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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 u5xU)l3  
    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! 6Cz7A  
     
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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)  p^pQZ6-  
    OCbQB5k3  
    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。 7AGZu?1]M  
    JEK%yMj  
    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) vZ_DG}n11  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. ZaNyNxbp>z  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Vne. HFXA  
    %   order N and frequency M, evaluated at R.  N is a vector of <750-d!  
    %   positive integers (including 0), and M is a vector with the %T,\xZ  
    %   same number of elements as N.  Each element k of M must be a U"%8"G0)  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) HkfSx rTgQ  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is `3>)BV<P  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix P5 <85t  
    %   with one column for every (N,M) pair, and one row for every O w($\,  
    %   element in R. +NzD/.gq  
    % 0()9vTY+  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- W(PW9J9  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 1CS]~1Yp:  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to bb O;AiHD  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 V"2AN3~&  
    %   for all [n,m]. qed!C  
    % 3$kv%uf{  
    %   The radial Zernike polynomials are the radial portion of the Me K\eZ\  
    %   Zernike functions, which are an orthogonal basis on the unit (W}i287  
    %   circle.  The series representation of the radial Zernike PU@U@  
    %   polynomials is M*T# 5  
    % *2m&?,nJ  
    %          (n-m)/2 !3X%5=#L4  
    %            __ Fb<\(#t  
    %    m      \       s                                          n-2s g6a3MJV`  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r u UVV>An  
    %    n      s=0 {L2Gb(YLW  
    % i_ODgc`H  
    %   The following table shows the first 12 polynomials. <1'X)n&Kw$  
    % yS.fe[  
    %       n    m    Zernike polynomial    Normalization }&C!^v o  
    %       --------------------------------------------- [%)B%h`XGf  
    %       0    0    1                        sqrt(2) {;z L[AgCg  
    %       1    1    r                           2 ae(]9VW  
    %       2    0    2*r^2 - 1                sqrt(6) BI]ut |Qw  
    %       2    2    r^2                      sqrt(6) k~9Ywf  
    %       3    1    3*r^3 - 2*r              sqrt(8) <2@<r t{  
    %       3    3    r^3                      sqrt(8) KxTYc  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) o}^vREO  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) W!Ct[t  
    %       4    4    r^4                      sqrt(10)  9jzLXym  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) S,<.!v57  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) \tw#p k  
    %       5    5    r^5                      sqrt(12) &40JN}  
    %       --------------------------------------------- ;|$]Qq  
    % e[ k;SSs  
    %   Example: 2DBFXhP  
    % pt|$bU7  
    %       % Display three example Zernike radial polynomials ~PAbLSL*u  
    %       r = 0:0.01:1; VV}fW"_ND  
    %       n = [3 2 5]; 4oa P"T@6  
    %       m = [1 2 1]; 7!yF5 +_d  
    %       z = zernpol(n,m,r); Nxs%~ wZ   
    %       figure w-Q 6 -  
    %       plot(r,z) Ef*.}gcU  
    %       grid on uA}FuOE6  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') +sbacMfq  
    % I(kIHjV|  
    %   See also ZERNFUN, ZERNFUN2. [Oy2&C  
    hpi_0lMkI  
    % A note on the algorithm. VflPNzixb!  
    % ------------------------ 7Caap/L:  
    % The radial Zernike polynomials are computed using the series E~O>m8hF  
    % representation shown in the Help section above. For many special xg5@;p  
    % functions, direct evaluation using the series representation can #&8pp8wd,}  
    % produce poor numerical results (floating point errors), because ]A<u eM  
    % the summation often involves computing small differences between czsoD) N  
    % large successive terms in the series. (In such cases, the functions Gt%?[  
    % are often evaluated using alternative methods such as recurrence tlxjs]{0E  
    % relations: see the Legendre functions, for example). For the Zernike 8RT0&[  
    % polynomials, however, this problem does not arise, because the OsSiBb,W79  
    % polynomials are evaluated over the finite domain r = (0,1), and waq_d.  
    % because the coefficients for a given polynomial are generally all x 3co?  
    % of similar magnitude. K[;,/:Y  
    % VKfHN_m*  
    % ZERNPOL has been written using a vectorized implementation: multiple Hf]}OvT>Z  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] /Ta0}Y(y  
    % values can be passed as inputs) for a vector of points R.  To achieve Ecl7=-y  
    % this vectorization most efficiently, the algorithm in ZERNPOL 5OqsnL_V  
    % involves pre-determining all the powers p of R that are required to 3bL2fsn5  
    % compute the outputs, and then compiling the {R^p} into a single PaI63 !  
    % matrix.  This avoids any redundant computation of the R^p, and TV>R(D3T/  
    % minimizes the sizes of certain intermediate variables. a|{<#<6n(  
    % ( 2(;u1  
    %   Paul Fricker 11/13/2006 ~map5@Kd  
    R/FV'qy]  
    >;U%~yy}qc  
    % Check and prepare the inputs: <@ex})su  
    % ----------------------------- CbaAnm1  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ^ J@i7FOb  
        error('zernpol:NMvectors','N and M must be vectors.') 90696v.  
    end WG=r? xE  
    @y2Bq['  
    if length(n)~=length(m) {ZI6!zh'  
        error('zernpol:NMlength','N and M must be the same length.') Vw@x  
    end ;j\$[4W.i  
    hpe s  
    n = n(:); %z_b/yG  
    m = m(:); bN %MT#X  
    length_n = length(n); Vk=<,<BB  
    A/6nV n  
    if any(mod(n-m,2)) n/Z =q?_  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') d#,V^  
    end r<H^%##,w  
    %ycT}Lu  
    if any(m<0) 05zdy-Fb  
        error('zernpol:Mpositive','All M must be positive.') <.XoC?j  
    end *"L:"i`*$  
    \>k#]4@rp  
    if any(m>n) aVL%-Il}  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') qiJ;v1  
    end Ybiz]1d  
    GB Un" _J  
    if any( r>1 | r<0 ) Bm>(m{sX>  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 9e*poG  
    end :iiTz$yk  
    32'9Ch.  
    if ~any(size(r)==1) *3oQS"8  
        error('zernpol:Rvector','R must be a vector.') wpMQ 7:j  
    end 8j +;Xlh  
    +/8?+1E ^  
    r = r(:); 3ZZI1_j  
    length_r = length(r); :dc J6  
    @D{[Hj`<  
    if nargin==4 \zDV|n~{w  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); m5g: Q  
        if ~isnorm )Em,3I/.l  
            error('zernpol:normalization','Unrecognized normalization flag.') HYa!$P3}[  
        end hzVO.Q*  
    else pDN,(Ip  
        isnorm = false; f}d@G/L  
    end GUZi }a|=  
    ( ~o+pp!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (jMp`4P  
    % Compute the Zernike Polynomials 3Or3@e5r  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j* ja)  
    YR#1[fe*_  
    % Determine the required powers of r: ~qxc!k!w4  
    % ----------------------------------- GoXHVUyp  
    rpowers = []; ^<b.j.$<z  
    for j = 1:length(n) ^el:)$  
        rpowers = [rpowers m(j):2:n(j)]; Onyq'  
    end I[C.iILL  
    rpowers = unique(rpowers); 0nn# U  
    Jrl xa3 [  
    % Pre-compute the values of r raised to the required powers, k{8N@&D  
    % and compile them in a matrix: N|d@B{a(  
    % -----------------------------  3".W  
    if rpowers(1)==0 p gi7 JQ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 9 f+7vCA  
        rpowern = cat(2,rpowern{:}); Yq.@7cJ  
        rpowern = [ones(length_r,1) rpowern]; :v48y.Ij7s  
    else 1Qkuxw  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7MfvU|D[d/  
        rpowern = cat(2,rpowern{:}); ?+_"2XY  
    end )E|Bb=%  
    4QDzG~N4)|  
    % Compute the values of the polynomials: 9bvd1bKEW  
    % -------------------------------------- nQC[[G*x  
    z = zeros(length_r,length_n); 3M`J.>  
    for j = 1:length_n Y6Q6--P  
        s = 0:(n(j)-m(j))/2; fA5# 2P{  
        pows = n(j):-2:m(j); !<'R%<E3 Q  
        for k = length(s):-1:1 J0o[WD$A x  
            p = (1-2*mod(s(k),2))* ... y3GIR f;>  
                       prod(2:(n(j)-s(k)))/          ... {^iV<>J  
                       prod(2:s(k))/                 ... bSzb! hT`  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... nwYeOa/t  
                       prod(2:((n(j)+m(j))/2-s(k))); 6<R U~Gh  
            idx = (pows(k)==rpowers); =n&83MYX  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 1owoh,V6  
        end &v88x s  
         #/6X44 *u  
        if isnorm +ZO*~.zZ  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); sa])^mkq(  
        end )c_ll;%  
    end 9EW 7,m{A  
    a1&^P1.  
    % 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) Lj#6K@u@Z  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. !.A>)+AK  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 4+0Zj+ q";  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive K`sm  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, H+Wd#7l,  
    %   and THETA is a vector of angles.  R and THETA must have the same a &j?"o  
    %   length.  The output Z is a matrix with one column for every P-value, B^Q#@[T   
    %   and one row for every (R,THETA) pair. e# DAa  
    % f\JyN@w+  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike  S_atEmQ  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) }\F>z  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) $}829<gh7  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 @QofsWC  
    %   for all p. }% =P(%-  
    % @QEV l  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 OkM>  
    %   Zernike functions (order N<=7).  In some disciplines it is K':f!sZ&2  
    %   traditional to label the first 36 functions using a single mode b< rM3P;  
    %   number P instead of separate numbers for the order N and azimuthal 4#T'Fy].  
    %   frequency M. &*}S 0  
    % * HVO  
    %   Example: fHiCuF  
    % UTz;Sw?~hw  
    %       % Display the first 16 Zernike functions VQCPgs  
    %       x = -1:0.01:1; ;%)i/MGEB  
    %       [X,Y] = meshgrid(x,x); oj/tim  
    %       [theta,r] = cart2pol(X,Y); :5(TOF  
    %       idx = r<=1; />?d 2?  
    %       p = 0:15; lZ|Ao0(  
    %       z = nan(size(X)); )c*~Y=f  
    %       y = zernfun2(p,r(idx),theta(idx)); C<pF13*4  
    %       figure('Units','normalized') Kr<O7t0X  
    %       for k = 1:length(p) cGD A0#r  
    %           z(idx) = y(:,k); W*)>Tr)o  
    %           subplot(4,4,k) l/]P6 @N  
    %           pcolor(x,x,z), shading interp >wn&+%i&  
    %           set(gca,'XTick',[],'YTick',[]) _ n>0!  
    %           axis square (- uk[["3  
    %           title(['Z_{' num2str(p(k)) '}']) 4xlsdq8`t  
    %       end `U1"WcN  
    % &sW/r::,  
    %   See also ZERNPOL, ZERNFUN. $KiA~l  
    [x&&N*>N  
    %   Paul Fricker 11/13/2006 gyPF!"!5dq  
    -vMP{,  
    yP@= x!$  
    % Check and prepare the inputs: F"q3p4-<>  
    % ----------------------------- 1+^c3Dd`  
    if min(size(p))~=1 k;)L-ge9  
        error('zernfun2:Pvector','Input P must be vector.') ]KfHuYjM  
    end 4]cOTXk9C  
    lfhB2^ ^  
    if any(p)>35 cc>h=%s`  
        error('zernfun2:P36', ... @{a(f;  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... E?;W@MJi  
               '(P = 0 to 35).']) 6};Sn/ 8  
    end mr*zl*  
    .RT5sj\d  
    % Get the order and frequency corresonding to the function number: -~5yl}  
    % ---------------------------------------------------------------- ScI9.{  
    p = p(:); wxoBq{r;  
    n = ceil((-3+sqrt(9+8*p))/2); 7S Qu  
    m = 2*p - n.*(n+2); wiutUb Y  
    OTRTa{TB  
    % Pass the inputs to the function ZERNFUN: h_cZ&P|  
    % ---------------------------------------- )a.U|[:y[+  
    switch nargin jQc0_F\  
        case 3 +n0y/0Au  
            z = zernfun(n,m,r,theta); ;c'jBi5W  
        case 4 =IUTU4!]  
            z = zernfun(n,m,r,theta,nflag); ur'A;B  
        otherwise /q>"">  
            error('zernfun2:nargin','Incorrect number of inputs.') 0$UE|yDs>  
    end JeO(sj$e  
    =.uE(L`]NA  
    % 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 s Ce{V*ua  
    function z = zernfun(n,m,r,theta,nflag) P'g$F<~V  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 8&3G|m1-2  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N n\d-^ml  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 2cww7z/B  
    %   unit circle.  N is a vector of positive integers (including 0), and TEY%OI zU+  
    %   M is a vector with the same number of elements as N.  Each element [Y5B$7|s<  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 9XS'5AXN  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, s:Memvf  
    %   and THETA is a vector of angles.  R and THETA must have the same 2?HLEiI1  
    %   length.  The output Z is a matrix with one column for every (N,M) oJ5V^.  
    %   pair, and one row for every (R,THETA) pair. {| Tl3  
    % R7vO,kZ6Q  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike O7E0{8  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), * c xYB  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral A9[l5E  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, c$>Tfa'H  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized / S]<MS  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. :]:q=1;c  
    % ,%Dn}mWu  
    %   The Zernike functions are an orthogonal basis on the unit circle. ]81P<Y(7  
    %   They are used in disciplines such as astronomy, optics, and @q|I$'K]x  
    %   optometry to describe functions on a circular domain. D;m>9{=  
    % F(mm0:lT  
    %   The following table lists the first 15 Zernike functions. I>:M1Yc0  
    % q&7J1  
    %       n    m    Zernike function           Normalization Yf<6[(6 O  
    %       -------------------------------------------------- _},u[+  
    %       0    0    1                                 1 =`u4xa#m  
    %       1    1    r * cos(theta)                    2 KYMz  
    %       1   -1    r * sin(theta)                    2 }ufH![|[r  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) .I<#i9Le  
    %       2    0    (2*r^2 - 1)                    sqrt(3) `Fnt#F}  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) u|i.6:/=  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) aO6w :IO  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) usX aT(K  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) e0qU2  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 66!cfpM  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) S}mqK|!  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 94\k++kc  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 8Y_wS&eB  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =UT*1-yh R  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) n}}$-xl  
    %       -------------------------------------------------- 7:<co  
    % +<7`Gn(n3  
    %   Example 1: z q _*)V  
    % E:!?A@Fy  
    %       % Display the Zernike function Z(n=5,m=1) { LZ` _1D  
    %       x = -1:0.01:1; wgp{P>oBX  
    %       [X,Y] = meshgrid(x,x); 6O>NDTd%  
    %       [theta,r] = cart2pol(X,Y); bC&*U|de  
    %       idx = r<=1; *;5P65:u$>  
    %       z = nan(size(X)); XcD$xFDZ  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 4'_PLOgnX  
    %       figure 7&-B6Y4  
    %       pcolor(x,x,z), shading interp tUaDwIu#  
    %       axis square, colorbar ^Q0%_V,  
    %       title('Zernike function Z_5^1(r,\theta)') 3+ JkV\AF  
    % Ahv%Q%m%2  
    %   Example 2: Q+YYj  
    % ]rY:C "#  
    %       % Display the first 10 Zernike functions jbZ%Y0km%  
    %       x = -1:0.01:1; 'So,*>]63  
    %       [X,Y] = meshgrid(x,x); VB=$D|Ll  
    %       [theta,r] = cart2pol(X,Y); FX}kH]  
    %       idx = r<=1; LpN_s#  
    %       z = nan(size(X)); bh V.uBH  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Hwiw:lPq`E  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; G6@XRib3  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; N/CL?Z>c  
    %       y = zernfun(n,m,r(idx),theta(idx)); #k?uYg8  
    %       figure('Units','normalized') \2]M &n GT  
    %       for k = 1:10 &![3{G"+>l  
    %           z(idx) = y(:,k); M5\$+Tu  
    %           subplot(4,7,Nplot(k)) W w\M3Q`h  
    %           pcolor(x,x,z), shading interp ~*NG~Kn"s  
    %           set(gca,'XTick',[],'YTick',[]) >JVdL\3  
    %           axis square x)GpNkx:  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) .0 }eg$d  
    %       end [C@ |q Ah  
    % $DS|jnpV  
    %   See also ZERNPOL, ZERNFUN2. *,az`U  
    lW6$v* s9  
    %   Paul Fricker 11/13/2006 ,y5,+:Y ~  
    we?# Dui  
    rHngYcjR  
    % Check and prepare the inputs: ^W#161&  
    % ----------------------------- =2J^ '7  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) FqwH:Fcr:  
        error('zernfun:NMvectors','N and M must be vectors.') I) ]"`2w2w  
    end |[./jg"  
    UmEc")3  
    if length(n)~=length(m) q# C;iK4  
        error('zernfun:NMlength','N and M must be the same length.') b';oFUU>Q  
    end ^L4"X~eM  
    P z< \q;  
    n = n(:); yX7P5c.   
    m = m(:); H;w8[ImK  
    if any(mod(n-m,2)) G1tua"Px  
        error('zernfun:NMmultiplesof2', ... 2e_m>I  
              'All N and M must differ by multiples of 2 (including 0).') ]Y;5U  
    end VPi*9(LS  
    z*,J0)<Q  
    if any(m>n) 9u0<$UY%  
        error('zernfun:MlessthanN', ... ks19e>'5Q  
              'Each M must be less than or equal to its corresponding N.') +Z7:(o<  
    end |X47&Y  
    e|1.-P@  
    if any( r>1 | r<0 ) " rVf{  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') a'!p^/6?  
    end 7ILb&JQ!%{  
    u; G-46  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) T;C0t9Yew  
        error('zernfun:RTHvector','R and THETA must be vectors.') (Q(=MEar  
    end 1[:tiTG|C  
    `=%mU/v  
    r = r(:); g>*P}r~;^b  
    theta = theta(:); +?9. &<?  
    length_r = length(r); O= 84ZP%  
    if length_r~=length(theta) i+@t_pxc  
        error('zernfun:RTHlength', ... A<p6]#t#X)  
              'The number of R- and THETA-values must be equal.') wGLSei-s  
    end +bdjZD3  
    2 Q}^<^r  
    % Check normalization: ~{cG"  
    % -------------------- NTV@,  
    if nargin==5 && ischar(nflag) CNM pyr  
        isnorm = strcmpi(nflag,'norm'); n?mV(?N  
        if ~isnorm |V-)3 #c  
            error('zernfun:normalization','Unrecognized normalization flag.') Jp 7m$D%  
        end 9 v 3%a3  
    else O>,Rsj!e  
        isnorm = false; Lq#$q>!K  
    end kO}Q OL4  
    k#"}oI{< 6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% HDQH7Bs  
    % Compute the Zernike Polynomials ItxC}qT  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Xpq=2`  
    jM[]Uh  
    % Determine the required powers of r: )-\[A<(  
    % ----------------------------------- \O=t5yS  
    m_abs = abs(m); 5: vy_e&  
    rpowers = []; l*-$H$  
    for j = 1:length(n) <IwfiI3y  
        rpowers = [rpowers m_abs(j):2:n(j)]; eh /QFm 4  
    end WUK{st.z  
    rpowers = unique(rpowers); "t&_!Rm  
    NR.YeKsBq  
    % Pre-compute the values of r raised to the required powers, L(`Rf0smt  
    % and compile them in a matrix: 'Ivr =-  
    % ----------------------------- D:#e;K  
    if rpowers(1)==0 4l~B/"}  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); `VXC*A   
        rpowern = cat(2,rpowern{:}); R4 AKp1Y  
        rpowern = [ones(length_r,1) rpowern]; X;QhK] Z  
    else L4!T  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); NsF8`r g  
        rpowern = cat(2,rpowern{:}); $E6bu4I  
    end VWT\wA L  
    Z"5ewU<?  
    % Compute the values of the polynomials: " "{#~X}  
    % -------------------------------------- Uu(FFd~3  
    y = zeros(length_r,length(n)); zrE Dld9  
    for j = 1:length(n) L@x#:s=  
        s = 0:(n(j)-m_abs(j))/2; v~KgCLo  
        pows = n(j):-2:m_abs(j); ~T:L0||.%9  
        for k = length(s):-1:1 M D,+>kh  
            p = (1-2*mod(s(k),2))* ... aP`V  
                       prod(2:(n(j)-s(k)))/              ... k*k 9hv?  
                       prod(2:s(k))/                     ... ^k}%k#)  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... =x-@-\m  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); KiYz]IM$4  
            idx = (pows(k)==rpowers); +&qj`hA-b  
            y(:,j) = y(:,j) + p*rpowern(:,idx); lQl  
        end Wer.VL  
         "2>_eZ#b  
        if isnorm W8Aii'Q8C/  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Kn4x _9  
        end u 4$$0 `  
    end *c' hmA s  
    % END: Compute the Zernike Polynomials We:b1sZR  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3ox 0-+_  
    m)"wd$O^w  
    % Compute the Zernike functions: b^C2<'  
    % ------------------------------ a6epew!2  
    idx_pos = m>0; 6+ C7vG`  
    idx_neg = m<0; [O\[,E"K  
    ![hVTZ,hyZ  
    z = y; PNG!q}(c  
    if any(idx_pos) 4 t< mX  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); i5CBLv  
    end /p7-D;  
    if any(idx_neg) xZ(f_Oy  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); jLCZ JSK  
    end ';Ew-u  
    Gb_y"rx?0  
    % 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