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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 z4 M1D9iPY  
    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! HDIk9WC^  
     
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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 m( 47s  
    function z = zernfun(n,m,r,theta,nflag) 8X7{vN_3K  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. pGWA\}'  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N R p.W,)i  
    %   and angular frequency M, evaluated at positions (R,THETA) on the f_6`tq m%  
    %   unit circle.  N is a vector of positive integers (including 0), and ]]uHM}l  
    %   M is a vector with the same number of elements as N.  Each element [ygF0-3ND  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) w2"]Pl  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, TZB+lj1  
    %   and THETA is a vector of angles.  R and THETA must have the same 1'KishHK=  
    %   length.  The output Z is a matrix with one column for every (N,M) :Jxh2  
    %   pair, and one row for every (R,THETA) pair. Z=$  T1|  
    % 2qj{n+  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike LtKB v 4  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), x8N|($1  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral %w"nDu2Gcv  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, >|udWd^$3  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized >SI<rR[~%  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. >1|g5  
    % \7 a4uc  
    %   The Zernike functions are an orthogonal basis on the unit circle. <+]f`c*Z  
    %   They are used in disciplines such as astronomy, optics, and  i g71/'D  
    %   optometry to describe functions on a circular domain. Kn}ub+ "J  
    % ^^?q$1k6r*  
    %   The following table lists the first 15 Zernike functions. Np,2j KF(  
    % cvo[s, p  
    %       n    m    Zernike function           Normalization =nxKttmU0  
    %       -------------------------------------------------- Z`_.x &Y  
    %       0    0    1                                 1 {BV4h%P]:  
    %       1    1    r * cos(theta)                    2 {=JF=8@A  
    %       1   -1    r * sin(theta)                    2 Ill[]O  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) fC<m^%*zgA  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Fwfo2   
    %       2    2    r^2 * sin(2*theta)             sqrt(6)  v[,Src  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) X;GfPw.m  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) i@$*Csj\9*  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) F:T GsV#  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) #@//7Bf%  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) t&RruwN_;  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) $ |<m9CW  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) !{%G0(Dv  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ]T<^{jG  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Qi=*1QAkr  
    %       -------------------------------------------------- S*t%RZ~a  
    % AFm1t2,+;  
    %   Example 1: hC<14  
    % b:MG@Hxc  
    %       % Display the Zernike function Z(n=5,m=1) ]7/gJ>g,  
    %       x = -1:0.01:1; NGTe4Crx  
    %       [X,Y] = meshgrid(x,x); AtHS@p  
    %       [theta,r] = cart2pol(X,Y); cyF4iG'M,y  
    %       idx = r<=1; La,QB3K/  
    %       z = nan(size(X)); yYTVXs`fVj  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); l5O=VqCj  
    %       figure R}{GwbF_\  
    %       pcolor(x,x,z), shading interp 8Qrpa o  
    %       axis square, colorbar (6qsKX  
    %       title('Zernike function Z_5^1(r,\theta)') nX5C< Ky  
    % HOPqxI(k  
    %   Example 2: ZF{~ih*^u  
    % ?[= U%sPu=  
    %       % Display the first 10 Zernike functions kX;$}7n  
    %       x = -1:0.01:1; )"u:ytK{  
    %       [X,Y] = meshgrid(x,x); ]0 ~qi@  
    %       [theta,r] = cart2pol(X,Y); |f' 8p8J  
    %       idx = r<=1; S@}4-\  
    %       z = nan(size(X)); z +VV}:Q  
    %       n = [0  1  1  2  2  2  3  3  3  3]; n[" 9|  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; _l&ucA  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; /1.rz{wpb  
    %       y = zernfun(n,m,r(idx),theta(idx)); OyVm(%Z   
    %       figure('Units','normalized') P Jo  
    %       for k = 1:10 kC$I2[t!  
    %           z(idx) = y(:,k); Ft-6m%  
    %           subplot(4,7,Nplot(k)) C0m\SNR  
    %           pcolor(x,x,z), shading interp BQNp$]5s  
    %           set(gca,'XTick',[],'YTick',[]) 77aX-e*=E  
    %           axis square 1f//wk|  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 3% vis\~^  
    %       end #A<"4#}  
    % J r*"V`  
    %   See also ZERNPOL, ZERNFUN2. X"/~4\tJ"  
    ;z>p8N  
    %   Paul Fricker 11/13/2006 jD9lz-Y@  
    ^gg!Me  
    z`#_F}v,m/  
    % Check and prepare the inputs: X;EJ&g/  
    % ----------------------------- +/N1_  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) z7=fDe -  
        error('zernfun:NMvectors','N and M must be vectors.') 80&D""  
    end ,wK 1=7  
    J/kH%_ >Ir  
    if length(n)~=length(m) o# {#r@,i  
        error('zernfun:NMlength','N and M must be the same length.') I'InZ0J2  
    end A ,<@m2  
    HdCk!Fv  
    n = n(:); &?T${*~  
    m = m(:); wn84?$BGd  
    if any(mod(n-m,2)) 0k1MKzi Q  
        error('zernfun:NMmultiplesof2', ... fPz=KoN  
              'All N and M must differ by multiples of 2 (including 0).') |- OHve4A  
    end !: |nI77|  
    AbY;H  
    if any(m>n) !-(J-45  
        error('zernfun:MlessthanN', ... ^5x4q  
              'Each M must be less than or equal to its corresponding N.') JQT4N[rEE  
    end l1RlYl5  
    0/Q5d,'Y[2  
    if any( r>1 | r<0 ) wAz,vq=x  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') `A{'s %$?!  
    end Z;J`5=TS  
    viV-e$s`.  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 3- )kwy6L  
        error('zernfun:RTHvector','R and THETA must be vectors.') ]h8/M7k  
    end  N|N/)  
    X[{\ 3Av  
    r = r(:); Pz {Ig  
    theta = theta(:); rC rr"O#j  
    length_r = length(r); %zQ2:iT5@=  
    if length_r~=length(theta) %kW3hQ<$  
        error('zernfun:RTHlength', ... Y_lCcu#OA  
              'The number of R- and THETA-values must be equal.') UJwq n"Q^  
    end Y[,U_GX/R  
    jl@K!=q  
    % Check normalization: 4 Q&mC"  
    % -------------------- y`+<X{V5L  
    if nargin==5 && ischar(nflag) V*uEJ6T  
        isnorm = strcmpi(nflag,'norm'); b,vL8*  
        if ~isnorm O 3}P07  
            error('zernfun:normalization','Unrecognized normalization flag.') !vrnoFVu  
        end 1eF@_Y^a!  
    else 44|03Ty  
        isnorm = false; + 1f{_v  
    end 4^BLSK~(  
    -W6V,+of  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yh$ ~*UV  
    % Compute the Zernike Polynomials o HRbAE^  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {5.?'vMp  
    )#mW7m9M#  
    % Determine the required powers of r: 10TSc j  
    % ----------------------------------- 4SBLu%=s%  
    m_abs = abs(m); : n`0)g[(  
    rpowers = []; (ai72#nFtb  
    for j = 1:length(n) cnYYs d{  
        rpowers = [rpowers m_abs(j):2:n(j)]; E =  ^-Z  
    end "mG!L$  
    rpowers = unique(rpowers); 8ZzU^x  
    -KA4Inn]5  
    % Pre-compute the values of r raised to the required powers, `F@f?*s:  
    % and compile them in a matrix: roL]v\tr  
    % ----------------------------- ]X4RnV55Q  
    if rpowers(1)==0 \O,j}O'  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); su%Z{f)#  
        rpowern = cat(2,rpowern{:}); ~.!?5(AH8z  
        rpowern = [ones(length_r,1) rpowern]; 5 u"nxT   
    else ),)Q{~&`  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 0-lPhnrp  
        rpowern = cat(2,rpowern{:}); 8Q)y%7 {6  
    end Mof)2Hbd:  
    B##C{^5A`  
    % Compute the values of the polynomials: ^M"HSewo  
    % -------------------------------------- 8L@UB6b\  
    y = zeros(length_r,length(n)); 64;oB_  
    for j = 1:length(n) #SK#k<&P  
        s = 0:(n(j)-m_abs(j))/2; Ds;Rb6WcnY  
        pows = n(j):-2:m_abs(j); Yoj~|qL  
        for k = length(s):-1:1 )lE3GDAPgZ  
            p = (1-2*mod(s(k),2))* ... d+1L5}Jn  
                       prod(2:(n(j)-s(k)))/              ... U8Cw7u2  
                       prod(2:s(k))/                     ... GF9ZL  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... av7q>NEZ!1  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); %y!   
            idx = (pows(k)==rpowers); 'aLPTVM^  
            y(:,j) = y(:,j) + p*rpowern(:,idx); e=YO.HT  
        end a  [0N,t  
         H@Kl  
        if isnorm xu0;a  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); dawVE O  
        end alWx=+d  
    end Cv gPIrl  
    % END: Compute the Zernike Polynomials F<H`8*q9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% bEEJVF0  
    cob9hj#&7  
    % Compute the Zernike functions: Z 5{*? 2  
    % ------------------------------ eimA *0Cq  
    idx_pos = m>0; ?Aj\1y4L1  
    idx_neg = m<0; O1l4gduN|i  
    ,dGFX]P  
    z = y; l;"ub^AH  
    if any(idx_pos) W ??;4  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); }A)^XZ/  
    end }7f 1(#{7  
    if any(idx_neg) v3iDh8.__  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ,APGPE}I[  
    end z{7,.S u  
    7"h=MB_  
    % 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) m bB\~n  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. A3<P li  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated * wQZ '  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive .q~,.yI&j  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Yg]FF`{p=  
    %   and THETA is a vector of angles.  R and THETA must have the same 'T #<OR  
    %   length.  The output Z is a matrix with one column for every P-value, bUZ&}(/  
    %   and one row for every (R,THETA) pair. *$*nY [/5  
    % &B{Jxc`VA  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike sf|_2sI  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) &~D.")Dz  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) h}c6+@w&-  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 10QNV=yK7s  
    %   for all p. tCF0Ah  
    % 4)c"@Zf  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 7Vof7Y <  
    %   Zernike functions (order N<=7).  In some disciplines it is }]Z,\lA  
    %   traditional to label the first 36 functions using a single mode l[x`*+ON:2  
    %   number P instead of separate numbers for the order N and azimuthal m+G0<E%  
    %   frequency M. 4 G68WBT  
    % AQ_#uxI'oa  
    %   Example: ]#WX|0''^  
    % ^.><t+tM  
    %       % Display the first 16 Zernike functions 7lBQd(  
    %       x = -1:0.01:1; ttJ:[ R'  
    %       [X,Y] = meshgrid(x,x); d/-0B<ts  
    %       [theta,r] = cart2pol(X,Y); FB^dp}  
    %       idx = r<=1; tpy :o(H  
    %       p = 0:15; A16-  
    %       z = nan(size(X)); NnSI)*%'  
    %       y = zernfun2(p,r(idx),theta(idx)); o<eWg  
    %       figure('Units','normalized') P PIG?fK)  
    %       for k = 1:length(p) SE7 (+r  
    %           z(idx) = y(:,k); V~%WKQ  
    %           subplot(4,4,k) z|4@nqqX  
    %           pcolor(x,x,z), shading interp ybuSqFy`$  
    %           set(gca,'XTick',[],'YTick',[]) mc[_> [m  
    %           axis square ^FpiQF  
    %           title(['Z_{' num2str(p(k)) '}'])  q;He:vX  
    %       end `HZHVV$~  
    % DIcyXZH<  
    %   See also ZERNPOL, ZERNFUN. @bVh?T0~F,  
    ^.$r1/U  
    %   Paul Fricker 11/13/2006 Wb-'E%K  
    ]|\>O5eeu  
    D} <o<Dk  
    % Check and prepare the inputs: f<t*#]<  
    % ----------------------------- 8Jly! =Qm5  
    if min(size(p))~=1 HT;QepY3  
        error('zernfun2:Pvector','Input P must be vector.') xhoLQD  
    end QIxJFr;>  
    ?@uK s4  
    if any(p)>35 '| Q*~Lh  
        error('zernfun2:P36', ... `^4>^  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 2/<WWfX'  
               '(P = 0 to 35).']) eVNBhR}HS  
    end Ga/\kO)x_  
    :!it7vZ  
    % Get the order and frequency corresonding to the function number: ObHz+qRG  
    % ---------------------------------------------------------------- :!*;0~#  
    p = p(:); 5]O LV1Xt  
    n = ceil((-3+sqrt(9+8*p))/2); eNk!pI7g  
    m = 2*p - n.*(n+2); CIs1*:Q9  
    SoON@h/  
    % Pass the inputs to the function ZERNFUN: n<(5B|~y  
    % ---------------------------------------- U\!LZ?gC  
    switch nargin kjYO0!C  
        case 3 #__'U6`(  
            z = zernfun(n,m,r,theta); 8 $*cfOC  
        case 4 /JY ph^3][  
            z = zernfun(n,m,r,theta,nflag); ?ia[KLt"  
        otherwise \#*;H|U.x  
            error('zernfun2:nargin','Incorrect number of inputs.') -,CndRKx  
    end Jj _+YfIM  
    L08;z  
    % 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) ]R.Vq\A%S  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. DB= cc  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of uN^qfJ'@ >  
    %   order N and frequency M, evaluated at R.  N is a vector of {qdhp_~^l  
    %   positive integers (including 0), and M is a vector with the Vy"^]5  
    %   same number of elements as N.  Each element k of M must be a xM"XNT6b  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) *:\9 T#h  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is H;8]GE2n  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix K`83C`w.  
    %   with one column for every (N,M) pair, and one row for every 5oyMR_yl  
    %   element in R. :s985sEv  
    % #)eJz1~  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 0'2{[xF  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is e:D9;`C  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to *bC^X'  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 HbVV]y  
    %   for all [n,m]. B{i;+[ase  
    % w!k4&Rb3  
    %   The radial Zernike polynomials are the radial portion of the 5'[X&r %#  
    %   Zernike functions, which are an orthogonal basis on the unit 1s\hJATfz  
    %   circle.  The series representation of the radial Zernike v-aq".XQ  
    %   polynomials is 31\^9w__8  
    % t# {>y1[29  
    %          (n-m)/2 M|]1}8d?  
    %            __ ee?ZkU#@  
    %    m      \       s                                          n-2s S; <?nz3  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ="e um7  
    %    n      s=0 L#N.pd  
    % &_^<B7aC'k  
    %   The following table shows the first 12 polynomials. _NW OSt  
    % f__WnW5h  
    %       n    m    Zernike polynomial    Normalization 6?x{-Zj ^?  
    %       --------------------------------------------- lR3^&d72?  
    %       0    0    1                        sqrt(2) 1 ![bu  
    %       1    1    r                           2 @z RB4d$  
    %       2    0    2*r^2 - 1                sqrt(6) \<>%_y'/)h  
    %       2    2    r^2                      sqrt(6) k.Nu(j"z  
    %       3    1    3*r^3 - 2*r              sqrt(8) E%:zE Q  
    %       3    3    r^3                      sqrt(8) "x^bl+_"  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) BC[d={_-  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Wm&f+{LO+K  
    %       4    4    r^4                      sqrt(10) $q+`GXc-  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) JNl+UH:.  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ;z=C]kI6M  
    %       5    5    r^5                      sqrt(12) A^pp'{ !.  
    %       --------------------------------------------- xT8"+}  
    % J8Db AB4X  
    %   Example: Kn\(Xd.>  
    % J>PV{N  
    %       % Display three example Zernike radial polynomials ,99G2E v4c  
    %       r = 0:0.01:1; m%\[1|N  
    %       n = [3 2 5]; 1dO8[5uM7a  
    %       m = [1 2 1]; jYZWf `X~  
    %       z = zernpol(n,m,r); !AHm+C_=Lg  
    %       figure Z.(x|Q9  
    %       plot(r,z) /)|y+<E]}  
    %       grid on 7rg[5hP T  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') F'*&-l  
    % 0G 1o3[F  
    %   See also ZERNFUN, ZERNFUN2. f}!26[_9{  
    #|i{#~gxM  
    % A note on the algorithm. o$ k$  
    % ------------------------ %Q~Lk]B?t  
    % The radial Zernike polynomials are computed using the series #4u; `j"4=  
    % representation shown in the Help section above. For many special *p!dd?8  
    % functions, direct evaluation using the series representation can \ChcJth@o<  
    % produce poor numerical results (floating point errors), because ge8zh/`  
    % the summation often involves computing small differences between NR@Tj]`k  
    % large successive terms in the series. (In such cases, the functions [40 YoVlfM  
    % are often evaluated using alternative methods such as recurrence TI  
    % relations: see the Legendre functions, for example). For the Zernike b1o(CG(}*  
    % polynomials, however, this problem does not arise, because the k 'b|#c9c  
    % polynomials are evaluated over the finite domain r = (0,1), and h`j gF  
    % because the coefficients for a given polynomial are generally all PEl]HI_H  
    % of similar magnitude. ,;18:  
    % _F E F+I  
    % ZERNPOL has been written using a vectorized implementation: multiple xw H`alu  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 20)Il:x  
    % values can be passed as inputs) for a vector of points R.  To achieve !W7ekPnK  
    % this vectorization most efficiently, the algorithm in ZERNPOL |L&V-f&K  
    % involves pre-determining all the powers p of R that are required to Uo ,3 lMr  
    % compute the outputs, and then compiling the {R^p} into a single N~d]}J8}gx  
    % matrix.  This avoids any redundant computation of the R^p, and <|iU+.j\  
    % minimizes the sizes of certain intermediate variables. < i|+p1t  
    % w%\;|y4+  
    %   Paul Fricker 11/13/2006 u{,^#I}  
    p]oo^  
    tPHiz%  
    % Check and prepare the inputs: ja2]VbB  
    % ----------------------------- 9g"H9)EZ^  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 5 S& >9l  
        error('zernpol:NMvectors','N and M must be vectors.') TW0^wSm  
    end 1<"kN^  
    /<Et   
    if length(n)~=length(m) IuF-bxA  
        error('zernpol:NMlength','N and M must be the same length.') c[$oR,2b13  
    end L\[jafb_`  
    MC@cT^Z^  
    n = n(:); zZHsS$/  
    m = m(:); |T%/d#b~  
    length_n = length(n); +h/$_5  
    _HQa3wj  
    if any(mod(n-m,2)) ~Zm(p*\T  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 2LdV=ifq2S  
    end 5;l_-0=  
    4E)[<%  
    if any(m<0) Q~kwUZ  
        error('zernpol:Mpositive','All M must be positive.') ;Z*RCuwg  
    end ($TxVFNT  
    oSoG&4  
    if any(m>n) TxWj gW~  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') L"a#Uu8  
    end |7-tUHMo[  
    S}/CzQ  
    if any( r>1 | r<0 ) ?H`LrL/k  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') wSK?mS6  
    end ,3j*D+  
    c#DTL/8"DO  
    if ~any(size(r)==1) ORoraEK  
        error('zernpol:Rvector','R must be a vector.') {~"=6iyj  
    end a lR}|ez  
    S;g~xo  
    r = r(:); V4H+m,R  
    length_r = length(r); eD3F%wxz  
    WJ*DWyd''  
    if nargin==4 h:;~)={"X  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); hmo?gD<  
        if ~isnorm L{ -w9(S`i  
            error('zernpol:normalization','Unrecognized normalization flag.') ^cNP ?7g7  
        end dXj.e4,m  
    else /d4xHt5a  
        isnorm = false; 4$^=1ax  
    end L0Cf@~k  
    [Dhc9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8/CGg_C1  
    % Compute the Zernike Polynomials vB p5&*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]~P?  
    KK+Mxoj,  
    % Determine the required powers of r: +CkK4<dF  
    % ----------------------------------- uFqH_04  
    rpowers = []; [D)A+  
    for j = 1:length(n) $I9zJ"*  
        rpowers = [rpowers m(j):2:n(j)]; p,+~dn;=  
    end + |,CIl+  
    rpowers = unique(rpowers); }?JO[Q +  
    -4]6tt'G  
    % Pre-compute the values of r raised to the required powers, tL~|/C)d R  
    % and compile them in a matrix: r\] WDX!`  
    % ----------------------------- ^:],JN k  
    if rpowers(1)==0 "`S61m_  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 1pK7EK3R  
        rpowern = cat(2,rpowern{:}); mf3G$=[  
        rpowern = [ones(length_r,1) rpowern]; N%-nxbI\  
    else uzo}?X#  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); C{) )T5G  
        rpowern = cat(2,rpowern{:}); o8,K1ic5#  
    end 5~kf:U%~  
    86_Zh5:  
    % Compute the values of the polynomials: Hq9(6w9w  
    % -------------------------------------- m0P5a%D  
    z = zeros(length_r,length_n); fq(e~Aqw$  
    for j = 1:length_n )_jO8 )jB  
        s = 0:(n(j)-m(j))/2; q=bXHtU  
        pows = n(j):-2:m(j); ";~#epPkX  
        for k = length(s):-1:1 n)0{mDf%  
            p = (1-2*mod(s(k),2))* ... E.}Zmr#H  
                       prod(2:(n(j)-s(k)))/          ... `/U:u9H9v  
                       prod(2:s(k))/                 ... >3bpa<M_  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... *M*k-Z':.*  
                       prod(2:((n(j)+m(j))/2-s(k))); (ZnA#%  
            idx = (pows(k)==rpowers); I/tzo(r  
            z(:,j) = z(:,j) + p*rpowern(:,idx); itP_Vxo/H  
        end -K0tK~%q  
         Jx}5`{\  
        if isnorm [bp"U*!9P  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); zumRbrz  
        end SlZu-4J.-  
    end JB-j@  
    p)oW'#@a  
    % 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*1UNQp@]O  
    1K`A.J:Uy  
    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。 4Y2>w  
    ra&C|"~E  
    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)