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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 (O3nL.  
    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! 2\{zmc}G-0  
     
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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 c-sfg>0^  
    function z = zernfun(n,m,r,theta,nflag) c7H^$_^=  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. YGNP53CU  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N `Urhy#LC  
    %   and angular frequency M, evaluated at positions (R,THETA) on the t%8BK>AHvw  
    %   unit circle.  N is a vector of positive integers (including 0), and wUJcmM;  
    %   M is a vector with the same number of elements as N.  Each element q!@4~plz  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) d&>^&>?$zh  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, "\yT7?},  
    %   and THETA is a vector of angles.  R and THETA must have the same 1< ?4\?j  
    %   length.  The output Z is a matrix with one column for every (N,M) R=\IEqqsi  
    %   pair, and one row for every (R,THETA) pair. 2&cT~ZX&'  
    % , W?VhO  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike "#g}ve,  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), n `Ac 3A  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ) )Za&S*<  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, JW&gJASGC  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized {_*yGK48n  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. E"IZ6)Q  
    % ~"A0Rs=  
    %   The Zernike functions are an orthogonal basis on the unit circle. c &c@M$  
    %   They are used in disciplines such as astronomy, optics, and 'Pbr v  
    %   optometry to describe functions on a circular domain. :k#HW6p  
    % 2~[juWbz  
    %   The following table lists the first 15 Zernike functions. uQzXfOq  
    % `WS&rmq&'  
    %       n    m    Zernike function           Normalization D2O~kN d  
    %       -------------------------------------------------- K (|}dl:  
    %       0    0    1                                 1 ;kKyksxlD  
    %       1    1    r * cos(theta)                    2 %a7$QF]  
    %       1   -1    r * sin(theta)                    2 k}rbim  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) F"mmLao  
    %       2    0    (2*r^2 - 1)                    sqrt(3) [#iz/q~}  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) N$tGQ@  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) cZ3v=ke^  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ia? c0xL  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Iga0 24KR  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) vih9 KBT  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 4^d?D!j  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) : rVnc =k  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) \{D" !e  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) zT{ VE+=  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) !5N.B|N t  
    %       -------------------------------------------------- )U# K  
    % s#GLJl\E_P  
    %   Example 1: l+b~KU7~l  
    % {4PwLCy  
    %       % Display the Zernike function Z(n=5,m=1) r mOj  
    %       x = -1:0.01:1; 1 -b_~DF  
    %       [X,Y] = meshgrid(x,x); `GLx#=Q  
    %       [theta,r] = cart2pol(X,Y); eJX#@`K  
    %       idx = r<=1; t#yuOUg  
    %       z = nan(size(X)); QsW/X0YBv  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); jb)ZLA;L_c  
    %       figure X wtqi@zlE  
    %       pcolor(x,x,z), shading interp )M^ gT}M  
    %       axis square, colorbar H"F29Pu2  
    %       title('Zernike function Z_5^1(r,\theta)') .S4u-  
    % 4&iCht =  
    %   Example 2: *K; ~!P  
    % +H2Qk4XFB  
    %       % Display the first 10 Zernike functions E(|>Ddv B&  
    %       x = -1:0.01:1; S8gs-gL#Og  
    %       [X,Y] = meshgrid(x,x); 6w77YTJ  
    %       [theta,r] = cart2pol(X,Y); eV~goj  
    %       idx = r<=1; i@'dH3-kO  
    %       z = nan(size(X)); W_ ZJ0GuE(  
    %       n = [0  1  1  2  2  2  3  3  3  3]; F:ELPs4"  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; wKHBAW[i]  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Ir]\|t  
    %       y = zernfun(n,m,r(idx),theta(idx)); `$NP> %J-  
    %       figure('Units','normalized') fc@A0Hf  
    %       for k = 1:10 B7%U_F|m  
    %           z(idx) = y(:,k); WEpoBP CL  
    %           subplot(4,7,Nplot(k)) M^I(OuRMeI  
    %           pcolor(x,x,z), shading interp [00m/fT6  
    %           set(gca,'XTick',[],'YTick',[]) D)Dr__x  
    %           axis square :hA#m[  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) yLcE X  
    %       end DTs;{c  
    % 0CvUc>Pj`"  
    %   See also ZERNPOL, ZERNFUN2. i6N',&jFU  
    {>;R?TG]$  
    %   Paul Fricker 11/13/2006 QSj]ZA  
    ItCv.yv35  
    92-I~ !d  
    % Check and prepare the inputs: Y^]rMK/;  
    % ----------------------------- h7@6T+#WoT  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) N uI9iU  
        error('zernfun:NMvectors','N and M must be vectors.') E)3NxmM#  
    end !o-@&q  
    'f|o{  
    if length(n)~=length(m) Dhv3jg;lq  
        error('zernfun:NMlength','N and M must be the same length.') We z 5N  
    end H']+L~j  
    |&jXp%4T  
    n = n(:); .8|X   
    m = m(:); Vz[C=_m  
    if any(mod(n-m,2)) 8EEuv-aeo  
        error('zernfun:NMmultiplesof2', ... "I TIhnE  
              'All N and M must differ by multiples of 2 (including 0).') qY#6SO`_iy  
    end )CyS#j#=  
    `,0}ZzaV&  
    if any(m>n) FgI3   
        error('zernfun:MlessthanN', ... {^\r`V p  
              'Each M must be less than or equal to its corresponding N.') bN88ua}k{  
    end j~QwV='S  
    ,2)6s\]/b  
    if any( r>1 | r<0 ) I O> yIU[  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') DeYV$W B  
    end E!AE4B1bd  
    $wU\Js`/S]  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) u-C)v*#L  
        error('zernfun:RTHvector','R and THETA must be vectors.') #D|p2L$  
    end [8*)8jP3  
    a}u Sm/S  
    r = r(:); l@:0e]8|o  
    theta = theta(:); [S W_C  
    length_r = length(r); s9d_GhT%-  
    if length_r~=length(theta) } d }lR  
        error('zernfun:RTHlength', ... hpJ-r  
              'The number of R- and THETA-values must be equal.') :j`s r  
    end D,ln)["xm  
     Mc}^LDX  
    % Check normalization: Tb-F]lg$  
    % -------------------- JMM W  
    if nargin==5 && ischar(nflag) MJrR[h]  
        isnorm = strcmpi(nflag,'norm'); Tac$LS\Q  
        if ~isnorm ,v&(YOd  
            error('zernfun:normalization','Unrecognized normalization flag.') ]0\MmAJRn  
        end 8KNZ](Dj  
    else 4H<lm*!^  
        isnorm = false; cFWc<55aX6  
    end V470C@  
    Qw)c$93  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% as_PoCoss  
    % Compute the Zernike Polynomials :,I:usW"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :a)u&g@G  
    9&ids!W~yx  
    % Determine the required powers of r: @ry_nKr9  
    % ----------------------------------- ?F;8Pa/  
    m_abs = abs(m); PiYxk+N  
    rpowers = []; .6'qoo_N  
    for j = 1:length(n) 6MkP |vr6  
        rpowers = [rpowers m_abs(j):2:n(j)]; B93+BwN>95  
    end K96<M);:g  
    rpowers = unique(rpowers); l/awS!Q/nF  
    0K2`-mL  
    % Pre-compute the values of r raised to the required powers, ,4oo=&  
    % and compile them in a matrix: 3%ZOKb"D*  
    % ----------------------------- ZQ0F$J)2~  
    if rpowers(1)==0 DDH:)=;z  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); '08=yqy4N  
        rpowern = cat(2,rpowern{:}); # Vha7  
        rpowern = [ones(length_r,1) rpowern]; '6Q =#:mc\  
    else Z)aUt Srf  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); z]9MM 2+  
        rpowern = cat(2,rpowern{:}); $p?aVO  
    end E+w<RNBmz  
    ]P?vdgEM&  
    % Compute the values of the polynomials: xK\d4 "  
    % -------------------------------------- xUistwq  
    y = zeros(length_r,length(n)); iW /}#  
    for j = 1:length(n) $ DSZO!pB  
        s = 0:(n(j)-m_abs(j))/2; ,nB5/Lx  
        pows = n(j):-2:m_abs(j); Per1IcN  
        for k = length(s):-1:1 & 9 ?\b7  
            p = (1-2*mod(s(k),2))* ... cpJ|w3x B  
                       prod(2:(n(j)-s(k)))/              ... A$:U'ZG_  
                       prod(2:s(k))/                     ... >&5DsV.B  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 0=E]cQwh  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 1PV'?tXp(  
            idx = (pows(k)==rpowers); s}% M4  
            y(:,j) = y(:,j) + p*rpowern(:,idx); >s?S+W[L  
        end `lt"[K<  
         2V;PYI  
        if isnorm :A'y+MnK<  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 7s{GbU\  
        end ?m? ::RH  
    end /CG"]!2 "  
    % END: Compute the Zernike Polynomials )f<z% :I+Z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }@+:\   
    exUu7& *:  
    % Compute the Zernike functions: *itUWpNhr  
    % ------------------------------ xx%j.zDI]  
    idx_pos = m>0; k{SAvKx=  
    idx_neg = m<0; -I,$_  
    ]F'e aR  
    z = y; 8C9-_Ng`  
    if any(idx_pos) @wNG{Stj  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); @'!SN\?W8  
    end D!-g&HBTC  
    if any(idx_neg) 8i#2d1O  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); xBi' X  
    end XXn67sF/  
    *]/zc1Q4M  
    % 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) ZyPVy  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. E GU 0)<  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Q%tXQP.r  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive ryUQU^v  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, a:IC)]j$_  
    %   and THETA is a vector of angles.  R and THETA must have the same f=gW]x7'R+  
    %   length.  The output Z is a matrix with one column for every P-value, $OkBg0  
    %   and one row for every (R,THETA) pair. RF4vtQC=  
    % CiLg]va   
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike x vl#w  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ZWU)\}}_R  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ]e>w }L(gV  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 FX`>J6l:X  
    %   for all p. gANuBWh8T  
    % Z|j>gq  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 *>'V1b4}  
    %   Zernike functions (order N<=7).  In some disciplines it is ?u=Fj_N_  
    %   traditional to label the first 36 functions using a single mode /saIs%(fU  
    %   number P instead of separate numbers for the order N and azimuthal '+!1Y o'G  
    %   frequency M. 6J6BF%  
    % I%X6T@P  
    %   Example: # 0Q]dO  
    % 6@ IXqKz  
    %       % Display the first 16 Zernike functions ju8q?Nyhs  
    %       x = -1:0.01:1; >xYpNtEs  
    %       [X,Y] = meshgrid(x,x); )<;Y-u.UW  
    %       [theta,r] = cart2pol(X,Y); KNpl:g3{<Q  
    %       idx = r<=1; J0\Fhe0'  
    %       p = 0:15; z] P SpUd  
    %       z = nan(size(X)); |[ k.ii6iO  
    %       y = zernfun2(p,r(idx),theta(idx)); `nv~NLkl  
    %       figure('Units','normalized') a#y;dK  
    %       for k = 1:length(p) q#ClnG*  
    %           z(idx) = y(:,k); LW'D?p#  
    %           subplot(4,4,k) T?soJ]A  
    %           pcolor(x,x,z), shading interp NAQAU *yP  
    %           set(gca,'XTick',[],'YTick',[]) Cc' 37~6~P  
    %           axis square fg!__Rdi  
    %           title(['Z_{' num2str(p(k)) '}']) ith 3 =`3  
    %       end ~tUl}  
    % ," Wr"  
    %   See also ZERNPOL, ZERNFUN. q?oP?cCw  
    x?p1 HUK  
    %   Paul Fricker 11/13/2006 st3l2Q  
    )=Z>#iH1  
    3<Zq ]jk?n  
    % Check and prepare the inputs: 7BjJhs  
    % ----------------------------- "9P>a=Y  
    if min(size(p))~=1 6?mibvK  
        error('zernfun2:Pvector','Input P must be vector.') C]eSizS.  
    end :W:K:lk  
    !N7s dY  
    if any(p)>35 &Gn 2tr  
        error('zernfun2:P36', ... t?ZI".>  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... M7a.8-!1  
               '(P = 0 to 35).']) o]` *M|  
    end ,o{9$H5{  
    S)k*?dQ##R  
    % Get the order and frequency corresonding to the function number: ?'#` nx(!  
    % ---------------------------------------------------------------- 3yY}04[9<  
    p = p(:); D},>mfzF  
    n = ceil((-3+sqrt(9+8*p))/2); D>@I+4{p  
    m = 2*p - n.*(n+2); +0%w ;'9z  
    tl4V7!U@^z  
    % Pass the inputs to the function ZERNFUN: 1N^[.=  
    % ---------------------------------------- -MO#]K3<  
    switch nargin `*["UER  
        case 3 pb?c$n$u*  
            z = zernfun(n,m,r,theta); Zq|I,l0+E  
        case 4 *vN-Vb^2i)  
            z = zernfun(n,m,r,theta,nflag); |zNX=mAV  
        otherwise ia~HQ$'+n  
            error('zernfun2:nargin','Incorrect number of inputs.') V>%rv'G8  
    end %~JJ.&  
    +L| ?~p`V  
    % 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) 2dcV"lY  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. [nG<[<0G;  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 9y8&9<#  
    %   order N and frequency M, evaluated at R.  N is a vector of 7Lc]HSZo,  
    %   positive integers (including 0), and M is a vector with the <X^@*79m  
    %   same number of elements as N.  Each element k of M must be a 4qbBc1,7y  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 4*#18<u5  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is UWJ8amA  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix B =T'5&  
    %   with one column for every (N,M) pair, and one row for every |t&>5HM  
    %   element in R. S_4?K)n #  
    % Ugt/rf5n  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- VUGmi]qd  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is $}q23  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to f#"J]p  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 "]x'PI 4J  
    %   for all [n,m]. JCzeXNY  
    % #PW9:_BE  
    %   The radial Zernike polynomials are the radial portion of the c(m<h+ 2VL  
    %   Zernike functions, which are an orthogonal basis on the unit !bx;Ta.  
    %   circle.  The series representation of the radial Zernike kGS;s B  
    %   polynomials is =tn)}Y.<e  
    % syj0.JD  
    %          (n-m)/2 o5O#vW2Il&  
    %            __ \ gGW8Q;  
    %    m      \       s                                          n-2s a=1@*ID  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r )5JFfp)#  
    %    n      s=0 2'\H\|  
    % aQcleTb  
    %   The following table shows the first 12 polynomials. ]t,BMu=%  
    % (9 GWbB?  
    %       n    m    Zernike polynomial    Normalization uc\Kg1{  
    %       --------------------------------------------- 7~ p@0)''  
    %       0    0    1                        sqrt(2) E\EsWb  
    %       1    1    r                           2 OU.6bmWy|  
    %       2    0    2*r^2 - 1                sqrt(6) ^j7Vt2-  
    %       2    2    r^2                      sqrt(6) }W8;=$jr  
    %       3    1    3*r^3 - 2*r              sqrt(8) nYSiS}?S .  
    %       3    3    r^3                      sqrt(8) cn3\kT*  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 3m)0z{n  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) gp?uHKsM  
    %       4    4    r^4                      sqrt(10) 2tEkj=fA-  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) EU;9 *W<  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) yu|8_<bq  
    %       5    5    r^5                      sqrt(12) :#ik. D  
    %       --------------------------------------------- L,`LN>  
    % k FD; i  
    %   Example: Yn Mvl  
    % "| g>'wM*  
    %       % Display three example Zernike radial polynomials E GS)b  
    %       r = 0:0.01:1; (OL4Ex']  
    %       n = [3 2 5]; T2W eE@o  
    %       m = [1 2 1]; j0aXyLNX  
    %       z = zernpol(n,m,r); m,w A:o$'  
    %       figure {9pZ)tB  
    %       plot(r,z) 5d^sA;c  
    %       grid on 9T9!kb  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') gO-  _  
    % ,PW'#U:  
    %   See also ZERNFUN, ZERNFUN2. iy!=6  
    2- h{N  
    % A note on the algorithm. gPO}d  
    % ------------------------  'KL0@l  
    % The radial Zernike polynomials are computed using the series JR21>;l#2  
    % representation shown in the Help section above. For many special @n /nH?L  
    % functions, direct evaluation using the series representation can I6av6t}  
    % produce poor numerical results (floating point errors), because ie95rZp  
    % the summation often involves computing small differences between 0i>5<ej,f  
    % large successive terms in the series. (In such cases, the functions ()?(I?II  
    % are often evaluated using alternative methods such as recurrence 4l'fCZhA}  
    % relations: see the Legendre functions, for example). For the Zernike f~R(D0@  
    % polynomials, however, this problem does not arise, because the 8/cX]J  
    % polynomials are evaluated over the finite domain r = (0,1), and F~W6Bp^W  
    % because the coefficients for a given polynomial are generally all fU}ub2_in  
    % of similar magnitude. [ l??A3G  
    % B dfwa  
    % ZERNPOL has been written using a vectorized implementation: multiple MJO-q $)c  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ltSU fI  
    % values can be passed as inputs) for a vector of points R.  To achieve 4k1xy##  
    % this vectorization most efficiently, the algorithm in ZERNPOL yx[/|nZDC4  
    % involves pre-determining all the powers p of R that are required to Qd{CMm x  
    % compute the outputs, and then compiling the {R^p} into a single AV]2 euyn  
    % matrix.  This avoids any redundant computation of the R^p, and U< fGGCw  
    % minimizes the sizes of certain intermediate variables. ec;o\erPG  
    % {dlXLx!B  
    %   Paul Fricker 11/13/2006 f'RX6$}\1X  
     |>^JRx  
    | YWD8 +  
    % Check and prepare the inputs: Ic<2QknmP  
    % ----------------------------- Dx?,=~W9  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) n( yn<  
        error('zernpol:NMvectors','N and M must be vectors.') a58H9w"u)  
    end uH^-R_tQ  
    &r /Mi%  
    if length(n)~=length(m) eo?bL$A[s  
        error('zernpol:NMlength','N and M must be the same length.') _|2:_N=   
    end vA{-{Q  
    Z5n1@a __  
    n = n(:); ?l{nk5,?-Y  
    m = m(:); t3_O H^  
    length_n = length(n); M|h3Wt~7  
    %sP*=5?vA  
    if any(mod(n-m,2)) 9cF[seE"0  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') (ZZ8L-s  
    end uGGt\.$]s  
    h438`  
    if any(m<0) \}b%E'+_T  
        error('zernpol:Mpositive','All M must be positive.') dZ@63a>>@  
    end YD6'#(  
    FW4<5~'  
    if any(m>n) 6nvz8f3*r]  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') C,r;VyW6BI  
    end Qw*|qGvy^  
    $6 f3F?y7  
    if any( r>1 | r<0 ) U iW>J  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') H7n>Vx:L-  
    end J*M>6Q.)  
    #;yZ  
    if ~any(size(r)==1) wi=v}R_  
        error('zernpol:Rvector','R must be a vector.') gwMNYMI  
    end P= NDS2  
    lL3U8}vn  
    r = r(:); "!^"[mX4  
    length_r = length(r); I\ob7X'Xu!  
    kDxFloK  
    if nargin==4 g) jYFfGfH  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); >kVz49j  
        if ~isnorm #X1ND  
            error('zernpol:normalization','Unrecognized normalization flag.') DTL.Bsc-.  
        end h2R::/2.  
    else ZFL~;_r  
        isnorm = false; #*Ctwl,T  
    end ;.980+i1  
    F JyT+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5 7c8xk[.2  
    % Compute the Zernike Polynomials 4tBYR9|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :vbW  
    e\L8oOk#r  
    % Determine the required powers of r: iYy1!\  
    % ----------------------------------- 26h21Z16q  
    rpowers = []; 7kE n \  
    for j = 1:length(n) 4kx N<]  
        rpowers = [rpowers m(j):2:n(j)]; I!K6o.|1  
    end KZf+MSq? B  
    rpowers = unique(rpowers); <LiPEo.R  
    lThB2/tV\  
    % Pre-compute the values of r raised to the required powers, 6'f;-2  
    % and compile them in a matrix: Q$"D]!G  
    % ----------------------------- |sE'XT4ag  
    if rpowers(1)==0 % pCTN P  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ;$g?T~v7  
        rpowern = cat(2,rpowern{:}); Nh44]*  
        rpowern = [ones(length_r,1) rpowern]; kAUymds;O  
    else 8quaXVj^a  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); S_H+WfIHV'  
        rpowern = cat(2,rpowern{:}); [nq@mc~<  
    end OjA,]Gv6  
    5b7RY V  
    % Compute the values of the polynomials: Ny/MJ#Lq  
    % -------------------------------------- z F;K  
    z = zeros(length_r,length_n); iy.\=Cs$N  
    for j = 1:length_n (TM,V!G+U~  
        s = 0:(n(j)-m(j))/2; @H8EWTZ  
        pows = n(j):-2:m(j); dWBA1p  
        for k = length(s):-1:1 GM<9p_ B  
            p = (1-2*mod(s(k),2))* ... jPkn[W# 6  
                       prod(2:(n(j)-s(k)))/          ... *o ix6  
                       prod(2:s(k))/                 ... E]r?{t`]  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... +=)+'q]S  
                       prod(2:((n(j)+m(j))/2-s(k))); lyhiFkO iH  
            idx = (pows(k)==rpowers); >9J:Uo1z  
            z(:,j) = z(:,j) + p*rpowern(:,idx); a 1*p*dM#  
        end MolgwVd  
         `Pnoxm'  
        if isnorm tZo} ;|~'  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); fc>L K7M  
        end G3v5KmT  
    end alb.g>LNPP  
    [2cD:JL  
    % 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)  n@[O|?S  
    MR.'t9m2L  
    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。 63x?MY6  
    u,Kly<0j  
    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)