % p_zVrlVb  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike lR F5/  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ^%6f%]_  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) iYdg1  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 'm<L}d  
    %   for all p. <Vh5`-J  
    % SEu:31k{o  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 C=K{;.  
    %   Zernike functions (order N<=7).  In some disciplines it is {65Y Tt%  
    %   traditional to label the first 36 functions using a single mode S,'ekWVD  
    %   number P instead of separate numbers for the order N and azimuthal " :[;}f;  
    %   frequency M. 3qV~C{ S  
    % @QYCoEU8J  
    %   Example: %Zfh6Bl\X  
    % b; vVlIG  
    %       % Display the first 16 Zernike functions n~jW  
    %       x = -1:0.01:1; JHg;2xm"<K  
    %       [X,Y] = meshgrid(x,x); 6Y.k<oem  
    %       [theta,r] = cart2pol(X,Y); hr&UD|E=  
    %       idx = r<=1; P;X0L{u0H  
    %       p = 0:15; %rl<%%T#.M  
    %       z = nan(size(X)); 3+ C;zDKa  
    %       y = zernfun2(p,r(idx),theta(idx)); ={p<|8`"  
    %       figure('Units','normalized') !af;5F  
    %       for k = 1:length(p) }`2+`w%uZ  
    %           z(idx) = y(:,k); tqt~F2u  
    %           subplot(4,4,k) >*|Eyv_  
    %           pcolor(x,x,z), shading interp %X\Rfn0J"  
    %           set(gca,'XTick',[],'YTick',[]) }\A 0g}  
    %           axis square e-xT.RnQ  
    %           title(['Z_{' num2str(p(k)) '}']) O|9Nl*rXz  
    %       end xkkG#n)  
    % 96gaun J  
    %   See also ZERNPOL, ZERNFUN. O!F"w !5@  
    ZY6%%7?1  
    %   Paul Fricker 11/13/2006 B>"-8#B[4  
    ;ae6h [  
    Fv nf;']q  
    % Check and prepare the inputs:  ?f5||^7  
    % ----------------------------- 6hFs{P7  
    if min(size(p))~=1 hig t(u  
        error('zernfun2:Pvector','Input P must be vector.') UU#$Kt*frR  
    end )Dcee@/7S  
    xKT;1(Mk  
    if any(p)>35 O=u.J8S2  
        error('zernfun2:P36', ... )%: W;H  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ((BdT:T\_  
               '(P = 0 to 35).']) u)D!RhV&  
    end  |>Pv2  
    LG?b]'#  
    % Get the order and frequency corresonding to the function number: !_&;#j](  
    % ---------------------------------------------------------------- V%^d~^m,H  
    p = p(:); h}+Gz={Q^  
    n = ceil((-3+sqrt(9+8*p))/2); Wx8 cK=  
    m = 2*p - n.*(n+2); Ax4;[K\Q  
    "nNT9 K|  
    % Pass the inputs to the function ZERNFUN: S! .N3ezn  
    % ---------------------------------------- E_xk8X~  
    switch nargin fKs3H?|  
        case 3 G<~P||Lu^  
            z = zernfun(n,m,r,theta); GlT/JZ9  
        case 4 /DSy/p0%  
            z = zernfun(n,m,r,theta,nflag); 7l'1  
        otherwise ,nw5 M.D_  
            error('zernfun2:nargin','Incorrect number of inputs.') <.@w%rvG  
    end gg Hl{cl)  
    1fh6A`c  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) g^k=z:n3,  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. zOL*XZ0c  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of J]&nZud`  
    %   order N and frequency M, evaluated at R.  N is a vector of 4 ..V  
    %   positive integers (including 0), and M is a vector with the \&s$?r  
    %   same number of elements as N.  Each element k of M must be a S`[r]msw  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) Wp= &nh  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is wq#'o9s,  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ;BEX|w xn  
    %   with one column for every (N,M) pair, and one row for every ,j[1!*Z_[  
    %   element in R. .wuRT>4G)G  
    % ">R`S<W  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- CSE!Abg  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is !p 70g0+  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to MPJ0>Ly  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 S%w67sGl4n  
    %   for all [n,m]. I8:G:s:  
    % zXeBUbVi  
    %   The radial Zernike polynomials are the radial portion of the |Fzt| \  
    %   Zernike functions, which are an orthogonal basis on the unit I,?!NzB  
    %   circle.  The series representation of the radial Zernike elb}] +  
    %   polynomials is zm^ 5WH  
    % _jk+$`[9PL  
    %          (n-m)/2 l8N5}!N  
    %            __ ^|%7}=e  
    %    m      \       s                                          n-2s j(Tk6S  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r  W o$UV  
    %    n      s=0 q%Lw#f  
    % "I45=nf  
    %   The following table shows the first 12 polynomials. >  ,P,{"  
    % x@<!#d+  
    %       n    m    Zernike polynomial    Normalization s Qa9M  
    %       --------------------------------------------- 7Y*Q)DDy  
    %       0    0    1                        sqrt(2) iSj.lW  
    %       1    1    r                           2 x/#.%Ga#T  
    %       2    0    2*r^2 - 1                sqrt(6) v7D3aWoe  
    %       2    2    r^2                      sqrt(6) _v=zFpR  
    %       3    1    3*r^3 - 2*r              sqrt(8) >tV:QP]Y  
    %       3    3    r^3                      sqrt(8) U{-[lpd  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ltXGm)+  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) T`|>oX  
    %       4    4    r^4                      sqrt(10) q3Y49d  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 7o`pNcabtz  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) coc :$Sr%  
    %       5    5    r^5                      sqrt(12) ' ui`EL%  
    %       --------------------------------------------- [E/^bM+  
    % 0,[- 4m  
    %   Example: C~VyM1inD  
    % d Y]i AJ  
    %       % Display three example Zernike radial polynomials x9H qc9q  
    %       r = 0:0.01:1; Y]HtO^T2  
    %       n = [3 2 5]; D$bJs O  
    %       m = [1 2 1]; bn"z&g   
    %       z = zernpol(n,m,r); * uZ'MS  
    %       figure M9sB2Ips<  
    %       plot(r,z) $*dY f  
    %       grid on t)*A#  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') }zV#?;}  
    % Dlu]4n[LB  
    %   See also ZERNFUN, ZERNFUN2. Q02:qn?T  
    U7Pn $l2!  
    % A note on the algorithm. fm^)u"  
    % ------------------------ 5%(xZ  6  
    % The radial Zernike polynomials are computed using the series I}7= \S/@  
    % representation shown in the Help section above. For many special d0@czNWIC  
    % functions, direct evaluation using the series representation can bR? $a+a)  
    % produce poor numerical results (floating point errors), because f%TP>)jag!  
    % the summation often involves computing small differences between rwep e5  
    % large successive terms in the series. (In such cases, the functions ZlsdO.G  
    % are often evaluated using alternative methods such as recurrence l j*J|%~  
    % relations: see the Legendre functions, for example). For the Zernike d9uT*5f  
    % polynomials, however, this problem does not arise, because the aQhr$aH  
    % polynomials are evaluated over the finite domain r = (0,1), and U})Z4>[bvt  
    % because the coefficients for a given polynomial are generally all N&+DhKw  
    % of similar magnitude. de?lO ;8  
    % ]$96#}7N  
    % ZERNPOL has been written using a vectorized implementation: multiple 9]TvL h3  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Z8_Q Kw>  
    % values can be passed as inputs) for a vector of points R.  To achieve yzmwNsu  
    % this vectorization most efficiently, the algorithm in ZERNPOL ^5![tTJ  
    % involves pre-determining all the powers p of R that are required to f:K`M W  
    % compute the outputs, and then compiling the {R^p} into a single H) &pay  
    % matrix.  This avoids any redundant computation of the R^p, and ;Cqjg.wkB  
    % minimizes the sizes of certain intermediate variables. vxC,8Z  
    % 66~]7w  
    %   Paul Fricker 11/13/2006 O1K~]Nt  
    j{a3AEmps  
    *]_GFixi  
    % Check and prepare the inputs: )|zna{g\  
    % ----------------------------- #T8$NZA  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7Y(ySW  
        error('zernpol:NMvectors','N and M must be vectors.') (KxL*gB  
    end zE NlL  
    x2g P, p-  
    if length(n)~=length(m) 13T0"}  
        error('zernpol:NMlength','N and M must be the same length.') Y4|g^>{<ni  
    end IW'2+EGc  
    &$</|F)y  
    n = n(:); f~D> *<L4-  
    m = m(:); N~An}QX|  
    length_n = length(n); ZXj;ymC'  
    2x*C1   
    if any(mod(n-m,2)) /y"Y o  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') t7p`A8&  
    end Soa5TM  
    [oQ&}3\XJ  
    if any(m<0) |cBpX+D  
        error('zernpol:Mpositive','All M must be positive.') W}5H'D  
    end qm)KO 4  
    sR. ecs+  
    if any(m>n) zz4A,XrD  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ep<2u x  
    end afMIqQ?  
    <IBzh_  
    if any( r>1 | r<0 ) :bA@ u>  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') n'#(iW)f  
    end 9)qjW&`  
    \k|ZbCWg  
    if ~any(size(r)==1) e1RtoNF^  
        error('zernpol:Rvector','R must be a vector.') %8V/QimHU  
    end Z<#beT6  
    THmX=K4=?  
    r = r(:); 4JHFn [%  
    length_r = length(r); \8!&X cA  
    ZC1U  
    if nargin==4 ?eb2T`\0Q  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); M9@#W"  
        if ~isnorm ;Db89Nc$  
            error('zernpol:normalization','Unrecognized normalization flag.') snBC +`-  
        end 6#k Ap+g7  
    else %L./U$  
        isnorm = false; pAV}hB  
    end Dc[Qu? ]LM  
    1b'1vp  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t _Q/v  
    % Compute the Zernike Polynomials y\Z-x  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i%W,Y8\uf*  
    t$=0  C  
    % Determine the required powers of r: AX%9k  
    % ----------------------------------- T] nZ3EZ  
    rpowers = []; e3wFi,/@  
    for j = 1:length(n) )G6]r$M>o0  
        rpowers = [rpowers m(j):2:n(j)]; h@z(yB j:0  
    end |Js96>B:  
    rpowers = unique(rpowers); 'Z7oPq6  
    'B"kUh%3$5  
    % Pre-compute the values of r raised to the required powers, t?v0ylN  
    % and compile them in a matrix: VYhZ0;' '  
    % ----------------------------- IrXC/?^h  
    if rpowers(1)==0 AO'B p5:Q  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ~ S R:,R  
        rpowern = cat(2,rpowern{:}); %f@VOSs  
        rpowern = [ones(length_r,1) rpowern]; 7;n'4LIa9  
    else ;1cX|N=  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); "$#x+|PyC  
        rpowern = cat(2,rpowern{:}); #4''Cs  
    end 79a{Zwdd9j  
    j~"X`:=  
    % Compute the values of the polynomials: $Tq-<FbM)  
    % -------------------------------------- "0g1'az}  
    z = zeros(length_r,length_n); nrA}36E  
    for j = 1:length_n Us YH#?|O  
        s = 0:(n(j)-m(j))/2; 9h$-:y3  
        pows = n(j):-2:m(j); 9r7QE&.  
        for k = length(s):-1:1  ?S0VtHQ  
            p = (1-2*mod(s(k),2))* ... _qmB PUx  
                       prod(2:(n(j)-s(k)))/          ... Gz!72H  
                       prod(2:s(k))/                 ... yaA9* k  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... g?i_10Xlp  
                       prod(2:((n(j)+m(j))/2-s(k))); -b(:kAwStk  
            idx = (pows(k)==rpowers); 5gV8=Ml"V  
            z(:,j) = z(:,j) + p*rpowern(:,idx); .d9VV&  
        end i[^?24~ c  
         _q=$L eO5  
        if isnorm 7 s5?^^  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); G_[|N>  
        end eF}Q8]da  
    end Qkk~{OuC  
    >9=Y(`  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  I \ vu?$w  
    Ms^Y:,;Hi  
    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。 YfxZ<  
    Am  kHVg  
    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)