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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 BY??X=  
    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! _u8d`7$*%  
     
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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 Z;nUS,?om  
    function z = zernfun(n,m,r,theta,nflag) hXz@ (cF  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. }uk]1M2=  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N HVK./y qy  
    %   and angular frequency M, evaluated at positions (R,THETA) on the sn.&|)?Fi  
    %   unit circle.  N is a vector of positive integers (including 0), and xl;0&/7e  
    %   M is a vector with the same number of elements as N.  Each element keL!;q|r-)  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Ld3!2g2y7&  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, B5fF\N^  
    %   and THETA is a vector of angles.  R and THETA must have the same mL[Y{t#N  
    %   length.  The output Z is a matrix with one column for every (N,M) \Yd 0oe82  
    %   pair, and one row for every (R,THETA) pair. Bwg\_:vq  
    % _f@, >l  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike &%`Y>\@f  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !`EhVV8u-_  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Z@b GLS  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, N"rZK/@}  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 7__?1n~{  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. #Ez+1  
    % u#`FkuE\}  
    %   The Zernike functions are an orthogonal basis on the unit circle. zCdzxb_h"  
    %   They are used in disciplines such as astronomy, optics, and ZP^7`q)6  
    %   optometry to describe functions on a circular domain. 2OQDG7#Kc  
    % '`fz|.|cbB  
    %   The following table lists the first 15 Zernike functions. A%c)=(,  
    % !_SIq`5]@  
    %       n    m    Zernike function           Normalization p7kH"j{xD  
    %       -------------------------------------------------- l9X\\uG&  
    %       0    0    1                                 1 nH % 1lD?:  
    %       1    1    r * cos(theta)                    2 Du."O]syD  
    %       1   -1    r * sin(theta)                    2 8'6$t@oT9w  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) "ZLujpZcG  
    %       2    0    (2*r^2 - 1)                    sqrt(3) dT*8I0\+  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) OGqsQ  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ~^R?HS  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ,,KGcDBj  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 0[T>UEI?  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) jJDY l([  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) lTn~VsoRZ  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) T^~9'KDd  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ^HasT4M+x  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Zc9j_.?*  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) }./_fFN@  
    %       -------------------------------------------------- )mbRG9P  
    % |ZnRr  
    %   Example 1: b[_${in:  
    % 8${Yu  
    %       % Display the Zernike function Z(n=5,m=1) r9d dVD  
    %       x = -1:0.01:1; @ dF]X  
    %       [X,Y] = meshgrid(x,x); qTl/bFD  
    %       [theta,r] = cart2pol(X,Y); Pqm)OZE?  
    %       idx = r<=1; 3!V$fl0  
    %       z = nan(size(X)); q"Z!}^{  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); OnKPD=<  
    %       figure OK^0,0kS3  
    %       pcolor(x,x,z), shading interp 5Si\hk:o  
    %       axis square, colorbar U.B=%S  
    %       title('Zernike function Z_5^1(r,\theta)') G]- wN7G  
    % A->y#KQ  
    %   Example 2: 5h4E>LB.B  
    % L!]~ J?)  
    %       % Display the first 10 Zernike functions ;dh8|ujh  
    %       x = -1:0.01:1; > \KVg(?D  
    %       [X,Y] = meshgrid(x,x); t9Nu4yl  
    %       [theta,r] = cart2pol(X,Y); fx783  
    %       idx = r<=1; Mn=5yU  
    %       z = nan(size(X)); S"z cSkF  
    %       n = [0  1  1  2  2  2  3  3  3  3]; WZ<kk T  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Hw"UJP  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; gxku3<S  
    %       y = zernfun(n,m,r(idx),theta(idx)); *KXg;777  
    %       figure('Units','normalized') k9^Vw+$m  
    %       for k = 1:10 M5Twulz/w  
    %           z(idx) = y(:,k); 6!3Jr  
    %           subplot(4,7,Nplot(k)) MK<VjpP0(  
    %           pcolor(x,x,z), shading interp .u_k?.8|  
    %           set(gca,'XTick',[],'YTick',[]) >Lo!8Hen  
    %           axis square G{cTQH|  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) weOzs]uc  
    %       end z]YP  
    % Gkr^uXNg#  
    %   See also ZERNPOL, ZERNFUN2. Q l$t  
    s\`Vr;R:|  
    %   Paul Fricker 11/13/2006 4P>tGO&*x  
    u%7a&1c  
    2 8j=q-9Z  
    % Check and prepare the inputs: Bn"r;pqWiT  
    % ----------------------------- WLAJqmC]  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 9 o7d3ir)  
        error('zernfun:NMvectors','N and M must be vectors.') Rro{A+[,X  
    end J\%<.S>  
    !7g E  
    if length(n)~=length(m) UEq;}4Bo  
        error('zernfun:NMlength','N and M must be the same length.') PSdH9ea  
    end 4nhe *ip  
    ZHs hg`I`  
    n = n(:); vl@t4\@3  
    m = m(:); 3"gifE  
    if any(mod(n-m,2)) 4JHQ^i-aY  
        error('zernfun:NMmultiplesof2', ... %;0w2W  
              'All N and M must differ by multiples of 2 (including 0).') sK:,c5^  
    end )Q\ZYCPOr  
    ."Yub];H  
    if any(m>n) @Y>3-,o,S  
        error('zernfun:MlessthanN', ... ;UgRm#  
              'Each M must be less than or equal to its corresponding N.') gkpNT)  
    end 1>*]jj}  
    |zu>G9m  
    if any( r>1 | r<0 ) xae rMr  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') NEO~|B*oDU  
    end lxK_+fj q  
    ~zz|U!TG  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Ar sMqb  
        error('zernfun:RTHvector','R and THETA must be vectors.') Yi[dS`,d  
    end l\^q7cXG  
    Q ;P~'  
    r = r(:); O#7ldF(  
    theta = theta(:); [ &*$!M  
    length_r = length(r); #{0DpSzE5  
    if length_r~=length(theta) (Df<QC`0v  
        error('zernfun:RTHlength', ... bE>3D#V<  
              'The number of R- and THETA-values must be equal.') $EJ*x$  
    end !9"R4~4  
    .Qh8I+Q%  
    % Check normalization: YeJ95\jf  
    % -------------------- 7o z(hO~  
    if nargin==5 && ischar(nflag) x#0C+cU  
        isnorm = strcmpi(nflag,'norm'); DuvP3(K  
        if ~isnorm ^@L[0Z`  
            error('zernfun:normalization','Unrecognized normalization flag.') <nsl`C~6g0  
        end 5?kA)!|UB  
    else (r[<g*+3  
        isnorm = false; ?<frU ,{  
    end +$>ut r  
    %Z{J=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |V 9%@ Y?  
    % Compute the Zernike Polynomials * Kzs(O  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \N#)e1.0P  
    e+R.0E  
    % Determine the required powers of r: 5,AQ~_,'\  
    % ----------------------------------- <Awx:lw.  
    m_abs = abs(m); J+*rjdI  
    rpowers = []; QrA8 KSLC  
    for j = 1:length(n)  (+]k{  
        rpowers = [rpowers m_abs(j):2:n(j)]; )N=b<%WD   
    end jPU# {Wo#  
    rpowers = unique(rpowers); /#G"'U/  
    u F*cS&'Z  
    % Pre-compute the values of r raised to the required powers, .;KupQ;*  
    % and compile them in a matrix: 4\OELU  
    % ----------------------------- hTG d Uw]  
    if rpowers(1)==0 3Xh&l[.  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @$+[IiP  
        rpowern = cat(2,rpowern{:}); $m=z87hX  
        rpowern = [ones(length_r,1) rpowern]; EhFhL4Xdn  
    else .V.N^8(:a  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 78QFaN$  
        rpowern = cat(2,rpowern{:}); ?^ErrlI_  
    end I(<G;ft<}  
    ai`:HhE  
    % Compute the values of the polynomials: )(L&+DDy  
    % -------------------------------------- f<;9q?0VF  
    y = zeros(length_r,length(n)); D1Sl+NOV  
    for j = 1:length(n) wKeqR$  
        s = 0:(n(j)-m_abs(j))/2; o7T|w~F~R  
        pows = n(j):-2:m_abs(j); _(z"l"l=$  
        for k = length(s):-1:1 j d8 1E  
            p = (1-2*mod(s(k),2))* ... z>0"T2W y  
                       prod(2:(n(j)-s(k)))/              ... )ED[cYGx  
                       prod(2:s(k))/                     ... _N:h&uw  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 0/gcSW b  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Td  F<  
            idx = (pows(k)==rpowers); p_AV3   
            y(:,j) = y(:,j) + p*rpowern(:,idx); +-nQ, fOV  
        end >eTlew<5  
         !qpu /  
        if isnorm -0X> y  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); bvx:R ~E$  
        end "XY?v8*c  
    end %KA/  
    % END: Compute the Zernike Polynomials X2uX+}h*tA  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,![=_d  
    R,\ r{@yrz  
    % Compute the Zernike functions: `-H:j:U{  
    % ------------------------------ C#~MR+;  
    idx_pos = m>0; +Y~+o-_  
    idx_neg = m<0; m#nxw  
    >&&xJ5  
    z = y; -"zu"H~t4  
    if any(idx_pos) }SV3PdE  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); `"H?nf0  
    end ]1&9~TL  
    if any(idx_neg) S0+zq<  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); QC4T=E]` j  
    end n{t',r50  
    1,j9(m2  
    % 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) 4QFOO sNp  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. @S7=6RKa[  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated HzV+g/8>A  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive h!K2F~i{P  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Z]TVH8%|k  
    %   and THETA is a vector of angles.  R and THETA must have the same DH9?2)aR  
    %   length.  The output Z is a matrix with one column for every P-value, 0xUj#)  
    %   and one row for every (R,THETA) pair. l :Nxl  
    % :WIf$P?X  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ZPsY0IzLo  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) kA<r:/  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) )_e"N d4  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 mzxvfXSF  
    %   for all p. 3c^=<i %  
    % Zk#i9[g9*  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 \a2oM$PX  
    %   Zernike functions (order N<=7).  In some disciplines it is 6r=)V$K <  
    %   traditional to label the first 36 functions using a single mode S1juAV=  
    %   number P instead of separate numbers for the order N and azimuthal HQ`nq~%&(  
    %   frequency M. q;../h]Ne  
    % 'lsq3!d.  
    %   Example: [y[v]'  
    % [l%fL9  
    %       % Display the first 16 Zernike functions Cn;H@!8<s  
    %       x = -1:0.01:1; T 0v@mXBQ  
    %       [X,Y] = meshgrid(x,x); m2uML*&O5K  
    %       [theta,r] = cart2pol(X,Y); P9i9<pR  
    %       idx = r<=1; :.[5('  
    %       p = 0:15; uxMy 1oy  
    %       z = nan(size(X)); ENXW#{N.v  
    %       y = zernfun2(p,r(idx),theta(idx)); ;=VK _3"  
    %       figure('Units','normalized') V@n(v\F  
    %       for k = 1:length(p) _Kl{50}]  
    %           z(idx) = y(:,k); m)|.:sj  
    %           subplot(4,4,k) (zJ$oRq  
    %           pcolor(x,x,z), shading interp Q`p}X&^a  
    %           set(gca,'XTick',[],'YTick',[]) g1 Wtu*K3  
    %           axis square ds$\vSd  
    %           title(['Z_{' num2str(p(k)) '}']) @7l=+`.i  
    %       end .A"T086  
    % t:"=]zUU  
    %   See also ZERNPOL, ZERNFUN. o(X90X  
    ?9zoQ[  
    %   Paul Fricker 11/13/2006 kk_9G -M  
    `YmI'  
    J'&B:PZObB  
    % Check and prepare the inputs: w] 5U  
    % ----------------------------- COan) <Ku  
    if min(size(p))~=1 Ro'4/{}+  
        error('zernfun2:Pvector','Input P must be vector.') \p@nH%@v  
    end |+;KhC  
    x)#<.DX  
    if any(p)>35 tU)r[2H2  
        error('zernfun2:P36', ... *@G(3 n  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... }lC64;yo  
               '(P = 0 to 35).']) K+ 7yUF8XP  
    end g=oeS%>E  
    wwK~H  
    % Get the order and frequency corresonding to the function number: cEdz;kbUM  
    % ---------------------------------------------------------------- :L [YmZ  
    p = p(:); +6#%P  
    n = ceil((-3+sqrt(9+8*p))/2); OHtgn  
    m = 2*p - n.*(n+2); >d27[%  
    #zSi/r/=1  
    % Pass the inputs to the function ZERNFUN: =hugnX<9  
    % ---------------------------------------- / UaNYv/  
    switch nargin 9o_ g_q  
        case 3 NDe[2  
            z = zernfun(n,m,r,theta); 4iYKW2a  
        case 4 e"o6C\c  
            z = zernfun(n,m,r,theta,nflag); V 4\^TO`q=  
        otherwise /]k ,,&  
            error('zernfun2:nargin','Incorrect number of inputs.') XC7Ty'#"KX  
    end 0$f_or9T  
    `b^#quz  
    % 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) vB{; N  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. }sTH.%  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of #uD)0zdw  
    %   order N and frequency M, evaluated at R.  N is a vector of $tDCS  
    %   positive integers (including 0), and M is a vector with the cotxo?)Zv  
    %   same number of elements as N.  Each element k of M must be a B&4fYpn  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) B91S h`  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is P S_3Oq)  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix iioct_7,g<  
    %   with one column for every (N,M) pair, and one row for every pPiYPfs  
    %   element in R. #L@} .Giz  
    % 9atjK4+o  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ]^yV`Z8  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is :"OZc7 ~  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Eu`2w%qz  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 c W81  
    %   for all [n,m]. * 1 |YLy  
    % ":UWowJO  
    %   The radial Zernike polynomials are the radial portion of the P3wU#qU  
    %   Zernike functions, which are an orthogonal basis on the unit LPq*ZZK  
    %   circle.  The series representation of the radial Zernike Cbgj@4H  
    %   polynomials is '2Q.~6   
    % u#a%(  
    %          (n-m)/2 blRY7  
    %            __ {f`lSu  
    %    m      \       s                                          n-2s $ 7U Dz  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Y=P9:unG  
    %    n      s=0 Ph(]?MG\_  
    % T7>4 8eH  
    %   The following table shows the first 12 polynomials. .DgoOo%?"  
    % yPf?"W  
    %       n    m    Zernike polynomial    Normalization pchQ#GU  
    %       --------------------------------------------- 2x7(}+eD  
    %       0    0    1                        sqrt(2) \]Y\P~n  
    %       1    1    r                           2 4)3g!o ?  
    %       2    0    2*r^2 - 1                sqrt(6) o/tVcv  
    %       2    2    r^2                      sqrt(6) h|J;6Sm@  
    %       3    1    3*r^3 - 2*r              sqrt(8) {c v;w  
    %       3    3    r^3                      sqrt(8) /~H[= Pf  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) fkdf~Vb  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) :`:xP  
    %       4    4    r^4                      sqrt(10) e+NWmu{<_  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) jo 7Hyw!g  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ,|e}Y [  
    %       5    5    r^5                      sqrt(12) tP}Xhn`  
    %       --------------------------------------------- 8ku? W  
    % bin6i2b  
    %   Example: e%PC e9  
    % 4^ c!_K&&  
    %       % Display three example Zernike radial polynomials #=X)Jx~  
    %       r = 0:0.01:1; R'S c  
    %       n = [3 2 5]; e(?:g@]-r  
    %       m = [1 2 1]; |$YyjYK  
    %       z = zernpol(n,m,r); sFbfFUd  
    %       figure !a' K &  
    %       plot(r,z) mZ`1JO9  
    %       grid on pwa.q  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') v*&Uk '4E  
    % 4st~3,lR$  
    %   See also ZERNFUN, ZERNFUN2. \9046An  
    6{5q@9F  
    % A note on the algorithm. Gh2#-~|cB  
    % ------------------------ ;l$9gD>R  
    % The radial Zernike polynomials are computed using the series =hJfL}&O3  
    % representation shown in the Help section above. For many special VT'0DQ!NIq  
    % functions, direct evaluation using the series representation can }$^]dn@  
    % produce poor numerical results (floating point errors), because [_j6cj]  
    % the summation often involves computing small differences between lo"j )Zt  
    % large successive terms in the series. (In such cases, the functions 6_W<hevI  
    % are often evaluated using alternative methods such as recurrence \|v`l{  
    % relations: see the Legendre functions, for example). For the Zernike {d| |q<.-  
    % polynomials, however, this problem does not arise, because the J cP~-cp  
    % polynomials are evaluated over the finite domain r = (0,1), and Kp8fh-4_  
    % because the coefficients for a given polynomial are generally all AnRlH  
    % of similar magnitude. -oU@D  
    % E^7C _JP  
    % ZERNPOL has been written using a vectorized implementation: multiple 7 n\mj\  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Y [4vRzc  
    % values can be passed as inputs) for a vector of points R.  To achieve : aHcPc:  
    % this vectorization most efficiently, the algorithm in ZERNPOL -UJ?L  
    % involves pre-determining all the powers p of R that are required to b2G2c L-(  
    % compute the outputs, and then compiling the {R^p} into a single Ud$Q0m&  
    % matrix.  This avoids any redundant computation of the R^p, and ~D*b3K 8X  
    % minimizes the sizes of certain intermediate variables. X2i*iW<  
    % |pBMrN+is  
    %   Paul Fricker 11/13/2006 &j3` )N  
    p=2zS.  
    {nTG~d  
    % Check and prepare the inputs: Sc$gnUYD{  
    % ----------------------------- DUqJ y*F(  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 4 ^4d9?c  
        error('zernpol:NMvectors','N and M must be vectors.') 7LG+$LEz  
    end b9`iZ  
    vuXS/ d  
    if length(n)~=length(m) p7s@%scp  
        error('zernpol:NMlength','N and M must be the same length.') Bw6L;Vu  
    end {wcO[bN  
    J6DnPaw-G  
    n = n(:); FtN}]@F  
    m = m(:); :"VujvFX  
    length_n = length(n); 6eM6[  
    z* RSMfRW  
    if any(mod(n-m,2)) c!mG1lwD.  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') <8f(eP\*F  
    end z /weit  
    {H+?z<BF<  
    if any(m<0) .?B{GnB>  
        error('zernpol:Mpositive','All M must be positive.') \<X2ns@Tf  
    end Ey'J]KVW  
    EA6t36|TX  
    if any(m>n) <>]1Y$^Y  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') _AprkI_  
    end 8`*`nQhWa  
    BMdSf(l  
    if any( r>1 | r<0 ) fSjs?zd`  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') {8 N=WZ  
    end :\ mRtVH  
    +ZclGchw  
    if ~any(size(r)==1) 7u::5W-q  
        error('zernpol:Rvector','R must be a vector.') n08; <  
    end zFywC-my@  
    7D   
    r = r(:); ocwE_dR{  
    length_r = length(r); %&tb9_T)d  
    Ew]<jF|.#  
    if nargin==4 1Fs-0)s8  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); Ssf+b!e]  
        if ~isnorm z{|LQt6q  
            error('zernpol:normalization','Unrecognized normalization flag.') F?cq'd  
        end Ib6(Bp9.L  
    else /=T H08  
        isnorm = false; 'y.JcS!|  
    end %l]Rh/VPn?  
    lufeieW  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1q] & 7R  
    % Compute the Zernike Polynomials 7TpRCq#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =*O=E@]  
    NF mc>0-  
    % Determine the required powers of r: ?Wa<AFXQ  
    % ----------------------------------- bK4&=#Zh  
    rpowers = []; f`?0WJ(M  
    for j = 1:length(n) !R6ApB4ZI  
        rpowers = [rpowers m(j):2:n(j)]; Gm A!Mo  
    end RLHYw@-j@  
    rpowers = unique(rpowers); +ubnx{VC  
    @\jQoaLT$_  
    % Pre-compute the values of r raised to the required powers, 5ITq?%{M  
    % and compile them in a matrix: r|fO7PD  
    % ----------------------------- ZdH1nX(Yh3  
    if rpowers(1)==0 oRq3 pO}f  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 76bc]o#  
        rpowern = cat(2,rpowern{:}); AF$\WWrB  
        rpowern = [ones(length_r,1) rpowern]; c+2sT3).D  
    else qjAh6Q/E`  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 2+:'0Krc  
        rpowern = cat(2,rpowern{:}); Xa,\EEmQ  
    end bi$VAYn.^  
    YE\K<T jH  
    % Compute the values of the polynomials: p411 `]Zf  
    % -------------------------------------- +s~.A_7)  
    z = zeros(length_r,length_n); [!~}S  
    for j = 1:length_n ="'- &  
        s = 0:(n(j)-m(j))/2; NXI[q 'y  
        pows = n(j):-2:m(j);  iSiDSeW8  
        for k = length(s):-1:1 /_*>d)  
            p = (1-2*mod(s(k),2))* ... "hPCQp`Tj  
                       prod(2:(n(j)-s(k)))/          ... lhO2'#]i  
                       prod(2:s(k))/                 ... {/|qjkT&W  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ($>XIb9f  
                       prod(2:((n(j)+m(j))/2-s(k))); /:p8I6;  
            idx = (pows(k)==rpowers); {G*OR,HN  
            z(:,j) = z(:,j) + p*rpowern(:,idx); S4bBafj[I  
        end p/*"4-S  
         @G*.1;jO  
        if isnorm OipqoI2  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); d~Mg vh'  
        end ^npJUa  
    end !h:  Q  
    jg_n7  
    % 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)  SZ!=`a]  
    :+&AY2`  
    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。 F?4(5 K  
    M8;lLcgu.  
    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)