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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 )v~]lk,o  
    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! s`RJl V  
     
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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 9X9zIh]JV  
    function z = zernfun(n,m,r,theta,nflag) jSp&mD*xv  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. U^BXCu1km  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N :b*`hWnQ  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Cf[F`pFM  
    %   unit circle.  N is a vector of positive integers (including 0), and ;<@6f@  
    %   M is a vector with the same number of elements as N.  Each element  O'|P|  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) `sy &dyM  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, OG7v'vmY  
    %   and THETA is a vector of angles.  R and THETA must have the same 5'Jh2r  
    %   length.  The output Z is a matrix with one column for every (N,M) O)%kl  
    %   pair, and one row for every (R,THETA) pair. ~PW}sN6ppG  
    % 7u5\#|yL  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike KGmc*Jwy  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 5|G3t`$pa  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral nvo1+W(%  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, #r)1<}_e#  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized gzCMJ<3!D  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. "4uUI_E9F;  
    % MI'l4<>u  
    %   The Zernike functions are an orthogonal basis on the unit circle. p6Dv;@)Yn  
    %   They are used in disciplines such as astronomy, optics, and qbq<O %g=  
    %   optometry to describe functions on a circular domain. uf'P9MA}>  
    % [j]J_S9jJ  
    %   The following table lists the first 15 Zernike functions. iz>y u[|  
    % y{Y+2}Dv/  
    %       n    m    Zernike function           Normalization J:Y|O-S!  
    %       -------------------------------------------------- .4re0:V  
    %       0    0    1                                 1 |\n)<r_  
    %       1    1    r * cos(theta)                    2 s8Ry}{  
    %       1   -1    r * sin(theta)                    2 W$Q)aA7  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ^Xy$is3  
    %       2    0    (2*r^2 - 1)                    sqrt(3) qvU$9cTY  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) j /dE6d  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) dF11Rj,~ 8  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) +<WRB\W  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ]n]uN~)9  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) &Dg)"Xji  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Y:!/4GF  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) wQ=yY$VP  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 1;:t~Y  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) T19rbL_  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) M|5]#2J_2  
    %       -------------------------------------------------- ?I2k6%a  
    % t#pqXY/;D  
    %   Example 1: -8Jl4F ,  
    % Z: lB:U'o  
    %       % Display the Zernike function Z(n=5,m=1) ]AZ\5C-J  
    %       x = -1:0.01:1; JdUz!=I  
    %       [X,Y] = meshgrid(x,x); {I9 N6BQ&  
    %       [theta,r] = cart2pol(X,Y); N~S[xS?  
    %       idx = r<=1; Uq]EJu  
    %       z = nan(size(X)); g t^]32$  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); MpIw^a3(r  
    %       figure mj~N]cxB  
    %       pcolor(x,x,z), shading interp Y = g>r]2  
    %       axis square, colorbar |IX`(  
    %       title('Zernike function Z_5^1(r,\theta)') | 2.e0Z]k  
    % /pIb@:Y1?  
    %   Example 2: ICl_ eb  
    % NM1cyZ  
    %       % Display the first 10 Zernike functions > 0Twr  
    %       x = -1:0.01:1; ua$k^m7m5  
    %       [X,Y] = meshgrid(x,x); p17|ld`  
    %       [theta,r] = cart2pol(X,Y); {GQ Aa  
    %       idx = r<=1; ~AC P%QM=  
    %       z = nan(size(X)); tFvgvx\:  
    %       n = [0  1  1  2  2  2  3  3  3  3]; KI Plb3oh  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; :,%J6Zh?  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 3Zaq#uA  
    %       y = zernfun(n,m,r(idx),theta(idx)); .YjrV+om1  
    %       figure('Units','normalized') WpJD=C%  
    %       for k = 1:10 RQo$iISwy  
    %           z(idx) = y(:,k); YV1a 3  
    %           subplot(4,7,Nplot(k)) iz9\D*or  
    %           pcolor(x,x,z), shading interp OC?Zw@  
    %           set(gca,'XTick',[],'YTick',[]) Sqdc1zC  
    %           axis square VA=#0w  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) + U+aWk  
    %       end LZUA+x(  
    % q?;*g@t  
    %   See also ZERNPOL, ZERNFUN2. Y/^[qD  
    i?a,^UM5n[  
    %   Paul Fricker 11/13/2006 sP6 ):h  
    %$ir a\ sM  
    @zr8%8n  
    % Check and prepare the inputs: '0CXHjZN  
    % ----------------------------- ^sT +5M^  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) !@^y)v  
        error('zernfun:NMvectors','N and M must be vectors.') %\X P:  
    end y$j1?7  
    5:*5j@/S  
    if length(n)~=length(m) `z3|M#r\;  
        error('zernfun:NMlength','N and M must be the same length.') f[JI/H>  
    end MfXt+c`r  
    tp1KP/2w[  
    n = n(:); Kf05<J!  
    m = m(:); ?JXBWB4  
    if any(mod(n-m,2)) C3 gZ6m  
        error('zernfun:NMmultiplesof2', ... #$rf-E5g-K  
              'All N and M must differ by multiples of 2 (including 0).') &\[Qm{lN  
    end 6P%<[Z  
    lFiq<3Nk  
    if any(m>n) ;f".'9 l^  
        error('zernfun:MlessthanN', ... < 72s7*Rv  
              'Each M must be less than or equal to its corresponding N.') F* 3G _V  
    end '^Pq(b~  
    wUru1_zjO  
    if any( r>1 | r<0 )  &7L~PZ  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') $:f.Krj  
    end ov\Ct%]  
    jo,6Aog|u  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 9nng}em>.  
        error('zernfun:RTHvector','R and THETA must be vectors.') z3^RUoGU  
    end WdTbt  
    $"Y3mD}?L  
    r = r(:); V.K70)]  
    theta = theta(:); l9_m>X~   
    length_r = length(r); W$z#ssr  
    if length_r~=length(theta) -!XrwQyk  
        error('zernfun:RTHlength', ... /J1S@-  
              'The number of R- and THETA-values must be equal.') H{j~ihq7  
    end ?JuX~{{. L  
    %$/=4f.j  
    % Check normalization: 6PiEa(  
    % -------------------- =:4 '  
    if nargin==5 && ischar(nflag) dzgs%qtK  
        isnorm = strcmpi(nflag,'norm'); zo_k\K`{@  
        if ~isnorm kk 8R  
            error('zernfun:normalization','Unrecognized normalization flag.') 2yl6~(JC+  
        end m5e\rMN~>\  
    else Z -pyFK\  
        isnorm = false; +DicP"~*  
    end rU; g0'4e  
    d>^~9X  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AU0$A403  
    % Compute the Zernike Polynomials S#P+B*v  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% utq.r_  
    (YAI,Xnw  
    % Determine the required powers of r: +7Sf8tg\  
    % ----------------------------------- M{*kB2jr  
    m_abs = abs(m); lN);~|IOv7  
    rpowers = []; U^B"|lc:[  
    for j = 1:length(n) '/Cg*o/  
        rpowers = [rpowers m_abs(j):2:n(j)]; s0gJ f[  
    end w|&,I4["  
    rpowers = unique(rpowers); B`LD7]ew  
    vz6SCGg,  
    % Pre-compute the values of r raised to the required powers, HvAE,0N  
    % and compile them in a matrix: kVWGDI$~  
    % ----------------------------- t G]N*%@  
    if rpowers(1)==0 P\.WXe#j  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); O-i4_YdVt  
        rpowern = cat(2,rpowern{:}); <"N:rn{Qq  
        rpowern = [ones(length_r,1) rpowern]; U%Dit  
    else $RpF xi  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); DD2adu^  
        rpowern = cat(2,rpowern{:}); /^d. &@*  
    end \.5F](:  
    :}^Rs9 '  
    % Compute the values of the polynomials: b([:,T7  
    % -------------------------------------- T0g0jr{  
    y = zeros(length_r,length(n)); ot^q}fRX  
    for j = 1:length(n) R_maNfS]Z  
        s = 0:(n(j)-m_abs(j))/2; |Es0[cU  
        pows = n(j):-2:m_abs(j); z|uOJ0uK  
        for k = length(s):-1:1 5xhM0 (  
            p = (1-2*mod(s(k),2))* ... j(&GVy^;?  
                       prod(2:(n(j)-s(k)))/              ... P2O\!'aEh  
                       prod(2:s(k))/                     ... O97VdNT8  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... `a-Bji?  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); wc"9A~  
            idx = (pows(k)==rpowers); j]AekI4I  
            y(:,j) = y(:,j) + p*rpowern(:,idx);  64SW  
        end ^#2xQ5h  
         '[%jjUU  
        if isnorm |0lLl^zp  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); nQ|GqU\oA  
        end wXz\NGW  
    end |ribWCv0  
    % END: Compute the Zernike Polynomials 5Wo5 n7o  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;;M"hI3@  
    \Ps5H5Qk;  
    % Compute the Zernike functions: tbg*_ZQO u  
    % ------------------------------ ^ s=*J=k  
    idx_pos = m>0; 2_ wv C  
    idx_neg = m<0; w:v=se"U  
    ka/nQ~_#<  
    z = y; { E^U6@  
    if any(idx_pos) Vu=] O/ =P  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); _FT6]I0  
    end h 5Hr[E1  
    if any(idx_neg) 7"#f!.E  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); a%v>eXc  
    end D'<$ g  
    "3wv:BL  
    % 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) PYu$1o9+N  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. v$;@0t:;#  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 9)Ly}Kzx  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 1 =^  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, /9Z!p  
    %   and THETA is a vector of angles.  R and THETA must have the same NZ+7p{&AN  
    %   length.  The output Z is a matrix with one column for every P-value, NCn`}QP  
    %   and one row for every (R,THETA) pair. 68p R:  
    % ] opto  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike $~G@   
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 0yaMe@&,  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) D#ZPq,f  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 sBU_Ft  
    %   for all p. V 9Hl1\j^  
    % "W5rx8a  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 !9D1 Fa  
    %   Zernike functions (order N<=7).  In some disciplines it is SB/3jH  
    %   traditional to label the first 36 functions using a single mode z0 \N{rP&  
    %   number P instead of separate numbers for the order N and azimuthal I|T7+{5z  
    %   frequency M. -=a[J;'q  
    % YQ7@D]#  
    %   Example: V'I T1~  
    % e1UITjy  
    %       % Display the first 16 Zernike functions </=3g>9Z  
    %       x = -1:0.01:1; '60//"9>k/  
    %       [X,Y] = meshgrid(x,x); xCq'[9oU  
    %       [theta,r] = cart2pol(X,Y); X$^JAZ09  
    %       idx = r<=1; #ua^{OrC/  
    %       p = 0:15; XXm'6xD-  
    %       z = nan(size(X)); ,rai%T/rL  
    %       y = zernfun2(p,r(idx),theta(idx)); |z*>ixK  
    %       figure('Units','normalized') mf9hFy* <4  
    %       for k = 1:length(p) WqQU@sA  
    %           z(idx) = y(:,k); Ha)np  
    %           subplot(4,4,k) lE'wfUb  
    %           pcolor(x,x,z), shading interp qnXTNs ?b  
    %           set(gca,'XTick',[],'YTick',[]) En\Z#0,V  
    %           axis square 6.19g'{sB  
    %           title(['Z_{' num2str(p(k)) '}']) Du!._  
    %       end (A4&k{C_  
    % ZLrHZhP-+  
    %   See also ZERNPOL, ZERNFUN. 2M&$Wuu.q  
    Mq'IkSt'  
    %   Paul Fricker 11/13/2006 ]]PE#DDg  
    V{ |[oIp  
    " #v%36U  
    % Check and prepare the inputs: x*q35K^PE  
    % ----------------------------- 3AeH7g4<  
    if min(size(p))~=1 J^:n* C  
        error('zernfun2:Pvector','Input P must be vector.') U=QA  e  
    end (NaK3_  
    E}YI WTX  
    if any(p)>35 hZU 1O  
        error('zernfun2:P36', ... M1{(OY(G  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... y/tSGkMv  
               '(P = 0 to 35).']) 12OlrU  
    end oKa>.e7.  
    ;==j|/ERe  
    % Get the order and frequency corresonding to the function number: vaB!R 0  
    % ---------------------------------------------------------------- RRzP* A%=  
    p = p(:); ;s_"{f`Y6  
    n = ceil((-3+sqrt(9+8*p))/2); 5v|EAjB6o  
    m = 2*p - n.*(n+2); MGaiTN^_<  
    K*+6`z#fMF  
    % Pass the inputs to the function ZERNFUN: L!y"d!6C  
    % ---------------------------------------- ;4kT?3$l  
    switch nargin W.[BPR  
        case 3 t9` Ed>a  
            z = zernfun(n,m,r,theta); ^P$7A]!  
        case 4 moG~S]  
            z = zernfun(n,m,r,theta,nflag); X"<|Z]w  
        otherwise $5A^'q  
            error('zernfun2:nargin','Incorrect number of inputs.') Mudrg[@ `  
    end q qvF-mDN  
    S>t>6&A  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) 6+PGwCS  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 0roCP=;  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ] .5O X84  
    %   order N and frequency M, evaluated at R.  N is a vector of '9q6aM/&  
    %   positive integers (including 0), and M is a vector with the m UgRm]  
    %   same number of elements as N.  Each element k of M must be a z_l. V/G)  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 'n4u-pM(nB  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is q-IWRb0j%a  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix VMHC/jlX@r  
    %   with one column for every (N,M) pair, and one row for every :-Wv>V\t  
    %   element in R. <[hz?:G"$  
    % GYoseqZM  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- zH=hI Vc  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is o , LK[Q  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to H[nz]s  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 t.U{Bu P  
    %   for all [n,m]. j-32S!  
    % _9kIRmT{  
    %   The radial Zernike polynomials are the radial portion of the VQ<5%+  
    %   Zernike functions, which are an orthogonal basis on the unit HcO5?{2  
    %   circle.  The series representation of the radial Zernike f:vD`Fz1  
    %   polynomials is  yekRwo|  
    % Euu ,mleM  
    %          (n-m)/2 #T"64%dX  
    %            __ 3cThu43c  
    %    m      \       s                                          n-2s !01i%W'  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r euZ I`*0  
    %    n      s=0 q[c Etp28h  
    % {D,RU8&  
    %   The following table shows the first 12 polynomials. $?f]ZyZr.  
    % X9~p4ys9{  
    %       n    m    Zernike polynomial    Normalization =]b9X7}  
    %       --------------------------------------------- >EIrw$V$  
    %       0    0    1                        sqrt(2) )4ncutb  
    %       1    1    r                           2 wd+O5Lr.R  
    %       2    0    2*r^2 - 1                sqrt(6) B.K4!/cF  
    %       2    2    r^2                      sqrt(6) *#h;c1aP  
    %       3    1    3*r^3 - 2*r              sqrt(8) 2.qpt'p[  
    %       3    3    r^3                      sqrt(8) kqf8=y  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) zlkWU  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) j38 6gL  
    %       4    4    r^4                      sqrt(10) ]l&'k23~p  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) =4>@8=JA  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) r*Z_+a8  
    %       5    5    r^5                      sqrt(12) iZC`z }  
    %       --------------------------------------------- Q|//Z  
    % !*bdG(pK  
    %   Example: qTy v.#{y  
    % 4J8Dh;a`  
    %       % Display three example Zernike radial polynomials <kbnu7?a*  
    %       r = 0:0.01:1; 'tuBuYD\  
    %       n = [3 2 5]; Yxq j -   
    %       m = [1 2 1]; aAcKwCGq\  
    %       z = zernpol(n,m,r); ZIW7_Y>_  
    %       figure ?O7iK<5N  
    %       plot(r,z) **HrWM%?8o  
    %       grid on gh61H:tkR  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') Z s73 ad  
    % 8&?p  
    %   See also ZERNFUN, ZERNFUN2. {(0Id!  
    XtzOFx/  
    % A note on the algorithm. {aIZFe}B  
    % ------------------------ qc&jd  
    % The radial Zernike polynomials are computed using the series w%2ziwgh  
    % representation shown in the Help section above. For many special r=\P!`{5  
    % functions, direct evaluation using the series representation can }.t^D|  
    % produce poor numerical results (floating point errors), because vX.]hp5~  
    % the summation often involves computing small differences between y/\ZAtnLo  
    % large successive terms in the series. (In such cases, the functions 5Pqt_ZWy  
    % are often evaluated using alternative methods such as recurrence ' &N20w  
    % relations: see the Legendre functions, for example). For the Zernike Gh iHA9.  
    % polynomials, however, this problem does not arise, because the |D+p$^L  
    % polynomials are evaluated over the finite domain r = (0,1), and |0]YA  
    % because the coefficients for a given polynomial are generally all >#?iO]).  
    % of similar magnitude. kQ[Jo%YT?E  
    % ==`Pb  
    % ZERNPOL has been written using a vectorized implementation: multiple C *Xik9n  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] -R8/`M8GbD  
    % values can be passed as inputs) for a vector of points R.  To achieve s9 &)Fv-#V  
    % this vectorization most efficiently, the algorithm in ZERNPOL FE}s#n_Pd  
    % involves pre-determining all the powers p of R that are required to b~9`]+  
    % compute the outputs, and then compiling the {R^p} into a single xD^wTtT  
    % matrix.  This avoids any redundant computation of the R^p, and q|Fjm]AF  
    % minimizes the sizes of certain intermediate variables. MYu`c[$jZ  
    % {83C,C-  
    %   Paul Fricker 11/13/2006 :mn(0 R~  
    1VGpq-4*j  
    8=pv/o  
    % Check and prepare the inputs: (gD Q\t@3-  
    % ----------------------------- ph+M3q(z  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) =-m(\ }  
        error('zernpol:NMvectors','N and M must be vectors.') sX!3_ '-  
    end X&.:H~xS+  
    <OIUyZS  
    if length(n)~=length(m) XJ O[[G`  
        error('zernpol:NMlength','N and M must be the same length.') LS{bg.e  
    end W7$s5G,  
    cB<O.@  
    n = n(:);  d~sJ=)  
    m = m(:); X'xUwT|_+  
    length_n = length(n); A`IHP{aB  
    dB@FI  
    if any(mod(n-m,2)) OelU D/[$  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 2+o!o  
    end i`R(7Z  
    N6WPTUQ1mF  
    if any(m<0) z>7=k`x`:  
        error('zernpol:Mpositive','All M must be positive.') 'OG{*TDPu  
    end 1z3]PA!R  
    hRa\1Jt>a  
    if any(m>n) =pyVn_dg  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') k?< i*;7  
    end eVrnVPkM  
    +;YE)~R?  
    if any( r>1 | r<0 ) Jb7iBQ2%  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') zUJx&5/  
    end IeH^Wm&^  
    |^ ?`Q.|c$  
    if ~any(size(r)==1) Bpm,mp4g\#  
        error('zernpol:Rvector','R must be a vector.') k&yQ98H$K"  
    end  8>ESD}(  
    '^e0Ud,  
    r = r(:); (VfwLo>#  
    length_r = length(r); - Sx0qi'%  
    l},dQ4R  
    if nargin==4 hH#lTye  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); eU`;L [  
        if ~isnorm gAj0ukX5  
            error('zernpol:normalization','Unrecognized normalization flag.') &ycjSBK  
        end Kq$:\B)<c  
    else UE\%e9<l  
        isnorm = false; Wa(W&]  
    end bAN10U  
    3'.! +#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $|rCrak;  
    % Compute the Zernike Polynomials nT_*EC<.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tsck|;v  
    keae.6[  
    % Determine the required powers of r: u$apH{  
    % ----------------------------------- L7X7Zt8%  
    rpowers = []; "v@);\-V  
    for j = 1:length(n) d:A\<F  
        rpowers = [rpowers m(j):2:n(j)]; 9*BoYFw92*  
    end hJecCOA)'  
    rpowers = unique(rpowers); mluW=fE  
    }*0,>w>  
    % Pre-compute the values of r raised to the required powers, 3Wjq>\  
    % and compile them in a matrix: TViBCed40  
    % ----------------------------- 4s[`yV  
    if rpowers(1)==0 "(Mvl1^BT  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); tX@ 0:RX%  
        rpowern = cat(2,rpowern{:}); ixIh T  
        rpowern = [ones(length_r,1) rpowern]; Y,3z-Pa=@  
    else 5P-K *C&  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); c]GQU  
        rpowern = cat(2,rpowern{:}); {k kAqJ  
    end eVJ= .?r  
    O 5g}2  
    % Compute the values of the polynomials: J>><o:~@  
    % -------------------------------------- !>CE(;E>z  
    z = zeros(length_r,length_n); 2O?Vr" A  
    for j = 1:length_n YI L'YNH  
        s = 0:(n(j)-m(j))/2; )C'G2RV  
        pows = n(j):-2:m(j); vEu Ka<5  
        for k = length(s):-1:1 -L+\y\F  
            p = (1-2*mod(s(k),2))* ... E%-Pyg*  
                       prod(2:(n(j)-s(k)))/          ... Rbx97(wK  
                       prod(2:s(k))/                 ... or?0PEx\  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... >`<Ued  
                       prod(2:((n(j)+m(j))/2-s(k))); >.a+:   
            idx = (pows(k)==rpowers); v]B0!k&4.  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ^RYn8I  
        end "2*G$\  
         ]ev*m&O  
        if isnorm T+`xr0  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); Hlz'a1\:O]  
        end $Sp*)A]E`  
    end D2Vb{%(4.  
    w%>aR_G  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  "xvV'&lQ  
    _H^^2#wc/  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 7.lK$J:  
    s]nGpA[!  
    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)