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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 WAJ KP"  
    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!  2.'hr/.  
     
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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 15OzO.Ud  
    function z = zernfun(n,m,r,theta,nflag) E6M*o+Y  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. z m]R76  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ZD4aT1|Q7  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 204"\ mv  
    %   unit circle.  N is a vector of positive integers (including 0), and &P"13]^@  
    %   M is a vector with the same number of elements as N.  Each element u"m TS&  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) kSEgq<i!  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ct<XKqbI  
    %   and THETA is a vector of angles.  R and THETA must have the same AQ,"):ofvT  
    %   length.  The output Z is a matrix with one column for every (N,M) C_yNSD  
    %   pair, and one row for every (R,THETA) pair. QL*RzFAD 3  
    % /IF?|71,m  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike tH#t8Tq5x  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 5rmQ:8_5  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral r! [Qpb-:  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ;#mm_*L%@  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized zGy+jeH:.  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. .`(YCn?\  
    % 'H#0-V"=  
    %   The Zernike functions are an orthogonal basis on the unit circle. .{|SKhXk  
    %   They are used in disciplines such as astronomy, optics, and YMVi7D~;Q$  
    %   optometry to describe functions on a circular domain. yYSoJqj Q  
    % L >)|l  
    %   The following table lists the first 15 Zernike functions. #Nad1C/]  
    % <$d2m6J  
    %       n    m    Zernike function           Normalization _>;{+XRX[  
    %       -------------------------------------------------- 'K01"`#  
    %       0    0    1                                 1 <PM.4B@  
    %       1    1    r * cos(theta)                    2 <j/wK]d*/  
    %       1   -1    r * sin(theta)                    2 e)m6xiZ  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) p<?lF   
    %       2    0    (2*r^2 - 1)                    sqrt(3) 2EYWX! Bx  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) {fjBa,o #  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) s_^N=3Si   
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) rhZ p  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 6/T/A+u  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) :qzh kKu  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ^bfU>02Q6p  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) H328I}7  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) \DWKG~r-%  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) MZxU)QW1  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) J^S!GG'gb  
    %       -------------------------------------------------- QpRk5NeLe  
    % Q laoa)d#  
    %   Example 1: 8&3& ^!I  
    % 5.DmMG[T^=  
    %       % Display the Zernike function Z(n=5,m=1) salDGsW^  
    %       x = -1:0.01:1; 3\{\ al   
    %       [X,Y] = meshgrid(x,x); s^4wn:*$zd  
    %       [theta,r] = cart2pol(X,Y); f.bwA x  
    %       idx = r<=1; 2aX$7E?  
    %       z = nan(size(X)); D,|TQ Q  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Q7{{r&|t&  
    %       figure C'{B  
    %       pcolor(x,x,z), shading interp ynZEJKo  
    %       axis square, colorbar S)W?W}*R\  
    %       title('Zernike function Z_5^1(r,\theta)') h9!4\{V;h  
    % ma!C:C9#J  
    %   Example 2: B9$pG  
    % f9 :=6  
    %       % Display the first 10 Zernike functions ~b0l?P*Ff  
    %       x = -1:0.01:1; vK+!m~kDu  
    %       [X,Y] = meshgrid(x,x); }2:q#}"  
    %       [theta,r] = cart2pol(X,Y); og~a*my3  
    %       idx = r<=1; 0c1=M|2  
    %       z = nan(size(X)); SuNc&e#(  
    %       n = [0  1  1  2  2  2  3  3  3  3]; :eT\XtxM~{  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ^)a:D KL  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; R y(<6u0  
    %       y = zernfun(n,m,r(idx),theta(idx)); cfRUVe  
    %       figure('Units','normalized') % tC[q   
    %       for k = 1:10 lj:.}+]r  
    %           z(idx) = y(:,k); |T/s>OW  
    %           subplot(4,7,Nplot(k)) i )$+#N  
    %           pcolor(x,x,z), shading interp ;!lwB  
    %           set(gca,'XTick',[],'YTick',[]) s{{8!Q  
    %           axis square )EQI>1_  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) VUP. \Vry  
    %       end ?^MH:o  
    % qFLt/ >  
    %   See also ZERNPOL, ZERNFUN2. nh80"Ny5  
    x]?V*Jz  
    %   Paul Fricker 11/13/2006 |1/8m/2Af.  
     vILB$%I  
    .F2"tt?'  
    % Check and prepare the inputs: 9`5.0**  
    % ----------------------------- v6 |[p  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ;]=@;? 9  
        error('zernfun:NMvectors','N and M must be vectors.') [eBt Dc*w  
    end W(?J,8>  
    u,}>I%21  
    if length(n)~=length(m) 2PUB@B' +  
        error('zernfun:NMlength','N and M must be the same length.') m=v.<+>  
    end l0qHoM,1Y[  
    +lZ-xU1  
    n = n(:); c* ~0R?  
    m = m(:); $:1/`m19  
    if any(mod(n-m,2)) #pPR>,4  
        error('zernfun:NMmultiplesof2', ... 0(9gTxdB  
              'All N and M must differ by multiples of 2 (including 0).') 4 >H0a  
    end e=IbEm{|  
    fCnwDT  
    if any(m>n) [D(JEO@ :  
        error('zernfun:MlessthanN', ... )8n?.keq  
              'Each M must be less than or equal to its corresponding N.') HU|qeSyel  
    end 8wZ $Hq  
    !{ _:k%B  
    if any( r>1 | r<0 ) .x/H2r'1  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') <7B;_3/  
    end B}* \ pdJ  
    z|Xt'?9&n  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) N1'Yo:_A  
        error('zernfun:RTHvector','R and THETA must be vectors.') 9$ VdYw7D  
    end -em3 #V  
    xdY'i0fh  
    r = r(:); =,i?8Fuz  
    theta = theta(:); PJe \PGh  
    length_r = length(r); eI|~neh  
    if length_r~=length(theta) J p%J02  
        error('zernfun:RTHlength', ...  2t  
              'The number of R- and THETA-values must be equal.') pCa~:q*85  
    end N"Y%* BkH  
    +|K,\ {'U  
    % Check normalization: )=aq j@v  
    % -------------------- Vhb~kI!x  
    if nargin==5 && ischar(nflag) Do^yer~  
        isnorm = strcmpi(nflag,'norm'); LW("/  
        if ~isnorm J4iu8_eH!D  
            error('zernfun:normalization','Unrecognized normalization flag.') |8x_Av0  
        end E)X_  
    else XuZgyt"=r  
        isnorm = false; 0TICv2l!  
    end 4j i#Q  
    (4`Tf*5hHa  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?V_v=X%w  
    % Compute the Zernike Polynomials >SYOtzg%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I<xcVY9L  
    KpS=oFX{}  
    % Determine the required powers of r: ZX{eggXl  
    % ----------------------------------- A,= R`m  
    m_abs = abs(m); |c-`XC2g  
    rpowers = []; CPP9=CoR37  
    for j = 1:length(n) oW(8bd)  
        rpowers = [rpowers m_abs(j):2:n(j)]; miCY?=N`  
    end OT)`)PZ"  
    rpowers = unique(rpowers); qPhVc9D#  
    b Hy<`p0  
    % Pre-compute the values of r raised to the required powers, *S4&V<W>  
    % and compile them in a matrix: T).}~i;!  
    % ----------------------------- [r'hX#  
    if rpowers(1)==0 JKCV >k  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Mz lE  
        rpowern = cat(2,rpowern{:}); 6e}T zc\@(  
        rpowern = [ones(length_r,1) rpowern]; <!|=_W6  
    else }2Im?Q  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); DAEWa Kui  
        rpowern = cat(2,rpowern{:}); Xa&:Hg<  
    end +ZBj_Vw*|  
    aIWpgUd`  
    % Compute the values of the polynomials: : R8+jO   
    % -------------------------------------- % %2~%FVb  
    y = zeros(length_r,length(n)); ;hFB]/.v  
    for j = 1:length(n) ~H]d9C  
        s = 0:(n(j)-m_abs(j))/2; y>RqA *J  
        pows = n(j):-2:m_abs(j); %U1HvmyK  
        for k = length(s):-1:1 >g[Wnzf  
            p = (1-2*mod(s(k),2))* ... g|!=@9[dv  
                       prod(2:(n(j)-s(k)))/              ... kYd=DY  
                       prod(2:s(k))/                     ... x_H"<-By  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... BTE&7/i 21  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 6b!1j,\Vx  
            idx = (pows(k)==rpowers); 0XL[4[LdA  
            y(:,j) = y(:,j) + p*rpowern(:,idx); \}Pr!tk!  
        end ,l\D@<F  
         .3 ^*_  
        if isnorm >rh<%55P`  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); o`}8ZtD  
        end 7G_lGV_  
    end D,uT#P  
    % END: Compute the Zernike Polynomials gti=GmL(L  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |e3YTLsI  
    8[8U49V9(  
    % Compute the Zernike functions: 27H4en; o=  
    % ------------------------------ / pR,l5  
    idx_pos = m>0; c;R .rV<  
    idx_neg = m<0; Y XxWu8  
    e}L(tXZ  
    z = y; l02aXxT)]  
    if any(idx_pos) .fY$$aD$4  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); j7HOh|q  
    end %E2C4UbY  
    if any(idx_neg) ra\|c>[%  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); i{>YQ  
    end WF <*rl  
    Q9t.*+  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) qk(u5Z  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. DZ|/#- k  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated yAVt[+0  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive OB ~74}3;  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ^KFwO=I@PV  
    %   and THETA is a vector of angles.  R and THETA must have the same 7kidPAhY  
    %   length.  The output Z is a matrix with one column for every P-value, pJwy ~ L  
    %   and one row for every (R,THETA) pair. >(a/K2$*1  
    % 2E3x=  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike q]t^6m&-  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) .w]S!=h  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) C/pu]%n@4  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 .DHRPel  
    %   for all p. NW;wy;;  
    % OKzk\F6  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 tA{<)T  
    %   Zernike functions (order N<=7).  In some disciplines it is &rxR"^x\  
    %   traditional to label the first 36 functions using a single mode =").W\,  
    %   number P instead of separate numbers for the order N and azimuthal 5($ '@u  
    %   frequency M. \(ZOt.3!J  
    % u8@>ThPD  
    %   Example: zL3'',Ha  
    % ^zaN?0%S33  
    %       % Display the first 16 Zernike functions bDPT1A`F  
    %       x = -1:0.01:1; 1YMu\(  
    %       [X,Y] = meshgrid(x,x); RpY#_\^hI  
    %       [theta,r] = cart2pol(X,Y); Yt;.Z$i ,  
    %       idx = r<=1; -n~VMLd?@  
    %       p = 0:15; Z?-l-s K  
    %       z = nan(size(X)); 7e&%R4{b  
    %       y = zernfun2(p,r(idx),theta(idx)); Zx]"2U#  
    %       figure('Units','normalized') K<+h/Ok  
    %       for k = 1:length(p)  3^zO G2  
    %           z(idx) = y(:,k); ) 4'@=q  
    %           subplot(4,4,k) ^UK6q2[  
    %           pcolor(x,x,z), shading interp nEm+cHHo?  
    %           set(gca,'XTick',[],'YTick',[]) FeFH_  
    %           axis square ?wx|n_3<:  
    %           title(['Z_{' num2str(p(k)) '}']) 07+Qai-]  
    %       end Wc$1Re{z  
    % hw&R .F  
    %   See also ZERNPOL, ZERNFUN. 4m6E~_:F  
    <tg>1,C  
    %   Paul Fricker 11/13/2006 u-. _;  
    Kq';[Yc  
    b=+'i  
    % Check and prepare the inputs: Sc*O_c3D  
    % ----------------------------- jM90 gPX>,  
    if min(size(p))~=1 uIvE~<  
        error('zernfun2:Pvector','Input P must be vector.') R@r"a&{/  
    end .gWYKZM  
    Xu:S h<:R  
    if any(p)>35 ;[@< ,  
        error('zernfun2:P36', ... ?J~(qaa;  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... pemb2HQ'4j  
               '(P = 0 to 35).'])  P-QZ=dm  
    end X}xy v  
    e? fFh,a  
    % Get the order and frequency corresonding to the function number: eJ#q! <   
    % ---------------------------------------------------------------- wvA@\-.+  
    p = p(:); #^v|u3^DD  
    n = ceil((-3+sqrt(9+8*p))/2); g%Ap<iT  
    m = 2*p - n.*(n+2); pVt8z|p_;{  
    x,z+l-y  
    % Pass the inputs to the function ZERNFUN: yA!#>u%g  
    % ---------------------------------------- vd9><W  
    switch nargin n-{G19?  
        case 3 ,r{\aW@  
            z = zernfun(n,m,r,theta); "el}@  
        case 4 N$H0o+9-Y  
            z = zernfun(n,m,r,theta,nflag); </|IgN$w`  
        otherwise _{6QvD3kg.  
            error('zernfun2:nargin','Incorrect number of inputs.') :'!,L0I|t  
    end C_Y^<  
    |[?"$g9v  
    % 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) xGJ{_M  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. &'UY V>  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Q9Wa@gi|  
    %   order N and frequency M, evaluated at R.  N is a vector of z)r)w?A  
    %   positive integers (including 0), and M is a vector with the % ^g BDlR^  
    %   same number of elements as N.  Each element k of M must be a }N1Z7G  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) "EQ-`b=I4  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is b}p0&%I  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix hp!UW  
    %   with one column for every (N,M) pair, and one row for every [: X  
    %   element in R. PWOV~ `^;  
    % |Z<NM#1  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 6yKr5tH4  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ;Id%{1  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 2Tt@2h_L  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 T&I*8 R~  
    %   for all [n,m]. c.Pyt  
    % JGp~A#H&  
    %   The radial Zernike polynomials are the radial portion of the !q! =VC  
    %   Zernike functions, which are an orthogonal basis on the unit |<P]yn  
    %   circle.  The series representation of the radial Zernike Hm4:m$=p4  
    %   polynomials is #vYdP#nWb  
    % q-3%.<LL  
    %          (n-m)/2 _K>cB<+d  
    %            __ w%)=`'s_  
    %    m      \       s                                          n-2s ] FvN*@lG  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r [XjJsk,  
    %    n      s=0 nk]jIR y^T  
    % ^_r8R__S:  
    %   The following table shows the first 12 polynomials. DHJh.Y@H  
    % {=j!2v#8~  
    %       n    m    Zernike polynomial    Normalization {e A4y~k  
    %       --------------------------------------------- Ps(3X@  
    %       0    0    1                        sqrt(2) KD*,u{v;  
    %       1    1    r                           2 oorit  
    %       2    0    2*r^2 - 1                sqrt(6) 4r`u@  
    %       2    2    r^2                      sqrt(6) .HF+JHIUu  
    %       3    1    3*r^3 - 2*r              sqrt(8) mF [w-<:.d  
    %       3    3    r^3                      sqrt(8) _`|Hk2O  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 3Ln~"HwP  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) &F*s.gL  
    %       4    4    r^4                      sqrt(10) <7/_Vs)F0  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Glcl7f"<^  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) meT~b  
    %       5    5    r^5                      sqrt(12) Id(o6j^J_  
    %       --------------------------------------------- {BKu'A  
    % hV])\t=yf  
    %   Example: ~mx me6"v  
    % DTk)Y-eQ  
    %       % Display three example Zernike radial polynomials Mb=vIk{B f  
    %       r = 0:0.01:1; ]d}Z2I'  
    %       n = [3 2 5]; >nkd U  
    %       m = [1 2 1]; So\(]S  
    %       z = zernpol(n,m,r); [WnX'R R  
    %       figure <Vm+Lt9  
    %       plot(r,z) 2@@OjeANsX  
    %       grid on -mOSB(#bo  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') }BS.OK?  
    % iXK.QktHw  
    %   See also ZERNFUN, ZERNFUN2. T>e!DOW;  
    Nt9M$?\P  
    % A note on the algorithm. ByoSwQ  
    % ------------------------ \:]Clvc  
    % The radial Zernike polynomials are computed using the series }$s#H{T!  
    % representation shown in the Help section above. For many special RrRrB"!8nR  
    % functions, direct evaluation using the series representation can p<*3mbgGO  
    % produce poor numerical results (floating point errors), because R<@s]xX_  
    % the summation often involves computing small differences between }20 Q`?  
    % large successive terms in the series. (In such cases, the functions !*ct3{m  
    % are often evaluated using alternative methods such as recurrence "qjkw f)\  
    % relations: see the Legendre functions, for example). For the Zernike ~UX@%0%)N  
    % polynomials, however, this problem does not arise, because the @ |v4B[/  
    % polynomials are evaluated over the finite domain r = (0,1), and I?LJXo\O  
    % because the coefficients for a given polynomial are generally all 4eK!1|1  
    % of similar magnitude. xQ9P'ru  
    % q%%8oaEI  
    % ZERNPOL has been written using a vectorized implementation: multiple z$$ E7i  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 2%i_SX[  
    % values can be passed as inputs) for a vector of points R.  To achieve 2W:R{dHE  
    % this vectorization most efficiently, the algorithm in ZERNPOL qZACX.Hw  
    % involves pre-determining all the powers p of R that are required to q,3_)ZOq  
    % compute the outputs, and then compiling the {R^p} into a single u+UtvzUC  
    % matrix.  This avoids any redundant computation of the R^p, and losm<  
    % minimizes the sizes of certain intermediate variables. Ppi/`X  
    % CRf!tsj@  
    %   Paul Fricker 11/13/2006 A}pmr  
    t+7h(?8L  
    \fIGMoy!  
    % Check and prepare the inputs: kIhP 73M  
    % ----------------------------- B/.+&AJw  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) JpqZVu"7  
        error('zernpol:NMvectors','N and M must be vectors.') |VxEW U/  
    end &NZl_7P L  
    $mOVo'2  
    if length(n)~=length(m) ivDmPHj{  
        error('zernpol:NMlength','N and M must be the same length.') IV*@}~BJ  
    end (T:OZmEO.  
    CZ"~N`  
    n = n(:); I.BsKB  
    m = m(:); =lY6v -MBw  
    length_n = length(n); |& OW_*l  
    idW=  
    if any(mod(n-m,2)) BK`NPC$a  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') FeOo;|a  
    end zmd,uhNc:  
    [mwJ*GJ-  
    if any(m<0) j06?Mm_c2  
        error('zernpol:Mpositive','All M must be positive.') yN}upYxp  
    end 1{D_30sG.  
    ^*JpdmVhu  
    if any(m>n) +@*}_%^l"  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ,ab_u@  
    end qYo"-D*  
    C+ibLS4i  
    if any( r>1 | r<0 ) !kCMw%[  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') *FhD%><  
    end Tk~RT<\Ab+  
    b{Srd3  
    if ~any(size(r)==1) xS.Rpx/8  
        error('zernpol:Rvector','R must be a vector.') vxuxfi8x  
    end ?^y%UIzf  
    }?[^q  
    r = r(:); I#lvaoeN  
    length_r = length(r); T}')QC&wQ  
    VG FWF3s  
    if nargin==4 Gt;@. jY&  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); @;>i3?  
        if ~isnorm ]j.=zQP?'  
            error('zernpol:normalization','Unrecognized normalization flag.') 81?7u!=ic+  
        end w&&uk[Gh/a  
    else 7C~qAI6Eg  
        isnorm = false; Arvxl(R\4  
    end <3?T^/8  
    ~9#x/EG/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _D{zB1d\0  
    % Compute the Zernike Polynomials WH:[Y7D  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Qdepzo>E  
    j5hM |\]  
    % Determine the required powers of r: tF:'Y ~3 p  
    % ----------------------------------- Hgu:*iYA  
    rpowers = []; jC_7cAsl  
    for j = 1:length(n) <(|No3jx  
        rpowers = [rpowers m(j):2:n(j)]; /rMxl(wD'  
    end \=n0@1Q=>  
    rpowers = unique(rpowers); aJh=4j~.  
    /48W]a}JS  
    % Pre-compute the values of r raised to the required powers, W40GW  
    % and compile them in a matrix: 7\.Ax  
    % ----------------------------- D+$k  
    if rpowers(1)==0 agQ5%t#  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); mX@Un9k  
        rpowern = cat(2,rpowern{:}); G`!ff  
        rpowern = [ones(length_r,1) rpowern]; Ub1?dk   
    else @\~qXz{6J  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); NF?FEUoxz  
        rpowern = cat(2,rpowern{:}); }h+_kRQ  
    end eFO+@  
    TF\<`}akX  
    % Compute the values of the polynomials: q;I`&JK  
    % -------------------------------------- 8 I'1~d%$  
    z = zeros(length_r,length_n); o;#{N~4[$  
    for j = 1:length_n e"jA#Y #  
        s = 0:(n(j)-m(j))/2; qF9rY)ifm  
        pows = n(j):-2:m(j); K?l1Gj  
        for k = length(s):-1:1 WA);Z=  
            p = (1-2*mod(s(k),2))* ... 9&'I?D&8  
                       prod(2:(n(j)-s(k)))/          ... *q5'~)W<  
                       prod(2:s(k))/                 ... E\M{/.4 4  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Q:iW k6  
                       prod(2:((n(j)+m(j))/2-s(k))); ?nm:e.S+?  
            idx = (pows(k)==rpowers); |pIA9/~Z  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ,v"/3Ff{,  
        end ^V^In-[!y:  
         WY@x2bBi  
        if isnorm vFfvvRda4x  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); S}"?#=Q.%O  
        end DdI7%?hK  
    end /)8 0@  
    ^q"wd?((h  
    % 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)  6AQ;P  
    [/#n+sz.A  
    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/2)R{  
    M.OWw#?p:_  
    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)