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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 h'+ swPh  
    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! g11K?3*%Q  
     
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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 eM}Xn^}  
    function z = zernfun(n,m,r,theta,nflag) ;%}  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Y`wi=(  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N e=U7w7(s9  
    %   and angular frequency M, evaluated at positions (R,THETA) on the <Ip}uy[Y  
    %   unit circle.  N is a vector of positive integers (including 0), and 6m9Z5:xG  
    %   M is a vector with the same number of elements as N.  Each element yI!K quMC  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) z|Xl%8  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, YG_3@`-<  
    %   and THETA is a vector of angles.  R and THETA must have the same GZ"O%: d  
    %   length.  The output Z is a matrix with one column for every (N,M) t0Uax-E(  
    %   pair, and one row for every (R,THETA) pair. ty ~U~  
    % <M=K!k  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 8mi IlB  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), *m2:iChY  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral UX6-{ RP  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, {pqm&PB04  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized .._wTOSq  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. %}@^[E)  
    % CzgLgh;:T  
    %   The Zernike functions are an orthogonal basis on the unit circle. \6o ~ i  
    %   They are used in disciplines such as astronomy, optics, and S}>rsg!  
    %   optometry to describe functions on a circular domain. jGt[[s  
    % I$YF55uB  
    %   The following table lists the first 15 Zernike functions. 1t6UI4U!$  
    % cla4%|kq3Y  
    %       n    m    Zernike function           Normalization Wl1%BN0>  
    %       -------------------------------------------------- _\[Zr.y  
    %       0    0    1                                 1 yuND0,e  
    %       1    1    r * cos(theta)                    2 /)|*Vzu  
    %       1   -1    r * sin(theta)                    2 G2mv6xK'  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) }Vt5].TA  
    %       2    0    (2*r^2 - 1)                    sqrt(3) {_ocW@@  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) )|:|.`H  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) W6Hiqu+  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) +f+\uObi:  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) )Aj~ xA  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) F](kU#3"S  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) %9IM|\ulp  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?wmr~j  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Cu}Rq!9i  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) M$w^g8F27H  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 8g<3J-7Mm  
    %       -------------------------------------------------- sGV%O=9?2  
    % b747eR 7E  
    %   Example 1: hI"I#(*jA%  
    % Ji=E 1R  
    %       % Display the Zernike function Z(n=5,m=1) 419t"1b  
    %       x = -1:0.01:1; IE3GM^7\  
    %       [X,Y] = meshgrid(x,x); il*bsnwpZv  
    %       [theta,r] = cart2pol(X,Y); Od!j+.OY<  
    %       idx = r<=1; `jP6;i  
    %       z = nan(size(X)); es.`:^A  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); C; ! )<(Vw  
    %       figure Fd2zvi  
    %       pcolor(x,x,z), shading interp k+&|*!j  
    %       axis square, colorbar JTVCaL3Z  
    %       title('Zernike function Z_5^1(r,\theta)') mWtwp-  
    % hd\iW7  
    %   Example 2: vQA: \!  
    % )4j#gHN\  
    %       % Display the first 10 Zernike functions KnlVZn[3t  
    %       x = -1:0.01:1; U|,VH-#  
    %       [X,Y] = meshgrid(x,x); 3dXyKi  
    %       [theta,r] = cart2pol(X,Y); B;^7Yu0,  
    %       idx = r<=1; m|'TPy  
    %       z = nan(size(X)); fuQ? @F  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ++xEMP)  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; &}rh+z  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ^G15]Pyw  
    %       y = zernfun(n,m,r(idx),theta(idx)); P\SE_*&  
    %       figure('Units','normalized') `6UW?1_Z5  
    %       for k = 1:10 aVd{XVE  
    %           z(idx) = y(:,k); 2OEO b,`  
    %           subplot(4,7,Nplot(k)) "Y4 tt0I  
    %           pcolor(x,x,z), shading interp xZBmQ:s',S  
    %           set(gca,'XTick',[],'YTick',[]) \07 s'W U  
    %           axis square /z6NJ2jb  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) d!!5'/tmS  
    %       end an.)2*u  
    % ]kR 93  
    %   See also ZERNPOL, ZERNFUN2. +,If|5>(  
    'H:lR1(,  
    %   Paul Fricker 11/13/2006 'R= r9_%  
    6X)8vQH  
    EY':m_7W  
    % Check and prepare the inputs: IeE+h-3p  
    % ----------------------------- ]x! vPIyq  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) amOBUD5Ld`  
        error('zernfun:NMvectors','N and M must be vectors.') "h\{PoG  
    end % `\8z  
    u[y>DPPx  
    if length(n)~=length(m) yjc:+Y{5'  
        error('zernfun:NMlength','N and M must be the same length.') >AV?g8B;  
    end WC0@g5;1[  
    Bx;bc  
    n = n(:); (',G Ako  
    m = m(:); u JGYXlLE  
    if any(mod(n-m,2)) XswEAz0=  
        error('zernfun:NMmultiplesof2', ... %=%jy  
              'All N and M must differ by multiples of 2 (including 0).') [[ H XOPaV  
    end ^<7)w2ns  
    > PfYHO  
    if any(m>n) }B^KV#_{S  
        error('zernfun:MlessthanN', ... L3'o2@$  
              'Each M must be less than or equal to its corresponding N.') gtJUQu p2  
    end >i-cR4=LL{  
    $D1Pk  
    if any( r>1 | r<0 ) 1P@&xcvS\  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') =#SKN\4  
    end U5%EQc-"P  
    e%o6s+"  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) BB>3Kj:|  
        error('zernfun:RTHvector','R and THETA must be vectors.') aV,>y"S  
    end !Tr +:SM  
    P%(pbG-X.  
    r = r(:); /EA4-#uw  
    theta = theta(:); D\bW' k]!  
    length_r = length(r); 6(VCQ{  
    if length_r~=length(theta) AS'a'x>8>,  
        error('zernfun:RTHlength', ... x/R|i%u-s  
              'The number of R- and THETA-values must be equal.') 8it|yK.G@&  
    end qJKD| =_  
    P10`X&  
    % Check normalization: O\-cLI<h2  
    % -------------------- JIQS'r  
    if nargin==5 && ischar(nflag) z{7&=$  
        isnorm = strcmpi(nflag,'norm'); zH.DyD5T;  
        if ~isnorm |r$Vb$z  
            error('zernfun:normalization','Unrecognized normalization flag.') -6aGcPq  
        end 8J7 xs6@  
    else   9Ld3  
        isnorm = false; &Dgho  
    end "n=`{~F  
    Da0E)  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :I1 )=8lO  
    % Compute the Zernike Polynomials (G*--+Gn  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YR=<xn;m.  
    n'U*8ID  
    % Determine the required powers of r: AM#VRRTU  
    % ----------------------------------- dyC: Mko=  
    m_abs = abs(m); l%oie1g l  
    rpowers = []; kzMCI)>"  
    for j = 1:length(n) Z;P[)q  
        rpowers = [rpowers m_abs(j):2:n(j)]; { %vX/Ek  
    end ~6Vs>E4G  
    rpowers = unique(rpowers); (&=-o(  
    P*BA  
    % Pre-compute the values of r raised to the required powers, 5rr7lw WZ  
    % and compile them in a matrix: ]3BTL7r  
    % ----------------------------- *1$rg?yGf  
    if rpowers(1)==0  S`)KC-  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); O$V 6QJ  
        rpowern = cat(2,rpowern{:}); W7c(] tg.  
        rpowern = [ones(length_r,1) rpowern]; F<M#T  
    else qH: ` O%,  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); N4}j,{#  
        rpowern = cat(2,rpowern{:}); .DMeW i  
    end $pyM<:*L&<  
    DGz'Dn  
    % Compute the values of the polynomials: H 0aDWFWS  
    % -------------------------------------- ]8NNxaE3(  
    y = zeros(length_r,length(n)); &.y:QVR,!  
    for j = 1:length(n) 5?&k? v@  
        s = 0:(n(j)-m_abs(j))/2; bc}U &X<  
        pows = n(j):-2:m_abs(j); . p^='Kz?  
        for k = length(s):-1:1 ;EP7q[  
            p = (1-2*mod(s(k),2))* ... RY8;bUSR  
                       prod(2:(n(j)-s(k)))/              ... H [wJ; l  
                       prod(2:s(k))/                     ... Py^F},?J  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... $W<H[k&(B  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); "CapP`:  
            idx = (pows(k)==rpowers); ^/47 *vcN5  
            y(:,j) = y(:,j) + p*rpowern(:,idx); vvU;55-  
        end "WdGY*r  
         ~}q"M[{  
        if isnorm dQVV0)z  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); cKEf- &~  
        end MUh )  
    end BNw^ _j1  
    % END: Compute the Zernike Polynomials #I|Vyufw  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% iNUisl  
    7L|w~l7R~  
    % Compute the Zernike functions: BC ]^BKP  
    % ------------------------------ hZ Gr/5f  
    idx_pos = m>0; 2f9~:.NgF  
    idx_neg = m<0; #O6SEK|Z  
    kj~)#KDN  
    z = y; (cAv :EKpo  
    if any(idx_pos) DmEmv/N=  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Oh9wBV  
    end 6a[D]46y,2  
    if any(idx_neg) ,>A9OTSN\  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ;{ u{F L  
    end iT1"Le/N  
    $v#Q'?jE  
    % 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) 58,_  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. \u ?z:mV  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated U>7"BpC  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive [7q~rcf,Z  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ^crk8O@Fw  
    %   and THETA is a vector of angles.  R and THETA must have the same 1dh_"/  
    %   length.  The output Z is a matrix with one column for every P-value, :BKY#uH~  
    %   and one row for every (R,THETA) pair. XL c&7  
    % 1fM= >Z  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike W-<E p<7{  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) d!7cIYVZ  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) q4@n pbx  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 A(X~pP &oF  
    %   for all p. hV#+joT8i  
    % #~*fZ|sq+3  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 uy)iB'st&  
    %   Zernike functions (order N<=7).  In some disciplines it is y K)7%j!  
    %   traditional to label the first 36 functions using a single mode ${0+LhST  
    %   number P instead of separate numbers for the order N and azimuthal ]Cnj=\'  
    %   frequency M. GQhzQM1HS  
    % gm~Ka%O|F  
    %   Example: zD}dvI}  
    % wr,X@y%(!  
    %       % Display the first 16 Zernike functions ZGK*]o =)  
    %       x = -1:0.01:1; cG1-.,r  
    %       [X,Y] = meshgrid(x,x); {c`kC]9  
    %       [theta,r] = cart2pol(X,Y); /f~ V(DK  
    %       idx = r<=1; 9Xo'U;J  
    %       p = 0:15; 2#~5[PtP^  
    %       z = nan(size(X)); N(q%|h<Z/=  
    %       y = zernfun2(p,r(idx),theta(idx)); :$."x '  
    %       figure('Units','normalized') Ug*:o d  
    %       for k = 1:length(p) 0^nnR7  
    %           z(idx) = y(:,k); pqFgi_2m  
    %           subplot(4,4,k) O&!>C7  
    %           pcolor(x,x,z), shading interp T V\21  
    %           set(gca,'XTick',[],'YTick',[]) 5jD2%"YUV  
    %           axis square s <Pk[7`*  
    %           title(['Z_{' num2str(p(k)) '}']) Bm2"} =  
    %       end qFp }+s  
    % gfG Mu0FjB  
    %   See also ZERNPOL, ZERNFUN. |_/q0#"  
    Zy _A3m{  
    %   Paul Fricker 11/13/2006 }eb}oK  
    iI ji[>qz  
    fiqeXE?E  
    % Check and prepare the inputs: .vYU4g]  
    % ----------------------------- ?RJ ) u  
    if min(size(p))~=1 L^uO.eI"m  
        error('zernfun2:Pvector','Input P must be vector.') 7y.$'<  
    end E9TWLB5A)(  
    ?ORG<11a  
    if any(p)>35 3Xyu`zS&   
        error('zernfun2:P36', ... )w_0lm'v{r  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Gh}sk-Xk=  
               '(P = 0 to 35).']) .)~IoIW=  
    end 37Ux2t  
    Ae R3wua  
    % Get the order and frequency corresonding to the function number: 1^^<6e  
    % ---------------------------------------------------------------- d?^bCf+<  
    p = p(:); `wz@l:e  
    n = ceil((-3+sqrt(9+8*p))/2); Lb;:<  
    m = 2*p - n.*(n+2); mlc0XDS%  
    H!mNHY_fA  
    % Pass the inputs to the function ZERNFUN: {^zieP!  
    % ---------------------------------------- _]:wltPv  
    switch nargin Z~)Bh~^A  
        case 3 [F{q.mZj  
            z = zernfun(n,m,r,theta); m[7@l  
        case 4 :@# '&(#~  
            z = zernfun(n,m,r,theta,nflag); 8$9<z  
        otherwise !j[Oy r|  
            error('zernfun2:nargin','Incorrect number of inputs.') Hh`x>{,|S  
    end de{@u<Y Zb  
    5/4N  Y  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) /NRdBN  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. '| (#^jAj  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Zn{,j0;  
    %   order N and frequency M, evaluated at R.  N is a vector of 6t@kft>Nv  
    %   positive integers (including 0), and M is a vector with the ajB4 Lj,:r  
    %   same number of elements as N.  Each element k of M must be a S%J$.ge  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) CqHCJ '  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is trD-qi  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix &Luq}^u  
    %   with one column for every (N,M) pair, and one row for every ]M%kt+u!  
    %   element in R. TY,5]*86I&  
    % O#Y;s;)i"  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- k~ Z9og  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 9w\ yWxl  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to b5WtL+Z  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 #+$pE@u7A  
    %   for all [n,m]. >a;0<Ui&Q  
    % pxC:VJ;  
    %   The radial Zernike polynomials are the radial portion of the /S9s%scAy  
    %   Zernike functions, which are an orthogonal basis on the unit fCg"tckE  
    %   circle.  The series representation of the radial Zernike K(bid0 Y  
    %   polynomials is es]S]}JV  
    % ErZYPl  
    %          (n-m)/2 ,au-g)IFZ  
    %            __  ?X{ul  
    %    m      \       s                                          n-2s &oi*]:<FNe  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Gp*U2LB  
    %    n      s=0 um.s :vj$  
    % Z*r;"WHB  
    %   The following table shows the first 12 polynomials. tR`'( *wh  
    % w]2tb  
    %       n    m    Zernike polynomial    Normalization $'m&RzZ  
    %       --------------------------------------------- eYSVAj  
    %       0    0    1                        sqrt(2) d3% 1 P)  
    %       1    1    r                           2 lJZ-*"9V  
    %       2    0    2*r^2 - 1                sqrt(6) }~/u%vI@M5  
    %       2    2    r^2                      sqrt(6) }<G"w 5.<  
    %       3    1    3*r^3 - 2*r              sqrt(8) F"2rX&W  
    %       3    3    r^3                      sqrt(8) oEfy{54  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) `2}H$D  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) H_3-"m&3  
    %       4    4    r^4                      sqrt(10) f( =3'wQ  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) kl4u]MyL#  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) snU $Na3  
    %       5    5    r^5                      sqrt(12) 2Mqac:L  
    %       --------------------------------------------- T2Duz,  
    % 8M9LY9C  
    %   Example: . Y@)3  
    % `8 Q3=^)3  
    %       % Display three example Zernike radial polynomials |n9q 4*dN  
    %       r = 0:0.01:1; s+mNr3  
    %       n = [3 2 5]; /%O+]#$`0  
    %       m = [1 2 1]; \TchRSe  
    %       z = zernpol(n,m,r); ds> V|}f[  
    %       figure u+ wKs`   
    %       plot(r,z) D)0pm?*5A  
    %       grid on :i{$p00 G  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') |q0MM^%"  
    % Ojea~Y]Sr  
    %   See also ZERNFUN, ZERNFUN2. }Z^r<-N  
    {u 7%Z}<0  
    % A note on the algorithm. X9:4oMux7  
    % ------------------------ -wA^ao   
    % The radial Zernike polynomials are computed using the series ^LaOl+;S  
    % representation shown in the Help section above. For many special 7*{9 2_M  
    % functions, direct evaluation using the series representation can ;|nC;D]  
    % produce poor numerical results (floating point errors), because Y$tgz)  
    % the summation often involves computing small differences between zxo0:dyw7  
    % large successive terms in the series. (In such cases, the functions ^ W/,Z`  
    % are often evaluated using alternative methods such as recurrence ,B^NH7A:  
    % relations: see the Legendre functions, for example). For the Zernike |dLA D4%  
    % polynomials, however, this problem does not arise, because the /3]b!lFZZ  
    % polynomials are evaluated over the finite domain r = (0,1), and g 0=Q>TzY  
    % because the coefficients for a given polynomial are generally all F0&BEJBkU  
    % of similar magnitude. ^;KL`  
    % C}})dL;(  
    % ZERNPOL has been written using a vectorized implementation: multiple CBj&8#8Z  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 1m$< %t.>  
    % values can be passed as inputs) for a vector of points R.  To achieve CO+[iJ,4C+  
    % this vectorization most efficiently, the algorithm in ZERNPOL SL( WE=H  
    % involves pre-determining all the powers p of R that are required to sg=mkkD!g  
    % compute the outputs, and then compiling the {R^p} into a single \I3={ii0  
    % matrix.  This avoids any redundant computation of the R^p, and 7mUpn:U  
    % minimizes the sizes of certain intermediate variables. ;t^8lC?>V  
    % k3:8T#N>!O  
    %   Paul Fricker 11/13/2006 =CCxY7)M+.  
    rSGt`#E-s.  
    M=HP!hn  
    % Check and prepare the inputs: p_K` `JE  
    % ----------------------------- !i"Z  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) d&!ZCq#_e  
        error('zernpol:NMvectors','N and M must be vectors.') $d@_R^]X  
    end i;'kQ  
    9)_fH6r  
    if length(n)~=length(m) i/Nd  
        error('zernpol:NMlength','N and M must be the same length.') 8Gw0;Uu8D  
    end O@n1E'S/  
    C>1fL6ct  
    n = n(:); |fQl0hL  
    m = m(:); q;XO1Se  
    length_n = length(n); +`@)87O  
    c(]NpH in  
    if any(mod(n-m,2)) O{B[iy(C  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') |~6X: M61  
    end hH=H/L_Z  
    {;iG}jK  
    if any(m<0) Hg~O0p}[  
        error('zernpol:Mpositive','All M must be positive.') f/_RtOSw  
    end `0]kRA8=  
    L}>XH*  
    if any(m>n) \P3[_kbf1  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') jK#[r[q{  
    end )^G&p[G  
    2J^jSgr50d  
    if any( r>1 | r<0 ) s@WF[S7D  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') [c{/0*  
    end > @Ux8#  
    G^Z SQ!  
    if ~any(size(r)==1) NlBnV  
        error('zernpol:Rvector','R must be a vector.') B%|cp+/  
    end 3C=|  
    W6b5elH@  
    r = r(:); rPk=9I  
    length_r = length(r); H;&^A5  
    ciq'fy  
    if nargin==4 ac/=%om8u  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); "IK QFt'  
        if ~isnorm F<KUVe  
            error('zernpol:normalization','Unrecognized normalization flag.') 9M$=X-  
        end NAy3Zd}  
    else :d&^//9  
        isnorm = false; ]5!}S-uJq  
    end L5E|1T  
    LD'eq\vO  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ' 9K4A'2[  
    % Compute the Zernike Polynomials *?k~n9n5U  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U%swqle4  
    %&c+} m  
    % Determine the required powers of r: jKOjw#N  
    % ----------------------------------- 8=]R6[,fD  
    rpowers = []; b*-g@S  
    for j = 1:length(n) :RJ=f  
        rpowers = [rpowers m(j):2:n(j)]; )PM&x   
    end ews4qP  
    rpowers = unique(rpowers); $"+ahS<?tC  
    EF7Y4lp  
    % Pre-compute the values of r raised to the required powers, (6xrs_ea  
    % and compile them in a matrix: U!GG8;4  
    % ----------------------------- ]F,mj-?4x  
    if rpowers(1)==0 ^%^~:<N  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); }CR@XD}[  
        rpowern = cat(2,rpowern{:}); CS:"F) at  
        rpowern = [ones(length_r,1) rpowern]; Kr$ w"]  
    else MKad 5gD*<  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); A4}6hG#  
        rpowern = cat(2,rpowern{:}); :R/szE*Ak  
    end "?I]h  
    '.n0[2>  
    % Compute the values of the polynomials: bt=%DMTn  
    % -------------------------------------- =Q % F~  
    z = zeros(length_r,length_n); @`qhQ  
    for j = 1:length_n |Rh%wJ  
        s = 0:(n(j)-m(j))/2; +V"t't7  
        pows = n(j):-2:m(j); vOb=>  
        for k = length(s):-1:1 F_m[EB  
            p = (1-2*mod(s(k),2))* ... (lDbArqy  
                       prod(2:(n(j)-s(k)))/          ...  ~ccwu  
                       prod(2:s(k))/                 ... ]fN\LY6p  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 83"Vh$&  
                       prod(2:((n(j)+m(j))/2-s(k))); F,Ls1  
            idx = (pows(k)==rpowers); 67/&AiS?  
            z(:,j) = z(:,j) + p*rpowern(:,idx); _m;#+`E  
        end P4{8pO]B  
         _z:7Dj#  
        if isnorm d" T">Og)  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); jU1([(?"  
        end ?GdoB7(%  
    end b)+;#m  
    fc'NU(70c  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  w$E8R[J~P  
    m%?+;V  
    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。 A.f!SYV6  
    K<BS%~,I  
    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)