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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 =h~\nTN  
    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! +^St"GWY  
     
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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)  EM+_c)d}  
    Aj06"ep  
    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。 K (yuL[p`  
    _zQ3sm  
    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) z%t>z9hU  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Nzz" w_#  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of |{HtY  
    %   order N and frequency M, evaluated at R.  N is a vector of D!TL~3d 1  
    %   positive integers (including 0), and M is a vector with the $(_Xt-6  
    %   same number of elements as N.  Each element k of M must be a .EGZv (rz&  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) &O(z|-&| x  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is :h1itn  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix `bO+3Y'5  
    %   with one column for every (N,M) pair, and one row for every {U4BPKof  
    %   element in R. 6R5) &L  
    % #X7fs5$&  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ZbCu -a{v  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is J\XYUs  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to P5Ms X~mT  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 3.B|uN  
    %   for all [n,m]. 5SFeJBS  
    % d}%GHvOi  
    %   The radial Zernike polynomials are the radial portion of the ~h?zK 1  
    %   Zernike functions, which are an orthogonal basis on the unit y!fV+S,  
    %   circle.  The series representation of the radial Zernike qR!SwG44+  
    %   polynomials is /r Zj=  
    % xlkEW&N&  
    %          (n-m)/2 @rkNx@[~  
    %            __ %v:9_nwO)  
    %    m      \       s                                          n-2s f&B&!&gZ  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r +LV~%?W  
    %    n      s=0 =dUeQ?>t=  
    % fLAOA9  
    %   The following table shows the first 12 polynomials. P-[6xu+]  
    % z+&mMP`-  
    %       n    m    Zernike polynomial    Normalization $d%m%SZxv  
    %       --------------------------------------------- fb4/LVg'J  
    %       0    0    1                        sqrt(2) 828E^Q"<  
    %       1    1    r                           2 Dms 6"x2  
    %       2    0    2*r^2 - 1                sqrt(6) B|gyr4]  
    %       2    2    r^2                      sqrt(6) FkR9-X<  
    %       3    1    3*r^3 - 2*r              sqrt(8) "n:z("Q*  
    %       3    3    r^3                      sqrt(8) &(-+?*A`E  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) " GkBX  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) G/\t<>O8o  
    %       4    4    r^4                      sqrt(10) |/[?]`  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) <i`Ipj  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) v/\l  
    %       5    5    r^5                      sqrt(12) I/rq@27o  
    %       --------------------------------------------- 3D[IZ^%VtM  
    % |Spy |,/  
    %   Example: _a"5[sG  
    % rq Dre`m  
    %       % Display three example Zernike radial polynomials U-#wFc2N  
    %       r = 0:0.01:1; ?kX$Y{M}  
    %       n = [3 2 5]; ".onev^(  
    %       m = [1 2 1]; 4!,x3H'  
    %       z = zernpol(n,m,r); )t{?7wy  
    %       figure ?d0I*bs)7  
    %       plot(r,z) lNowH0K!D  
    %       grid on j8WnXp_  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') cU?A|'  
    % rR&;2  
    %   See also ZERNFUN, ZERNFUN2. Z\D!'FX  
    <5rp$AzT  
    % A note on the algorithm. 5ycccMx0V  
    % ------------------------ CEq0ZL-W  
    % The radial Zernike polynomials are computed using the series <Qu]m.z[  
    % representation shown in the Help section above. For many special F\F_">5  
    % functions, direct evaluation using the series representation can 9'faH  
    % produce poor numerical results (floating point errors), because UUc{1"z{  
    % the summation often involves computing small differences between !#` .Mv Z  
    % large successive terms in the series. (In such cases, the functions Vb az#I  
    % are often evaluated using alternative methods such as recurrence .z4 fJx  
    % relations: see the Legendre functions, for example). For the Zernike s'qd%JxD  
    % polynomials, however, this problem does not arise, because the O@6iG  
    % polynomials are evaluated over the finite domain r = (0,1), and {Y6U%HG{{r  
    % because the coefficients for a given polynomial are generally all u6Fm qK]Dj  
    % of similar magnitude. k#NIY4%.  
    % "MQy>mD6  
    % ZERNPOL has been written using a vectorized implementation: multiple SB0Cq  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] $!%/Kk4M  
    % values can be passed as inputs) for a vector of points R.  To achieve '!^5GSP3&  
    % this vectorization most efficiently, the algorithm in ZERNPOL A-x; ai]  
    % involves pre-determining all the powers p of R that are required to ^0p y  
    % compute the outputs, and then compiling the {R^p} into a single uOUgU$%zqH  
    % matrix.  This avoids any redundant computation of the R^p, and ad1I2  
    % minimizes the sizes of certain intermediate variables. m1H_kJ  
    % P|HxD0c^u  
    %   Paul Fricker 11/13/2006 ?~c=Sa-  
    FOVghq@  
    8Yc'4v#}  
    % Check and prepare the inputs: y:u7*%"  
    % ----------------------------- Evu`e=LaG  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 6RV42r^pf  
        error('zernpol:NMvectors','N and M must be vectors.') Lgp{  hK  
    end +A%|.;  
    &0cfTb)dG  
    if length(n)~=length(m) 5IE3[a%X  
        error('zernpol:NMlength','N and M must be the same length.') ?_)b[-N!  
    end (37dD!  
    7niZ`doBA  
    n = n(:); uqy&P S  
    m = m(:); ._'AJhU$0  
    length_n = length(n); v6=pV4k9  
    9MP_#M7  
    if any(mod(n-m,2)) #$W02L8  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 9/2VU< K  
    end -([ ipg(r  
    y0s=yN_  
    if any(m<0) Z 0&=Lw  
        error('zernpol:Mpositive','All M must be positive.') ? 1Os%9D*  
    end 9#(Nd, m})  
    JC6?*R  
    if any(m>n) mD@*vq  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') xb{G:v  
    end rSu+zS7`X  
    (~S=DFsP  
    if any( r>1 | r<0 ) #<h//<  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') x -;tV=E}  
    end +/O3L=QyJ  
    9u[^9tL+D  
    if ~any(size(r)==1) ($QQuM=  
        error('zernpol:Rvector','R must be a vector.') RvQa&r5l  
    end 7slpj8  
    7pPaHX8  
    r = r(:); *G rYB6MT  
    length_r = length(r); $?P5A E  
    lV%oIf[OB  
    if nargin==4  kg &R  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); Lf 0X(tC  
        if ~isnorm zTBf.A;e7  
            error('zernpol:normalization','Unrecognized normalization flag.') Cb}I-GtO  
        end m3T=x =  
    else 3uXRS,C  
        isnorm = false; Gxhr0'  
    end sdp3geBYo  
    !d.bCE~  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [?>\]  
    % Compute the Zernike Polynomials 7l(GBr  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Rf4}((y7Y\  
    .9NYa|+0  
    % Determine the required powers of r: X+2uM+  
    % ----------------------------------- *Jwx,wF}4  
    rpowers = []; -UB XWl  
    for j = 1:length(n) { )g $  
        rpowers = [rpowers m(j):2:n(j)]; 0u) m9eg  
    end OLS/3c z  
    rpowers = unique(rpowers); !i>d04u`%  
    c7~'GXxQ2  
    % Pre-compute the values of r raised to the required powers, rP{Jep!  
    % and compile them in a matrix: [s{ B vn  
    % ----------------------------- WQ+ xS!ba  
    if rpowers(1)==0 sOJXloeO[6  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); emSky-{$u  
        rpowern = cat(2,rpowern{:}); gNHS:k\"  
        rpowern = [ones(length_r,1) rpowern]; b#nI#!p'  
    else pC~ M5(F_  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ^UCH+C yl  
        rpowern = cat(2,rpowern{:}); Xj9\:M-  
    end 9-+N;g!q  
    4mHk,Dd9,  
    % Compute the values of the polynomials: hmkm^2  
    % -------------------------------------- \Up~ "q>Kb  
    z = zeros(length_r,length_n); )+Y"4?z~  
    for j = 1:length_n l6*MiX]q  
        s = 0:(n(j)-m(j))/2; ?$K.*])e  
        pows = n(j):-2:m(j); W{%X1::q$  
        for k = length(s):-1:1 'NMO>[.  
            p = (1-2*mod(s(k),2))* ... 4/ WKR3X  
                       prod(2:(n(j)-s(k)))/          ... M5a&eO  
                       prod(2:s(k))/                 ... p#%*z~ui  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... O dbXna  
                       prod(2:((n(j)+m(j))/2-s(k))); kRjNz~g  
            idx = (pows(k)==rpowers); G?v!Uv8O  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Q=6 1.lP6  
        end 5Gs>rq" #  
         7B<,nKd  
        if isnorm _7z]zy@PC5  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); +%f6{&q$  
        end r+d+gO.  
    end riL|B 3  
    's.e"F#  
    % 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) 6J\ 2 =c`  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Rc:}%a%e  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated D w/vXyZ  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive R`Fgne$4  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, S6}_N/;6~  
    %   and THETA is a vector of angles.  R and THETA must have the same ZH|q#< {l  
    %   length.  The output Z is a matrix with one column for every P-value, o5j6(`#;  
    %   and one row for every (R,THETA) pair. ",&QO 7_  
    % zrqI^i"c  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike $OG){'X  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) t=X=",)f  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) P6Y+ u  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 h (q,T$7 W  
    %   for all p. :._Igjj$=  
    %  ?HRS*  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 er5!n e  
    %   Zernike functions (order N<=7).  In some disciplines it is qFK.ULgP`  
    %   traditional to label the first 36 functions using a single mode OX'V  
    %   number P instead of separate numbers for the order N and azimuthal  =   
    %   frequency M. 5+11J[~{  
    % 7)]boW~Q  
    %   Example: Pjk2tf0j`  
    % V'e%%&g~N  
    %       % Display the first 16 Zernike functions cE:s\hG  
    %       x = -1:0.01:1; 5q*s_acQ  
    %       [X,Y] = meshgrid(x,x); l;KrFJ6  
    %       [theta,r] = cart2pol(X,Y); >I0;MNX  
    %       idx = r<=1; p:TE##  
    %       p = 0:15; \c<;!vkZ04  
    %       z = nan(size(X)); WKiP0~  
    %       y = zernfun2(p,r(idx),theta(idx)); $cIaLq  
    %       figure('Units','normalized') |,@D <  
    %       for k = 1:length(p) $1 "gFg  
    %           z(idx) = y(:,k); F \ls]luN  
    %           subplot(4,4,k) R;!@ xy  
    %           pcolor(x,x,z), shading interp YTFU# F  
    %           set(gca,'XTick',[],'YTick',[]) &:5*^1oP  
    %           axis square McN[  
    %           title(['Z_{' num2str(p(k)) '}']) ;  ?f+  
    %       end 5\# F5s}  
    % pHmqwB~|  
    %   See also ZERNPOL, ZERNFUN. t$(#$Z,RS  
    j &,Gv@  
    %   Paul Fricker 11/13/2006 _,!0_\+i  
    COsmVQ.  
    ,\J 8(,%L  
    % Check and prepare the inputs: 2=- .@,6  
    % ----------------------------- ed`"xm  
    if min(size(p))~=1 g%l ,a3"  
        error('zernfun2:Pvector','Input P must be vector.') L4Zt4Yuw  
    end ,eBC]4)B6  
    V\Gs&>  
    if any(p)>35 =4MTb_  
        error('zernfun2:P36', ... <HoCt8>U  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... F| eWHw?t  
               '(P = 0 to 35).']) TL2E|@k1]  
    end 9tJ0O5  
    !nSa4U,$w<  
    % Get the order and frequency corresonding to the function number:  >9!J?HA  
    % ---------------------------------------------------------------- r@3-vLI!u  
    p = p(:); 9 Gd6/2  
    n = ceil((-3+sqrt(9+8*p))/2);  I8?  
    m = 2*p - n.*(n+2); T4] 2R  
    O3@DU#N&s  
    % Pass the inputs to the function ZERNFUN: "G [Nb:,CR  
    % ---------------------------------------- a*y9@RC}  
    switch nargin ;.uYWP|9  
        case 3 -n.m "O3  
            z = zernfun(n,m,r,theta); gSwV:hm  
        case 4 )]j3-#  
            z = zernfun(n,m,r,theta,nflag); J)YlG*  
        otherwise 4%J0e'iN  
            error('zernfun2:nargin','Incorrect number of inputs.') q13fmK(n-5  
    end Q4m> 3I  
    UE'=9{o`  
    % 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 8X,6U_>#a  
    function z = zernfun(n,m,r,theta,nflag) G$>?UQ[  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 5`.CzQVb  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ~o!- [  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Q-w# !<L.  
    %   unit circle.  N is a vector of positive integers (including 0), and XLn9NBT4K  
    %   M is a vector with the same number of elements as N.  Each element .J75bX5  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ~A=zjkm  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, i\6CE|  
    %   and THETA is a vector of angles.  R and THETA must have the same }*6BaB  
    %   length.  The output Z is a matrix with one column for every (N,M) PyQ .B*JJ  
    %   pair, and one row for every (R,THETA) pair. op}!1y$9P  
    % *DPX4 P  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike *SNdU^!  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), h9Far8}  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral TN0KS]^A3  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ~</FF'Xz  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized N]+6<  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. vUpAW[[  
    % Ocdy;|&  
    %   The Zernike functions are an orthogonal basis on the unit circle. }/a%-07R  
    %   They are used in disciplines such as astronomy, optics, and 5.6tVr  
    %   optometry to describe functions on a circular domain. yNns6  
    % %)lp]Y33  
    %   The following table lists the first 15 Zernike functions. [7@ g*!+d  
    % 3NpB1lgh&:  
    %       n    m    Zernike function           Normalization ^o3,YH  
    %       -------------------------------------------------- =npE?wK  
    %       0    0    1                                 1 <T_3s\  
    %       1    1    r * cos(theta)                    2 e#C v*i_<  
    %       1   -1    r * sin(theta)                    2 z+"$G  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 072C!F  
    %       2    0    (2*r^2 - 1)                    sqrt(3) }emUpju<C  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) {fXkbMO|  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) vXDs/,`r  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) <VxA&bb7c  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ^~H}N$W"-q  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) KOy{?  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) i|^Q{3?o#  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6iU&9Z<%  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) #%E`~&[  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~tp]a]yV  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) K}l3t2uk  
    %       -------------------------------------------------- >G5aFk  
    % ~~/,2^   
    %   Example 1: ]M5~p^ RB  
    % :TQp,CEa  
    %       % Display the Zernike function Z(n=5,m=1) ;\RV C 7  
    %       x = -1:0.01:1; TWRP|i!i  
    %       [X,Y] = meshgrid(x,x); H+[?{+"#@l  
    %       [theta,r] = cart2pol(X,Y); 60+zoL'  
    %       idx = r<=1; B(W~]i  
    %       z = nan(size(X)); Av.tr&ZNb  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); lCU clD  
    %       figure 3:WqUb\QK  
    %       pcolor(x,x,z), shading interp ['mpxtG  
    %       axis square, colorbar V+`kB3GV  
    %       title('Zernike function Z_5^1(r,\theta)') x4q}xwH  
    % P =X]'m_B  
    %   Example 2: tRoSq;VrS  
    % d {!P c<  
    %       % Display the first 10 Zernike functions O=o}uB-*6  
    %       x = -1:0.01:1; W>pe-  
    %       [X,Y] = meshgrid(x,x); W3.[d->X  
    %       [theta,r] = cart2pol(X,Y); O\=Z;}<N  
    %       idx = r<=1; {lds?AuK  
    %       z = nan(size(X)); Orq/38:4G  
    %       n = [0  1  1  2  2  2  3  3  3  3]; _5p$#U`  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Vh\_Ko\V5  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; wo`.sB&T  
    %       y = zernfun(n,m,r(idx),theta(idx)); [K4cxqlfk  
    %       figure('Units','normalized') x/s:/YN'  
    %       for k = 1:10 OWvblEBF  
    %           z(idx) = y(:,k); ^OY$ W  
    %           subplot(4,7,Nplot(k)) :4{ `c.S  
    %           pcolor(x,x,z), shading interp >e Gg 1  
    %           set(gca,'XTick',[],'YTick',[]) [edF'7La  
    %           axis square )O[8 D  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) u2#q7}  
    %       end qR@ES J_  
    % Dge#e  
    %   See also ZERNPOL, ZERNFUN2. |P5dv>tb F  
    !`{?qQ[=  
    %   Paul Fricker 11/13/2006 N?@^BZ  
    9~UR(Ts}l  
    Km5_P##  
    % Check and prepare the inputs: ,Q"'q0hM=  
    % ----------------------------- #ZZe*B!s_  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ei= 4u'  
        error('zernfun:NMvectors','N and M must be vectors.') FF8jW1  
    end :BxO6@>Xc  
    s@L ;3WdO  
    if length(n)~=length(m) )q<VZ|V  
        error('zernfun:NMlength','N and M must be the same length.') Y(,RJ&7  
    end B!&5*f}*  
    I=L[ "]  
    n = n(:); \92M\S  
    m = m(:); t!\aDkxo %  
    if any(mod(n-m,2)) #eJfwc1JY  
        error('zernfun:NMmultiplesof2', ... vC,FE )'  
              'All N and M must differ by multiples of 2 (including 0).') #4AU&UM+i  
    end 6/;YS[jX  
    6[t<g=  
    if any(m>n) NCk-[I?R  
        error('zernfun:MlessthanN', ... ranem0KQ)]  
              'Each M must be less than or equal to its corresponding N.') ]>~.U ~  
    end "==c  
    f,ro1Nke  
    if any( r>1 | r<0 ) 1:eWZ]B5"  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') j}Tv/O,f  
    end z_'^=9m  
    Oem1=QpaC  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) l4RqQ+[KA;  
        error('zernfun:RTHvector','R and THETA must be vectors.') @JSWqi>  
    end T.#_v# oM  
    >"/TiQt  
    r = r(:); 0s`6d;  
    theta = theta(:); k)knyEUi  
    length_r = length(r); t3$cX_  
    if length_r~=length(theta) S*Ea" vBA  
        error('zernfun:RTHlength', ... 6O/L~Z*t  
              'The number of R- and THETA-values must be equal.') cs2-jbRn  
    end `6rLd>=R  
    7O)ATb#up  
    % Check normalization: ~ T}D#}  
    % -------------------- #Shy^58$  
    if nargin==5 && ischar(nflag) 7Ydqg&  
        isnorm = strcmpi(nflag,'norm'); g(P7CX+y  
        if ~isnorm m !:F/?B  
            error('zernfun:normalization','Unrecognized normalization flag.') 9?Bh8%$  
        end UW":&`i  
    else Z( :\Vj"  
        isnorm = false; 3~`\FuHHe  
    end +Vg(2Xt  
    =F[M>o  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% og$dv 23  
    % Compute the Zernike Polynomials uhq6dhhR  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0084`&Ki  
    f.ua,,P.  
    % Determine the required powers of r: h6~xz0,u  
    % ----------------------------------- 0of:tZU  
    m_abs = abs(m); UVXruH  
    rpowers = []; 70avr)OM  
    for j = 1:length(n) p YCMJK-H  
        rpowers = [rpowers m_abs(j):2:n(j)]; a/E(GQ,,  
    end ="T}mc  
    rpowers = unique(rpowers); uEPm[oyX  
    fe4/[S{a   
    % Pre-compute the values of r raised to the required powers, a\-5tYo`u  
    % and compile them in a matrix: fCa lR7!  
    % ----------------------------- [GyPwb-  
    if rpowers(1)==0 +4t \j<T  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); =YYqgNz+\w  
        rpowern = cat(2,rpowern{:}); )cN=/i  
        rpowern = [ones(length_r,1) rpowern]; ~i-n_7+  
    else H|s Iw:  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %.[AZ>  
        rpowern = cat(2,rpowern{:}); !h&h;m/c  
    end 5UL5C:3R9  
    !/2kJOSp  
    % Compute the values of the polynomials: L_Z`UhD3{  
    % -------------------------------------- TbMlYf]It  
    y = zeros(length_r,length(n)); Q-_;.xy#4  
    for j = 1:length(n) {w>ofyqfp&  
        s = 0:(n(j)-m_abs(j))/2; 6mZpyt  
        pows = n(j):-2:m_abs(j); 6#d+BBKIc  
        for k = length(s):-1:1 k="w EZ;Q  
            p = (1-2*mod(s(k),2))* ... }8.$)&O$^  
                       prod(2:(n(j)-s(k)))/              ... Pw|/PfG  
                       prod(2:s(k))/                     ... a6T!)g  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 9MRe?  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Xa8_kv_  
            idx = (pows(k)==rpowers); N}bZdE9F  
            y(:,j) = y(:,j) + p*rpowern(:,idx); vO{[P# L}  
        end Gd&G*x  
         '@^<c#h]=  
        if isnorm DKQQZ` PF  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); UL8"{-`_\  
        end ^YPw'cZZ&  
    end c_?!V  
    % END: Compute the Zernike Polynomials tV.96P;)/9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ky7-6$  
    K!jau|FS  
    % Compute the Zernike functions: M>Ws}Y  
    % ------------------------------ XK l3B=h  
    idx_pos = m>0; 9 LEUj  
    idx_neg = m<0; @(st![i+  
    =>C3IR/  
    z = y; UJX5}36  
    if any(idx_pos) xI=[=;L  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); p t{/|P  
    end 9NC6q-2  
    if any(idx_neg) cMt , 80  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 4s&koH(x  
    end ]kkH|b$[T  
    ;S>ml   
    % 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