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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 v"yu7tZ3N  
    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! <lIm==U<-  
     
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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)  6;Z`9PGp  
    (=u!E+N  
    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。 &8i$`6wY  
    a_+3, fP  
    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) O251. hXK  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. *^{j!U37s  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of )Z4iM;4]  
    %   order N and frequency M, evaluated at R.  N is a vector of OB=bRLd.IR  
    %   positive integers (including 0), and M is a vector with the CTg79 ITYk  
    %   same number of elements as N.  Each element k of M must be a P}Mu|AEG  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) G2n. NW#d4  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is '6\w4J(  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 7Cz=;  
    %   with one column for every (N,M) pair, and one row for every xa_ IdkV  
    %   element in R. XD6Kp[s  
    % Z3wdk6%:}  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- :0%[u(  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 2 7dS.6  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to IY!.j5q8  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 {%('|(57  
    %   for all [n,m]. >_]Ov:5  
    %  )D+eWo  
    %   The radial Zernike polynomials are the radial portion of the %kkDitmI{  
    %   Zernike functions, which are an orthogonal basis on the unit KU/QEeqbrp  
    %   circle.  The series representation of the radial Zernike {_4Hsw?s6  
    %   polynomials is Y@UW\d*'%I  
    % )iIsnM  
    %          (n-m)/2 oMM@{Jp  
    %            __ 01 +#2~S  
    %    m      \       s                                          n-2s BUi,+NdIk  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r NODg_J~T  
    %    n      s=0 RJ4=AA|  
    % @pJ;L1sn  
    %   The following table shows the first 12 polynomials. 9ec#'i=  
    % 2XUIC^<@s  
    %       n    m    Zernike polynomial    Normalization "\~>[on  
    %       --------------------------------------------- fCs{%-6cP  
    %       0    0    1                        sqrt(2) c?c"|.-<p  
    %       1    1    r                           2 =*-a c  
    %       2    0    2*r^2 - 1                sqrt(6) XF3lS#pt  
    %       2    2    r^2                      sqrt(6) 7r(c@4yPI  
    %       3    1    3*r^3 - 2*r              sqrt(8) b/T k$&  
    %       3    3    r^3                      sqrt(8) h;(mb2[R  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) &432/=QSm0  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ) .V,zmI  
    %       4    4    r^4                      sqrt(10) &C9)%5 O)  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) QDK }e:4q  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) if1)AE-  
    %       5    5    r^5                      sqrt(12) (Cti,g~  
    %       --------------------------------------------- y^X]q[-?  
    % VyIJ)F.c  
    %   Example: -~~R?,H'Z_  
    % 2=7[r-*E  
    %       % Display three example Zernike radial polynomials ?u{Mz9:?HT  
    %       r = 0:0.01:1; PK{FQ3b2{  
    %       n = [3 2 5]; "K|':3n|  
    %       m = [1 2 1]; HmsXV_B8[Y  
    %       z = zernpol(n,m,r); N/2WUp  
    %       figure .[:WMCc\  
    %       plot(r,z) Qe9}%k6@E  
    %       grid on WwKpZ67$R  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') a9 S&n5  
    % .",BLuce  
    %   See also ZERNFUN, ZERNFUN2. >*l2]3' `  
    fgo3Gy*#  
    % A note on the algorithm. t B}W )Eb  
    % ------------------------ pZc`!f"  
    % The radial Zernike polynomials are computed using the series fo9V&NE  
    % representation shown in the Help section above. For many special g+&wgyq5  
    % functions, direct evaluation using the series representation can WdJeh:h  
    % produce poor numerical results (floating point errors), because 3][   
    % the summation often involves computing small differences between p[!9objU  
    % large successive terms in the series. (In such cases, the functions $['`H)z  
    % are often evaluated using alternative methods such as recurrence /Vv)00  
    % relations: see the Legendre functions, for example). For the Zernike Mp J3*$Dr  
    % polynomials, however, this problem does not arise, because the #aP;a-Q|k  
    % polynomials are evaluated over the finite domain r = (0,1), and O15~\8#'  
    % because the coefficients for a given polynomial are generally all *li5/=UC5*  
    % of similar magnitude. *TxR2pC}  
    % S->Sp  
    % ZERNPOL has been written using a vectorized implementation: multiple ?IO3w{fmH  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] )4YtdAV  
    % values can be passed as inputs) for a vector of points R.  To achieve !83 N#Y_Mz  
    % this vectorization most efficiently, the algorithm in ZERNPOL p+2%LYR u  
    % involves pre-determining all the powers p of R that are required to ^(qR({cX  
    % compute the outputs, and then compiling the {R^p} into a single 2:[G4  
    % matrix.  This avoids any redundant computation of the R^p, and p5nrPL  
    % minimizes the sizes of certain intermediate variables. z5f3T D6,  
    %  )Z:maz  
    %   Paul Fricker 11/13/2006 G1,u{d-_  
    ;O .;i,#Z  
    $M4C4_oPy  
    % Check and prepare the inputs: xaIe7.Z"xo  
    % ----------------------------- bh5C  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) DxP65wU  
        error('zernpol:NMvectors','N and M must be vectors.') /w*HxtwFmD  
    end p,)pz_M  
    SR/ "{\C  
    if length(n)~=length(m) m O0#xY_z  
        error('zernpol:NMlength','N and M must be the same length.') ~~,#<g[  
    end 5XNFu C9E  
    aU] nh. a  
    n = n(:);  A1jA$  
    m = m(:); 3"6-X_  
    length_n = length(n); yyjgPbLN=  
    ,z$ U=u o  
    if any(mod(n-m,2)) lYrW"(2  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') yMb.~A^$J  
    end ':T"nORC  
    7<F{a"5P  
    if any(m<0) YQ}IE[J}v  
        error('zernpol:Mpositive','All M must be positive.') =XUt?5  
    end QnH~' k  
    _^w^tfH]  
    if any(m>n) tlmfDQD  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 3.04Toq!  
    end ]=5D98B  
    _M[T8"e(  
    if any( r>1 | r<0 ) *3y:Wv T>  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') I}/-zyx>=  
    end pW2-RHGJY  
    @0%^\Qf2  
    if ~any(size(r)==1) kc"SUiy/  
        error('zernpol:Rvector','R must be a vector.') Ktf lbI!  
    end G^w:c]  
    F:2V;  
    r = r(:); TSP#.QY  
    length_r = length(r); z Q11dLjs  
    (w, Gv-S  
    if nargin==4 h&t9CpTfeJ  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ^:m7Qd?Z[  
        if ~isnorm N1z:9=(I  
            error('zernpol:normalization','Unrecognized normalization flag.') <o_(,,P%  
        end f.u+({"ql  
    else ^WIGd"^  
        isnorm = false; z_ia3k<  
    end +C9 l7 q  
    }tH6E  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O8w|!$Q.  
    % Compute the Zernike Polynomials Z|$OPMLX  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C?VNkBJ>\  
    t9m08K:Y  
    % Determine the required powers of r: l=t$ XWh!  
    % ----------------------------------- M!b"c4|<  
    rpowers = []; Q|:qs\6q5  
    for j = 1:length(n) !5[5l!{x  
        rpowers = [rpowers m(j):2:n(j)]; 8 gzf$Oc  
    end 2t"&>1  
    rpowers = unique(rpowers); ioS(;2F  
    ;_= +h,n  
    % Pre-compute the values of r raised to the required powers, 8Ir = @  
    % and compile them in a matrix: +`~6Weay  
    % ----------------------------- A<s9c=d6  
    if rpowers(1)==0 nJ~5ICyd  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); K)r|oW=6Y  
        rpowern = cat(2,rpowern{:}); vTUhIFa{  
        rpowern = [ones(length_r,1) rpowern]; ;R{ffS6  
    else d,caOE8N  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 'nh^'i&0.  
        rpowern = cat(2,rpowern{:}); o@tc   
    end H{j jA+0  
    E >lW'  
    % Compute the values of the polynomials: ;B !u=_'  
    % -------------------------------------- c0u1L@tj  
    z = zeros(length_r,length_n); 8P3"$2q  
    for j = 1:length_n ^5BQ=  
        s = 0:(n(j)-m(j))/2; [}t^+^/  
        pows = n(j):-2:m(j); C{8(ew  
        for k = length(s):-1:1 uiIS4S_  
            p = (1-2*mod(s(k),2))* ... El#"vIg(\  
                       prod(2:(n(j)-s(k)))/          ... "s5[w+,R  
                       prod(2:s(k))/                 ... -7:_Dy  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... %T'<vw0  
                       prod(2:((n(j)+m(j))/2-s(k))); r:Rk!z*  
            idx = (pows(k)==rpowers); ~zT743  
            z(:,j) = z(:,j) + p*rpowern(:,idx); E+e:UBeUV  
        end YPNG9^Y  
         &pZn cm  
        if isnorm mJL=H  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); -{rUE +  
        end A 2Rp  
    end C4^o= 6{  
    !omf>CW;ud  
    % EOF zernpol
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) {YigB  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. %29lDd(<  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated N>Q~WXvV#  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive oZwu`~h Y  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, G24 Ov&H  
    %   and THETA is a vector of angles.  R and THETA must have the same -h8@B+  
    %   length.  The output Z is a matrix with one column for every P-value, ]<Kkq !  
    %   and one row for every (R,THETA) pair. #$0*Gd-N  
    % h"$)[k~  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike qe<aJn  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) tqXr6+!Q  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) hxe X6  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 yHYK,3/C,  
    %   for all p. Vc*"Q8aZ~  
    % ,zVS}!jRhy  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ^2)<H7p  
    %   Zernike functions (order N<=7).  In some disciplines it is 7w51UmO  
    %   traditional to label the first 36 functions using a single mode ^LAnR>mz^r  
    %   number P instead of separate numbers for the order N and azimuthal Ssg1p#0J  
    %   frequency M. }NpN<C+  
    % $QB/n63  
    %   Example: ~6Pv5DKq  
    % 7*\Cf qrU  
    %       % Display the first 16 Zernike functions It:,8  
    %       x = -1:0.01:1; )/cf%  
    %       [X,Y] = meshgrid(x,x); s&7TARd  
    %       [theta,r] = cart2pol(X,Y); l#bE_PD;  
    %       idx = r<=1; JBk >|q"  
    %       p = 0:15; 8@A}.:  
    %       z = nan(size(X)); 34<k)0sO  
    %       y = zernfun2(p,r(idx),theta(idx)); gJBw6'Z  
    %       figure('Units','normalized') /^hc8X  
    %       for k = 1:length(p) jT=fq'RK  
    %           z(idx) = y(:,k); Xb2.t^ ]f  
    %           subplot(4,4,k) TY;%nT  
    %           pcolor(x,x,z), shading interp *%CDQx0}  
    %           set(gca,'XTick',[],'YTick',[]) %Hu?syo  
    %           axis square ex6 QHUQ  
    %           title(['Z_{' num2str(p(k)) '}']) F4DJML-(  
    %       end lsA?|4`mn  
    % 4t,f$zk  
    %   See also ZERNPOL, ZERNFUN. hg2UZ% Y  
    *BHp?cn;F2  
    %   Paul Fricker 11/13/2006 R4vf  
    QWwdtk  
    {5 Sy=Y  
    % Check and prepare the inputs: ~@mNR^W-W  
    % ----------------------------- 9";qR,  
    if min(size(p))~=1 N"8'=wB  
        error('zernfun2:Pvector','Input P must be vector.') _E2W%N  
    end # 1 1<=3Yj  
    L<k(stx~  
    if any(p)>35 EGVS8YP>h  
        error('zernfun2:P36', ... Y1G/1Z# 2  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... B&m6N,  
               '(P = 0 to 35).']) ~s*kuj'%+  
    end )F+wk"`+6  
    r;_*.|AH  
    % Get the order and frequency corresonding to the function number: w@WPp0mny  
    % ---------------------------------------------------------------- X`28?  
    p = p(:); *$Y_ %}  
    n = ceil((-3+sqrt(9+8*p))/2); Ug  )eyu  
    m = 2*p - n.*(n+2); apjoIO-<  
    W. BX6  
    % Pass the inputs to the function ZERNFUN: <:4b4Nl  
    % ---------------------------------------- C#n.hgo>I  
    switch nargin Y<h6m]H  
        case 3 A|YiSwyy  
            z = zernfun(n,m,r,theta); fd$nAE  
        case 4 $8}'h  
            z = zernfun(n,m,r,theta,nflag); OlP1Zd/l  
        otherwise p z\8Bp}yo  
            error('zernfun2:nargin','Incorrect number of inputs.') HCT+.n6  
    end '^`iF,rg  
    t;V^OGflv  
    % EOF zernfun2
    离线niuhelen
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    只看该作者 3楼 发表于: 2011-03-12
    回 2楼(phility) 的帖子
    非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 L;WFHIE  
    function z = zernfun(n,m,r,theta,nflag) \-SC-c  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ZW4$Ks2]Y  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 6F5g2hBz  
    %   and angular frequency M, evaluated at positions (R,THETA) on the nk;^sq4M:  
    %   unit circle.  N is a vector of positive integers (including 0), and ;iW>i8  
    %   M is a vector with the same number of elements as N.  Each element 1Tr%lO5?6  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Ym.{ {^=  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, "T*1C=  
    %   and THETA is a vector of angles.  R and THETA must have the same gVrfZ&XF84  
    %   length.  The output Z is a matrix with one column for every (N,M) tSe[*V4{'  
    %   pair, and one row for every (R,THETA) pair. Ri\\Yb  
    % H>o \C  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike %j/pln&  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), > `mV^QD  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral  /PTq.  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, BwrX.!M  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized WrS>^\:  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. {$#88Qa\-  
    % 'j-U=2,n  
    %   The Zernike functions are an orthogonal basis on the unit circle. t1NGs-S3  
    %   They are used in disciplines such as astronomy, optics, and ?C- ju8]|  
    %   optometry to describe functions on a circular domain. DIfQ~O+u  
    % 4Y1dkg1y  
    %   The following table lists the first 15 Zernike functions. Z;,G:@,  
    % }1%%`  
    %       n    m    Zernike function           Normalization e ^,IZ{  
    %       -------------------------------------------------- tfD7!N{  
    %       0    0    1                                 1 =dsEt\ j  
    %       1    1    r * cos(theta)                    2 iXq*EZb"R  
    %       1   -1    r * sin(theta)                    2 OL%}C*Zq  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) MiR$N  
    %       2    0    (2*r^2 - 1)                    sqrt(3) wWSo+40  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ns *:mGh  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 3 q J00A  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 81C;D`!K  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) @biU@[D  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 9aNOfs8(  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Ql%B=vgKL  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Zd88+GS,#  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) t2YB(6w+xg  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) tfu`_6  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) )8oN$2 0  
    %       -------------------------------------------------- d!4TwpIgx  
    % *l;S"}b*,_  
    %   Example 1: #6v357-5  
    % !XM<`H/  
    %       % Display the Zernike function Z(n=5,m=1) z>\l%_w  
    %       x = -1:0.01:1; cGR)$:  
    %       [X,Y] = meshgrid(x,x); gwdAf%|f  
    %       [theta,r] = cart2pol(X,Y); SF9NS*mr  
    %       idx = r<=1; W#E(?M[r  
    %       z = nan(size(X)); _RUL$Ds  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ijUu{PG`X  
    %       figure >{9VXSc  
    %       pcolor(x,x,z), shading interp D.Cn`O}  
    %       axis square, colorbar 5Zd oem  
    %       title('Zernike function Z_5^1(r,\theta)') QnP?j&  
    % l($ 8H AJ  
    %   Example 2: j S[#R_  
    % <QO1Yg7}  
    %       % Display the first 10 Zernike functions \*'@F+  
    %       x = -1:0.01:1; dJ#go*Gn  
    %       [X,Y] = meshgrid(x,x); ck%YEMs  
    %       [theta,r] = cart2pol(X,Y); @}:E{J#g  
    %       idx = r<=1; RwYFBc  
    %       z = nan(size(X)); $(+xhn(O  
    %       n = [0  1  1  2  2  2  3  3  3  3]; /zb/ am1#  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; YM6 J:89  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; MBU|<tc  
    %       y = zernfun(n,m,r(idx),theta(idx)); TET=>6  
    %       figure('Units','normalized') |Olz h63k:  
    %       for k = 1:10 v|\#wrCT?  
    %           z(idx) = y(:,k); ~, E }^  
    %           subplot(4,7,Nplot(k)) qp/1 tC`  
    %           pcolor(x,x,z), shading interp L6DYunh}^N  
    %           set(gca,'XTick',[],'YTick',[]) S89j:KRXH%  
    %           axis square vz>9jw:Y  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) OzD\* ,{7  
    %       end *9uNM@7&0  
    %  <7SE|  
    %   See also ZERNPOL, ZERNFUN2. K;WQV,  
    4hLk+z<n  
    %   Paul Fricker 11/13/2006 ~[dL:=?c  
    HfgTc h  
    !02y'JS1  
    % Check and prepare the inputs: c"-X: m"  
    % ----------------------------- c*.  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) U._fb=  
        error('zernfun:NMvectors','N and M must be vectors.') dNNXMQ0"  
    end Du65>O  
    24k]X`/n  
    if length(n)~=length(m) A%?c1`ZxF  
        error('zernfun:NMlength','N and M must be the same length.') TfT^.p*  
    end /RMtCa~  
    TukhGgmF  
    n = n(:); f<iK%  
    m = m(:); [5!}+8]W  
    if any(mod(n-m,2)) ygj%VG  
        error('zernfun:NMmultiplesof2', ... c0o Z7)*}  
              'All N and M must differ by multiples of 2 (including 0).') VevG 64o  
    end yj#FO'UY  
    \8!CKnfs  
    if any(m>n) Q~qM;l\i  
        error('zernfun:MlessthanN', ... DbLo{mFEIj  
              'Each M must be less than or equal to its corresponding N.') dor1(@no|  
    end j5" L  
    M!5=3>Z  
    if any( r>1 | r<0 ) #b;k+<n[X  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') utuWFAGn A  
    end ymqv@Byi8A  
    vs[!B-  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) /g!ZU2&l  
        error('zernfun:RTHvector','R and THETA must be vectors.') 6H: fg  
    end *]NfT}}  
    W_E^+Wl@  
    r = r(:); pZopdEFDK|  
    theta = theta(:); hU-FSdR  
    length_r = length(r); T9& {s-3*  
    if length_r~=length(theta) IqFcrU$4  
        error('zernfun:RTHlength', ... cZ|NGkZ  
              'The number of R- and THETA-values must be equal.') `ovMfL.u  
    end "qF/7`e[  
    du$M  
    % Check normalization: H`fJ< So?  
    % -------------------- F nXm;k,9*  
    if nargin==5 && ischar(nflag) L&)e}"  
        isnorm = strcmpi(nflag,'norm'); !J<Xel {  
        if ~isnorm bRyxP2  
            error('zernfun:normalization','Unrecognized normalization flag.') }q]*aADe  
        end E56  
    else (}6\_k[}m  
        isnorm = false; i 0/QfB%O  
    end aT Izf qCM  
    HVoP J!K3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MXfyj5K  
    % Compute the Zernike Polynomials / 7\q#qIm:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &8l?$7S"_/  
    $;G<!]& s  
    % Determine the required powers of r: 9ghzK?Yc  
    % ----------------------------------- ,'HjL:r  
    m_abs = abs(m); qhvT,"  
    rpowers = []; ]tT=jN&(  
    for j = 1:length(n) LYL_Ah'=  
        rpowers = [rpowers m_abs(j):2:n(j)]; ; 8DtnnE  
    end 0+op|bdj  
    rpowers = unique(rpowers); kN1R8|pv  
    ,M?8s2?  
    % Pre-compute the values of r raised to the required powers, |[iO./ zP  
    % and compile them in a matrix: 5o 5DG  
    % ----------------------------- aWJ BYw6{L  
    if rpowers(1)==0 NYP3u_ QX  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); cL*oO@I&_  
        rpowern = cat(2,rpowern{:}); Mz(?_7  
        rpowern = [ones(length_r,1) rpowern]; Q &{C%j~N  
    else Z3c\}HLY  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 5j.@)XXe  
        rpowern = cat(2,rpowern{:}); UakVmVN/P  
    end syg{qtBz^  
    O&aD]~|  
    % Compute the values of the polynomials: Z]Ud x  
    % -------------------------------------- 8%u|[Si;  
    y = zeros(length_r,length(n)); /{hT3ncb  
    for j = 1:length(n) Xw'sh#i2  
        s = 0:(n(j)-m_abs(j))/2; R[l`# I  
        pows = n(j):-2:m_abs(j); T^#d;A  
        for k = length(s):-1:1 Cq/u$G  
            p = (1-2*mod(s(k),2))* ... \8<[P(!3  
                       prod(2:(n(j)-s(k)))/              ... rQ_cH  
                       prod(2:s(k))/                     ... #tHYCSr]  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... /cx'(AT  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); O>h h  
            idx = (pows(k)==rpowers); B,_K mHItd  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 5EQ)pH+  
        end .hxFFk%5  
         VT4 >6u}  
        if isnorm H.XyNtJ  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); }]dzY(   
        end k"gm;,`  
    end hy;V~J#  
    % END: Compute the Zernike Polynomials eDP&W$s#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% iOhX\@&  
    xLFMC?I  
    % Compute the Zernike functions:  u? >x  
    % ------------------------------ =J)-#|eZG  
    idx_pos = m>0; R'tvF$3=i  
    idx_neg = m<0; >f Hu  
    z7XI`MZN^  
    z = y; *2-b&PQR{  
    if any(idx_pos) +ug2p;<B  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); HU/4K7e`  
    end hG~.Sc:G  
    if any(idx_neg) wAW{{ p  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); $Bc3| `K1v  
    end `a[fC9  
    H1q,w|O9j  
    % EOF zernfun
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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
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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