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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 :Ry 24X  
    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! |)_R bqZ  
     
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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 6hK"k  
    function z = zernfun(n,m,r,theta,nflag) HqBPY[;s  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. (Y)h+}n5N  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N %#9~V  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 0Q>|s_  
    %   unit circle.  N is a vector of positive integers (including 0), and .M2&ad :  
    %   M is a vector with the same number of elements as N.  Each element MF}Lv1/[-J  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Ba0D"2CgY  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, kVnyX@  
    %   and THETA is a vector of angles.  R and THETA must have the same |vz;bJG  
    %   length.  The output Z is a matrix with one column for every (N,M) "S`wwl  
    %   pair, and one row for every (R,THETA) pair. --`LP[ll  
    % !d .>r 7w  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ]4mj 1g&C  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), G3+a+=e  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ;|QR-m2/  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, QV$dKjMS  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized q&Wwt qc9  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. RCYbRR4y  
    % B:4qW[U#  
    %   The Zernike functions are an orthogonal basis on the unit circle. 0t?<6-3`/  
    %   They are used in disciplines such as astronomy, optics, and 9Fx z!-9m  
    %   optometry to describe functions on a circular domain. t[,T}BCy.  
    % YO$b#  
    %   The following table lists the first 15 Zernike functions. sDm},=X}  
    % XhAcC  
    %       n    m    Zernike function           Normalization ws>Iyw.u  
    %       -------------------------------------------------- sFCs_u1tNN  
    %       0    0    1                                 1 I%>]!X  
    %       1    1    r * cos(theta)                    2 FR^wDm$  
    %       1   -1    r * sin(theta)                    2 |~LjH|*M  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) s4`*0_n  
    %       2    0    (2*r^2 - 1)                    sqrt(3) !9LAXM  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ]#q7}Sd  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) L_ qv<iM$  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Z?c=t-yqp  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) W?J*9XQ`  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) [pgkY!R?)  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) dzNaow*0&V  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) I+?$4SC  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) _AHB|P I  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `[` *@O(y  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) #u5;utY:F  
    %       -------------------------------------------------- 9hLmrYNM1  
    % < Gy!i/  
    %   Example 1: M(WOxZ8  
    %  ~uZLe\>K  
    %       % Display the Zernike function Z(n=5,m=1) K<  
    %       x = -1:0.01:1; &Yks,2:P  
    %       [X,Y] = meshgrid(x,x); `{Di*  
    %       [theta,r] = cart2pol(X,Y); +fCyR  
    %       idx = r<=1; X`v79`g_  
    %       z = nan(size(X)); u:H 3.5)%  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); y#Za|nt  
    %       figure i 2} =/  
    %       pcolor(x,x,z), shading interp <\9Ijuq}k  
    %       axis square, colorbar UNcJ=   
    %       title('Zernike function Z_5^1(r,\theta)') 9Glfi@.  
    % ah"MzU)  
    %   Example 2: O{cGk: y  
    % \ [^) WQ  
    %       % Display the first 10 Zernike functions q,,>:]f#  
    %       x = -1:0.01:1; - Zoo)  
    %       [X,Y] = meshgrid(x,x); Hs`#{W{.  
    %       [theta,r] = cart2pol(X,Y); I1 R\Ts@  
    %       idx = r<=1; (VXx G/E3  
    %       z = nan(size(X)); Or#+E2%1E  
    %       n = [0  1  1  2  2  2  3  3  3  3]; `ToRkk&&>{  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; a. `JS  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ZJI1NCBZ  
    %       y = zernfun(n,m,r(idx),theta(idx)); qqt.nrQ^  
    %       figure('Units','normalized') cM<hG:4%wX  
    %       for k = 1:10 MHr0CYyb.  
    %           z(idx) = y(:,k); -2jBs-z  
    %           subplot(4,7,Nplot(k)) Zc\h15+P  
    %           pcolor(x,x,z), shading interp CMxjX  
    %           set(gca,'XTick',[],'YTick',[]) {cyo0-9nv  
    %           axis square EBDC'^  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) vu'!-K=0  
    %       end +?5Uy*$  
    % gC_s\WU  
    %   See also ZERNPOL, ZERNFUN2. >upXt?  
    l;{N/cS  
    %   Paul Fricker 11/13/2006 p`<e~[]a  
    B-ri}PA  
    e"s{_V  
    % Check and prepare the inputs: Th;gps%b  
    % ----------------------------- kG;eOp16R  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 9p\wTzA  
        error('zernfun:NMvectors','N and M must be vectors.') x+*L5$;h  
    end "U5Ln2X{J  
    0q>NE <L  
    if length(n)~=length(m) K@j^gF/0B  
        error('zernfun:NMlength','N and M must be the same length.') mb~=Xyk&  
    end MNf@HG  
    & L.PU@  
    n = n(:); 6PQJgki  
    m = m(:); mcz(,u}  
    if any(mod(n-m,2)) =6Kv`  
        error('zernfun:NMmultiplesof2', ... kO,VayjT  
              'All N and M must differ by multiples of 2 (including 0).') l`M5'r]l  
    end ]g8i>,G  
    VNxpOoV=S  
    if any(m>n) =N@)CB7a  
        error('zernfun:MlessthanN', ... ZXsY-5$#d-  
              'Each M must be less than or equal to its corresponding N.') u .pKK  
    end vzohq1r5  
    \\2k}TsB  
    if any( r>1 | r<0 ) =UB*xm%!  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Oj4u!SY\j  
    end 7i+!^Qj?y  
    m>abK@5na  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 0x>/6 <<  
        error('zernfun:RTHvector','R and THETA must be vectors.') C$'D]fX  
    end 68J 9T^84  
    iKF$J3a\2f  
    r = r(:); =;k+g?.@I  
    theta = theta(:); ^ =/?<C4  
    length_r = length(r); >TlW]st  
    if length_r~=length(theta) O7'<I|aD  
        error('zernfun:RTHlength', ... B \_d5WJ<  
              'The number of R- and THETA-values must be equal.') V &mH#k  
    end Mf ;|z0UX  
    j5,^9'  
    % Check normalization: 56bud3CVs  
    % -------------------- ]e@0T{!  
    if nargin==5 && ischar(nflag) c4ZuW_&:  
        isnorm = strcmpi(nflag,'norm'); 5M<' A=  
        if ~isnorm $IxU6=ajn  
            error('zernfun:normalization','Unrecognized normalization flag.') S/nj5Lh  
        end \ifK~?  
    else B0b[p*g Il  
        isnorm = false; "W &:j:o  
    end |b$>68:  
    WNn[L=f  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *]}CSZ[>  
    % Compute the Zernike Polynomials cQ3W;F8|n  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +{")E)  
    (xZr ]v ]U  
    % Determine the required powers of r: ,?xLT2>J_  
    % ----------------------------------- Ci7P%]9  
    m_abs = abs(m); O6m.t%*  
    rpowers = []; {) :%Wn M9  
    for j = 1:length(n) %]a @A8o0  
        rpowers = [rpowers m_abs(j):2:n(j)]; X$7Oo^1;  
    end vU_d=T%$  
    rpowers = unique(rpowers); }J ei$0x  
    .>mH]/]m  
    % Pre-compute the values of r raised to the required powers, zb>f;[  
    % and compile them in a matrix: V;h=8C5J  
    % ----------------------------- oUJj5iu}  
    if rpowers(1)==0 Vs#"SpH{'  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); mR{CVU  
        rpowern = cat(2,rpowern{:}); @4IW=V  
        rpowern = [ones(length_r,1) rpowern]; YSR mt/  
    else sU) TXL'_!  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); (C8 U   
        rpowern = cat(2,rpowern{:}); ]pW86L%  
    end H~A"C'P3#  
    A}t%;V2  
    % Compute the values of the polynomials: C`\9c ej  
    % -------------------------------------- "+=Pp  
    y = zeros(length_r,length(n)); Y/. AUN Z  
    for j = 1:length(n) FJP< bREQ  
        s = 0:(n(j)-m_abs(j))/2; HXQ e\r  
        pows = n(j):-2:m_abs(j); +c^_^Z$_4o  
        for k = length(s):-1:1 Iz DG&c  
            p = (1-2*mod(s(k),2))* ... "j{i,&Y$_  
                       prod(2:(n(j)-s(k)))/              ... xK(IS:HJ*  
                       prod(2:s(k))/                     ... O^5UB~  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... T4mv%zzS  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); >^a$  
            idx = (pows(k)==rpowers); 1EVfowIl  
            y(:,j) = y(:,j) + p*rpowern(:,idx); <f N; xIB  
        end "jMqt9ysN  
         C:]s;0$3'9  
        if isnorm KQ&Y2l1*>>  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); "`pNH'   
        end aF!Ex  
    end Q"40#RFA  
    % END: Compute the Zernike Polynomials et=7}K]l  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .%q$d d>>  
    ]kx-,M(  
    % Compute the Zernike functions: Yc^%zxub  
    % ------------------------------ I%oRvg|q  
    idx_pos = m>0; o]Gguw5W{  
    idx_neg = m<0; 5iVQc-m&  
    l^\(ss0~  
    z = y; v:E;^$6Vn  
    if any(idx_pos) "e 1wr  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ( S`6Q  
    end NUCiY\td  
    if any(idx_neg) cFK @3a  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); GcT;e5D  
    end F/>*If s  
    lwc5S `"  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) jfD1  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. aBhV3Fd[B  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated kZLMtj-   
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive U3E&n1AA  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, !nd*W"_gQ/  
    %   and THETA is a vector of angles.  R and THETA must have the same SHV4!xP-V  
    %   length.  The output Z is a matrix with one column for every P-value, 8Hi!kc;f6>  
    %   and one row for every (R,THETA) pair. 7\ypW$Ot  
    % a8pY[)^c  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike [ %}u=}@  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 7G<t"'  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) T-gk<V  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 Ts9ktPlm  
    %   for all p. pXu/(&?  
    % TJ(K3/)Z  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 m} 3gZu]  
    %   Zernike functions (order N<=7).  In some disciplines it is eS'yGY0b  
    %   traditional to label the first 36 functions using a single mode S{#cD1>.  
    %   number P instead of separate numbers for the order N and azimuthal AQss4[\Dx  
    %   frequency M. P:C2G(V1AR  
    % AVl~{k|  
    %   Example: !2tW$BP^  
    % c+kU o$  
    %       % Display the first 16 Zernike functions >\2:\wI  
    %       x = -1:0.01:1; ;apzAF  
    %       [X,Y] = meshgrid(x,x); 8z2Rry w  
    %       [theta,r] = cart2pol(X,Y); ?+0GfIV  
    %       idx = r<=1; e5?PkFV^a1  
    %       p = 0:15; n6MM5h/#r  
    %       z = nan(size(X)); C [uOReo  
    %       y = zernfun2(p,r(idx),theta(idx)); g&Vcg`  
    %       figure('Units','normalized') uH@FU60  
    %       for k = 1:length(p) WG7k(Sp ]  
    %           z(idx) = y(:,k); XL$* _c <)  
    %           subplot(4,4,k) zR;X*q"T$4  
    %           pcolor(x,x,z), shading interp  k5`OH8G  
    %           set(gca,'XTick',[],'YTick',[]) G8Z4J7^  
    %           axis square ]m4OIst  
    %           title(['Z_{' num2str(p(k)) '}']) FT.,%2  
    %       end _[K"gu  
    % &a,OfSz  
    %   See also ZERNPOL, ZERNFUN. MZ> 6o5K|  
    /) 4GSC}Gg  
    %   Paul Fricker 11/13/2006 4|?{VQ  
    *sw7niw  
    S4^N^lQ]  
    % Check and prepare the inputs: 23!;}zHp  
    % ----------------------------- uI~S=;o  
    if min(size(p))~=1 Iu@y(wyg  
        error('zernfun2:Pvector','Input P must be vector.') D*PYr{z'  
    end n5-)/R[z  
    +rXF{@ l  
    if any(p)>35 kq=V4-a[  
        error('zernfun2:P36', ... pd[ncL  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... V'Kgdj  
               '(P = 0 to 35).']) )D&M2CUw"f  
    end V/d/L3p  
    )E#2J$TD  
    % Get the order and frequency corresonding to the function number: :O<bA& :d  
    % ---------------------------------------------------------------- wC_l@7 t  
    p = p(:); I<td1Y1q  
    n = ceil((-3+sqrt(9+8*p))/2); %+>s#Q2d  
    m = 2*p - n.*(n+2); @Ky> 9m{  
    b2,mCfLsv  
    % Pass the inputs to the function ZERNFUN: $2^`Uca  
    % ---------------------------------------- (>4aibA'P  
    switch nargin 2& PPz}Sw  
        case 3 51C2u)HE  
            z = zernfun(n,m,r,theta); WEg6Kz  
        case 4 }9HmTr|  
            z = zernfun(n,m,r,theta,nflag); LVJI_O{fH  
        otherwise f3j{VN  
            error('zernfun2:nargin','Incorrect number of inputs.') %@a8P  
    end [D-Q'"'A  
    T<3BT  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) BqDKT  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. *<N3_tx"  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ;6@r-r  
    %   order N and frequency M, evaluated at R.  N is a vector of V.ht, ~l  
    %   positive integers (including 0), and M is a vector with the F' U 50usV  
    %   same number of elements as N.  Each element k of M must be a y@2epY?{  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) dzK{ Z  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is b|Q)[y]  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix o1&:ry  
    %   with one column for every (N,M) pair, and one row for every 4'$g(+z  
    %   element in R. mk7&<M  
    % } VJfJ/  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- .=m,hu~  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is +3s%E{  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to +^*iZ6{+7  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 DeR='7n  
    %   for all [n,m]. !O\;Nua  
    % [E#UGJ@  
    %   The radial Zernike polynomials are the radial portion of the [."[pY  
    %   Zernike functions, which are an orthogonal basis on the unit 8WE{5#oi  
    %   circle.  The series representation of the radial Zernike %Qg+R26U  
    %   polynomials is 5es[Ph|K5  
    % :o:e,WKxb  
    %          (n-m)/2 dz~co Z9  
    %            __ iAT)VQ&  
    %    m      \       s                                          n-2s 2G$SpfeIu  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 7+x? " 4  
    %    n      s=0 rc+C?)S  
    % 8/)qTUx:  
    %   The following table shows the first 12 polynomials. $/6;9d^  
    % QwhRNnE=  
    %       n    m    Zernike polynomial    Normalization l5l>d62  
    %       --------------------------------------------- w9 w%&{j  
    %       0    0    1                        sqrt(2) e><5Pr)  
    %       1    1    r                           2 BBcV9CGU  
    %       2    0    2*r^2 - 1                sqrt(6) q+B&orp  
    %       2    2    r^2                      sqrt(6) 0$7.g!h?  
    %       3    1    3*r^3 - 2*r              sqrt(8) "[}O"LTQ  
    %       3    3    r^3                      sqrt(8) PtqJ*Z  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) Sa19q.~%  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) XZw6Xtn  
    %       4    4    r^4                      sqrt(10) d #jK=:eK  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 7ugZE93!  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 42>Ge>#F  
    %       5    5    r^5                      sqrt(12) ~=R SKyzt  
    %       --------------------------------------------- `jS T  
    % 2lL,zFAq  
    %   Example: k6=nO?$  
    % "UwH\T4I  
    %       % Display three example Zernike radial polynomials eT2*W$  
    %       r = 0:0.01:1; fO#vF.k%  
    %       n = [3 2 5]; |yo\R{&6  
    %       m = [1 2 1]; +a^F\8H  
    %       z = zernpol(n,m,r); 7)h[Zy,A  
    %       figure p}[zt#v  
    %       plot(r,z) pRSOYTebP  
    %       grid on !|c|o*t{  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') Ts~L:3oaQ  
    % };'\~g,1  
    %   See also ZERNFUN, ZERNFUN2. dL|+d:v  
    d#2$!z#  
    % A note on the algorithm. Fs[aa#v4B  
    % ------------------------ {mB0rKVm  
    % The radial Zernike polynomials are computed using the series d;n."+=[x  
    % representation shown in the Help section above. For many special >vo=]c w  
    % functions, direct evaluation using the series representation can " vtCTl~t  
    % produce poor numerical results (floating point errors), because !'LW_@  
    % the summation often involves computing small differences between TIvRhbu  
    % large successive terms in the series. (In such cases, the functions %v2R.?F8  
    % are often evaluated using alternative methods such as recurrence \=>H6x]q  
    % relations: see the Legendre functions, for example). For the Zernike #nh|=X  
    % polynomials, however, this problem does not arise, because the Ytgj|@jsp  
    % polynomials are evaluated over the finite domain r = (0,1), and UwC=1g U  
    % because the coefficients for a given polynomial are generally all G9JAcO1  
    % of similar magnitude. u+{a8=  
    % ^] kF{ o?  
    % ZERNPOL has been written using a vectorized implementation: multiple &HSq(te  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] :T9< d er,  
    % values can be passed as inputs) for a vector of points R.  To achieve }`+B=h-dW  
    % this vectorization most efficiently, the algorithm in ZERNPOL /r_~: 3F  
    % involves pre-determining all the powers p of R that are required to U4G`ZK v(!  
    % compute the outputs, and then compiling the {R^p} into a single .KdyJ6o  
    % matrix.  This avoids any redundant computation of the R^p, and %\i9p]=  
    % minimizes the sizes of certain intermediate variables. 10H)^p%3+  
    % H:"ma S\I  
    %   Paul Fricker 11/13/2006 z3uW)GQ.  
    `O'`eY1f  
    P (S>=,Y&  
    % Check and prepare the inputs: NzNA>[$[  
    % ----------------------------- %w7]@VZ  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) @g G<le6  
        error('zernpol:NMvectors','N and M must be vectors.') "*})3['n  
    end ur$l Z0  
    t]X w{)T  
    if length(n)~=length(m) jMpD+Mb  
        error('zernpol:NMlength','N and M must be the same length.') H<1WbM:w  
    end b:w?PC~O  
    ]n-:Yv5 W  
    n = n(:); @}kv-*  
    m = m(:); V <bd;m  
    length_n = length(n); zz& ?{vJ  
    Gm\/Y:U  
    if any(mod(n-m,2)) D.mHIsX6\  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') _2N$LLbg  
    end a+Ac[>  
    ]Zmj4vK J  
    if any(m<0) MQ"xOcD*F  
        error('zernpol:Mpositive','All M must be positive.') 8.[SU  
    end +a*tO@HG  
    Q~T$N  
    if any(m>n) :kGU,>BN  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') D::rGB?.b  
    end H<3I 5Kgt  
    M|R b&6O  
    if any( r>1 | r<0 ) k-}b{  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 7.`fJf?  
    end Phke`3tth  
    7nuU^wc  
    if ~any(size(r)==1) y:6; LZ9[  
        error('zernpol:Rvector','R must be a vector.') KGg3 !jY  
    end Z4\=*ic@  
    6R^^.tCs  
    r = r(:); C 9t4#"  
    length_r = length(r); [Vma^B$7Vj  
    Sy 'Dp9!|  
    if nargin==4  s~Te  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); rvw)-=qR[  
        if ~isnorm Gh}*q|Lz  
            error('zernpol:normalization','Unrecognized normalization flag.') !@v7Zu43,  
        end |vw"[7_aS  
    else #{\%rWnCm  
        isnorm = false; Er{>p|n =  
    end 5D'\b}*lJ}  
    ctGL-kp  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :eCU/BC4  
    % Compute the Zernike Polynomials LSRk7'0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a0#J9O_  
    hyFyP\u]  
    % Determine the required powers of r: c??mL4$'N  
    % ----------------------------------- ( U xW;  
    rpowers = []; Pjc Tx +  
    for j = 1:length(n) RVQh2'w  
        rpowers = [rpowers m(j):2:n(j)]; .Fp4: e  
    end r% +V8o  
    rpowers = unique(rpowers); {Ja!~N;3  
    - RU=z!{  
    % Pre-compute the values of r raised to the required powers, _/tHD]um  
    % and compile them in a matrix: aSnF KB  
    % ----------------------------- i,/0/?)*_  
    if rpowers(1)==0 B]l)++~  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); HKUn`ng  
        rpowern = cat(2,rpowern{:}); sdo [D  
        rpowern = [ones(length_r,1) rpowern]; ;N?]eM}yf  
    else $F5 b  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); #%h-[/  
        rpowern = cat(2,rpowern{:}); K>@+m  
    end Bn &Ws  
    >: g3k  
    % Compute the values of the polynomials: Zo~  
    % -------------------------------------- ?o|f':  
    z = zeros(length_r,length_n); jJPGrkr  
    for j = 1:length_n fd.^h*'mU  
        s = 0:(n(j)-m(j))/2; TJR:vr  
        pows = n(j):-2:m(j); /PSd9N*=y  
        for k = length(s):-1:1 JVSA&c%3  
            p = (1-2*mod(s(k),2))* ... Y<%@s}zc  
                       prod(2:(n(j)-s(k)))/          ... @pRlxkvV  
                       prod(2:s(k))/                 ... d\gJ$ ~^K  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... )`O~f_pIC  
                       prod(2:((n(j)+m(j))/2-s(k))); 7V!*NBsl  
            idx = (pows(k)==rpowers); @X;!92i  
            z(:,j) = z(:,j) + p*rpowern(:,idx); /~$WUAh  
        end IHv[v*4:  
         fy@<&U5rg  
        if isnorm S(*sw 0O@+  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ^Xq 6:  
        end LQRQA[^  
    end :Ra,Eu  
    $m-2Hh qZ  
    % 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)  _X^1IaL  
    q-[@$9AS  
    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。 R0P iv:  
    ~bM4[*Q7  
    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)