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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 z+c'-!e/  
    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! n T7]PhJ  
     
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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 ;1k_J~Qei  
    function z = zernfun(n,m,r,theta,nflag) [-\DC*6  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. V/ZWyYxjLi  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N V/)3d  
    %   and angular frequency M, evaluated at positions (R,THETA) on the R%JEx3)0m  
    %   unit circle.  N is a vector of positive integers (including 0), and mG%cE(j*D  
    %   M is a vector with the same number of elements as N.  Each element nTsPX Tat  
    %   k of M must be a positive integer, with possible values M(k) = -N(k)  <JZa  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, `Mo%)I<`=  
    %   and THETA is a vector of angles.  R and THETA must have the same ,88%eX|  
    %   length.  The output Z is a matrix with one column for every (N,M) 7>gW2 m  
    %   pair, and one row for every (R,THETA) pair. >P6U0  
    % SNV;s,  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ve4 QS P  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !)c0  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral R~bLEo  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, xH-} <7  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ^1ks`1  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. CF5%&B  
    % ;8gODj:dO  
    %   The Zernike functions are an orthogonal basis on the unit circle. w$Mb+b$  
    %   They are used in disciplines such as astronomy, optics, and P2)g%$ME  
    %   optometry to describe functions on a circular domain. %;`3I$  
    % 5JZZvc$au  
    %   The following table lists the first 15 Zernike functions. ,7e 2M@=  
    % *oIKddZh  
    %       n    m    Zernike function           Normalization #elaz8 5  
    %       -------------------------------------------------- s3M#ua#mX  
    %       0    0    1                                 1 :Czvwp{z  
    %       1    1    r * cos(theta)                    2 cH7D@p}  
    %       1   -1    r * sin(theta)                    2 FRTvo  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) B^1Io9  
    %       2    0    (2*r^2 - 1)                    sqrt(3) F,XJGD*  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) r3.v^  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) tWdP5vfp  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 4_S%K&  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) z yI4E\  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) l1RFn,Tzr  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Jaf=qwZ/`  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) &S# bLE  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) POQ1K O  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ..^,*  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) g? \pH:|79  
    %       -------------------------------------------------- ~#[ ZuMO?  
    % v aaZ  
    %   Example 1: [g*]u3s  
    % jdVdz,Y  
    %       % Display the Zernike function Z(n=5,m=1) Q_a%$a.rV  
    %       x = -1:0.01:1; ?rV c}  
    %       [X,Y] = meshgrid(x,x); SHPZXJ{  
    %       [theta,r] = cart2pol(X,Y); fKT(.VN q5  
    %       idx = r<=1; fI0L\^b%  
    %       z = nan(size(X)); YJwz*@l  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 6UJBE<ntj  
    %       figure -OP5v8c f  
    %       pcolor(x,x,z), shading interp +<I1@C  
    %       axis square, colorbar B6vmBmN  
    %       title('Zernike function Z_5^1(r,\theta)') d_Vwjv&@/"  
    % ^A$~8?f  
    %   Example 2: c[0$8F>  
    % v]27+/a$c  
    %       % Display the first 10 Zernike functions oAp I/o  
    %       x = -1:0.01:1; WJL,L[XC  
    %       [X,Y] = meshgrid(x,x); yc5n   
    %       [theta,r] = cart2pol(X,Y); #Ryu`b  
    %       idx = r<=1; P^LOrLmo8  
    %       z = nan(size(X)); B[MZ Pv)  
    %       n = [0  1  1  2  2  2  3  3  3  3]; |wj/lX7y  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ]R{=|  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; )u&_}6z  
    %       y = zernfun(n,m,r(idx),theta(idx)); Bf88f<Z  
    %       figure('Units','normalized') w02HSQ  
    %       for k = 1:10 ;7<a0HZ5!  
    %           z(idx) = y(:,k); Ic&t_B*i}]  
    %           subplot(4,7,Nplot(k)) UwQ3q  
    %           pcolor(x,x,z), shading interp Xc5[d`]  
    %           set(gca,'XTick',[],'YTick',[]) _.06^5o  
    %           axis square fhn0^Qc"+  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) RN:#+S(8  
    %       end U>x2'B v  
    % z_l3=7R  
    %   See also ZERNPOL, ZERNFUN2. 0QIocha  
    .^.UJo;4G  
    %   Paul Fricker 11/13/2006 T[q-$8U  
    @4B2O"z`  
    {Q(6 .0R  
    % Check and prepare the inputs: a\m10Ih:  
    % ----------------------------- nZ7v9o9  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ,em6wIq,  
        error('zernfun:NMvectors','N and M must be vectors.') $'FPst8Q<  
    end =3SL& :8  
    [Iihk5TT  
    if length(n)~=length(m) = xX^  
        error('zernfun:NMlength','N and M must be the same length.') Ft.BfgJ$  
    end Dfhs@ z  
    OEwfNZQ-  
    n = n(:); q=1SP@;\6  
    m = m(:); 47K5[R  
    if any(mod(n-m,2)) rw\4KI@ L  
        error('zernfun:NMmultiplesof2', ... r&3fSx9  
              'All N and M must differ by multiples of 2 (including 0).') <7] z'  
    end #=.h:_9  
    ^:)&KV8D|  
    if any(m>n) Xp?Z;$r$  
        error('zernfun:MlessthanN', ... c\b>4 &n  
              'Each M must be less than or equal to its corresponding N.') 3MzY]J y(  
    end rzBWk  
    :A{-^qd(  
    if any( r>1 | r<0 ) ? sewU9*  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') "DN`@  
    end _5Ll L#)  
    #EM'=Q%TO  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) )  zm.2L  
        error('zernfun:RTHvector','R and THETA must be vectors.') 4 z`5W,  
    end pq&c]8H  
    `WW0~Tp3  
    r = r(:); SA7,]&Zb  
    theta = theta(:);  Fszk?0T  
    length_r = length(r); Cp* n2  
    if length_r~=length(theta) <C{5(=X{  
        error('zernfun:RTHlength', ... y d$37G|n  
              'The number of R- and THETA-values must be equal.') j&mL]'Zy  
    end =% JDo  
    Bm7GU`j"  
    % Check normalization: Ji[w; [qL  
    % -------------------- FT enXJ/c  
    if nargin==5 && ischar(nflag) ^,5.vfES  
        isnorm = strcmpi(nflag,'norm'); >lW*%{|b$^  
        if ~isnorm T:&+#0<  
            error('zernfun:normalization','Unrecognized normalization flag.') <FK><aA_i*  
        end a wK'XFk  
    else nJya1AH;  
        isnorm = false; ]xG4T>S  
    end T7Ac4LA  
    \nyFN  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ({9!P30:  
    % Compute the Zernike Polynomials Y"jDZG?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;~bn@T-  
    `+o.w#cl  
    % Determine the required powers of r: ;hvXFU  
    % ----------------------------------- yi?&^nX@9,  
    m_abs = abs(m); {EUH#':  
    rpowers = []; :qp"Ao{M  
    for j = 1:length(n) `IoX'|C[h  
        rpowers = [rpowers m_abs(j):2:n(j)]; lBdF9F<  
    end h,+=h;!  
    rpowers = unique(rpowers); _2Z3?/Y  
    K?je(t^  
    % Pre-compute the values of r raised to the required powers, ]`XuE-Uh  
    % and compile them in a matrix: hrD6r=JT<~  
    % ----------------------------- v^pP& <G  
    if rpowers(1)==0 -!cAr <  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ,f+5x]F?m  
        rpowern = cat(2,rpowern{:}); "/fs%F  
        rpowern = [ones(length_r,1) rpowern]; TH!8G,(w  
    else g4 X,*H  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); wVOL7vh  
        rpowern = cat(2,rpowern{:}); `RcNqPY#S  
    end $hQg+nY.  
    H=#Jg;_w  
    % Compute the values of the polynomials: }j1Zk4}[x  
    % -------------------------------------- R6XMBYK^  
    y = zeros(length_r,length(n)); tl5IwrF6;  
    for j = 1:length(n) 7]j-zv  
        s = 0:(n(j)-m_abs(j))/2; h$k3MhYDes  
        pows = n(j):-2:m_abs(j); Vcq?>mH&T  
        for k = length(s):-1:1 Zg&\K~OC  
            p = (1-2*mod(s(k),2))* ... {UBQ?7.jE  
                       prod(2:(n(j)-s(k)))/              ... Ekme62Q>u  
                       prod(2:s(k))/                     ... ef;L|b%pp  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... :(`>bY  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); B7MW" y  
            idx = (pows(k)==rpowers); *h:EE6|  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 1>VS/H`  
        end 0Zh _Q  
         Y0\\(0j64  
        if isnorm Q; /F0JDH  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); U]0)$OH5e  
        end Q;O)>K  
    end |S:!+[  
    % END: Compute the Zernike Polynomials ~!F4JRf  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% WnzPPh3PJ  
     MK"  
    % Compute the Zernike functions: Bq]O &>\hX  
    % ------------------------------ l6c%_<P|  
    idx_pos = m>0; 4E\ntufo  
    idx_neg = m<0; 6QXQ<ah"  
    t}k'Ba3]:Y  
    z = y; t} i97;  
    if any(idx_pos) {IHK<aW  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); lp-Zx[#`}C  
    end oz6+rM6MY  
    if any(idx_neg) YG~ o  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Ygi1"X}  
    end RIEv*2_O  
    .l=*R7~EU  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) !x$ :8R  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. .k,Jt+  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated qD;v/,?  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive n_aNs]C9R  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ~8xh0TSi  
    %   and THETA is a vector of angles.  R and THETA must have the same vk)0n=  
    %   length.  The output Z is a matrix with one column for every P-value, (vAv^A*i}  
    %   and one row for every (R,THETA) pair. L;M^>{>  
    % [TK? P0  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike bV$8 >[`  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) {jM<t  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) *V6QB e  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ,3]?%t0xe  
    %   for all p. C8qTz".5$  
    % mKq<'t]^k  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 xw+<p  
    %   Zernike functions (order N<=7).  In some disciplines it is KT];SF ^Y  
    %   traditional to label the first 36 functions using a single mode :-u-hO5*8  
    %   number P instead of separate numbers for the order N and azimuthal w,![;wG  
    %   frequency M. }-{b$6]  
    % ";_K x={  
    %   Example: #K#Mv /  
    % ] `B,L*m6  
    %       % Display the first 16 Zernike functions jj3Pf>D+k  
    %       x = -1:0.01:1; 6c2ThtL  
    %       [X,Y] = meshgrid(x,x); y Tw',N{  
    %       [theta,r] = cart2pol(X,Y); 6mBDd>`0  
    %       idx = r<=1; nGx ~) T  
    %       p = 0:15; ByhOK}u;P4  
    %       z = nan(size(X)); ]D{c4)\7C|  
    %       y = zernfun2(p,r(idx),theta(idx)); 4\1wyN /}M  
    %       figure('Units','normalized') WbQhl sc:  
    %       for k = 1:length(p) 8K.s@<  
    %           z(idx) = y(:,k); i<#h]o C}  
    %           subplot(4,4,k) Eg`R|CF  
    %           pcolor(x,x,z), shading interp ApG'jN  
    %           set(gca,'XTick',[],'YTick',[]) $v:gBlj%"  
    %           axis square ?-8y4 Ex  
    %           title(['Z_{' num2str(p(k)) '}']) Sf'i{xye  
    %       end (F,(]71Z+  
    % m|[\F#+C  
    %   See also ZERNPOL, ZERNFUN. [_ M6/  
    n5egKAgA  
    %   Paul Fricker 11/13/2006 gb=80s0  
    |"CJ  
    $/[Gys3"  
    % Check and prepare the inputs: _\,rX\  
    % ----------------------------- ;48P vw>g}  
    if min(size(p))~=1 \;-=ODC  
        error('zernfun2:Pvector','Input P must be vector.') ;'=VrE6  
    end 7.Ml9{M/i  
    t7|MkX1  
    if any(p)>35 9m\)\/V  
        error('zernfun2:P36', ... 7szls71/=  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... # #k #q=4  
               '(P = 0 to 35).']) M'cJ)-G  
    end a9#W9eP  
    -l-AToO4  
    % Get the order and frequency corresonding to the function number: M8 iEVJ  
    % ---------------------------------------------------------------- a3O nW\N  
    p = p(:); m !#_CQ:  
    n = ceil((-3+sqrt(9+8*p))/2); cs K>iN  
    m = 2*p - n.*(n+2); rD0k%-{{  
    @Pxw hlxa  
    % Pass the inputs to the function ZERNFUN: PM~bM3Ei  
    % ---------------------------------------- e> ar  
    switch nargin Q&u>7_, Du  
        case 3 '1CD- Bu  
            z = zernfun(n,m,r,theta); GhqgRzX  
        case 4 `GvA241  
            z = zernfun(n,m,r,theta,nflag); [urH a  
        otherwise 3AvVU]@&Z@  
            error('zernfun2:nargin','Incorrect number of inputs.') L3B8IDq  
    end QjQ4Z'.r>  
    =a?a@+  
    % 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)  "\`>2  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. xC)7eQn/R  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of h=hoV5d@  
    %   order N and frequency M, evaluated at R.  N is a vector of 6yN" l Q7  
    %   positive integers (including 0), and M is a vector with the R @"`~#$$  
    %   same number of elements as N.  Each element k of M must be a >/b^fAG  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) e:qo_eSC^-  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is j\! e9M  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix GL/  KB  
    %   with one column for every (N,M) pair, and one row for every Y\.DQ  
    %   element in R. aJI>FTdK  
    % iNt 4>  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ;JYoW{2  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ?3[tJreVj  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to V R"8Di&)  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Bk+{}  
    %   for all [n,m]. o5>/}wIf  
    % *h2`^Z  
    %   The radial Zernike polynomials are the radial portion of the D^n xtuT*  
    %   Zernike functions, which are an orthogonal basis on the unit  ?J<T  
    %   circle.  The series representation of the radial Zernike NNgK:YibD  
    %   polynomials is fc3 Fi'^  
    % {h,_"g\V  
    %          (n-m)/2 gTnS[  
    %            __ 0k%hY{  
    %    m      \       s                                          n-2s &1=g A.ZR  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r t7&Dwmck9  
    %    n      s=0 `y'aH 'EEd  
    % )R~a;?T_c0  
    %   The following table shows the first 12 polynomials. am2a#4`  
    % A hR0zg  
    %       n    m    Zernike polynomial    Normalization ikr7DBLt  
    %       --------------------------------------------- =9(tsB gTX  
    %       0    0    1                        sqrt(2) vuZf#\zh}  
    %       1    1    r                           2 )PwQ^||{  
    %       2    0    2*r^2 - 1                sqrt(6)  *=TYVM9  
    %       2    2    r^2                      sqrt(6) <E SvvTf  
    %       3    1    3*r^3 - 2*r              sqrt(8) oQ{cSThj  
    %       3    3    r^3                      sqrt(8) qT$ )Rb&  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) uNy!< u  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) r4EoJyt  
    %       4    4    r^4                      sqrt(10) V7BsEw  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) +ZX .1[O  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) RYH)AS4w'  
    %       5    5    r^5                      sqrt(12) n6/fan;  
    %       --------------------------------------------- AO $Wy@  
    % ! j{CuA/  
    %   Example: W=3? x  
    % IYLZ +>  
    %       % Display three example Zernike radial polynomials LaclC]yLU  
    %       r = 0:0.01:1; 0TmZ*?3!4  
    %       n = [3 2 5]; JxHv<p[  
    %       m = [1 2 1]; ~?CS_B *  
    %       z = zernpol(n,m,r); ,aWCiu}  
    %       figure -n-Z/5~ X  
    %       plot(r,z) ;7L;  
    %       grid on FJ}gUs{m  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') \ZsP]};*  
    % !O8.#+  
    %   See also ZERNFUN, ZERNFUN2. Gp8psH  
    gX_SKy  
    % A note on the algorithm. RbKwO} z$q  
    % ------------------------ Sj@15 W  
    % The radial Zernike polynomials are computed using the series [<Q4U{F  
    % representation shown in the Help section above. For many special 6P^hN%0  
    % functions, direct evaluation using the series representation can AC'lS >7s  
    % produce poor numerical results (floating point errors), because \WX@PfL  
    % the summation often involves computing small differences between &*A:[b\  
    % large successive terms in the series. (In such cases, the functions 1LX)4TCC  
    % are often evaluated using alternative methods such as recurrence R B%:h-t4  
    % relations: see the Legendre functions, for example). For the Zernike c@9##DPn  
    % polynomials, however, this problem does not arise, because the oBC]UL;8xJ  
    % polynomials are evaluated over the finite domain r = (0,1), and 6^ab@GrN\  
    % because the coefficients for a given polynomial are generally all >x*)GPDa  
    % of similar magnitude. 3 1k  
    % G1/Gq.<  
    % ZERNPOL has been written using a vectorized implementation: multiple y] y9'5_  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 8+ hhdy*b  
    % values can be passed as inputs) for a vector of points R.  To achieve F )7j@h^  
    % this vectorization most efficiently, the algorithm in ZERNPOL ~2(]ZfO?>H  
    % involves pre-determining all the powers p of R that are required to h9jc,X u5X  
    % compute the outputs, and then compiling the {R^p} into a single c})wD+1  
    % matrix.  This avoids any redundant computation of the R^p, and op.d;lO@  
    % minimizes the sizes of certain intermediate variables. F<gMUDB  
    % T0Q51Q  
    %   Paul Fricker 11/13/2006 \C7q4p?8  
    7gr^z)${J  
    Q vJZkGX  
    % Check and prepare the inputs: iu|v9+  
    % ----------------------------- 2LC w*eT{)  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ]w-.|vx  
        error('zernpol:NMvectors','N and M must be vectors.') +|0 t  
    end |Qr:!MA  
    c$A@T~$  
    if length(n)~=length(m) *Dc@CmBr  
        error('zernpol:NMlength','N and M must be the same length.') tpGCrn2w>  
    end TL@mM  
    TRy^hr8~  
    n = n(:); 1yS&~ y?a  
    m = m(:); >jKjh!`)!e  
    length_n = length(n); h7*O.Opm=  
    _*n)mlLln  
    if any(mod(n-m,2)) G1[(F`t>  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') y;QQ| =,  
    end cJ/4G l  
    ]!faA\1  
    if any(m<0) ST% T =_q  
        error('zernpol:Mpositive','All M must be positive.') xl,ryc3J  
    end * :S~C  
    5*+I M*c  
    if any(m>n) JqSr[q  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') A_F0\ EN*  
    end tah }^  
    K_&_z  
    if any( r>1 | r<0 ) %6HX*_Mr&  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') CIy^`2wq  
    end 61>f(?s  
    }LQ\a8]<  
    if ~any(size(r)==1) MQ9vPgh  
        error('zernpol:Rvector','R must be a vector.') R"{l[9j4>  
    end `M0YAiG  
    v2=/[E@  
    r = r(:); ,5. <oDH  
    length_r = length(r); 6klD22b2$  
    ZPvf-Pq Jl  
    if nargin==4 yzg9I  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); p&O8qAaO  
        if ~isnorm :q;R6-|.  
            error('zernpol:normalization','Unrecognized normalization flag.') OfJd/D  
        end 63C(Tp"  
    else 0 Az/fzJlz  
        isnorm = false; gmDR{loX  
    end j.B>v\b_3  
    x=vK EyS@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x{?sn  
    % Compute the Zernike Polynomials N6q5`Ry  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?S9Nm~vlt  
    :*cHA  
    % Determine the required powers of r: Y$+QNi  
    % ----------------------------------- eo ?Oir)  
    rpowers = []; ^9=4iXd  
    for j = 1:length(n) :~er h}~ps  
        rpowers = [rpowers m(j):2:n(j)]; <r3Jf}%tT  
    end \ j:AR4  
    rpowers = unique(rpowers); 7*MU2gb  
    vzcz<i )  
    % Pre-compute the values of r raised to the required powers, o X@nP?\  
    % and compile them in a matrix: >j:|3atb  
    % ----------------------------- UO1$UF! QC  
    if rpowers(1)==0 ]) =H  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); kF?S 2(vH  
        rpowern = cat(2,rpowern{:}); yfC2^#9 Zu  
        rpowern = [ones(length_r,1) rpowern]; ,jTPg/r  
    else W }Zb~[,  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); NnP.k7m)  
        rpowern = cat(2,rpowern{:}); 1@@y]s_.a  
    end # \<P]<C  
    "f<#.}8  
    % Compute the values of the polynomials: g}YToOs  
    % -------------------------------------- 3; A$<s  
    z = zeros(length_r,length_n); {KsVK4\r  
    for j = 1:length_n )tvc/)&A}  
        s = 0:(n(j)-m(j))/2; &urb!tQ>&  
        pows = n(j):-2:m(j); NA\x<  
        for k = length(s):-1:1 0$l&i=L  
            p = (1-2*mod(s(k),2))* ... JSRg?p\  
                       prod(2:(n(j)-s(k)))/          ... s&0*'^'O[S  
                       prod(2:s(k))/                 ... V_lGj  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... U1jSUkqb  
                       prod(2:((n(j)+m(j))/2-s(k))); Kk`<f d  
            idx = (pows(k)==rpowers); jzQ I>u  
            z(:,j) = z(:,j) + p*rpowern(:,idx); c8q G\\t[  
        end ]| z")gOE  
         %_)b>C18 y  
        if isnorm /3s@6Ex}E  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); C[g&F 0 6  
        end 'f8 p7 _F  
    end Lg b  
    QU,?}w'?d  
    % 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)  81%qM7v9H  
    j};pv2  
    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。 fS8XuT  
    v)v{QNQp^  
    07年就写过这方面的计算程序了。
    让光学不再神秘,让光学变得容易,快速实现客户关于光学的设想与愿望。