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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 vg5_@7  
    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! CGzu(@dd\  
     
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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 j}eb _K+I  
    function z = zernfun(n,m,r,theta,nflag) @)uV Fw"\  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 26rg-?;V^  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N x lS*9>Ij  
    %   and angular frequency M, evaluated at positions (R,THETA) on the wCB*v<*  
    %   unit circle.  N is a vector of positive integers (including 0), and z'_Fg0kR{  
    %   M is a vector with the same number of elements as N.  Each element ur\6~'l4  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) nYj rEy)Q  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, HDhISPg  
    %   and THETA is a vector of angles.  R and THETA must have the same YE{ [f@i0  
    %   length.  The output Z is a matrix with one column for every (N,M) fk5'v   
    %   pair, and one row for every (R,THETA) pair. Td|u@l4B  
    % P,{Q k~iu  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike )6C+0b*  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), $M 8& &M  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 'R79,)|;[  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, eKvr1m- -  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized GDL/5m#  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2URGd#{VQ  
    % %S^hqC  
    %   The Zernike functions are an orthogonal basis on the unit circle. &sWr)>vs  
    %   They are used in disciplines such as astronomy, optics, and 8m?(* [[  
    %   optometry to describe functions on a circular domain. A~bSB n: '  
    % LJGpa )(  
    %   The following table lists the first 15 Zernike functions. k.ou$mIY  
    % lx%c&~.DiB  
    %       n    m    Zernike function           Normalization U`ttT5;  
    %       -------------------------------------------------- I?3b}#&V9  
    %       0    0    1                                 1 T,pr&1]Lw  
    %       1    1    r * cos(theta)                    2 FfJp::|ddr  
    %       1   -1    r * sin(theta)                    2 B>^6tdz  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 'K?h6?#  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 0\tac/  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) \5r^D|Rp}  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 5-|!mSd   
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ?ZlXh51  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Fvl\.  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) z4:!*:.Asu  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10)  j%Au0k  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) X3:z=X&Zd  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 1_] X  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9&eY<'MgP  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) [<R haZz  
    %       -------------------------------------------------- L1SKOM$  
    % N>H@vt~  
    %   Example 1: sN[}B{+  
    % j ~-N2b6z  
    %       % Display the Zernike function Z(n=5,m=1) O2{["c e  
    %       x = -1:0.01:1; |IcW7(  
    %       [X,Y] = meshgrid(x,x); [gmov)\c  
    %       [theta,r] = cart2pol(X,Y); XHk"nbj  
    %       idx = r<=1; */;7Uv7  
    %       z = nan(size(X)); ttsR`R1.k  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); \Ggh 95y  
    %       figure jq,M1  
    %       pcolor(x,x,z), shading interp %} `` :  
    %       axis square, colorbar 9Y:I)^ek  
    %       title('Zernike function Z_5^1(r,\theta)') !/XNpQP  
    % @Lnv  
    %   Example 2: b w P=f.  
    % PlkZ)S7C  
    %       % Display the first 10 Zernike functions p3=Py7iz  
    %       x = -1:0.01:1; 1Toiqb/  
    %       [X,Y] = meshgrid(x,x); ~S8:xG+s  
    %       [theta,r] = cart2pol(X,Y); J?8Mo=UZz  
    %       idx = r<=1; O k`}\NZL  
    %       z = nan(size(X)); eP-|3$  
    %       n = [0  1  1  2  2  2  3  3  3  3]; o9eOp3w30  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; VHD+NY/  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; GTP'js  
    %       y = zernfun(n,m,r(idx),theta(idx)); GmH DG-  
    %       figure('Units','normalized') Z3S+")^  
    %       for k = 1:10 Z}+}X|  
    %           z(idx) = y(:,k); dR S:S_  
    %           subplot(4,7,Nplot(k)) _i05' _  
    %           pcolor(x,x,z), shading interp ^9Pr`\   
    %           set(gca,'XTick',[],'YTick',[]) w|9 >4  
    %           axis square 1+FVM\<&  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 6gV*G  
    %       end Dkz/hg:q  
    % PK[mf\G\  
    %   See also ZERNPOL, ZERNFUN2. su%(!XJQpg  
    B0@ Tz39=  
    %   Paul Fricker 11/13/2006 >w S'z]T9  
    W8d-4')|  
    eY<<Hld  
    % Check and prepare the inputs: S^rf^%  
    % ----------------------------- k1wIb']m]z  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ukiWNF/  
        error('zernfun:NMvectors','N and M must be vectors.') xb;{<~`71  
    end I1<WHq  
    dQ`Tt- n  
    if length(n)~=length(m) ;st0Ekni)  
        error('zernfun:NMlength','N and M must be the same length.') 7:jLZ!mgi  
    end #t ;`  
    d0(zB5'}  
    n = n(:); E5ce=$o  
    m = m(:); uM2@&)u  
    if any(mod(n-m,2)) k =! Q  
        error('zernfun:NMmultiplesof2', ... :?Ns>#6t  
              'All N and M must differ by multiples of 2 (including 0).') _?~%+Oz/  
    end n28JWkK8  
    Q~N,QMr)k&  
    if any(m>n) jWrU'X  
        error('zernfun:MlessthanN', ... w(oK   
              'Each M must be less than or equal to its corresponding N.') 5XKTb  
    end eAy,T<#  
    &QHJ%c  
    if any( r>1 | r<0 ) ~5KcbGD~  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 'UlVc2%{  
    end Uy?jVPL  
    meX2Y;  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) QG5WsuT  
        error('zernfun:RTHvector','R and THETA must be vectors.') U{2xgN J  
    end e*:K79 y  
    LF7- ?? '  
    r = r(:); (]]hSkE  
    theta = theta(:); c*IrZm  
    length_r = length(r); @:>"VP<(  
    if length_r~=length(theta) mnpk9x}m  
        error('zernfun:RTHlength', ... 8 .%0JJ.3  
              'The number of R- and THETA-values must be equal.') TLwxP"  
    end &;@L] o  
    +z;*r8d<X  
    % Check normalization: H>TO8;5(  
    % -------------------- !Z$d<~Mq q  
    if nargin==5 && ischar(nflag) UoT`/.  
        isnorm = strcmpi(nflag,'norm'); :HY$x  
        if ~isnorm :&BPKqKp  
            error('zernfun:normalization','Unrecognized normalization flag.') v=llg ^  
        end t13V>9to  
    else \g}]u(zg%  
        isnorm = false; y7HFmGM  
    end f?5>V   
    (?4%Xtul1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6GxLaI  
    % Compute the Zernike Polynomials (`.# n3{  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% noWF0+ %  
    _b&|0j:Ud  
    % Determine the required powers of r: $)M3fZ$#  
    % ----------------------------------- D+7xMT8pqH  
    m_abs = abs(m); 0*{(R#  
    rpowers = []; 9X*N k~}Y  
    for j = 1:length(n) F[ E'R.:  
        rpowers = [rpowers m_abs(j):2:n(j)]; tMl y*E  
    end  Vl_6nY;  
    rpowers = unique(rpowers); :>&q?xvA  
    tq L(H25z  
    % Pre-compute the values of r raised to the required powers, GHv6UIe&  
    % and compile them in a matrix: *6}M.`.-  
    % ----------------------------- t'44X  
    if rpowers(1)==0 5Kzt8Tv[  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5H3o?x   
        rpowern = cat(2,rpowern{:}); @|Pm%K`1  
        rpowern = [ones(length_r,1) rpowern]; 3%POTAw%  
    else !5*VBE\  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); dseI~}  
        rpowern = cat(2,rpowern{:}); j yHa}OT  
    end :;%Jm  
    Wb}-H-O  
    % Compute the values of the polynomials: aT0~C.vT  
    % -------------------------------------- _pdKcE\X  
    y = zeros(length_r,length(n)); @ m`C%7<  
    for j = 1:length(n) L.;b( bFe  
        s = 0:(n(j)-m_abs(j))/2; Myc-lCE  
        pows = n(j):-2:m_abs(j); h#0n2o#  
        for k = length(s):-1:1 SAm%$v z%M  
            p = (1-2*mod(s(k),2))* ... opa/+V3E4  
                       prod(2:(n(j)-s(k)))/              ... %1#\LRA(  
                       prod(2:s(k))/                     ... UQ0!tFx  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... mb*Yw 6q  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); s<k[<  
            idx = (pows(k)==rpowers); 0+\725DJ  
            y(:,j) = y(:,j) + p*rpowern(:,idx); B!'K20"gF  
        end do" m=y  
         lelmX  
        if isnorm kQ+y9@=/g  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Jn hdZa  
        end .tRm1&Qi  
    end m H:Un{,  
    % END: Compute the Zernike Polynomials 6))":<J  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kK5&?)3Y:  
    {K|?i9K  
    % Compute the Zernike functions: @GQe-04W`  
    % ------------------------------ DAw1S$dM  
    idx_pos = m>0; W|< c[S  
    idx_neg = m<0; +^7cS6"L  
    Dl>tF?=  
    z = y; 'o&d!  
    if any(idx_pos) \;7U:Y$v  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); h8V*$  
    end wUvE  
    if any(idx_neg) u|<?m A!  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); S_7]_GQ9  
    end l,|Llb  
    4X=VNORlU0  
    % 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)  ~&~4{  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ~ZafTCa;  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated !oyo_h  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive jt oS{B,  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, E$%v);u  
    %   and THETA is a vector of angles.  R and THETA must have the same Zv yZ5UA  
    %   length.  The output Z is a matrix with one column for every P-value, th{J;a  
    %   and one row for every (R,THETA) pair. 7nPg2K&  
    % 1ZF KLI`V  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike DT-.Gdb8  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) /3)\^Pof  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) RZW=z}T+H  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 Hec8pL  
    %   for all p. }8^qb5+!3  
    % -#I]/7^  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 vapC5,W"2-  
    %   Zernike functions (order N<=7).  In some disciplines it is w,P@@Q E  
    %   traditional to label the first 36 functions using a single mode 8YZ9  
    %   number P instead of separate numbers for the order N and azimuthal )Q1aAS3  
    %   frequency M. 6zM:p/  
    % ?EUg B\  
    %   Example: \zU<o~gs  
    % 0=,Nz  
    %       % Display the first 16 Zernike functions }/J"/ T  
    %       x = -1:0.01:1;  Ht.P670  
    %       [X,Y] = meshgrid(x,x); J>R $K  
    %       [theta,r] = cart2pol(X,Y); M XW1 :  
    %       idx = r<=1; ~Jf(M ^E  
    %       p = 0:15; `NhG|g  
    %       z = nan(size(X)); R.i ]6H!  
    %       y = zernfun2(p,r(idx),theta(idx)); e,C c.T\o  
    %       figure('Units','normalized') :G6CWE  
    %       for k = 1:length(p) 09 McUR@  
    %           z(idx) = y(:,k); >-c?+oy  
    %           subplot(4,4,k) pYcs4f!?p  
    %           pcolor(x,x,z), shading interp zgFL/a<  
    %           set(gca,'XTick',[],'YTick',[]) p+b/k2 Q  
    %           axis square QE/kR!r  
    %           title(['Z_{' num2str(p(k)) '}']) l|+$4 Nb2  
    %       end _L=-z*a\  
    % ;):;H?WS|A  
    %   See also ZERNPOL, ZERNFUN. a )lCp  
    `BZ|[ q3  
    %   Paul Fricker 11/13/2006 >}wFePl  
    n U=  
    BqJ|l7+  
    % Check and prepare the inputs: OM.-apzC  
    % ----------------------------- {_tq6ja-<  
    if min(size(p))~=1 b81^756  
        error('zernfun2:Pvector','Input P must be vector.') }alq~jY  
    end :UT \L2 q=  
    4wv0~T$;x  
    if any(p)>35 V9/2y9u  
        error('zernfun2:P36', ... ~NE`Ad.G  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... O-.G("  
               '(P = 0 to 35).']) qbQdx Kk  
    end h}|.#!C3  
    h>>~Bi  
    % Get the order and frequency corresonding to the function number: ADF<5#I  
    % ---------------------------------------------------------------- aAqM)T83  
    p = p(:); pj7a l;  
    n = ceil((-3+sqrt(9+8*p))/2); 7 2i&-`&4  
    m = 2*p - n.*(n+2); {|$kI`h,3-  
    s Y4w dG  
    % Pass the inputs to the function ZERNFUN: :CqR1_n%  
    % ---------------------------------------- ]Ky`AG`2~  
    switch nargin Z;NaIJiL-  
        case 3 yOD=Vc7i  
            z = zernfun(n,m,r,theta); k]AL\) &W  
        case 4 e{5O>RO  
            z = zernfun(n,m,r,theta,nflag); ^d# AU7V|  
        otherwise qOmL\'8  
            error('zernfun2:nargin','Incorrect number of inputs.') w^cQL%  
    end <8~c7kT'  
    <k3KCt  
    % 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) DD  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 2zbn8tO  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of K[?@nl?,z  
    %   order N and frequency M, evaluated at R.  N is a vector of YnU*MC}  
    %   positive integers (including 0), and M is a vector with the mm*nXJ  
    %   same number of elements as N.  Each element k of M must be a C2b<is=H:  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) H}&JrT95  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 0,&] 2YJ  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix :_F 8O  
    %   with one column for every (N,M) pair, and one row for every I\-M`^@  
    %   element in R. O`!XW8  
    % xY.?OHgG/  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 9:3`LY3wW  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is (]?M=?0\  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to JbitRV@a  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1  `2\:b^h  
    %   for all [n,m]. 6~>h;wC  
    % hW~UJ/$  
    %   The radial Zernike polynomials are the radial portion of the 4\ElMb[]  
    %   Zernike functions, which are an orthogonal basis on the unit j%ZBAk)}  
    %   circle.  The series representation of the radial Zernike ~$//4kES  
    %   polynomials is )Pc>+} D  
    % Q t>|TGz  
    %          (n-m)/2 q-@&n6PEOZ  
    %            __ X$~T*l0  
    %    m      \       s                                          n-2s wi%ls8F  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Gr}NgyT<!D  
    %    n      s=0 AuM}L&`i^  
    % A42!%>PB  
    %   The following table shows the first 12 polynomials. _d^d1Q}V  
    % \J#&]o)Y  
    %       n    m    Zernike polynomial    Normalization FI$ -."F  
    %       --------------------------------------------- ^xScVOdP  
    %       0    0    1                        sqrt(2) >[nR$8_J-l  
    %       1    1    r                           2 F|'u0JQ)$  
    %       2    0    2*r^2 - 1                sqrt(6) GJU9[  
    %       2    2    r^2                      sqrt(6) I#]pk!  
    %       3    1    3*r^3 - 2*r              sqrt(8) De{ZQg)  
    %       3    3    r^3                      sqrt(8) X f;R'a,$  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 0DnOO0Nc  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) =HV${+K=~  
    %       4    4    r^4                      sqrt(10) O~?d;.b  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) '-XO;{,-R  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) o9C# 5%9  
    %       5    5    r^5                      sqrt(12) c/j+aj0.v  
    %       --------------------------------------------- %2B1E( r%M  
    % |TQ4:P1T  
    %   Example: %<p/s;eu  
    % k=d0%} `M(  
    %       % Display three example Zernike radial polynomials $Xu/P5  
    %       r = 0:0.01:1; z yp3 +|  
    %       n = [3 2 5]; K3WaBcm  
    %       m = [1 2 1]; Ejf5M\o  
    %       z = zernpol(n,m,r); k!bJ&} Q(b  
    %       figure #, 1)@[  
    %       plot(r,z) D6fd(=t1Z  
    %       grid on *(5T?p[7  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 2}* 8( 32  
    % D dCcsYm,  
    %   See also ZERNFUN, ZERNFUN2. ~6O~Fth  
    &m\Uc  
    % A note on the algorithm. 5:5d=7WX  
    % ------------------------ %]4=D)Om  
    % The radial Zernike polynomials are computed using the series LCzeE7x  
    % representation shown in the Help section above. For many special '{.8tT ?tJ  
    % functions, direct evaluation using the series representation can 6hcK%0z  
    % produce poor numerical results (floating point errors), because Bga4kjfmk  
    % the summation often involves computing small differences between m6}_kzFz  
    % large successive terms in the series. (In such cases, the functions l[%=S!  
    % are often evaluated using alternative methods such as recurrence bR:hu}YS  
    % relations: see the Legendre functions, for example). For the Zernike %~>-nqS  
    % polynomials, however, this problem does not arise, because the p=kt+H&;  
    % polynomials are evaluated over the finite domain r = (0,1), and {9Ok^O  
    % because the coefficients for a given polynomial are generally all knpdECq&k  
    % of similar magnitude. BnDCK@+|Q  
    % 6V@_?a-K  
    % ZERNPOL has been written using a vectorized implementation: multiple F!4V!VWA}  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] hd(TKFL^y  
    % values can be passed as inputs) for a vector of points R.  To achieve a<E9@  
    % this vectorization most efficiently, the algorithm in ZERNPOL tTub W=H  
    % involves pre-determining all the powers p of R that are required to OQKc_z'"  
    % compute the outputs, and then compiling the {R^p} into a single ^|hVFM2  
    % matrix.  This avoids any redundant computation of the R^p, and >LH}A6dUC  
    % minimizes the sizes of certain intermediate variables. k\dPF@~Hvl  
    % / ;[x3}[  
    %   Paul Fricker 11/13/2006 23,pVo  
    xD~r Q$6sI  
    BgDWl{pm  
    % Check and prepare the inputs: cnw+^8  
    % ----------------------------- gf9U<J#&C  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Je2&7uR0  
        error('zernpol:NMvectors','N and M must be vectors.') `CBXz!v!O  
    end L 8;H_:~_'  
    $ e,r>tgD  
    if length(n)~=length(m) YTTi j|(  
        error('zernpol:NMlength','N and M must be the same length.') nII#uI /!q  
    end 02NVdpo[wU  
    <r>Sj /w<D  
    n = n(:); G%zJ4W%  
    m = m(:); K)+]as  
    length_n = length(n); \DBEs02  
    q"DHMZB  
    if any(mod(n-m,2)) 19pFNg'kA  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ,`k6 @4  
    end v]H9`s#,  
    YU)%-V\  
    if any(m<0) N^PkSf[)h5  
        error('zernpol:Mpositive','All M must be positive.') s1#A0%gx  
    end I3'UrKKO  
    jt9- v-  
    if any(m>n) {cO8q }L  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') '$'a .q1q9  
    end "O|.e`C%^  
    SyT{k\[  
    if any( r>1 | r<0 ) G!G:YVWXP  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ~Lg ;7i1L  
    end uqa pj("  
    W\5 -Yg(@  
    if ~any(size(r)==1) G`Z<a  
        error('zernpol:Rvector','R must be a vector.') wPu.hVz  
    end %X}vuE[[UC  
    v,z~#$T&  
    r = r(:); KhX)maQ  
    length_r = length(r); =n_z`I  
    AkdO:hVtG  
    if nargin==4 ~gOZ\jm}  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); x>tm[k  
        if ~isnorm ~-uf%=  
            error('zernpol:normalization','Unrecognized normalization flag.') c#1kg@q@  
        end ^8]7  
    else Am*lx  
        isnorm = false; LHu  
    end ?D_iib7  
    %hw4IcWJ|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LpSF*xm  
    % Compute the Zernike Polynomials |%fNLUJ)  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5} ur,0{  
    XP:fL NpQ  
    % Determine the required powers of r: @]8flb )T  
    % ----------------------------------- Bt5 P][<  
    rpowers = []; $>uUn3hSx\  
    for j = 1:length(n) &(l.jgqg&  
        rpowers = [rpowers m(j):2:n(j)]; < 3*q) VT  
    end PApr8Xe  
    rpowers = unique(rpowers); 6#K.n&=*  
    e6tU8`z  
    % Pre-compute the values of r raised to the required powers,  "^BA5  
    % and compile them in a matrix: >.9V`m|  
    % ----------------------------- R^sgafGl=  
    if rpowers(1)==0 { D|ST2:E  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); gd#?rc*f<3  
        rpowern = cat(2,rpowern{:}); H\E%.QIx  
        rpowern = [ones(length_r,1) rpowern]; &m[}%e%~0  
    else ' S,g3  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); -MsL>F.]  
        rpowern = cat(2,rpowern{:}); ;mz#$"(  
    end j)@{_tv6;  
    >SziRm>Y7  
    % Compute the values of the polynomials: ZGI<L  
    % -------------------------------------- ) R5j?6}xF  
    z = zeros(length_r,length_n); tlg}"lY  
    for j = 1:length_n .f;@O qU  
        s = 0:(n(j)-m(j))/2; :pz@'J  
        pows = n(j):-2:m(j); #Cy3x-!  
        for k = length(s):-1:1 |22vNt_  
            p = (1-2*mod(s(k),2))* ... qdKqc,R1{  
                       prod(2:(n(j)-s(k)))/          ... h05FR[</  
                       prod(2:s(k))/                 ... 61&{I>~1  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Lc[TIX  
                       prod(2:((n(j)+m(j))/2-s(k))); I*(kv7(c0  
            idx = (pows(k)==rpowers); yD.(j*bMK;  
            z(:,j) = z(:,j) + p*rpowern(:,idx); G,^ ?qbHg  
        end @p^EXc*|  
         _5(p=Zc  
        if isnorm h"Wpb}FT  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ,)J>8eV  
        end S%Ky+0  
    end 1=sL[I7<  
    u;1[_~  
    % 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)  In1n.oRFn^  
    f>4|>kS  
    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。 G%>[7]H  
    T{m) = (q  
    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)