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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;' 37o#q  
    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! { wx!~K  
     
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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 Wi Mi0?$.  
    function z = zernfun(n,m,r,theta,nflag) ?[}r& f  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. i[_WO2  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1>1&NQ#}  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 25RFi24>D  
    %   unit circle.  N is a vector of positive integers (including 0), and B`x rdtW  
    %   M is a vector with the same number of elements as N.  Each element ^-9g_5  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ruG5~dm>  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, tk@ T-;  
    %   and THETA is a vector of angles.  R and THETA must have the same _h2axXFhT  
    %   length.  The output Z is a matrix with one column for every (N,M) P\B ]><!ep  
    %   pair, and one row for every (R,THETA) pair. h|tdK;)  
    % zU;%s<(p  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 'a`cK;X9F  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), $ \j/s:Y  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral `<1o}r 7i  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, "#d>3M_  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized K]{Y >w  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. J|-X?V;ZW  
    % *"\QR>n   
    %   The Zernike functions are an orthogonal basis on the unit circle. (,wIbwa  
    %   They are used in disciplines such as astronomy, optics, and 5G"DgG*<  
    %   optometry to describe functions on a circular domain.  $^F L*w  
    % bhqBFiuhH  
    %   The following table lists the first 15 Zernike functions. 88]V6Rm9[*  
    % AM4lAq_  
    %       n    m    Zernike function           Normalization \a+.~_iL|  
    %       -------------------------------------------------- SW!lSIk  
    %       0    0    1                                 1 4NaL#3  
    %       1    1    r * cos(theta)                    2 #1-,s.)  
    %       1   -1    r * sin(theta)                    2 9?5'>WO  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) fk5xIW  
    %       2    0    (2*r^2 - 1)                    sqrt(3) OT[&a6_  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 1]Q;fe  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) WZ\bm$  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) R_IUuz$e  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) N?Byp&rqI<  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) % ~eIx=s  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 9K]Li\  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f;AQw_{  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Ah5`Cnv  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) x3j)'`=15  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) TPjElBh  
    %       -------------------------------------------------- 'MLp*3djF,  
    % $T.u Iq  
    %   Example 1: |$*1!pL-QP  
    % w;@NYMK)  
    %       % Display the Zernike function Z(n=5,m=1) |]--sUx:  
    %       x = -1:0.01:1; *$K_Tii  
    %       [X,Y] = meshgrid(x,x); e[<vVe!  
    %       [theta,r] = cart2pol(X,Y); a8D7n Ea  
    %       idx = r<=1; us j:I`>  
    %       z = nan(size(X)); >KPxksFR8  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 7Gwn,&)  
    %       figure aQjs5RbP~  
    %       pcolor(x,x,z), shading interp ;gS)o#v0  
    %       axis square, colorbar muh[wo  
    %       title('Zernike function Z_5^1(r,\theta)') &8p]yo2zO  
    % w ]8+ OP  
    %   Example 2: :1>h,NKC>  
    % yx0wR  
    %       % Display the first 10 Zernike functions 63'Rw'g^|2  
    %       x = -1:0.01:1; bSa%?laS  
    %       [X,Y] = meshgrid(x,x); cQg:yoF  
    %       [theta,r] = cart2pol(X,Y); 6pJFrWe{  
    %       idx = r<=1; 4eF qD;  
    %       z = nan(size(X)); R;mA2:W)x  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 73Zx`00  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; <{ZDD]UGs0  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; s fD@lW3  
    %       y = zernfun(n,m,r(idx),theta(idx)); 0d>|2QV   
    %       figure('Units','normalized') 0m2%ucKw  
    %       for k = 1:10 &>nB@SQZ  
    %           z(idx) = y(:,k); I /2{I  
    %           subplot(4,7,Nplot(k)) eILdq*  
    %           pcolor(x,x,z), shading interp )RUx  
    %           set(gca,'XTick',[],'YTick',[]) JM&`&fsOC{  
    %           axis square <M){rce  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) %#go9H(K  
    %       end xG_LEk( zD  
    % nXU`^<nA  
    %   See also ZERNPOL, ZERNFUN2. W;Y"J_  
    4{PN9i E  
    %   Paul Fricker 11/13/2006 ;H' ,PjU  
    ys/U.e|)!  
    kAV4V;ydh  
    % Check and prepare the inputs: qjr:(x/  
    % ----------------------------- 1k)31GEQw  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) aB_~V h  
        error('zernfun:NMvectors','N and M must be vectors.') ZVX1@p  
    end hkpS}*L9o  
    :9H`O!VF  
    if length(n)~=length(m) ;*c8,I;  
        error('zernfun:NMlength','N and M must be the same length.') ltWEA  
    end |*fi!nvk@  
    AU$<W"%R  
    n = n(:); eoj(zY3  
    m = m(:); =67ab_V  
    if any(mod(n-m,2)) (G6lr%d  
        error('zernfun:NMmultiplesof2', ... R$Rub/b6  
              'All N and M must differ by multiples of 2 (including 0).') "lV bla4b  
    end /wi*OZ7R  
    ge#0Q L0K  
    if any(m>n)  2S  
        error('zernfun:MlessthanN', ... }j)][{i*x  
              'Each M must be less than or equal to its corresponding N.') *Uw"`l  
    end PIHix{YR  
    8l>7=~Egp  
    if any( r>1 | r<0 ) ul-O3]\'@  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') w#d7  
    end 9oj#5Hq  
    N,bH@Q.Ci  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 7VIfRN{5n  
        error('zernfun:RTHvector','R and THETA must be vectors.') j'uzjs[  
    end ]."t  
    {i<L<Y(3  
    r = r(:); M7fPaJKL  
    theta = theta(:); Vl^p3f[  
    length_r = length(r); "8$Muwm  
    if length_r~=length(theta) `t7z LC^c  
        error('zernfun:RTHlength', ... w-"tA`F4  
              'The number of R- and THETA-values must be equal.') t`- [  
    end &c^tJ-s  
    5oe{i/#di  
    % Check normalization: 5yL\@7u`  
    % -------------------- Wh)>E!~ 9  
    if nargin==5 && ischar(nflag) P(b ds  
        isnorm = strcmpi(nflag,'norm'); r,<p#4(>_  
        if ~isnorm I]z4}#+cX  
            error('zernfun:normalization','Unrecognized normalization flag.') -]Ny-[P  
        end s7(1|}jh  
    else !lL~#l:F  
        isnorm = false; gXj3=N(l  
    end OI,F,4e  
    ~}_S]^br  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I.gF38Mx  
    % Compute the Zernike Polynomials WR9-HPF  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #z61 I"kU  
    D4T42L  
    % Determine the required powers of r: V)fF|E~0  
    % ----------------------------------- rMoz+{1A  
    m_abs = abs(m); v*kX?J#]5  
    rpowers = []; ~#dfZa&   
    for j = 1:length(n) SN 4JX  
        rpowers = [rpowers m_abs(j):2:n(j)]; Cb6K!5[q]  
    end Gb4p "3  
    rpowers = unique(rpowers); Mn 8| K nh  
    0Q~\1D 9g  
    % Pre-compute the values of r raised to the required powers, <Zo{D |hW  
    % and compile them in a matrix: !ir%Pz ^)  
    % ----------------------------- ?jU 3%"  
    if rpowers(1)==0 dbg%n 0h  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); `k7X|  
        rpowern = cat(2,rpowern{:}); /&E]qc*-p  
        rpowern = [ones(length_r,1) rpowern]; I%jlM0ZUI"  
    else y\n#`*5k  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); WQ9VcCY  
        rpowern = cat(2,rpowern{:}); *|^|| bd  
    end $k+XH+1CW  
    ,E8g~ZUY9  
    % Compute the values of the polynomials: w ^ X@PpP  
    % -------------------------------------- ,uD}1 G<u  
    y = zeros(length_r,length(n)); ~"Su2{"8B  
    for j = 1:length(n) E;YD5^B  
        s = 0:(n(j)-m_abs(j))/2; SB:z[kfz|  
        pows = n(j):-2:m_abs(j); (ylZ[M&B:  
        for k = length(s):-1:1 +fHqGZ]  
            p = (1-2*mod(s(k),2))* ... h(i_'P?  
                       prod(2:(n(j)-s(k)))/              ... Lie= DD  
                       prod(2:s(k))/                     ... '8LHX6FXK  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 4O4}C#6(4  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); PB #EU 9  
            idx = (pows(k)==rpowers); IH"_6s#$&  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 9Qq%Fw_  
        end }vZTiuzC  
         u}7r\MnwK,  
        if isnorm ;+n25_9  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); (\4YBaGd  
        end &-KQ m20n  
    end X=VaBy4#  
    % END: Compute the Zernike Polynomials GXR7Ug}k  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U8eU[|-8O/  
    ]_hXg*?  
    % Compute the Zernike functions: 0{u#{_  
    % ------------------------------ R4XcWx*pQ  
    idx_pos = m>0; XeozRfk%J|  
    idx_neg = m<0; { /Gm|*e{  
    OQ _wsAA  
    z = y; T_qh_L3  
    if any(idx_pos) V6b)  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); mZ.E;X& ,*  
    end ^ |>)H  
    if any(idx_neg) G EAVc9V  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); %/wfYRp*  
    end e0<L^|S  
    tUs{/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) *!+?%e{;b  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. E %> ){Y)  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated |p+ xM  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive f%Bmx{Ttq  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, As*59jkB  
    %   and THETA is a vector of angles.  R and THETA must have the same "a >a "Ei  
    %   length.  The output Z is a matrix with one column for every P-value, veGRwir  
    %   and one row for every (R,THETA) pair. cx(b5Z  
    % Gex%~';+q  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike -\=kd {*B  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) qxglA*/ [  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) %D}]Z=gp  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ]#oqum@Yf1  
    %   for all p. :n<<hR0d  
    % 8fs::}0  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 4U1"F 7'  
    %   Zernike functions (order N<=7).  In some disciplines it is RJZ4fl  
    %   traditional to label the first 36 functions using a single mode 79Vp^GG7  
    %   number P instead of separate numbers for the order N and azimuthal })}-K7v1+  
    %   frequency M. 18U CZ;)>  
    % OQh(qa  
    %   Example: nxh9'"th  
    % ;}gS8I|  
    %       % Display the first 16 Zernike functions D>Ph))QI  
    %       x = -1:0.01:1; yasKU6^R'  
    %       [X,Y] = meshgrid(x,x); L`{EXn[  
    %       [theta,r] = cart2pol(X,Y); c/E6}OWA  
    %       idx = r<=1; 0UT2sM$  
    %       p = 0:15; 6?c(ueiL[  
    %       z = nan(size(X)); Zcn,_b7  
    %       y = zernfun2(p,r(idx),theta(idx)); ,*@6NK,.  
    %       figure('Units','normalized') A">A@`}  
    %       for k = 1:length(p) 8TnByKZz  
    %           z(idx) = y(:,k); tJ9i{TS  
    %           subplot(4,4,k) _*Z2</5  
    %           pcolor(x,x,z), shading interp SggS8$a`  
    %           set(gca,'XTick',[],'YTick',[]) URD<KIN>  
    %           axis square Kr]`.@/.S  
    %           title(['Z_{' num2str(p(k)) '}']) *u%4]q  
    %       end Ng3MfbFG  
    % DHV#PLbN$  
    %   See also ZERNPOL, ZERNFUN. i XI:yE;  
    3q.O^`y FU  
    %   Paul Fricker 11/13/2006 PDcZno?  
    It@ak6u?  
    *:}NS8hP  
    % Check and prepare the inputs: 6"W~%FSJX  
    % ----------------------------- }9xEA[@;  
    if min(size(p))~=1 DN@T4!  
        error('zernfun2:Pvector','Input P must be vector.') 6Hn3  
    end /IC7q?avQN  
    +)fl9>Mb  
    if any(p)>35 0iX;%SPYz  
        error('zernfun2:P36', ... pc w^W  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... /*bS~7f1  
               '(P = 0 to 35).']) [$Ld>`3  
    end Hs+VA$$*  
    /0mbG!Ac  
    % Get the order and frequency corresonding to the function number: NVMhbpX6  
    % ---------------------------------------------------------------- ^V~r S8]gj  
    p = p(:); YGObTIGJvf  
    n = ceil((-3+sqrt(9+8*p))/2); {qCmZn5  
    m = 2*p - n.*(n+2); B;?"R  
    3~4e\xL  
    % Pass the inputs to the function ZERNFUN: E VBB:*q6  
    % ----------------------------------------  wNW9xmS  
    switch nargin i(JBBE"  
        case 3 z2&SZ.mk  
            z = zernfun(n,m,r,theta); RTNUHz;{L  
        case 4 ?s("@dz_  
            z = zernfun(n,m,r,theta,nflag); ^Q]*CU+C  
        otherwise x aWmwsym  
            error('zernfun2:nargin','Incorrect number of inputs.') _n(NPFV  
    end Z2WAVSw  
    $@t-Oor;  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) \CL |=8[2  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. y5.Z<Y  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of mk1;22o{TX  
    %   order N and frequency M, evaluated at R.  N is a vector of &eT)c<yhyK  
    %   positive integers (including 0), and M is a vector with the vt[4"eU  
    %   same number of elements as N.  Each element k of M must be a _`L,}=um'  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) f8)D|  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is sf]y\_zU  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix LKG],1n-  
    %   with one column for every (N,M) pair, and one row for every #JGy2Hk$^  
    %   element in R. l0g#&V--  
    % Wy,DA^\ef  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ]6</{b  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is *~fZ9EkD  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to %FQMB  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 >/EmC3?b!  
    %   for all [n,m]. xmTa$tR+  
    % LGPy>,!  
    %   The radial Zernike polynomials are the radial portion of the ,-t3gc1~X  
    %   Zernike functions, which are an orthogonal basis on the unit Y*O7lZuF%  
    %   circle.  The series representation of the radial Zernike XZA3T Z  
    %   polynomials is iqghcY)  
    % e%j+,)Ry  
    %          (n-m)/2 b/'fC%o,  
    %            __ q~r )B}  
    %    m      \       s                                          n-2s )ye[R^!}  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 5SK{^hw  
    %    n      s=0 ji`N1e,l  
    % SZ~Ti|^  
    %   The following table shows the first 12 polynomials. Xcicqywe?  
    % {Zjnf6d]  
    %       n    m    Zernike polynomial    Normalization =lS~2C  
    %       --------------------------------------------- #18H Z4N  
    %       0    0    1                        sqrt(2) }?#<)|_5  
    %       1    1    r                           2 $.cNY+  k  
    %       2    0    2*r^2 - 1                sqrt(6) ^M  PU?k  
    %       2    2    r^2                      sqrt(6) >ALU}o/  
    %       3    1    3*r^3 - 2*r              sqrt(8) B>t$Z5Q^X  
    %       3    3    r^3                      sqrt(8) oGly|L>  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 95aa  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) T%KZV/  
    %       4    4    r^4                      sqrt(10) $uawQf+S  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) r`i<XGPJ%  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ss%ahs  
    %       5    5    r^5                      sqrt(12) F^IYx~:  
    %       --------------------------------------------- J+[&:]=P  
    % vd SV6p.d  
    %   Example: @W=#gRqQPy  
    % FsY}mql  
    %       % Display three example Zernike radial polynomials IQoz8!guh:  
    %       r = 0:0.01:1; X7{ueP#L  
    %       n = [3 2 5]; wtetB')yD  
    %       m = [1 2 1]; VCcLS3  
    %       z = zernpol(n,m,r); )}=`Gx5+  
    %       figure . 3=WE@M  
    %       plot(r,z) 8Cs)_bj#!  
    %       grid on lOPCM1Se  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') $f*N  
    % @YMef `T:  
    %   See also ZERNFUN, ZERNFUN2. 0lF.!\9  
    4!d&Zc>C4  
    % A note on the algorithm. v6HBO#F'V{  
    % ------------------------ 26yv w  
    % The radial Zernike polynomials are computed using the series #c'yAa  
    % representation shown in the Help section above. For many special n(/(F `  
    % functions, direct evaluation using the series representation can ?a7PxD.  
    % produce poor numerical results (floating point errors), because :f ybH)*  
    % the summation often involves computing small differences between 0V"r$7(}  
    % large successive terms in the series. (In such cases, the functions 3)T'&HKQ  
    % are often evaluated using alternative methods such as recurrence  3p"VmO  
    % relations: see the Legendre functions, for example). For the Zernike CK 3]]{  
    % polynomials, however, this problem does not arise, because the rm;'/l8Y-E  
    % polynomials are evaluated over the finite domain r = (0,1), and "L|Ew#  
    % because the coefficients for a given polynomial are generally all U voX\  
    % of similar magnitude. y!6B Gz  
    % H`njKKdR  
    % ZERNPOL has been written using a vectorized implementation: multiple 7!#x-KR~5  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] {x W? v;  
    % values can be passed as inputs) for a vector of points R.  To achieve 36*"oD=@  
    % this vectorization most efficiently, the algorithm in ZERNPOL @R_a'v-  
    % involves pre-determining all the powers p of R that are required to Q'~kWmLf  
    % compute the outputs, and then compiling the {R^p} into a single &v Lz{  
    % matrix.  This avoids any redundant computation of the R^p, and (#BkL:dg  
    % minimizes the sizes of certain intermediate variables. Y _m4:9p  
    % _~&6Kb^*  
    %   Paul Fricker 11/13/2006 2S&e!d-  
    xKWqDt  
    :@rE&  
    % Check and prepare the inputs: 2BXpk^d5y  
    % ----------------------------- u01 'f-h  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) =\B{)z7@6D  
        error('zernpol:NMvectors','N and M must be vectors.') 4[ M!x  
    end Jor >YB`X  
    !lKO|Y  
    if length(n)~=length(m) n`2 d   
        error('zernpol:NMlength','N and M must be the same length.') wOOBW0tj  
    end A07g@3n  
    J_C<Erx[O  
    n = n(:); );_g2=:#  
    m = m(:); F^ 7qLvh  
    length_n = length(n); o%i^t4J$e  
    !jN}n)FSq  
    if any(mod(n-m,2)) mv O!Y  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Bq.@CxK  
    end g34<0%6jd  
    {E-.W"t4  
    if any(m<0) SG_^Rd9 D  
        error('zernpol:Mpositive','All M must be positive.') ((Ak/qz  
    end _T&?H&#  
    mcy\nAf5%  
    if any(m>n) Y (x_bJ  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 9 [v=`  
    end =dx!R ,Bw  
    'A;G[(SYy  
    if any( r>1 | r<0 ) "~(qp_AI  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') btHN  
    end 7Ab&C&3  
    DZ92;m  
    if ~any(size(r)==1) C8rD54A'M  
        error('zernpol:Rvector','R must be a vector.') &PVos|G  
    end A-^[4&rb  
    -$**/~0zU  
    r = r(:); b6:A-jb*I  
    length_r = length(r); T6h-E^Z  
    '9c`[^  
    if nargin==4 X1&Ug ^  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); _*7h1[,{f  
        if ~isnorm . <`i!Ls  
            error('zernpol:normalization','Unrecognized normalization flag.') ^u&oS1U  
        end #no~g( !o  
    else 4e~^G  
        isnorm = false; gD10C,{  
    end  N-`Vb0;N  
    dE19_KPm[j  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h6<abT@I  
    % Compute the Zernike Polynomials Jz7a|pgep  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6w(Mb~[n  
    s0uI;WMg  
    % Determine the required powers of r: wI><kdz  
    % ----------------------------------- '?=SnjMX  
    rpowers = []; ma9q?H#X  
    for j = 1:length(n) Yv k Qh{  
        rpowers = [rpowers m(j):2:n(j)]; ;iR( Ir  
    end K]ob>wPf  
    rpowers = unique(rpowers); 4 AZ~<e\  
    $&~/`MxE  
    % Pre-compute the values of r raised to the required powers, A]ZCQ49  
    % and compile them in a matrix: oNQ;9&Z,^2  
    % ----------------------------- W&CQ87b  
    if rpowers(1)==0 ,Tc3koi  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); oJa6)+b(3  
        rpowern = cat(2,rpowern{:}); W,"|([t4.\  
        rpowern = [ones(length_r,1) rpowern]; nfpkWyIu{  
    else _J(n~"eR  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); kR$>G2$!  
        rpowern = cat(2,rpowern{:}); D,q=?~  
    end 1{+x >Pv:  
    n X4R  
    % Compute the values of the polynomials: BC*vG=a  
    % -------------------------------------- (uW/t1  
    z = zeros(length_r,length_n); j(^ot001%v  
    for j = 1:length_n ^,u0kMG5l  
        s = 0:(n(j)-m(j))/2; ALvj)I`Al  
        pows = n(j):-2:m(j); \Zc$X^}vN  
        for k = length(s):-1:1 *z[G+JX  
            p = (1-2*mod(s(k),2))* ... [M>Md-pj  
                       prod(2:(n(j)-s(k)))/          ... x{4Rm,Dxn  
                       prod(2:s(k))/                 ... *uHL'Pe;m  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 6BM[RL?T  
                       prod(2:((n(j)+m(j))/2-s(k))); !OWPwBm;  
            idx = (pows(k)==rpowers); y_O[r1MF  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Dy6uWv,P  
        end h'VN& T,  
         =|>CB  
        if isnorm :$k':0 n  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); @(>XSTh9  
        end vSty.:bY\p  
    end s2f9 5<B  
    b`"E(S/  
    % 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)  mOwgk7s[ J  
    _dqjRhu  
    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。 @_YEK3l]l  
    Ff>Y<7CQ v  
    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)