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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 &Pk #v  
    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! }le}Vuy\s  
     
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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)  A<s zY92&5  
    1 ORA6  
    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。 BjSd\Ul  
    .&i_~?1[N  
    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) YB1Jv[  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. hTQ8y10a  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of fuU 3?SG  
    %   order N and frequency M, evaluated at R.  N is a vector of - -\eYVh[  
    %   positive integers (including 0), and M is a vector with the N*f ]NCSi  
    %   same number of elements as N.  Each element k of M must be a dsn(h5,Q'  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) _;,"!'R`f  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is d%K&  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix }` YtXD-o  
    %   with one column for every (N,M) pair, and one row for every mX%T"_^  
    %   element in R. TQtHU6  
    % Iqci}G%r  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- Nwo*tb:  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is rvacCwI  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to \S_A e;  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 >K<cc#Aa  
    %   for all [n,m]. 3a[LM!  
    % rJ{k1H>  
    %   The radial Zernike polynomials are the radial portion of the ]XASim:A  
    %   Zernike functions, which are an orthogonal basis on the unit R7 rO7M !  
    %   circle.  The series representation of the radial Zernike "rrw~  
    %   polynomials is ]K'OH&  
    % n?>|2>  
    %          (n-m)/2 /:v}Ni"6nF  
    %            __ &]tm 'N25  
    %    m      \       s                                          n-2s 1En:QQ4/  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r :NL[NbQYt  
    %    n      s=0 0 ;].q*|#  
    % h1)ny1;  
    %   The following table shows the first 12 polynomials. / */"gz%  
    % V,% K"b=  
    %       n    m    Zernike polynomial    Normalization QUm[7<"  
    %       --------------------------------------------- icQQLSU5  
    %       0    0    1                        sqrt(2) rp4{lHw>C/  
    %       1    1    r                           2 P:WxhO/  
    %       2    0    2*r^2 - 1                sqrt(6) RG=i74a  
    %       2    2    r^2                      sqrt(6) $o. ;}  
    %       3    1    3*r^3 - 2*r              sqrt(8) )gD2wk(  
    %       3    3    r^3                      sqrt(8) dOK]Su  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) a)*(**e$*i  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) lvRTy|%[  
    %       4    4    r^4                      sqrt(10) 2r!- zEV  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) *+k yuY J  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) @ M4m!;rM  
    %       5    5    r^5                      sqrt(12) +^jm_+  
    %       --------------------------------------------- ^ p7z3ng  
    % p({Lp}'  
    %   Example: w5yX~8UzJ  
    % 505ejO|  
    %       % Display three example Zernike radial polynomials K"[\)&WBG  
    %       r = 0:0.01:1; 8;"9A  
    %       n = [3 2 5]; iJeo d fC  
    %       m = [1 2 1]; dq%C~j{v  
    %       z = zernpol(n,m,r); x+TdTe;p  
    %       figure %O!TS_~9  
    %       plot(r,z) Xy./1`X  
    %       grid on yVQW|D0,j  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') &grvlK  
    % >4q6  
    %   See also ZERNFUN, ZERNFUN2. E#3tkFF0Z[  
    #k1IrqUp  
    % A note on the algorithm. t%O)Ti  
    % ------------------------ b@Dt]6_ UL  
    % The radial Zernike polynomials are computed using the series R )4,f~@"  
    % representation shown in the Help section above. For many special ?p/}eRgi  
    % functions, direct evaluation using the series representation can DAg*  
    % produce poor numerical results (floating point errors), because Pe-rwM  
    % the summation often involves computing small differences between cq5^7.  
    % large successive terms in the series. (In such cases, the functions mfF `K2R  
    % are often evaluated using alternative methods such as recurrence x}O,xquY  
    % relations: see the Legendre functions, for example). For the Zernike cs _  
    % polynomials, however, this problem does not arise, because the TyA1Qk\  
    % polynomials are evaluated over the finite domain r = (0,1), and *2}f $8  
    % because the coefficients for a given polynomial are generally all +J~%z*A  
    % of similar magnitude. >$yA ,N  
    % :xTm- L  
    % ZERNPOL has been written using a vectorized implementation: multiple o~W,VhCP  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] B'mUDW8\D  
    % values can be passed as inputs) for a vector of points R.  To achieve k ]T  
    % this vectorization most efficiently, the algorithm in ZERNPOL azNv(|eeJL  
    % involves pre-determining all the powers p of R that are required to (`_fP.Ogb  
    % compute the outputs, and then compiling the {R^p} into a single yye5GVY$  
    % matrix.  This avoids any redundant computation of the R^p, and 2#00<t\  
    % minimizes the sizes of certain intermediate variables. z,hBtq:-$  
    % Qg]A^{.1  
    %   Paul Fricker 11/13/2006 -E3cS  
    uix/O*^  
    DF>tQ  
    % Check and prepare the inputs: ,t;US.s([.  
    % ----------------------------- *0?@/2&  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /2hRL yeAZ  
        error('zernpol:NMvectors','N and M must be vectors.') ^16zZ*  
    end ycwkF$7  
    fYzP4  
    if length(n)~=length(m) o2hk!#5[4  
        error('zernpol:NMlength','N and M must be the same length.') 3Ijs V5a  
    end Vy|4k2  
    s? Xgo&rS_  
    n = n(:); : 2$*'{mM  
    m = m(:); ?=^\kXc[  
    length_n = length(n); VXlAK(   
    ho B[L}<c  
    if any(mod(n-m,2)) BX6kn/i  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Hq,@j{($  
    end ,!LY:pMK  
    '\+"3!$  
    if any(m<0) fL d2{jI,  
        error('zernpol:Mpositive','All M must be positive.') H3`.Y$z  
    end |W$|og'wC  
    2_6ON   
    if any(m>n) qX; F+~  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') _ WPt zL  
    end \x\N?$`ANc  
    GQJ4d-w  
    if any( r>1 | r<0 ) 80 T2EN:$  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') kM1N4N7  
    end (fr=N5   
    _ h1eW9q  
    if ~any(size(r)==1) "wg$ H1K  
        error('zernpol:Rvector','R must be a vector.') h^qZi@L  
    end :vx<m_  
    Q$ Dx:  
    r = r(:); A%7f;&x!  
    length_r = length(r); Iu~<Y(8^q#  
    V82I%gPF  
    if nargin==4 "frioi`a2  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); wHQ$xO;vD'  
        if ~isnorm =J]EVD   
            error('zernpol:normalization','Unrecognized normalization flag.') 4zt:3bW U  
        end D/ sYH0.V$  
    else z}.6yHS  
        isnorm = false; 4\p%|G^hU  
    end JR xY#k  
    O^ui+44wp  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /988K-5k  
    % Compute the Zernike Polynomials W,[QK~  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% H?M:<q0|G  
    GCiG50Z=  
    % Determine the required powers of r: fA?v\'Qq/  
    % ----------------------------------- V/#J>-os}W  
    rpowers = []; 2<p@G#(  
    for j = 1:length(n) aaw[ia_EL  
        rpowers = [rpowers m(j):2:n(j)]; vu91" 4Fa  
    end TXXG0 G  
    rpowers = unique(rpowers); s :BW}PM  
    @1gURx&2_  
    % Pre-compute the values of r raised to the required powers, yzT1Zg_ER  
    % and compile them in a matrix: frDMFEXXP  
    % ----------------------------- *| W*Mu  
    if rpowers(1)==0 -$:*!55:j  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); $w <R".4  
        rpowern = cat(2,rpowern{:}); <_Z.fdUA  
        rpowern = [ones(length_r,1) rpowern]; |r,})o>  
    else jb,a>9 ]p  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); p~=z)7% e'  
        rpowern = cat(2,rpowern{:}); E8"&gblg  
    end Im!b-1  
    Bos} `S![  
    % Compute the values of the polynomials: 2#3`[+g<n  
    % -------------------------------------- V_D wHq2  
    z = zeros(length_r,length_n); =EM<LjO  
    for j = 1:length_n G3+e5/0  
        s = 0:(n(j)-m(j))/2; ts@Z5Yw*!  
        pows = n(j):-2:m(j); tc)Md]S  
        for k = length(s):-1:1 im9EV|;  
            p = (1-2*mod(s(k),2))* ... k\;D;e{  
                       prod(2:(n(j)-s(k)))/          ... +r//8&  
                       prod(2:s(k))/                 ... T+zhj++  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... aXQAm$/ >  
                       prod(2:((n(j)+m(j))/2-s(k))); $gz8! f?  
            idx = (pows(k)==rpowers); GD d'{qE6  
            z(:,j) = z(:,j) + p*rpowern(:,idx); =q)+_@24>d  
        end z;2& d<h  
         vO&X<5?Qc  
        if isnorm \9tJ/~   
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); V9}\0joM  
        end rr\9HA  
    end %mU$]^Tw(  
    2-N7%]h  
    % 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) ~vA8I#.  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. He4HI Z  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated KehM.c^  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive X"`[&l1  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, V#.pi zb  
    %   and THETA is a vector of angles.  R and THETA must have the same gg^iYTpt  
    %   length.  The output Z is a matrix with one column for every P-value, O43"-  
    %   and one row for every (R,THETA) pair. .o]I^3tf c  
    % yih|6sd$F  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ~}d\sQF .  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ml^=y~J[  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) fJ5mKN  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 x\~ <8o  
    %   for all p. qrj f  
    % M=ag\1S&ZF  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 YU+P+m2X  
    %   Zernike functions (order N<=7).  In some disciplines it is vL[IVBG^  
    %   traditional to label the first 36 functions using a single mode _[(EsIqc(F  
    %   number P instead of separate numbers for the order N and azimuthal : OjmaP  
    %   frequency M. 0QPY+6  
    % 'm |T"Ym~  
    %   Example: cfv: Ld m  
    % mS;WNlm\  
    %       % Display the first 16 Zernike functions ^q/$a2<4  
    %       x = -1:0.01:1; ntPj9#lf  
    %       [X,Y] = meshgrid(x,x); +e*C`uP!  
    %       [theta,r] = cart2pol(X,Y); 8mRZ(B>% X  
    %       idx = r<=1; (;05=DsO  
    %       p = 0:15; 3]lq#p:  
    %       z = nan(size(X)); )F&.0 '  
    %       y = zernfun2(p,r(idx),theta(idx)); :BV$3]y  
    %       figure('Units','normalized') <*^|Aj|#  
    %       for k = 1:length(p) ._A4 :  
    %           z(idx) = y(:,k); LY)Wwl*wc  
    %           subplot(4,4,k) ?q Q.Wj6Mj  
    %           pcolor(x,x,z), shading interp fJ _MuAv  
    %           set(gca,'XTick',[],'YTick',[]) _yH">x<  
    %           axis square >7cj. %  
    %           title(['Z_{' num2str(p(k)) '}']) 5izpQ'>  
    %       end j1->w8  
    % -}sMOy`  
    %   See also ZERNPOL, ZERNFUN. B:UPSX)A  
    ZlE=P4`X:  
    %   Paul Fricker 11/13/2006 d_&pxy? >  
    >Je$WE3  
    hJ[keaO  
    % Check and prepare the inputs: 6|n3Q$p  
    % ----------------------------- 6(htpT%J  
    if min(size(p))~=1 R)$]r>YZF  
        error('zernfun2:Pvector','Input P must be vector.') (6mw@gzr  
    end h:C:opa-=  
    c2 :,  
    if any(p)>35 _dAn/rj   
        error('zernfun2:P36', ... ~l] w=[ z  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... JgP%4)]LV  
               '(P = 0 to 35).']) 4Wa$>vz  
    end 0LzS #J+  
    DoO ;VF  
    % Get the order and frequency corresonding to the function number: dQ_'8 )  
    % ---------------------------------------------------------------- . uGne  
    p = p(:); gN(kRhp  
    n = ceil((-3+sqrt(9+8*p))/2); 5%V(eR  
    m = 2*p - n.*(n+2); ('j'>"1H  
    5?Q5cD2]\6  
    % Pass the inputs to the function ZERNFUN: x30|0EHYl[  
    % ---------------------------------------- jgXr2JQ<  
    switch nargin ,-k?"|tQ  
        case 3 .`J*l=u$  
            z = zernfun(n,m,r,theta); 7.2!g}E  
        case 4 IQ~Anp^R  
            z = zernfun(n,m,r,theta,nflag); 0;=]MEk?  
        otherwise YKayaI\*  
            error('zernfun2:nargin','Incorrect number of inputs.') (;9fkqm%m  
    end ;"EDFH#W  
    N.E{6_{S  
    % 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 {IBbN05 ;  
    function z = zernfun(n,m,r,theta,nflag) [rAi9LSO"  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. /Hm/%os  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N P$AHw;n[R  
    %   and angular frequency M, evaluated at positions (R,THETA) on the +@8, uL  
    %   unit circle.  N is a vector of positive integers (including 0), and (o{x*';i4  
    %   M is a vector with the same number of elements as N.  Each element K~^o06 Y  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) <bhJ>  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, + <w6sPm  
    %   and THETA is a vector of angles.  R and THETA must have the same @V Tw>=94  
    %   length.  The output Z is a matrix with one column for every (N,M) k@n L(2  
    %   pair, and one row for every (R,THETA) pair. 3w[uc~f  
    % 3qNuv];2  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike UaQW<6+  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ]PL\;[b>  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral $SFreyI;Uf  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, SjJ$Oinc  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized F60m]NUM)c  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. }PDtx:T-  
    % -(%Xq{  
    %   The Zernike functions are an orthogonal basis on the unit circle. c1*^ \   
    %   They are used in disciplines such as astronomy, optics, and hA&m G33  
    %   optometry to describe functions on a circular domain. YCzH@94QeV  
    % ~\u>jel  
    %   The following table lists the first 15 Zernike functions. ^$oEM0h  
    % 9 v ,y  
    %       n    m    Zernike function           Normalization E J6|y'  
    %       -------------------------------------------------- iQCs 8hIR  
    %       0    0    1                                 1 QOJ5  
    %       1    1    r * cos(theta)                    2 Xo.3OER  
    %       1   -1    r * sin(theta)                    2 %^"i\- *|S  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) f|s,%AU"i  
    %       2    0    (2*r^2 - 1)                    sqrt(3) += gU`<\  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) i8R 2Y9Q*O  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) pm=s  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Yc5) ^v  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 1mfB6p1Z(  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) `VglE?M  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) = P$7 "  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) />PH{ l  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) EWVn*xl?  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) EzCi%>q  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) oMq:4W,  
    %       -------------------------------------------------- p8&rl|z|  
    % >DzW  OB  
    %   Example 1: poi39B/Vt  
    % kCoEdQ_  
    %       % Display the Zernike function Z(n=5,m=1) \[B#dw#  
    %       x = -1:0.01:1; i(q a'*  
    %       [X,Y] = meshgrid(x,x); akgvV~5  
    %       [theta,r] = cart2pol(X,Y); SvQj'5~<  
    %       idx = r<=1; H3ob 8+J  
    %       z = nan(size(X)); ET6}V"UD  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 2)q$HUIX  
    %       figure i^}DIx{  
    %       pcolor(x,x,z), shading interp 0{Zwg0&  
    %       axis square, colorbar _]+ \ B  
    %       title('Zernike function Z_5^1(r,\theta)') D;DI8.4`N  
    % #CB`7 }jq  
    %   Example 2: 09Z\F^*$F  
    % 3.?oG5 P#  
    %       % Display the first 10 Zernike functions Hegj_FQ  
    %       x = -1:0.01:1; +/#Lm#*nu%  
    %       [X,Y] = meshgrid(x,x); DwXSlsN3v  
    %       [theta,r] = cart2pol(X,Y); Rd1I$| Y  
    %       idx = r<=1; @CCDe`R*  
    %       z = nan(size(X)); a 0qDRB  
    %       n = [0  1  1  2  2  2  3  3  3  3]; D CSTp2  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ,L(q/#p  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; G`u";w_  
    %       y = zernfun(n,m,r(idx),theta(idx)); nN[QUg  
    %       figure('Units','normalized') >#xIqxV,  
    %       for k = 1:10 rPJbbV",+^  
    %           z(idx) = y(:,k); O-<nL B!Wf  
    %           subplot(4,7,Nplot(k)) Aq&H-g]s  
    %           pcolor(x,x,z), shading interp MrS~u  
    %           set(gca,'XTick',[],'YTick',[]) 6 &MATMR  
    %           axis square <\\,L@  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) mS-{AK  
    %       end ? y^t  
    % 2&:w_KJ  
    %   See also ZERNPOL, ZERNFUN2. "nn>I}jK  
    7{u1ynt   
    %   Paul Fricker 11/13/2006 |%Ssb;M  
    D{, b|4  
    /2]=.bLwz  
    % Check and prepare the inputs: X&|y|  
    % ----------------------------- V#d8fRm  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) !}&f2!?.W  
        error('zernfun:NMvectors','N and M must be vectors.') Z E},x U%  
    end ! d" i  
    ,Je9]XT  
    if length(n)~=length(m) ADlLodG  
        error('zernfun:NMlength','N and M must be the same length.') EY.Z.gMZI(  
    end ?C|b>wM/  
    +"SYG  
    n = n(:); vsCy?  
    m = m(:); *Zk$P.]  
    if any(mod(n-m,2)) $N17GqoC  
        error('zernfun:NMmultiplesof2', ... !" 7ip9a  
              'All N and M must differ by multiples of 2 (including 0).') |),3`*N  
    end eTY" "EWU  
    bl`vT3  
    if any(m>n) )R9QJSe  
        error('zernfun:MlessthanN', ... c *]6>50  
              'Each M must be less than or equal to its corresponding N.') ;,jms~ik  
    end 4qLH3I[Y  
    ){,v&[  
    if any( r>1 | r<0 ) PLDp=T%  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ]$ iqJL  
    end VA@t8H,  
    SRpPLY{:F  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) <+${gu?^  
        error('zernfun:RTHvector','R and THETA must be vectors.') '/H+  
    end I?IAZa)  
    $56Z/*  
    r = r(:); -PH!U Hg  
    theta = theta(:); D= LLm$y  
    length_r = length(r); -c'~0g]<  
    if length_r~=length(theta) \>GHc}  
        error('zernfun:RTHlength', ... XCU>b[Cj,  
              'The number of R- and THETA-values must be equal.') CLX!qw]@ +  
    end dd@-9?6M  
    ~xP4}gs1  
    % Check normalization: p:8&&v~I  
    % -------------------- $ -n?q w  
    if nargin==5 && ischar(nflag) ]2o?Gnn@  
        isnorm = strcmpi(nflag,'norm'); I~P]_D mM  
        if ~isnorm SLMnEtyTS  
            error('zernfun:normalization','Unrecognized normalization flag.') s.uV,E*wu  
        end c2fbqM~  
    else bQu1L>c,Uw  
        isnorm = false; &^!vi2$5}  
    end nq"U`z@R  
    A5LTgGzaW  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R#i{eE*WF  
    % Compute the Zernike Polynomials n%Gk {h5  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {E+o+2L  
    l^*'W(%  
    % Determine the required powers of r: [N4#R  
    % ----------------------------------- Y$ To)qo  
    m_abs = abs(m); UL   
    rpowers = []; 8KrqJN0\  
    for j = 1:length(n) \9GJa"xA`  
        rpowers = [rpowers m_abs(j):2:n(j)]; QCvz|)  
    end F 7~T=X)1  
    rpowers = unique(rpowers); 1:r8p6  
    y.'5*08S0  
    % Pre-compute the values of r raised to the required powers, [ym ynr3M  
    % and compile them in a matrix: .W)%*~ O!;  
    % ----------------------------- P,/=c(5\}  
    if rpowers(1)==0 u= u#6%  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); r#CQCq  
        rpowern = cat(2,rpowern{:}); >SR! *3$5  
        rpowern = [ones(length_r,1) rpowern]; C VyE5w  
    else OcWzo#q4[  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7 P$>T  
        rpowern = cat(2,rpowern{:}); Ckc4U. t|  
    end 4)XZ'~|  
    R8U?s/*  
    % Compute the values of the polynomials: fxKhe[;  
    % -------------------------------------- bdUe,2Yin  
    y = zeros(length_r,length(n)); ?i8a)!U  
    for j = 1:length(n) i)pAFv<$,  
        s = 0:(n(j)-m_abs(j))/2; CtO`t5  
        pows = n(j):-2:m_abs(j); hH|moj]  
        for k = length(s):-1:1 #M5R>&?Jqz  
            p = (1-2*mod(s(k),2))* ... 1D/9lR,  
                       prod(2:(n(j)-s(k)))/              ... r(#]Z   
                       prod(2:s(k))/                     ... ^t'mfG|DV  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 'nO%1BZj+  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); X  !vBD  
            idx = (pows(k)==rpowers); D|`I"N[<  
            y(:,j) = y(:,j) + p*rpowern(:,idx); dO{a!Ca  
        end A*r6  
         "DniDA  
        if isnorm MvLmEmKb}\  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); p*_^JU(<p  
        end >)sB# <e  
    end Vk>m/"  
    % END: Compute the Zernike Polynomials 9Rg|oCP_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4._ U  
    0A}'@N@G)  
    % Compute the Zernike functions: ?7Y6: zo$^  
    % ------------------------------ O~1vX9  
    idx_pos = m>0; B?c n5  
    idx_neg = m<0; <^APq8>  
    2!u4nxZ.  
    z = y; <oc"!c;T  
    if any(idx_pos) M+akD  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); #H;yXsR `  
    end (")IU{>c6  
    if any(idx_neg) >*hY1@N1  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); GjmPpKIu\  
    end VX!UT=;  
    gW[(gf.oo  
    % 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