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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 n{sk  
    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! |`#fX(=  
     
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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 HXKM<E{j  
    function z = zernfun(n,m,r,theta,nflag) f X[xZGV,  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 4)w,gp  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N /=p[k^A  
    %   and angular frequency M, evaluated at positions (R,THETA) on the y<FC7  
    %   unit circle.  N is a vector of positive integers (including 0), and P! +Gwm{  
    %   M is a vector with the same number of elements as N.  Each element @\?ub F  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) RD:G 9[  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, a -Pz<*  
    %   and THETA is a vector of angles.  R and THETA must have the same 0!3. .5==  
    %   length.  The output Z is a matrix with one column for every (N,M) "++\6 H<  
    %   pair, and one row for every (R,THETA) pair. 6AJk6 W^Z  
    % 4}m9,  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike JW[6 ^Rw  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 6U !P8q  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral d78 [(;  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, v'S]g^  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Wz' !stcp  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. F `o9GLxM}  
    % $$m0mK  
    %   The Zernike functions are an orthogonal basis on the unit circle. 1jd{AqHl  
    %   They are used in disciplines such as astronomy, optics, and kAEq +{h  
    %   optometry to describe functions on a circular domain. v](Y n) #  
    % 0fewMS*  
    %   The following table lists the first 15 Zernike functions. E_=F' sP?  
    % \~*<[.8~  
    %       n    m    Zernike function           Normalization 9PKXQp  
    %       -------------------------------------------------- ~g=& wT11  
    %       0    0    1                                 1 0]SWyC :  
    %       1    1    r * cos(theta)                    2 3FR(gr$X  
    %       1   -1    r * sin(theta)                    2 (O+d6oT=Z2  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) `*vO8v  
    %       2    0    (2*r^2 - 1)                    sqrt(3) h]s6)tI I  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) $Lj ]NtO  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) UAF$bR  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) b y>%}#M  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 8S#$'2sT  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 7_ix&oVI  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ooJxE\L  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) rtS cQ  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ~k&b  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?IAu,s*u  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) DD=X{{;D\"  
    %       -------------------------------------------------- oUnb-,8n  
    % 4JK6<Pk  
    %   Example 1: 4N,[Gs<7  
    % <}WSYK,zUY  
    %       % Display the Zernike function Z(n=5,m=1) 9wR D=a  
    %       x = -1:0.01:1; @LI;q  
    %       [X,Y] = meshgrid(x,x); #[M^Q h  
    %       [theta,r] = cart2pol(X,Y); Q$U.vF7BnP  
    %       idx = r<=1; PFp!T [)  
    %       z = nan(size(X)); T@ESMPeU:X  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Nmx\qJUR(  
    %       figure ~:JAWs$\V  
    %       pcolor(x,x,z), shading interp E}4{{{r  
    %       axis square, colorbar Mi.2 >  
    %       title('Zernike function Z_5^1(r,\theta)') A]m*~Vj]  
    % R7rM$|n=o  
    %   Example 2: |5(un#  
    % q}Po)IUT`5  
    %       % Display the first 10 Zernike functions 4Vi*Qa_,y  
    %       x = -1:0.01:1; D-@6 hWh~  
    %       [X,Y] = meshgrid(x,x); gWHY7rv  
    %       [theta,r] = cart2pol(X,Y); s;P _LaIp)  
    %       idx = r<=1; #8t=vb3  
    %       z = nan(size(X)); 9K}DmS  
    %       n = [0  1  1  2  2  2  3  3  3  3]; WrwbLlE  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; MX~h>v3_R4  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; N_:!uR  
    %       y = zernfun(n,m,r(idx),theta(idx)); by9UwM=gp  
    %       figure('Units','normalized') &kd W(;`  
    %       for k = 1:10 xb[yy}>"L  
    %           z(idx) = y(:,k); gAvNm[=wD2  
    %           subplot(4,7,Nplot(k)) $o+@}B0)  
    %           pcolor(x,x,z), shading interp Q-h< av9  
    %           set(gca,'XTick',[],'YTick',[]) @}UOm- M  
    %           axis square Wp = ]YO  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) RoHX0   
    %       end M GC=L .  
    % _@\-`>J  
    %   See also ZERNPOL, ZERNFUN2. Wx/PD=Sf&  
    8B6(SQp%  
    %   Paul Fricker 11/13/2006 q) 5s'(  
    T^8`ji  
    'GW~~UhdW  
    % Check and prepare the inputs: }:?_/$};  
    % ----------------------------- rr1,Ijh{D  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }}Q h_(  
        error('zernfun:NMvectors','N and M must be vectors.') y1Br4K5C  
    end #?M[Q:  
    KxmB$x5-=8  
    if length(n)~=length(m) sFfargl  
        error('zernfun:NMlength','N and M must be the same length.') )MN6\v  
    end V+' zuX  
    "5,Cy3  
    n = n(:); B_c-@kl   
    m = m(:); Jk<b#SZ[b  
    if any(mod(n-m,2)) o9D#d\G  
        error('zernfun:NMmultiplesof2', ... +^,&z}( Ak  
              'All N and M must differ by multiples of 2 (including 0).') {R~L7uR @O  
    end U&+lw=  
    l>Zp#+I-  
    if any(m>n) /ubGa6N  
        error('zernfun:MlessthanN', ... W}^>lM\8  
              'Each M must be less than or equal to its corresponding N.') 5n2}|V$VqP  
    end "8[Vb#=*e  
    Xs4G#QsA J  
    if any( r>1 | r<0 ) Ag]Hk %  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') XKBQH(  
    end rYyEs I#qo  
    A@EUH  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) F>q%~  
        error('zernfun:RTHvector','R and THETA must be vectors.') 4y9n,~Qgw  
    end aj]%c_])(  
    (@*#Pn|A  
    r = r(:); rI1;>/Ir  
    theta = theta(:); x6~`{N1N M  
    length_r = length(r); CY8=prC  
    if length_r~=length(theta) wW;!L =j  
        error('zernfun:RTHlength', ... +(2mHS0_a  
              'The number of R- and THETA-values must be equal.')  N5GQ2V  
    end 5zI I4ukn*  
    Qte'f+  
    % Check normalization: FBK6{rLMc  
    % -------------------- uJHf6Ye  
    if nargin==5 && ischar(nflag) 9L xa?Y1  
        isnorm = strcmpi(nflag,'norm'); 7b[vZNi_  
        if ~isnorm yn5yQ;  
            error('zernfun:normalization','Unrecognized normalization flag.') JS1''^G&.  
        end W 7Y5~%@  
    else {p(.ck ze+  
        isnorm = false; hGvuA9d~  
    end 0 /JusQ  
    B?J #NFUb  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g"sW_y_O  
    % Compute the Zernike Polynomials Gv w:h9v  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  OL|UOG  
    miZ&9m  
    % Determine the required powers of r: Ey!+rq}  
    % ----------------------------------- Cuq=>J  
    m_abs = abs(m); p M:lg  
    rpowers = []; %g4G&My@J  
    for j = 1:length(n) H`;q@  
        rpowers = [rpowers m_abs(j):2:n(j)]; r4h4A w{  
    end #;6YADk2_  
    rpowers = unique(rpowers); 4b B)t#  
    BVX6  
    % Pre-compute the values of r raised to the required powers, %P2GQS-N  
    % and compile them in a matrix: c _li.]P  
    % ----------------------------- T8 ,?\7)S9  
    if rpowers(1)==0 :!\?yj{{  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); E}d@0C:  
        rpowern = cat(2,rpowern{:}); .>0j<|~  
        rpowern = [ones(length_r,1) rpowern]; ?6F\cl0.  
    else ^b]h4z$  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);  s=&&gC1  
        rpowern = cat(2,rpowern{:}); 9"3 7va  
    end ]4m;NId  
    )B86  
    % Compute the values of the polynomials: bZ0mK$B  
    % -------------------------------------- RG9YA&1ce  
    y = zeros(length_r,length(n)); O9#8%p% )  
    for j = 1:length(n) /G`'9cD  
        s = 0:(n(j)-m_abs(j))/2; dBKL_'@@}  
        pows = n(j):-2:m_abs(j); WleE$ ,  
        for k = length(s):-1:1 [=[>1<L>  
            p = (1-2*mod(s(k),2))* ... owDp?Sy}E  
                       prod(2:(n(j)-s(k)))/              ... ]_6w(>A@3#  
                       prod(2:s(k))/                     ... 18ApHp  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... >YwvM=b"V  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); RjC3wO::  
            idx = (pows(k)==rpowers); OT[&a6_  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ,%>]  
        end 4PtRTb0<i3  
         ,Jm2|WKH  
        if isnorm V2As 5  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); =hZ#Z]f  
        end v?Z30?_&h  
    end e :(7$jo  
    % END: Compute the Zernike Polynomials |]--sUx:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b.mcP@  
    gEghDO_G  
    % Compute the Zernike functions:  GtR!a  
    % ------------------------------ aQjs5RbP~  
    idx_pos = m>0; YfRjr  
    idx_neg = m<0; ENZjRf4  
    +,7nsWV  
    z = y; 63'Rw'g^|2  
    if any(idx_pos) _"_ 21uB  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); _C`&(?}  
    end /g/]Q^  
    if any(idx_neg) srzlr-J  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); p **Sd[|  
    end {5 V@O_*{  
    O`?qnNmc;  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) ?4`f@=}'K  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. $~3?nib"j  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ] /"!J6(e  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive MZrLLnl6\  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, zQxTPd  
    %   and THETA is a vector of angles.  R and THETA must have the same 9^?2{aP%  
    %   length.  The output Z is a matrix with one column for every P-value, 9]L4`.HM  
    %   and one row for every (R,THETA) pair. $6l^::U  
    % +xL' LC x  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Gh5 3 Pne  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) {i<L<Y(3  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) IKrojK8-?  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 {_G_YL[  
    %   for all p. 5E#8F  
    % 1f+z[ad&^  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 F2>W{-H+  
    %   Zernike functions (order N<=7).  In some disciplines it is /g>]J70  
    %   traditional to label the first 36 functions using a single mode {,=U]^A  
    %   number P instead of separate numbers for the order N and azimuthal % !>@m6JK  
    %   frequency M. !lL~#l:F  
    % j.yh>"de  
    %   Example: \;+TZ1i_  
    % d" =)=hm!  
    %       % Display the first 16 Zernike functions Obx!>mI^6  
    %       x = -1:0.01:1; hZ|8mV  
    %       [X,Y] = meshgrid(x,x); RGLJaEl !  
    %       [theta,r] = cart2pol(X,Y); M4n0GWHLy  
    %       idx = r<=1; Gb4p "3  
    %       p = 0:15; 9JqT"zj  
    %       z = nan(size(X)); k]9y+WC2  
    %       y = zernfun2(p,r(idx),theta(idx)); gDjAnz#  
    %       figure('Units','normalized') K}DrJ/s  
    %       for k = 1:length(p) B2:GGZ|jS  
    %           z(idx) = y(:,k); p@?ud%  
    %           subplot(4,4,k) I%jlM0ZUI"  
    %           pcolor(x,x,z), shading interp !iL6/  
    %           set(gca,'XTick',[],'YTick',[]) mYqLqezAA  
    %           axis square jFl!<ooCo  
    %           title(['Z_{' num2str(p(k)) '}']) Jv8VM\ *  
    %       end ?jsgBol  
    % hG}gKs  
    %   See also ZERNPOL, ZERNFUN. P\h1%a/D  
    braI MIQ`  
    %   Paul Fricker 11/13/2006 w3;T]R*  
    9 RC:-d;;_  
    Y=/;7T  
    % Check and prepare the inputs: Y}h&dAr  
    % ----------------------------- '8LHX6FXK  
    if min(size(p))~=1 HP=5 a.  
        error('zernfun2:Pvector','Input P must be vector.') EL6<%~,V"I  
    end yQq|!'MKk  
    >ktekO:H  
    if any(p)>35 D77$aCt  
        error('zernfun2:P36', ... q8& ^E.K  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Gw#z:gX2  
               '(P = 0 to 35).']) (\4YBaGd  
    end sUZ2A1J}  
    b~?3HY:t~K  
    % Get the order and frequency corresonding to the function number: \,G19o}`Es  
    % ---------------------------------------------------------------- ;(s.G-9S  
    p = p(:); JY9hD;`6y  
    n = ceil((-3+sqrt(9+8*p))/2); RPkOtRKL=w  
    m = 2*p - n.*(n+2); &>Z p}.V  
    { /Gm|*e{  
    % Pass the inputs to the function ZERNFUN: UO' X"`  
    % ---------------------------------------- G 'CYvV  
    switch nargin % %QAC4  
        case 3 ?e23[  
            z = zernfun(n,m,r,theta); 30h1)nQ$h}  
        case 4 &_Z8:5e  
            z = zernfun(n,m,r,theta,nflag); &uJ7[m19z  
        otherwise BtP*R,>  
            error('zernfun2:nargin','Incorrect number of inputs.') Q?Uk%t\hwc  
    end 3M{b:|3/q  
    g%d&>y?1r  
    % 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) g,cl|]/\d  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M.  W,)qE^+  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ^<O:`c6_  
    %   order N and frequency M, evaluated at R.  N is a vector of ;u: }rA)  
    %   positive integers (including 0), and M is a vector with the ^!>o5Y)  
    %   same number of elements as N.  Each element k of M must be a [8.w2\<?  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) H"> }y D  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is {pNf& '  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Y+I`XeY  
    %   with one column for every (N,M) pair, and one row for every Sud5F4S  
    %   element in R. sGD b<  
    % I|?Z.!I|  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- <U]#722  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is :S5B3S@|  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Ka\%kB>*`  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 URD<KIN>  
    %   for all [n,m]. 0BTLIV$d;  
    % yegTKoY  
    %   The radial Zernike polynomials are the radial portion of the fw1g;;E  
    %   Zernike functions, which are an orthogonal basis on the unit nP>*0Fq  
    %   circle.  The series representation of the radial Zernike ]='E&=nc  
    %   polynomials is *tda_B 2  
    % <99Xg_e  
    %          (n-m)/2 \}e1\MiZ  
    %            __ -)tu$W*  
    %    m      \       s                                          n-2s V4OhdcW{  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r [$Ld>`3  
    %    n      s=0 ##!) }i  
    % M"]~}*  
    %   The following table shows the first 12 polynomials. YGObTIGJvf  
    % WKQVT I&A.  
    %       n    m    Zernike polynomial    Normalization 451r!U1Z  
    %       --------------------------------------------- {7)D/WY5  
    %       0    0    1                        sqrt(2) 4cql?W(D  
    %       1    1    r                           2 lV-7bZ  
    %       2    0    2*r^2 - 1                sqrt(6) vvLm9Tw  
    %       2    2    r^2                      sqrt(6) @U%I 6 t  
    %       3    1    3*r^3 - 2*r              sqrt(8) /)xG%J7H  
    %       3    3    r^3                      sqrt(8) !c\d(u  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) H'$g!Pg  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ;PJWd|3  
    %       4    4    r^4                      sqrt(10) 8ltHR]v  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) B3'qmi<  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 0CxQ@~ttl  
    %       5    5    r^5                      sqrt(12) >'/G:\M>A  
    %       --------------------------------------------- mk1;22o{TX  
    % d+%1q  
    %   Example: uRKCvsisX  
    %  pFGK-J  
    %       % Display three example Zernike radial polynomials E) >~0jv  
    %       r = 0:0.01:1; rB|D^@mG  
    %       n = [3 2 5]; Gu<3*@Ng  
    %       m = [1 2 1]; ~ -Rr[O=E  
    %       z = zernpol(n,m,r); r^ &{0c&o  
    %       figure {!xPq%  
    %       plot(r,z) xUPM-eF=  
    %       grid on }#q9>gx  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') (hd^  
    % )ye[R^!}  
    %   See also ZERNFUN, ZERNFUN2. qI<6% ^i  
    BXaA#} ;e  
    % A note on the algorithm. J& +s  
    % ------------------------ 'rRo2oTN  
    % The radial Zernike polynomials are computed using the series jwTb09  
    % representation shown in the Help section above. For many special 0kpRvdEr-  
    % functions, direct evaluation using the series representation can ? 8S0  
    % produce poor numerical results (floating point errors), because oGly|L>  
    % the summation often involves computing small differences between 2;5EH 0  
    % large successive terms in the series. (In such cases, the functions WhSQ>h!@s  
    % are often evaluated using alternative methods such as recurrence Mvrc[s+o  
    % relations: see the Legendre functions, for example). For the Zernike 0(Z ER sP  
    % polynomials, however, this problem does not arise, because the <dD}4c+/t  
    % polynomials are evaluated over the finite domain r = (0,1), and R(=Lhz6R4  
    % because the coefficients for a given polynomial are generally all W cPDPu~/  
    % of similar magnitude. OTL=(k  
    %  ^P~%^?(  
    % ZERNPOL has been written using a vectorized implementation: multiple B1JdkL 3h  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] c7q1;X{:  
    % values can be passed as inputs) for a vector of points R.  To achieve B+iVK(j'[v  
    % this vectorization most efficiently, the algorithm in ZERNPOL R?(0:f  
    % involves pre-determining all the powers p of R that are required to ?a7PxD.  
    % compute the outputs, and then compiling the {R^p} into a single  ^vYH"2  
    % matrix.  This avoids any redundant computation of the R^p, and _jR%o1Y}  
    % minimizes the sizes of certain intermediate variables. \ZigG{  
    % "'4R _R  
    %   Paul Fricker 11/13/2006 MmI4J$F  
    n %"q>  
     &xgMqv2/  
    % Check and prepare the inputs: Ha@'%<gFe  
    % ----------------------------- wxkCmrV  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2^bq4c4J  
        error('zernpol:NMvectors','N and M must be vectors.') e ,/I}W  
    end hq6fDRO/4  
    XpdDIKMmE  
    if length(n)~=length(m) GRB/N1=  
        error('zernpol:NMlength','N and M must be the same length.') \6-x~%xK  
    end !lKO|Y  
    N#Y%+1  
    n = n(:); dQYb)4ir  
    m = m(:); ;gY W!rM  
    length_n = length(n); WW{5[;LYiB  
    !jN}n)FSq  
    if any(mod(n-m,2)) l_hM,]T0  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') QM2Y?."#  
    end nT..+ J)  
    uz-,)  
    if any(m<0) B]L5K~d  
        error('zernpol:Mpositive','All M must be positive.') >G$8\&]j  
    end vH%AXz IA  
    'iA#lKG  
    if any(m>n) 'MRvH lCM  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') -$**/~0zU  
    end <`k\kZM  
    Aayh'xQ  
    if any( r>1 | r<0 ) 8] skAh  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') +%Q:  
    end \ZX5dFu0  
    %tul(Z~<1  
    if ~any(size(r)==1) Se<]g$eK?5  
        error('zernpol:Rvector','R must be a vector.') I=o[\?u*_  
    end `bT!_Ru  
    Ko_Sx.  
    r = r(:); ma9q?H#X  
    length_r = length(r); d0Xb?- }3M  
    >}Qj|05G  
    if nargin==4 9pUvw_9MY  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); E9z^#@s  
        if ~isnorm <k?ofE1o  
            error('zernpol:normalization','Unrecognized normalization flag.') ZycV?ob8}  
        end 28FC@&'H  
    else N`XJA-DE  
        isnorm = false; ?PVJeFH  
    end ie|I*;#  
    :MeshzWK  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M2HO!btf  
    % Compute the Zernike Polynomials AzAD76iNv  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *z[G+JX  
    dp`xyBQ3  
    % Determine the required powers of r:  o&uO]  
    % ----------------------------------- $$ %4,\{l  
    rpowers = []; Dy6uWv,P  
    for j = 1:length(n) ?_mcg8A@@*  
        rpowers = [rpowers m(j):2:n(j)]; [-o`^;  
    end 5\93-e  
    rpowers = unique(rpowers); @sQ^6FK0G  
    ",/3PT  
    % Pre-compute the values of r raised to the required powers, P^m+SAAB  
    % and compile them in a matrix: Ow7NOhw  
    % ----------------------------- z_qy >  
    if rpowers(1)==0 GC?X>AC:  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); OLXkiesK{  
        rpowern = cat(2,rpowern{:}); F) w.q  
        rpowern = [ones(length_r,1) rpowern]; *M5 : \+  
    else m87,N~DP  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); zJ{?'kp  
        rpowern = cat(2,rpowern{:}); whb|N2  
    end \s<7!NAE4  
    JFaxxW  
    % Compute the values of the polynomials: ^NJ]~h{n$  
    % -------------------------------------- mv@cGdxu  
    z = zeros(length_r,length_n); bc}X.IC  
    for j = 1:length_n O v3W;jD  
        s = 0:(n(j)-m(j))/2; }.Eq_wP<  
        pows = n(j):-2:m(j); [?)=3Pp  
        for k = length(s):-1:1 8r[ZGUV  
            p = (1-2*mod(s(k),2))* ...  Q(SVJ  
                       prod(2:(n(j)-s(k)))/          ... F-}-/N]o q  
                       prod(2:s(k))/                 ... wUzQ`h2  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... V~LZ%NZ8  
                       prod(2:((n(j)+m(j))/2-s(k))); iO=xx|d  
            idx = (pows(k)==rpowers); kg[u@LgvoN  
            z(:,j) = z(:,j) + p*rpowern(:,idx); r}) 2-3ZA9  
        end ~ZU;0#  
         Tks;,C  
        if isnorm ^}; 4r  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 8?] :>  
        end Z:f0>  
    end \V@SCA'  
    !- f>*|@  
    % 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)  ;<MaCtDt  
    ~Yr.0i.W  
    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。 +m+HC(Z  
    /4T%&#6s  
    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)