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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 X-#&]^d  
    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! b` va\ '&3  
     
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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 G}9f/$'3  
    function z = zernfun(n,m,r,theta,nflag) >6HGh#0(p  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 6(rN(C  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N "ayV8{m^3  
    %   and angular frequency M, evaluated at positions (R,THETA) on the I<ohh`.  
    %   unit circle.  N is a vector of positive integers (including 0), and t>/x-{bH\  
    %   M is a vector with the same number of elements as N.  Each element brs`R#e \  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) b5LToy:  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 7J!s"|VS  
    %   and THETA is a vector of angles.  R and THETA must have the same 5l 3PAG  
    %   length.  The output Z is a matrix with one column for every (N,M) f?51sr  
    %   pair, and one row for every (R,THETA) pair. [&PF ;)i  
    % Dzf\m>H[  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Dws) 4hH  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), RYjK4xT?Y/  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ]i@73h YT  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, S`U8\KTi  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized UZ2_FP  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m].  2Y23!hw  
    % 6UuN-7z!"  
    %   The Zernike functions are an orthogonal basis on the unit circle. CV.|~K0O  
    %   They are used in disciplines such as astronomy, optics, and xdgAu  
    %   optometry to describe functions on a circular domain. ,>h"~X  
    % k#Sr;"  
    %   The following table lists the first 15 Zernike functions. C| ~ A]wc=  
    % .i I{  
    %       n    m    Zernike function           Normalization >&KH!:OX|  
    %       -------------------------------------------------- rZJJ\ , |  
    %       0    0    1                                 1 45 sEhs[$  
    %       1    1    r * cos(theta)                    2 $R/@8qnP W  
    %       1   -1    r * sin(theta)                    2 |HD>m'e  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 3HpqMz  
    %       2    0    (2*r^2 - 1)                    sqrt(3) hm! J@  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) c 'wRGMP  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Sv{n?BYq  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) )>:~XA|?  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) jRU: un4  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 1 >j,v+  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) W~;Jsd=f  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !SW0iq[7j  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 1vj@ qw3  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -je} PwT  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) XNWtX-[ ^@  
    %       -------------------------------------------------- OW4j!W  
    % =wdh# {  
    %   Example 1: 0BlEt1e2T  
    % 7,+eG">0  
    %       % Display the Zernike function Z(n=5,m=1) W3tin3__  
    %       x = -1:0.01:1; E5n7 <  
    %       [X,Y] = meshgrid(x,x); kk-<+R2  
    %       [theta,r] = cart2pol(X,Y); ;rT'~?q  
    %       idx = r<=1; E=ijt3  
    %       z = nan(size(X)); /B@{w-N  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); KHML!f=mu  
    %       figure P);s0Y|@H  
    %       pcolor(x,x,z), shading interp 5lG\ Z?  
    %       axis square, colorbar 0]|`*f&p;  
    %       title('Zernike function Z_5^1(r,\theta)') YQ G<Q  
    % :@[\(:  
    %   Example 2: MF4 (  
    % LUMbRrD-  
    %       % Display the first 10 Zernike functions ?n `m  
    %       x = -1:0.01:1; ~~OFymQ%?q  
    %       [X,Y] = meshgrid(x,x); q5SPyfE[  
    %       [theta,r] = cart2pol(X,Y); Kq3c Kp4  
    %       idx = r<=1; &L+uu',M0c  
    %       z = nan(size(X)); u]IbTJ'  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 8@m$(I +  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 5 3%>)gk:  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; kL*P 3 0  
    %       y = zernfun(n,m,r(idx),theta(idx)); .d1ff] ;  
    %       figure('Units','normalized') u[b |QR=5  
    %       for k = 1:10 sE% $]Jp  
    %           z(idx) = y(:,k); RhE~-b[X  
    %           subplot(4,7,Nplot(k)) :snO*Zg  
    %           pcolor(x,x,z), shading interp (SBhU:^h  
    %           set(gca,'XTick',[],'YTick',[]) nnv|GnQST  
    %           axis square &W@2n&U.q  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Q M0B6F  
    %       end d&j  
    % ,0W^"f.g{m  
    %   See also ZERNPOL, ZERNFUN2. ^<CVQ8R7  
    7Bp7d/R-  
    %   Paul Fricker 11/13/2006 'E_~ |C  
    1D*=ZkA)  
    LORcf1X/  
    % Check and prepare the inputs: Z10Vx2B  
    % ----------------------------- 8z#Qp(he  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) )  z% wh|q  
        error('zernfun:NMvectors','N and M must be vectors.') 4nsJZo#S/  
    end ~5N}P>4 *  
    nqyD>>  
    if length(n)~=length(m)  'o-4'  
        error('zernfun:NMlength','N and M must be the same length.') 7)lEZJK&T  
    end j]BRfA  
    5?7AzJl>  
    n = n(:); =u<:'\_  
    m = m(:); nq M7Is  
    if any(mod(n-m,2)) ==dKC;  
        error('zernfun:NMmultiplesof2', ... FH~:&;  
              'All N and M must differ by multiples of 2 (including 0).') {~U3|_"[pX  
    end bF"l0 jS  
    yajdRU  
    if any(m>n) `L'g<VK;  
        error('zernfun:MlessthanN', ... 3 _  
              'Each M must be less than or equal to its corresponding N.') -'&/7e6>y  
    end )'djqpM.  
    vY4sU@+V  
    if any( r>1 | r<0 ) npdljLN  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') sK}AS;:  
    end Qm%PpQ^Lz3  
    !zA@{gvEc  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Hb)FeGsd).  
        error('zernfun:RTHvector','R and THETA must be vectors.') %fj5 ;}E.  
    end %2\Hj0JQQ  
    2d&F<J<sU  
    r = r(:); C~ 1]  
    theta = theta(:); cM#rus?)+  
    length_r = length(r); b:dN )m  
    if length_r~=length(theta) p#@#$u-  
        error('zernfun:RTHlength', ... 9kL,69d2  
              'The number of R- and THETA-values must be equal.') FZHA19Kb  
    end JVc{vSa!rm  
    #EPC]jFk  
    % Check normalization: zPby+BP  
    % -------------------- @aIgif+v  
    if nargin==5 && ischar(nflag) R/vHq36d  
        isnorm = strcmpi(nflag,'norm'); nKx)R^]k  
        if ~isnorm +,76|oMsQ%  
            error('zernfun:normalization','Unrecognized normalization flag.') }%|ewy9|CW  
        end GcBqe=/B!  
    else s4|\cY`b-  
        isnorm = false; l=~!'1@L}  
    end '75T2Ud  
    WK{`_c U^  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^tB 1Nu %  
    % Compute the Zernike Polynomials Tc WCr  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $V~@w.-Z#  
    EQ1**[$  
    % Determine the required powers of r: >e;-$$e  
    % ----------------------------------- +?_!8N8  
    m_abs = abs(m); G@8)3 @  
    rpowers = []; #HUn~r  
    for j = 1:length(n) 5ya9VZ5#  
        rpowers = [rpowers m_abs(j):2:n(j)]; vSgT36ZF  
    end ? #K|l*  
    rpowers = unique(rpowers); /v{+V/'+  
    /_C2O"h  
    % Pre-compute the values of r raised to the required powers, P'W} ]mCD  
    % and compile them in a matrix: 4V+bE$Wu  
    % ----------------------------- \[MAa:/  
    if rpowers(1)==0 M(-)\~9T  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); =xI;D,@S  
        rpowern = cat(2,rpowern{:}); ;ArwEzo(  
        rpowern = [ones(length_r,1) rpowern]; !_Lmrs  
    else RZa/la*  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 1Viz`y)^  
        rpowern = cat(2,rpowern{:}); ~ ld.I4  
    end qmrT d G  
    SDnl^a  
    % Compute the values of the polynomials: @^,q/%;  
    % -------------------------------------- LF dvz0  
    y = zeros(length_r,length(n)); AxEyXT(h5  
    for j = 1:length(n) 5zl+M`  
        s = 0:(n(j)-m_abs(j))/2; 8!_jZf8  
        pows = n(j):-2:m_abs(j); T+Oqd\05.+  
        for k = length(s):-1:1 ,-UF5U  
            p = (1-2*mod(s(k),2))* ... vW+6_41ZM  
                       prod(2:(n(j)-s(k)))/              ... Z\!,f.>g  
                       prod(2:s(k))/                     ... g3^s_*A  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ,.,8-In^  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); j\y;~ V  
            idx = (pows(k)==rpowers); =ZgueUz,  
            y(:,j) = y(:,j) + p*rpowern(:,idx); pBsb>wvej  
        end 3?93Pj3oPt  
         !<[+u  
        if isnorm 'Y?-."eKh  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); RY-iFydPc  
        end jv)+qmqo!  
    end 9CD ei~  
    % END: Compute the Zernike Polynomials ipSMmpB  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `4"8@>D  
    T4eJ:u*;  
    % Compute the Zernike functions: 'xW=qboOp  
    % ------------------------------ u6|C3,!z"  
    idx_pos = m>0; ;n&95t1$  
    idx_neg = m<0; xT*'p&ap  
    J Enjc/  
    z = y; ]N>ZOV,>  
    if any(idx_pos) Y=S0|!u  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); IwyA4Ak Ru  
    end ]*0zir/  
    if any(idx_neg) QkrQM&Im  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); v+ $3  
    end Q):#6|u+  
    ?ANW I8'_j  
    % 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) +gl\l?>sr  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. =s\$i0A2  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated $|$@?H>K  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive y0%@^^-Ru  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, d4y#n=HnnV  
    %   and THETA is a vector of angles.  R and THETA must have the same :H}iL*  
    %   length.  The output Z is a matrix with one column for every P-value, j0l,1=^>l  
    %   and one row for every (R,THETA) pair. xm m,- u  
    % @c~Z0+Ji  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ing'' _  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 2Kxb(q"  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 91R# /i  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 1*O|[W  
    %   for all p. ;c 7I "?@z  
    % "Q:Gd6?h;  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Q{/z>-X\x  
    %   Zernike functions (order N<=7).  In some disciplines it is H,> }t S  
    %   traditional to label the first 36 functions using a single mode rto?*^N?  
    %   number P instead of separate numbers for the order N and azimuthal tU4#7b:Y  
    %   frequency M. 54 $^ldD  
    % jz!I +  
    %   Example: K^WDA])  
    % @A[)\E1  
    %       % Display the first 16 Zernike functions 6[m~xegG  
    %       x = -1:0.01:1; BM :x`JY  
    %       [X,Y] = meshgrid(x,x); d1~#@6CIz  
    %       [theta,r] = cart2pol(X,Y); F^l1WX6  
    %       idx = r<=1; \h ~_<)  
    %       p = 0:15; =, XCjiBeC  
    %       z = nan(size(X)); 86#l$QaK{  
    %       y = zernfun2(p,r(idx),theta(idx)); 6,t6~Uo/  
    %       figure('Units','normalized') Du_5iuMh  
    %       for k = 1:length(p) tZ]gVgZg  
    %           z(idx) = y(:,k); }Rh\JDiQ  
    %           subplot(4,4,k) 6uE20O<z]  
    %           pcolor(x,x,z), shading interp @eYpARF  
    %           set(gca,'XTick',[],'YTick',[]) a`wjZ"}'[  
    %           axis square Xi="gxp$%  
    %           title(['Z_{' num2str(p(k)) '}']) 9p_?t'&>q  
    %       end p?gm=b#  
    % L;V 8c  
    %   See also ZERNPOL, ZERNFUN. n Bm ]?  
    n/9afIN  
    %   Paul Fricker 11/13/2006 h&4s%:_4  
    a>j}@8[J  
    dIC\U  
    % Check and prepare the inputs: ^4sfVpD2!  
    % ----------------------------- 1I%u)[;>  
    if min(size(p))~=1 w=GMQ8  
        error('zernfun2:Pvector','Input P must be vector.') H-_gd.VD  
    end =-sTV\  
    B.N#9u-vW  
    if any(p)>35 EL,k z8  
        error('zernfun2:P36', ... 7|}4UXr7y  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... #*h\U]=VS  
               '(P = 0 to 35).']) < $zJi V  
    end qO{z{@jo55  
    ZSf &M  
    % Get the order and frequency corresonding to the function number: 5 kHaZ Q  
    % ---------------------------------------------------------------- (3n "a'  
    p = p(:); :FAPH8]  
    n = ceil((-3+sqrt(9+8*p))/2); CX]1I|T5  
    m = 2*p - n.*(n+2); ?L<B]!9HZt  
    }nrjA0WN  
    % Pass the inputs to the function ZERNFUN: =Jm[1Mgt  
    % ---------------------------------------- t:10  
    switch nargin sq$v6x sl  
        case 3 xXJ*xYn "}  
            z = zernfun(n,m,r,theta); Ph3;;,v '  
        case 4 TOwqr T/  
            z = zernfun(n,m,r,theta,nflag); oSCaP,P  
        otherwise vF^d40gV  
            error('zernfun2:nargin','Incorrect number of inputs.') /A{ Zf'DI  
    end UhH#> 2r_  
    um,f!ho-U  
    % 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) 4l 67B]o  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. W3le)&  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of qFmw9\Fn  
    %   order N and frequency M, evaluated at R.  N is a vector of W,4!"*+  
    %   positive integers (including 0), and M is a vector with the v#,queGi  
    %   same number of elements as N.  Each element k of M must be a ?[JP[ qS  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) {SV/AN  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is /DAR'9@h  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix $INB_/R E  
    %   with one column for every (N,M) pair, and one row for every &jJu=6 U B  
    %   element in R. "D'e  
    % ?X@!jB,Pv  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- S f?;j{?G  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is /2p*uv }IP  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to !Gmnck&+  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 z:\9t[e4  
    %   for all [n,m]. i<Z%  
    % v]k-x n|$j  
    %   The radial Zernike polynomials are the radial portion of the r `PJb5^\|  
    %   Zernike functions, which are an orthogonal basis on the unit KQr+VQdq>  
    %   circle.  The series representation of the radial Zernike Z|t=t"6"  
    %   polynomials is I!hh_  
    % rqjq}L)  
    %          (n-m)/2 B%tF|KKj  
    %            __ w9'>&W8T  
    %    m      \       s                                          n-2s OHndZ$'fI  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r +4 k=Y  
    %    n      s=0 .~i|kc]Ue  
    % |Y uf/G%/  
    %   The following table shows the first 12 polynomials. P) vD?)Q  
    % c {= ; lT  
    %       n    m    Zernike polynomial    Normalization HPphTu}`  
    %       --------------------------------------------- 3m>YR-n$  
    %       0    0    1                        sqrt(2) NzT &K7v  
    %       1    1    r                           2 Jxvh;  
    %       2    0    2*r^2 - 1                sqrt(6) .8YxEnXw)(  
    %       2    2    r^2                      sqrt(6) AQ)gj$ m3  
    %       3    1    3*r^3 - 2*r              sqrt(8) 6*Z7JiQ 0  
    %       3    3    r^3                      sqrt(8) 'WW:'[Syn'  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) .I^4Fc}&4  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) QoYEWXT|g  
    %       4    4    r^4                      sqrt(10) Wj.t4XG!  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) %5e|  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) fOi Rstci  
    %       5    5    r^5                      sqrt(12) p]kEH\ sh  
    %       --------------------------------------------- X /c8XLe"  
    % ]^ R':YE  
    %   Example: YdhV a!Y  
    % .`IhxE~mN  
    %       % Display three example Zernike radial polynomials Y:DopKRD  
    %       r = 0:0.01:1; W]po RTJ:  
    %       n = [3 2 5]; T]\1gs41  
    %       m = [1 2 1]; GxhE5f;  
    %       z = zernpol(n,m,r); 'ma X  
    %       figure ~uhW~bT  
    %       plot(r,z) Y#os6|MV#  
    %       grid on vm@V5oH  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') %6[,a  
    % P&Ke slk  
    %   See also ZERNFUN, ZERNFUN2. rn-bfzoDS  
    7Yg1z%%U  
    % A note on the algorithm. "AagTFs(i  
    % ------------------------ 5|rBb[  
    % The radial Zernike polynomials are computed using the series PSJj$bt;<+  
    % representation shown in the Help section above. For many special Z1,rN#p9  
    % functions, direct evaluation using the series representation can z@ `o(gh  
    % produce poor numerical results (floating point errors), because iDDJJ>F26  
    % the summation often involves computing small differences between 5qM$ahN3wH  
    % large successive terms in the series. (In such cases, the functions ^0zfQu+!  
    % are often evaluated using alternative methods such as recurrence $2uC%er"H  
    % relations: see the Legendre functions, for example). For the Zernike RL` jaS?V  
    % polynomials, however, this problem does not arise, because the no+ m.B  
    % polynomials are evaluated over the finite domain r = (0,1), and j/aJDE(+  
    % because the coefficients for a given polynomial are generally all @@H/q  
    % of similar magnitude. h3 H Udu  
    % o<5`uV!f  
    % ZERNPOL has been written using a vectorized implementation: multiple .'`aX 7{\  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 2q[pOT'k  
    % values can be passed as inputs) for a vector of points R.  To achieve _WNbuk0  
    % this vectorization most efficiently, the algorithm in ZERNPOL 9JdJn>  
    % involves pre-determining all the powers p of R that are required to ;87PP7~  
    % compute the outputs, and then compiling the {R^p} into a single SA'g`  
    % matrix.  This avoids any redundant computation of the R^p, and 1KUjb@"  
    % minimizes the sizes of certain intermediate variables. L]MWdD  
    % gs1yWnSv5  
    %   Paul Fricker 11/13/2006 G/JGb2I/7|  
    $T K*w8@:  
    ;?{^LiD+F  
    % Check and prepare the inputs: a &tWMxBr  
    % ----------------------------- -Y524   
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) '=;e# C`<{  
        error('zernpol:NMvectors','N and M must be vectors.') ( K[e=0Rf  
    end UnDCC_ud  
    pJFn 8&!J  
    if length(n)~=length(m) T8m]f<  
        error('zernpol:NMlength','N and M must be the same length.') = 9Yf o,F  
    end }36AeJ7L  
    V<5. 4{[G  
    n = n(:); *w;?&)8%  
    m = m(:); 6-\Mf:%B  
    length_n = length(n); >\K=)/W2  
    -n 7 @r  
    if any(mod(n-m,2)) h 8$.mQr  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') yhgGvyD  
    end ovN3.0tAI  
    uNI&U7_"  
    if any(m<0) BP j?l  
        error('zernpol:Mpositive','All M must be positive.') ,~cK]!:>s  
    end P?q HzNGi7  
    3chx 4  
    if any(m>n) WT\wV\Pu  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') oQ,n?on  
    end xp1/@Pw?  
    GDCp@%xW  
    if any( r>1 | r<0 ) D`QMlRzXy  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') W!HjO;  
    end K^o$uUBe  
    "<|KR{/+  
    if ~any(size(r)==1) hYyIC:PXR  
        error('zernpol:Rvector','R must be a vector.') u)h {"pP  
    end GI{EP&C  
    z(c8]Wu#  
    r = r(:); I+Fy)=DO9  
    length_r = length(r); 7Wef[N\x  
    &FmTT8"l  
    if nargin==4 wxB HlgK4z  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); lO\HchG zB  
        if ~isnorm f-#:3k*7S  
            error('zernpol:normalization','Unrecognized normalization flag.') |?jgjn&RQ  
        end GTB\95j]  
    else 0(d!w*RpG  
        isnorm = false; w &YUb,{Y  
    end N^B7<~ bD  
    $MvKwQ/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P 4)Q5r  
    % Compute the Zernike Polynomials 5 `TMqrk  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f'ld6jt|%  
    VEa"^{,w  
    % Determine the required powers of r: &(<>} r  
    % ----------------------------------- +h-% {  
    rpowers = []; [[_>D M  
    for j = 1:length(n) \roJf&O }  
        rpowers = [rpowers m(j):2:n(j)]; >7@,,~3  
    end :o` <CO  
    rpowers = unique(rpowers); Ib{#dhV  
    CV HKP[-  
    % Pre-compute the values of r raised to the required powers, $-^& AKc  
    % and compile them in a matrix: +D @B eQu  
    % ----------------------------- sh,4n{+  
    if rpowers(1)==0 enxb pq#  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); aVHID{Gf Z  
        rpowern = cat(2,rpowern{:}); U}HSL5v  
        rpowern = [ones(length_r,1) rpowern]; 7 `~0j6FY  
    else u0) O Fz  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);  ]LsT  
        rpowern = cat(2,rpowern{:}); /)v+|%U  
    end a(IE8:yU`  
    0-OKbw5%=b  
    % Compute the values of the polynomials: [,st: Y  
    % -------------------------------------- OS%[SHs  
    z = zeros(length_r,length_n); JkR%o #>5  
    for j = 1:length_n V O1   
        s = 0:(n(j)-m(j))/2; @Wd (>*"zw  
        pows = n(j):-2:m(j); ox<6qW  
        for k = length(s):-1:1 nGTGX  
            p = (1-2*mod(s(k),2))* ... CUTjRWQ  
                       prod(2:(n(j)-s(k)))/          ... 3;b)pQ~6CJ  
                       prod(2:s(k))/                 ... _3u3b/%J?  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 4T52vM  
                       prod(2:((n(j)+m(j))/2-s(k))); yS K81`  
            idx = (pows(k)==rpowers); ?.ObHV*k  
            z(:,j) = z(:,j) + p*rpowern(:,idx); `B&E?x  
        end P$Y w'3v/  
         > mCH!ey  
        if isnorm |,KsJ2hD  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 0 -M i q  
        end b!MN QGs  
    end d8 ~%(I9  
    GLub5GrxR  
    % 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)  iP9Dr<P  
    _?$')P|  
    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。 b+dmJ]c  
    , d HAD  
    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)