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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 正序阅读 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 K/(QR_@?  
    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! 7'At_oG  
     
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    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)  G,B4=[Y  
    X<$DNRN  
    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。 bWlY Q  
    01&E.A  
    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)
    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niuhelen
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) b l+g7g;  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 5dE=M};v  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of >{:hadUH  
    %   order N and frequency M, evaluated at R.  N is a vector of $of2lA  
    %   positive integers (including 0), and M is a vector with the |K-`  
    %   same number of elements as N.  Each element k of M must be a CAs:>s '8  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) hKWWN`;b !  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 8,!Oup  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix R(VOHFvW6  
    %   with one column for every (N,M) pair, and one row for every /}L2LMIm  
    %   element in R. 3z2 OW@zL$  
    % 8 p[n>qV9  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- S 593wfc  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is v}V[sIs}  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to V(DY!f_%  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 2xX:Q'\2  
    %   for all [n,m]. kV5)3%?  
    % "2sk1  
    %   The radial Zernike polynomials are the radial portion of the Q1?*+]  
    %   Zernike functions, which are an orthogonal basis on the unit 9jEH"`qqk  
    %   circle.  The series representation of the radial Zernike rZaO^}u]  
    %   polynomials is .#|?-5q/iN  
    % *tEqu%N1'  
    %          (n-m)/2 ^ W?cuJ8  
    %            __ "z3rH~q72  
    %    m      \       s                                          n-2s )hj:Xpj9#  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 9_O4 yTL  
    %    n      s=0 V8TdtGB.|h  
    % =uAy/S  
    %   The following table shows the first 12 polynomials. m Jk\$/Kh  
    % Jut&J]{h  
    %       n    m    Zernike polynomial    Normalization E8}evi  
    %       --------------------------------------------- (A6~mi r!  
    %       0    0    1                        sqrt(2) /kkUEo+  
    %       1    1    r                           2 _"yA1D0d_  
    %       2    0    2*r^2 - 1                sqrt(6) fTvm2+.nX  
    %       2    2    r^2                      sqrt(6) 'EAskA] *  
    %       3    1    3*r^3 - 2*r              sqrt(8) wL 4Y%g  
    %       3    3    r^3                      sqrt(8) V<H9KA  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 9iZio3m  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) n%J=!z3  
    %       4    4    r^4                      sqrt(10) p T8?z  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) u%)gnj_  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ty0P9.Q  
    %       5    5    r^5                      sqrt(12) p,$N-22a  
    %       --------------------------------------------- &.*UVc2+Y  
    % tMiIlf!>p  
    %   Example: K=v:qY4Z  
    % !!^z6jpvn  
    %       % Display three example Zernike radial polynomials =ZIT!B?4  
    %       r = 0:0.01:1; AT~,  
    %       n = [3 2 5]; &o;0%QgF  
    %       m = [1 2 1]; j"69uj` R  
    %       z = zernpol(n,m,r); \BXzmok  
    %       figure CG=c@-"n/  
    %       plot(r,z) ls]N&!/hq  
    %       grid on 3$k#bC  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') +)jll#}?  
    % 0Lxz?R x]<  
    %   See also ZERNFUN, ZERNFUN2. IL %]4,  
    qM(}|fMbN  
    % A note on the algorithm. x^ f)I|t  
    % ------------------------ .w.:o2L  
    % The radial Zernike polynomials are computed using the series =79R;|5  
    % representation shown in the Help section above. For many special |0y#} |/  
    % functions, direct evaluation using the series representation can ns8s2kYcm  
    % produce poor numerical results (floating point errors), because !-f Bw  
    % the summation often involves computing small differences between Pj-INc96  
    % large successive terms in the series. (In such cases, the functions 3N+lWuE}K  
    % are often evaluated using alternative methods such as recurrence !rM~   
    % relations: see the Legendre functions, for example). For the Zernike K}R+~<bIY  
    % polynomials, however, this problem does not arise, because the Lqj Qv$  
    % polynomials are evaluated over the finite domain r = (0,1), and )O~[4xV~  
    % because the coefficients for a given polynomial are generally all 5XZ! yYB?  
    % of similar magnitude. y!77gx?-  
    % xLz=)k[''  
    % ZERNPOL has been written using a vectorized implementation: multiple (hzN(Dh  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Yv;s3>r  
    % values can be passed as inputs) for a vector of points R.  To achieve 1q;v|F  
    % this vectorization most efficiently, the algorithm in ZERNPOL G:=hg6 '  
    % involves pre-determining all the powers p of R that are required to ?0VR2Yb${b  
    % compute the outputs, and then compiling the {R^p} into a single LmF,en5  
    % matrix.  This avoids any redundant computation of the R^p, and #dA$k+3  
    % minimizes the sizes of certain intermediate variables. vjGQ!xF  
    % )#}>,,S  
    %   Paul Fricker 11/13/2006 -1g :3'% P  
    3yZmW$E.  
    dw bR,K  
    % Check and prepare the inputs: @LKQ-<dZG  
    % ----------------------------- yLX $SR  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) $RaN@& Wm  
        error('zernpol:NMvectors','N and M must be vectors.') 5yy:JTAH5  
    end i<m(neX[H  
    FRBu8WW0L  
    if length(n)~=length(m) N6U d(8*  
        error('zernpol:NMlength','N and M must be the same length.') >1~ /:DJ  
    end fGo4&( U  
    ~?Q sr  
    n = n(:); 0M_oFx  
    m = m(:); &v{Ehkr*  
    length_n = length(n); 5</$dcG  
    &_ekA44E  
    if any(mod(n-m,2)) I &t~o  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') g{65QP  
    end ,fVD`RR(W?  
    11[lc2  
    if any(m<0) :S+K\  
        error('zernpol:Mpositive','All M must be positive.') \*xB<mq  
    end o\IMYT  
    :HkBP90o  
    if any(m>n) 7RAB"T;?Q  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 5'~_d@M  
    end 0lfK} a  
    f!Q\M1t)  
    if any( r>1 | r<0 ) yB*,)x0 @  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') )+E[M!34  
    end 0+1wi4wy/  
    6o 3 bq|  
    if ~any(size(r)==1) La26"C"X  
        error('zernpol:Rvector','R must be a vector.') VM"cpC_8  
    end _'u]{X\k{J  
    >:;dNVz  
    r = r(:); <j'V}|3  
    length_r = length(r); Lsmcj{1d  
    RpHlq  
    if nargin==4 3^ Yc%  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); \oQ]=dDCd%  
        if ~isnorm yBIlwN`kB  
            error('zernpol:normalization','Unrecognized normalization flag.') 5,RUPaE  
        end UkV?,P@l  
    else |y\Km  
        isnorm = false; m o0\t#jA  
    end (B7G'h.?  
    f^WTsh]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Wgq|Q*  
    % Compute the Zernike Polynomials fV(3RG  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !d* [QD8  
    GXNf@&  
    % Determine the required powers of r: J~Xv R  
    % ----------------------------------- US6_5>/  
    rpowers = []; )s2] -n}W  
    for j = 1:length(n) TOYK'|lwM  
        rpowers = [rpowers m(j):2:n(j)]; ]Z JoC!u  
    end P:qmg"i@3  
    rpowers = unique(rpowers);  6 K $mW  
    YdY-Jg Xm  
    % Pre-compute the values of r raised to the required powers, wucdXj{%  
    % and compile them in a matrix: bQAznd0  
    % ----------------------------- mYBEjZ B  
    if rpowers(1)==0 7p&jSOY  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); B[vj X"yg  
        rpowern = cat(2,rpowern{:}); da{]B5p\  
        rpowern = [ones(length_r,1) rpowern]; -+9[X*VCc  
    else $6QIYF""  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8#I>`z^F  
        rpowern = cat(2,rpowern{:}); I-7LT?r  
    end T X`X5j  
    snV*gSUH  
    % Compute the values of the polynomials: e5 =d Ev  
    % -------------------------------------- uFd$*`jS  
    z = zeros(length_r,length_n); MfTLa)Rz  
    for j = 1:length_n Z5{a7U4z_  
        s = 0:(n(j)-m(j))/2; D-3[# ~MV  
        pows = n(j):-2:m(j); aD/Rr3v>  
        for k = length(s):-1:1 @nxo Bc !P  
            p = (1-2*mod(s(k),2))* ... ijI/z5  
                       prod(2:(n(j)-s(k)))/          ... -DDA b(2*  
                       prod(2:s(k))/                 ... bP,<^zA|X  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... mp|pz%U  
                       prod(2:((n(j)+m(j))/2-s(k))); kH!Z|P s?R  
            idx = (pows(k)==rpowers); <?jd NM  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ~Eut_d  
        end dWY%bb  
         Vla,avON  
        if isnorm E'5*w6  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); vN v?trw  
        end  K +7  
    end Ku;fZN[g  
    l =^A41L_  
    % EOF zernpol
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) 8(3(kZxS  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 5Iu5N0cn  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated r)(5,*v  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive *[MWvs:,  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ]A4=/6`g?b  
    %   and THETA is a vector of angles.  R and THETA must have the same N_t,n^i9>*  
    %   length.  The output Z is a matrix with one column for every P-value, lED!}h'4  
    %   and one row for every (R,THETA) pair. 8K8u|]i  
    % ;w{<1NH2+.  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike I?KN7(9u?  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) %lKw+D  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) GR,2^]<{  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 -15e  
    %   for all p. \u=d`}E  
    % {sTf4S\S  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ,CE/o7.FG  
    %   Zernike functions (order N<=7).  In some disciplines it is =4y gbk  
    %   traditional to label the first 36 functions using a single mode LPs%^*8(2  
    %   number P instead of separate numbers for the order N and azimuthal ?2<QoS  
    %   frequency M. HKN|pO3v  
    % _S!^=9bJ  
    %   Example: }"Y<<e<z:  
    % Bz+oM N#XJ  
    %       % Display the first 16 Zernike functions uH_KOiF  
    %       x = -1:0.01:1; 0Q3U\cDr  
    %       [X,Y] = meshgrid(x,x); B`pBIUu  
    %       [theta,r] = cart2pol(X,Y); ; :e7Z^\/k  
    %       idx = r<=1; g`EZLDjt  
    %       p = 0:15; 1=VyD<dNG6  
    %       z = nan(size(X)); QE]@xLz   
    %       y = zernfun2(p,r(idx),theta(idx)); LUbhTc  
    %       figure('Units','normalized') 3 ML][|TR  
    %       for k = 1:length(p) eSPS3|YYn  
    %           z(idx) = y(:,k); vrn4yHoZ  
    %           subplot(4,4,k) SA, ~q&  
    %           pcolor(x,x,z), shading interp '2,~'Zk  
    %           set(gca,'XTick',[],'YTick',[]) /4{WT?j  
    %           axis square ]&'!0'3`  
    %           title(['Z_{' num2str(p(k)) '}']) :@w~*eK~  
    %       end Gz`Jzh j  
    % [d`Jw/4n  
    %   See also ZERNPOL, ZERNFUN. --HDEc|  
    g&RpE41x  
    %   Paul Fricker 11/13/2006 3j#VKj+Uc  
    #1YMpL  
    ODJ"3 J  
    % Check and prepare the inputs: 4+olyBht  
    % ----------------------------- :kZ]Swi 5  
    if min(size(p))~=1 'r'=%u$1C  
        error('zernfun2:Pvector','Input P must be vector.') g$(Y\`zw  
    end b"g^Jm! j  
    0%xktf  
    if any(p)>35 V[ UOlJ  
        error('zernfun2:P36', ... 3s\UU2yr  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... %p7 ?\>  
               '(P = 0 to 35).']) mR}8}K]L  
    end ,>|tQ'  
    1q}32^>+o  
    % Get the order and frequency corresonding to the function number: a[ULSYEi  
    % ---------------------------------------------------------------- R P{pEd  
    p = p(:); QPy h.9:N  
    n = ceil((-3+sqrt(9+8*p))/2); E2hsSqsu=  
    m = 2*p - n.*(n+2); BKFO^  
     &_)P)L  
    % Pass the inputs to the function ZERNFUN: g5BL"Dn  
    % ---------------------------------------- [[T7s(3  
    switch nargin oKGH|iVEe  
        case 3 r$<!?Z  
            z = zernfun(n,m,r,theta); |:)Bo<8  
        case 4 d% EdvM|)  
            z = zernfun(n,m,r,theta,nflag); J~x]~}V&  
        otherwise fb f&bJT  
            error('zernfun2:nargin','Incorrect number of inputs.') R6~6b&-8  
    end tmGhJZ2j  
    /.:1Da  
    % EOF zernfun2
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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 ^|/<e?~I  
    function z = zernfun(n,m,r,theta,nflag) qJ" (:~  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. zDg*ds\  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N R/u0,  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 4n#u?)  
    %   unit circle.  N is a vector of positive integers (including 0), and mjOxmwo  
    %   M is a vector with the same number of elements as N.  Each element l(Y32]Z   
    %   k of M must be a positive integer, with possible values M(k) = -N(k) $y;w@^  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, s-#@t  
    %   and THETA is a vector of angles.  R and THETA must have the same ImQ -kz?b  
    %   length.  The output Z is a matrix with one column for every (N,M) mR.j8pi  
    %   pair, and one row for every (R,THETA) pair. n6[shXH  
    % ~ESw* 6s9  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike U["<f`z4\  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), iBWzxPv:z  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral d{TcjZ  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, E[hSL#0  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized M_O$]^I3w  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. l>jrY1u  
    % )g=mv*9>  
    %   The Zernike functions are an orthogonal basis on the unit circle. Fpo}UQQbc  
    %   They are used in disciplines such as astronomy, optics, and v~RxtTu  
    %   optometry to describe functions on a circular domain. BTsvL>Wy  
    % H28-;>'`  
    %   The following table lists the first 15 Zernike functions. !/`AM<`o  
    % VK4UhN2  
    %       n    m    Zernike function           Normalization i<&z'A6&]*  
    %       -------------------------------------------------- f$</BND  
    %       0    0    1                                 1 Sw$&E  
    %       1    1    r * cos(theta)                    2 QVn2`hr  
    %       1   -1    r * sin(theta)                    2 2U%t  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) +%CXc%  
    %       2    0    (2*r^2 - 1)                    sqrt(3) kW+>"3  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ym p*:lH(  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) FoIK, MdJ  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) `?:{aOI  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) w2$ L;q  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) r:xg#&"*  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) @"cnPLh&  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1`II%mf[  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) zt((TD2  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) mj9|q8v{+  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 4o''C |ND  
    %       -------------------------------------------------- ( *26aMp  
    % `zs@W  
    %   Example 1: ~+\A4BW  
    % 5m;pHgkb  
    %       % Display the Zernike function Z(n=5,m=1) X:FyNUa  
    %       x = -1:0.01:1; h1)+QLI  
    %       [X,Y] = meshgrid(x,x); <-d-. 8  
    %       [theta,r] = cart2pol(X,Y); X"8$,\wX,  
    %       idx = r<=1; +=`w  
    %       z = nan(size(X)); W OYZ  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); F0m[ls$  
    %       figure rI)&.5^  
    %       pcolor(x,x,z), shading interp yl<=_Q  
    %       axis square, colorbar YU87l  
    %       title('Zernike function Z_5^1(r,\theta)') aF=;v*  
    % WUDXx %  
    %   Example 2: 5W{|? l{  
    % _#<l -R`  
    %       % Display the first 10 Zernike functions p<VW;1bt5  
    %       x = -1:0.01:1; J(~xU0gd'  
    %       [X,Y] = meshgrid(x,x); B^1jd!m  
    %       [theta,r] = cart2pol(X,Y); 8Z@O%\1x6  
    %       idx = r<=1; Rlr[uU_  
    %       z = nan(size(X)); 3,+Us B%  
    %       n = [0  1  1  2  2  2  3  3  3  3]; +SRM?av  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Mi!ak  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; I1ibrn  
    %       y = zernfun(n,m,r(idx),theta(idx)); 'u [cT$  
    %       figure('Units','normalized') jaTCRn3|<  
    %       for k = 1:10 a 0FU[*q  
    %           z(idx) = y(:,k); OUHd@up@n  
    %           subplot(4,7,Nplot(k))  GwD"j]  
    %           pcolor(x,x,z), shading interp %MfT5*||f  
    %           set(gca,'XTick',[],'YTick',[]) ^w RD|  
    %           axis square YkV-]%c  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @XF/hhGE_y  
    %       end ,g)9ZP.F  
    % KrECAc  
    %   See also ZERNPOL, ZERNFUN2. =2wy;@f  
    &kOb#\11u  
    %   Paul Fricker 11/13/2006 FLlL0Gu  
    J0Y-e39 `  
    nYY'hjZ  
    % Check and prepare the inputs: V> eJ  
    % ----------------------------- A`1/g{Ha  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) DB1Y`l  
        error('zernfun:NMvectors','N and M must be vectors.') y /?;s]>b  
    end *Oe;JqQkK  
    -E!V;Tgc%U  
    if length(n)~=length(m) #KSB%  
        error('zernfun:NMlength','N and M must be the same length.') X?"Ro`S  
    end r(=3yd/G$  
    "Zicac@N  
    n = n(:); K[|d7e  
    m = m(:); v3jx2Z  
    if any(mod(n-m,2)) -Kf'02  
        error('zernfun:NMmultiplesof2', ... Neb%D8/Kn  
              'All N and M must differ by multiples of 2 (including 0).') 4VL]v9  
    end kA:cz$ )  
    5h(] S[Zf3  
    if any(m>n) Ib4 8`  
        error('zernfun:MlessthanN', ... u RNc9  
              'Each M must be less than or equal to its corresponding N.') k@R)_,2HH  
    end W,n0'";')  
    My'6 yQL  
    if any( r>1 | r<0 ) ?3i-wpzMp  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') hAZ"M:f  
    end ]pA}h. R#-  
    k4-C*Gx$h  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) {=d\t<p*n  
        error('zernfun:RTHvector','R and THETA must be vectors.') <BN)>NqM  
    end ~ #~Kxh  
    86@@j*c(@k  
    r = r(:); p>p=nLK  
    theta = theta(:); f&>Q 6 {*]  
    length_r = length(r); = %7:[#n  
    if length_r~=length(theta) 3'6>zp  
        error('zernfun:RTHlength', ... ',* 6vbII  
              'The number of R- and THETA-values must be equal.') {4{ACp  
    end \*w*Q(&3  
    |3g:q  
    % Check normalization: 7QRtNYo#\  
    % -------------------- UkL'h&J~  
    if nargin==5 && ischar(nflag) Fx0<!_tY-  
        isnorm = strcmpi(nflag,'norm'); O@_)]z?jUc  
        if ~isnorm (#. )~poZ  
            error('zernfun:normalization','Unrecognized normalization flag.') m5LP~Gb  
        end _hLM\L  
    else ni]gS0/  
        isnorm = false; Tr}c]IP*  
    end S*CRVs  
    aARm nV  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% XH2g:$  
    % Compute the Zernike Polynomials HWGlC <  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !/$BXUrd  
    ^fb4g+Au  
    % Determine the required powers of r: }qXi;u))  
    % ----------------------------------- PHD$E s  
    m_abs = abs(m); F:M3^I  
    rpowers = []; >UuLSF}  
    for j = 1:length(n) W#0pFofXw  
        rpowers = [rpowers m_abs(j):2:n(j)]; 5kJ>pb$/  
    end te'<xfG  
    rpowers = unique(rpowers); /gHRJ$2|Sx  
    Oy[t}*Ik  
    % Pre-compute the values of r raised to the required powers, +3t(kQ  
    % and compile them in a matrix: ./ib{ @A.  
    % ----------------------------- Fu m1w  
    if rpowers(1)==0 tL;;Yt  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); K)0 6][ ,  
        rpowern = cat(2,rpowern{:}); Z!|nc.  
        rpowern = [ones(length_r,1) rpowern]; w];t]q|  
    else L1"X`Pz[}  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); lTdYPqMi  
        rpowern = cat(2,rpowern{:}); Mi 'eViH  
    end )WEyB~'o  
    m.EWYO0XQ  
    % Compute the values of the polynomials: XUUS N  
    % -------------------------------------- v?h#Ym3e<  
    y = zeros(length_r,length(n)); fwxyZBr  
    for j = 1:length(n) %r~TMU2"  
        s = 0:(n(j)-m_abs(j))/2; *Xl&N- 04  
        pows = n(j):-2:m_abs(j); z6FG^  
        for k = length(s):-1:1 o*I-~k  
            p = (1-2*mod(s(k),2))* ... Vv=d*  
                       prod(2:(n(j)-s(k)))/              ... 1/w['d4l!  
                       prod(2:s(k))/                     ... o &LNtl;  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... S)$ES6]9/  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); |TEf? <"c  
            idx = (pows(k)==rpowers); ^X0<ZI  
            y(:,j) = y(:,j) + p*rpowern(:,idx); /2?GRwU~P  
        end &g@?{5FP  
         18ci-W#p  
        if isnorm R^_/iy  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); {My/+{eS!?  
        end 6eK18*j%H  
    end 0Km{fZYq7;  
    % END: Compute the Zernike Polynomials Ty#L%k}-t  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jr@<-.  
    a*S4rq@  
    % Compute the Zernike functions: WGVvBX7#  
    % ------------------------------ ga~rllm;i  
    idx_pos = m>0; &Cdk%@Tj]B  
    idx_neg = m<0; ]eP&r?B  
    S4`uNB#Ht  
    z = y; LfrS:g  
    if any(idx_pos) $N5}N\C:a  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); M.!U;U<?  
    end xk.\IrB_  
    if any(idx_neg)  @;d(>_n  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); H-0A&oG  
    end ;9 XM s)  
    *wyaBV?*K  
    % EOF zernfun
    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