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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 e[Ul"pMvS`  
    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! lVK F^-i  
     
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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)  s~'C'B?  
    z+`)|c4-  
    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。 =*G'.D /*  
    Tp.iRFFkP  
    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) 2J;CiEB  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Mb!^_cS(  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 1MSu ]) W  
    %   order N and frequency M, evaluated at R.  N is a vector of p$ <qT^]&  
    %   positive integers (including 0), and M is a vector with the TD9`S SpP  
    %   same number of elements as N.  Each element k of M must be a z#/*LP#oY  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) |0mI3r  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is )T_ #X!  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 1=.?KAXR  
    %   with one column for every (N,M) pair, and one row for every ,:{+ H  
    %   element in R. *RM'0[1F4  
    % 3!W&J  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- '+wTrW m~j  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is z w9r0bG  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to _Y=yR2O  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 xx8na8  
    %   for all [n,m]. EUqG"h5#A{  
    % E]Q)pZ{Jb  
    %   The radial Zernike polynomials are the radial portion of the 0rUf'S ?K  
    %   Zernike functions, which are an orthogonal basis on the unit * vD<6qf  
    %   circle.  The series representation of the radial Zernike aoS1Yt'@  
    %   polynomials is G.T1rUh=  
    % .EwK>ro4  
    %          (n-m)/2 7a net  
    %            __ ?CDq^)T[  
    %    m      \       s                                          n-2s ],|B4\b;  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r PMcyQ2R->  
    %    n      s=0 5c8x: e@  
    % (#qVtN`t  
    %   The following table shows the first 12 polynomials. " cg>g/  
    % r MlNp?{_  
    %       n    m    Zernike polynomial    Normalization 7(Kc9sJC%%  
    %       --------------------------------------------- WTx;,TNG  
    %       0    0    1                        sqrt(2) 1\uS~RR  
    %       1    1    r                           2 5JXLfYTUI  
    %       2    0    2*r^2 - 1                sqrt(6) 7j8_O@_  
    %       2    2    r^2                      sqrt(6) =UY@,*q:c  
    %       3    1    3*r^3 - 2*r              sqrt(8) dGe  
    %       3    3    r^3                      sqrt(8) ;U&VPIX$  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) X*Zv,Wm  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) =B1!em|  
    %       4    4    r^4                      sqrt(10) ix;8S=eP~{  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ?%(*bRV -  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) f`e.c_n(  
    %       5    5    r^5                      sqrt(12) g:yK/1@Hk}  
    %       --------------------------------------------- z?xd\x  
    % ;f Gi5=-  
    %   Example: VbjW$?  
    % 2Z~o frj  
    %       % Display three example Zernike radial polynomials 9tO_hhEQ@  
    %       r = 0:0.01:1; 26A#X  
    %       n = [3 2 5]; ZUycJ-[  
    %       m = [1 2 1]; Y i`.zm  
    %       z = zernpol(n,m,r); [Wc 73-  
    %       figure Nsq%b?#  
    %       plot(r,z) 4~4Hst#^  
    %       grid on *O~D lf  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') uY,FugWbl  
    % mwxJ#  
    %   See also ZERNFUN, ZERNFUN2. vq}V0- <  
    aF:LL>H  
    % A note on the algorithm. novZ<?7 5;  
    % ------------------------ DVd/OU  
    % The radial Zernike polynomials are computed using the series Dts:$PlCk  
    % representation shown in the Help section above. For many special W2RS G~|  
    % functions, direct evaluation using the series representation can P\JpE  
    % produce poor numerical results (floating point errors), because PLD!BD  
    % the summation often involves computing small differences between CJ_B.  
    % large successive terms in the series. (In such cases, the functions N@}U;x}  
    % are often evaluated using alternative methods such as recurrence >qCT#TY  
    % relations: see the Legendre functions, for example). For the Zernike SDkN  
    % polynomials, however, this problem does not arise, because the 4.8,&{w<m  
    % polynomials are evaluated over the finite domain r = (0,1), and Rjf |  
    % because the coefficients for a given polynomial are generally all 7_RU*U^  
    % of similar magnitude. PA E)3  
    % r"+ WUU  
    % ZERNPOL has been written using a vectorized implementation: multiple 'py k  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 4?Io@[7A)  
    % values can be passed as inputs) for a vector of points R.  To achieve vzL>ZBe Z  
    % this vectorization most efficiently, the algorithm in ZERNPOL x{m)I <.:  
    % involves pre-determining all the powers p of R that are required to ,3wo  
    % compute the outputs, and then compiling the {R^p} into a single f] Vz!hM~  
    % matrix.  This avoids any redundant computation of the R^p, and 99 [ "I:  
    % minimizes the sizes of certain intermediate variables. z,)Fvs4U.  
    % HwHI$IB  
    %   Paul Fricker 11/13/2006 v[x`I;  
    Hl#o& *Ui"  
    &IcDUr]L  
    % Check and prepare the inputs: XP'KgTF  
    % ----------------------------- -Jhf]  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) I ?1E}bv  
        error('zernpol:NMvectors','N and M must be vectors.') $hL0/T-m  
    end 0t) IW D  
    X_h+\ 7N>  
    if length(n)~=length(m) L@/+u+j0  
        error('zernpol:NMlength','N and M must be the same length.') >,DR{A2hSB  
    end )YZ41K5N  
    2dC)%]aLme  
    n = n(:); L2 I/h`n"  
    m = m(:); '&"7(8E} *  
    length_n = length(n); vjHbg#0%  
    k z#DBh!&  
    if any(mod(n-m,2)) BjA|H  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') txi m|)  
    end 8w{V[@QLn  
    k=LY 6  
    if any(m<0) ?B-aj  
        error('zernpol:Mpositive','All M must be positive.') {S|uQgs6j  
    end eN/Jb;W  
    4EOu)#  
    if any(m>n) PgVM>_nHk  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') !l'Zar  
    end C. Sb4i*  
    8} U/fQ~  
    if any( r>1 | r<0 ) 7B'0(70  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') }RD,JgmV  
    end R)#"Ab Z'  
    S`NH6?/uH  
    if ~any(size(r)==1) 5vS'Qhc  
        error('zernpol:Rvector','R must be a vector.') !XK p_v  
    end =14pEe  
    3m x7[Q  
    r = r(:); FCg,p2  
    length_r = length(r); y_\d[  
    [9w8oNg0  
    if nargin==4 3c6<JW  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 2^|*M@3r  
        if ~isnorm A0'Yfuie  
            error('zernpol:normalization','Unrecognized normalization flag.') k(T/yd rw  
        end ^f4qs  
    else vwP83b0ov"  
        isnorm = false; akaQ6DIdG  
    end AR&u9Y)I  
    ,#s}nJ4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z{%h6""  
    % Compute the Zernike Polynomials %R}}1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P,,@&* :  
    _v_ak4m>  
    % Determine the required powers of r: XrYz[h*)!  
    % ----------------------------------- /H}83 C  
    rpowers = []; S]k<Ixvf  
    for j = 1:length(n) La9dFe-uu{  
        rpowers = [rpowers m(j):2:n(j)]; nL\BB&  
    end ).xQ~A\.  
    rpowers = unique(rpowers); KpSHf9!&[  
    'DVPx%p  
    % Pre-compute the values of r raised to the required powers, !sUo+Y  
    % and compile them in a matrix: ^ng?+X>mP  
    % ----------------------------- $LKniK  
    if rpowers(1)==0 q4Q1Ib-<2  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 75gE>:f  
        rpowern = cat(2,rpowern{:}); P.LMu  
        rpowern = [ones(length_r,1) rpowern]; ao Y "uT+  
    else 0&Zm3(}  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ]Rz]"JZ\S  
        rpowern = cat(2,rpowern{:}); $n!saPpxS  
    end _8kZ>w(L  
    GBN^ *I  
    % Compute the values of the polynomials: 1H%LUA  
    % -------------------------------------- Fj|C+;Q.  
    z = zeros(length_r,length_n); 7)z^*;x  
    for j = 1:length_n EZao\,t  
        s = 0:(n(j)-m(j))/2; jeC3}BL }  
        pows = n(j):-2:m(j); CsXIq.9  
        for k = length(s):-1:1 {Dqf.w>t  
            p = (1-2*mod(s(k),2))* ... 8IbHDDS  
                       prod(2:(n(j)-s(k)))/          ... , LX]  
                       prod(2:s(k))/                 ... _z~|*7@  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... tyNT1F{  
                       prod(2:((n(j)+m(j))/2-s(k))); a*! wiTGf  
            idx = (pows(k)==rpowers); ^lf{IM-Y  
            z(:,j) = z(:,j) + p*rpowern(:,idx); BG/M3  
        end ;i>|5tEy  
         dFK/  
        if isnorm ~[t%g9  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); yY[N\*P  
        end =rGjOb3+  
    end ]^p6db zWe  
    YR^J7b\  
    % 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) y}Oc^Fc  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. sFuB[ JJ}  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 6=0"3%jn@  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive $i;%n1VBg  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Z.ky=vCt  
    %   and THETA is a vector of angles.  R and THETA must have the same }w}2'P'T  
    %   length.  The output Z is a matrix with one column for every P-value, |VQ17*4ff1  
    %   and one row for every (R,THETA) pair. HN]roSt~  
    % wsYvbI!  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ~7IXJeon  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) tN&4t xB  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) w9Bbvr6  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 YzNSZJPD  
    %   for all p. ,4M7:=gf  
    % XvETys@d  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ' @i0~  
    %   Zernike functions (order N<=7).  In some disciplines it is B+:/!_  
    %   traditional to label the first 36 functions using a single mode Ii FeO  
    %   number P instead of separate numbers for the order N and azimuthal dZ K /v  
    %   frequency M. 7&;M"?m&  
    % fP# !ywgr%  
    %   Example: LX2rg\a+%  
    % F!(Vg  
    %       % Display the first 16 Zernike functions ^YiGvZJ  
    %       x = -1:0.01:1; K[r<-6TS  
    %       [X,Y] = meshgrid(x,x); 3}~.#`QeY  
    %       [theta,r] = cart2pol(X,Y); %? -E)n[  
    %       idx = r<=1; cNOtfn6?F  
    %       p = 0:15; jwhc;y  
    %       z = nan(size(X)); d 5jZ?  
    %       y = zernfun2(p,r(idx),theta(idx)); /enlkZx=8  
    %       figure('Units','normalized') BQTZt'p  
    %       for k = 1:length(p) 3Z/_}5%"  
    %           z(idx) = y(:,k); RC?gozBFJ  
    %           subplot(4,4,k) :+#$=4  
    %           pcolor(x,x,z), shading interp W>W b|W  
    %           set(gca,'XTick',[],'YTick',[]) v,]-;V~<  
    %           axis square AH-B/c5  
    %           title(['Z_{' num2str(p(k)) '}']) In13crr4!  
    %       end y``[CBj  
    % U1nObA  
    %   See also ZERNPOL, ZERNFUN. c[VVCN8dA  
    t@r>GHO  
    %   Paul Fricker 11/13/2006 F/p/&9  
    x9\z^GU%H  
    3ScOJo  
    % Check and prepare the inputs: pY.R?\  
    % ----------------------------- +;,65j+n   
    if min(size(p))~=1 .Nk'yow  
        error('zernfun2:Pvector','Input P must be vector.') 4Ys\<\~d  
    end WAq! _xE  
    +q*WY*gX  
    if any(p)>35 vo (riHH  
        error('zernfun2:P36', ... Z;/QB6|%  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... !U::kr=t  
               '(P = 0 to 35).']) ' _ZiZ4O  
    end +%Z#!1u  
    N W]zMU{c  
    % Get the order and frequency corresonding to the function number: nMM:Tr  
    % ---------------------------------------------------------------- *? V boyU  
    p = p(:); (>49SOu;$\  
    n = ceil((-3+sqrt(9+8*p))/2); >G9YYt~  
    m = 2*p - n.*(n+2); &ci;0P#Q  
    !#y_vz9  
    % Pass the inputs to the function ZERNFUN: 5]f6YlJZ  
    % ---------------------------------------- b I"+b\K  
    switch nargin CH9Psr78  
        case 3 Tfq7<<0$N  
            z = zernfun(n,m,r,theta); k%D|17I  
        case 4 :MaP58dhh  
            z = zernfun(n,m,r,theta,nflag); w`YN#G  
        otherwise M "\Iw'5$  
            error('zernfun2:nargin','Incorrect number of inputs.') q!;u4J  
    end :_8Nf1B+T  
    F:7 d}Jx  
    % 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 7!F -.kG  
    function z = zernfun(n,m,r,theta,nflag) cY^'Cj  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. icK$W2<8mg  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ^ 0.`1$  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 6Vgxfic  
    %   unit circle.  N is a vector of positive integers (including 0), and :i3 W U%  
    %   M is a vector with the same number of elements as N.  Each element 8kLHQ0pmu  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 7#&e0fw/I  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1,  "F=ta  
    %   and THETA is a vector of angles.  R and THETA must have the same }U'VVPh _  
    %   length.  The output Z is a matrix with one column for every (N,M) +!Q*ie+q  
    %   pair, and one row for every (R,THETA) pair. vRh)o1u)  
    % cJE4uL<  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike XL7||9,(h  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), SM8f"H28  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral + )n}n5  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, !bIE%cq  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Mt4*`CxtH;  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. k4PXH  
    % I5@8=rFk  
    %   The Zernike functions are an orthogonal basis on the unit circle. "m%EFWUOl  
    %   They are used in disciplines such as astronomy, optics, and d#HlO}  
    %   optometry to describe functions on a circular domain. G<-<>)zO!  
    % X[!S7[d-y  
    %   The following table lists the first 15 Zernike functions. GG`j9"t4  
    % 3bRW]mP8  
    %       n    m    Zernike function           Normalization Cg(&WJw(ep  
    %       -------------------------------------------------- sXmP<c  
    %       0    0    1                                 1 ?bPW*A82{q  
    %       1    1    r * cos(theta)                    2 &5[B\yv  
    %       1   -1    r * sin(theta)                    2 '#C5m#v  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) .}5qi;CA  
    %       2    0    (2*r^2 - 1)                    sqrt(3) D*>#]0X  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 6zi 5#23  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Z,tHyyF?j  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 1Va=.#<  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 34QW^{dgE  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ^T*!~K8A  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Vr@tSc&  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Qz89=#W  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ^/VnRpU  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) u* G+=aV.6  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) *aJO5&w<T  
    %       -------------------------------------------------- \a4X},h\  
    % JZK93R  
    %   Example 1: S['cX ~  
    % /ykc`E?f  
    %       % Display the Zernike function Z(n=5,m=1) 1?yj<^"  
    %       x = -1:0.01:1; z%1e>`\E  
    %       [X,Y] = meshgrid(x,x); h@z0 x4_])  
    %       [theta,r] = cart2pol(X,Y); q65]bs4M  
    %       idx = r<=1; MsZx 0]  
    %       z = nan(size(X)); CG95ScrX  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ~%2yDhdQ  
    %       figure i&8|@CACb  
    %       pcolor(x,x,z), shading interp l,~`o$ _  
    %       axis square, colorbar :+ mULUi  
    %       title('Zernike function Z_5^1(r,\theta)') }'?qUy3x  
    % eY-h<K)y  
    %   Example 2: d"@ /{O^1  
    % {kBsiSvsA;  
    %       % Display the first 10 Zernike functions tJ7F.}\;C  
    %       x = -1:0.01:1; `!spi=f  
    %       [X,Y] = meshgrid(x,x); VR .t  
    %       [theta,r] = cart2pol(X,Y); 4AKr.a0q  
    %       idx = r<=1; as'yYn8  
    %       z = nan(size(X)); ?"^{:~\N  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Mna yiJl  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; [Y~~C J  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 4"H *hKp  
    %       y = zernfun(n,m,r(idx),theta(idx)); g*(z .  
    %       figure('Units','normalized') ZyDNtX%  
    %       for k = 1:10 a]P w:lT  
    %           z(idx) = y(:,k); a#{"3Z2|  
    %           subplot(4,7,Nplot(k)) Zk/ejhy0  
    %           pcolor(x,x,z), shading interp F,A+O+  
    %           set(gca,'XTick',[],'YTick',[]) qpMcVJL  
    %           axis square Bz <I7h  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) =36fS/Gb  
    %       end 7{(UiQbf  
    % L N Fe7<y  
    %   See also ZERNPOL, ZERNFUN2. ; o Y|~  
    U[|5:qWs  
    %   Paul Fricker 11/13/2006 <R+?>kz6  
    kz1#"8Zd!  
    "\O7_od-  
    % Check and prepare the inputs: o[}Dj6e\t  
    % ----------------------------- Jfk#E^1  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) @0s' (  
        error('zernfun:NMvectors','N and M must be vectors.') 934j5D  
    end jpO0dtn3=  
    j}tM0Ug.U  
    if length(n)~=length(m) IG# wY  
        error('zernfun:NMlength','N and M must be the same length.') hRRxOr#*$  
    end cc*?4C/t  
    8'L:D  
    n = n(:); K#N9N@WjR  
    m = m(:); bhGRD{=  
    if any(mod(n-m,2)) RRPPojKZ  
        error('zernfun:NMmultiplesof2', ... >Oj$ Dn=  
              'All N and M must differ by multiples of 2 (including 0).') 9 " t;6  
    end 4r `I)  
    6)ibXbH  
    if any(m>n) VBQAkl?(}4  
        error('zernfun:MlessthanN', ... Xz^k.4 Y{4  
              'Each M must be less than or equal to its corresponding N.') -(F} =o'  
    end Q,JH/X  
    E0Q6Ryn  
    if any( r>1 | r<0 ) #^r-D[/m  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') wM4{\  f\  
    end }~|`h1JF  
    v@OELJX  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) _AFje  
        error('zernfun:RTHvector','R and THETA must be vectors.') 4K'U}W  
    end Dk a8[z7  
    km C0.\  
    r = r(:); eOiH7{OA,  
    theta = theta(:); -&`_bf%M  
    length_r = length(r); :d9GkC  
    if length_r~=length(theta) p0 X%^A,4  
        error('zernfun:RTHlength', ... pP1DR'  
              'The number of R- and THETA-values must be equal.') iAQ[;M 3p  
    end Iy49o!  
    Y @'do)  
    % Check normalization: oA[`| ji  
    % -------------------- yQUrHxm  
    if nargin==5 && ischar(nflag) s`H|o'0  
        isnorm = strcmpi(nflag,'norm'); n]Yz<#  
        if ~isnorm 3))CD,|  
            error('zernfun:normalization','Unrecognized normalization flag.') &_-=(rK  
        end p?>J86%[  
    else fcEm :jEZ*  
        isnorm = false; v~Dobk/n  
    end |v%$Q/zp&  
    -rI7ihr*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fsPNxy"_  
    % Compute the Zernike Polynomials 8v2Wi.4T  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Cip|eM&l  
    DJgM>&Y6,  
    % Determine the required powers of r: B=K<k+{6"  
    % ----------------------------------- #*qV kPX  
    m_abs = abs(m); zO\_^A|8H  
    rpowers = []; z+;$cfN  
    for j = 1:length(n) }v2p]D5n.  
        rpowers = [rpowers m_abs(j):2:n(j)]; Xe\}(O  
    end ~&p]kmwXSX  
    rpowers = unique(rpowers); AZhI~QWo  
    9C,gJp}P  
    % Pre-compute the values of r raised to the required powers, }NwmZ w>_  
    % and compile them in a matrix: mfI[9G  
    % ----------------------------- guYP|  
    if rpowers(1)==0 _ps4-<ugC  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); sj&(O@~R  
        rpowern = cat(2,rpowern{:}); ]kmAN65c  
        rpowern = [ones(length_r,1) rpowern]; #e-7LmO~  
    else uKXU.u*C  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9NVtvBA  
        rpowern = cat(2,rpowern{:}); 89D`!`Ah]  
    end ym6Emf]  
    /];N1  
    % Compute the values of the polynomials: T+P{,,a/]  
    % -------------------------------------- )E=B;.FH  
    y = zeros(length_r,length(n)); ,Aq, f$5V  
    for j = 1:length(n) um]*nXIr  
        s = 0:(n(j)-m_abs(j))/2; jWxa [ >  
        pows = n(j):-2:m_abs(j); ld(_+<e  
        for k = length(s):-1:1 2BOH8Mp9  
            p = (1-2*mod(s(k),2))* ... Q$.CtECo  
                       prod(2:(n(j)-s(k)))/              ... `_Iyr3HAf  
                       prod(2:s(k))/                     ... ~oSA&v4V  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... i=b'_SZ '  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 7YTO{E6]d\  
            idx = (pows(k)==rpowers); E5P.x^  
            y(:,j) = y(:,j) + p*rpowern(:,idx); t"%~r3{  
        end -M]/Xv]  
         ZT&[:>upR  
        if isnorm j^ 8Hjg  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Y(rQ032s  
        end x?{l<mc  
    end =u9e5n  
    % END: Compute the Zernike Polynomials S?v;+3TG  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SP2";,%/9  
    ~rOvVi&4  
    % Compute the Zernike functions: {yf, :5  
    % ------------------------------ 8[^b8^  
    idx_pos = m>0; [C 7X#|  
    idx_neg = m<0; A;C4>U Y  
    Sb?v5  
    z = y; ?=iy 6q  
    if any(idx_pos) i0x[w>\-  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); E(S$Q^  
    end 0\ j)!b  
    if any(idx_neg) fH ,h\0  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); @d3yqA  
    end yyVJb3n5:!  
    bsc b  
    % 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