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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 倒序阅读 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 VV 54$a  
    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! j0Cj&x%qF}  
     
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    离线phility
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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
    离线phility
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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
    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 f#7=N{wm  
    function z = zernfun(n,m,r,theta,nflag) bR:hu}YS  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. K:Z(jF!j  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N IGly x'\_  
    %   and angular frequency M, evaluated at positions (R,THETA) on the B[#n,ay  
    %   unit circle.  N is a vector of positive integers (including 0), and ;kR=vv  
    %   M is a vector with the same number of elements as N.  Each element tGbx/$Y   
    %   k of M must be a positive integer, with possible values M(k) = -N(k) BJ'pe[Xa5  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, zKaj<Og  
    %   and THETA is a vector of angles.  R and THETA must have the same 5j0 Ib>\  
    %   length.  The output Z is a matrix with one column for every (N,M) ?4aW^l6/  
    %   pair, and one row for every (R,THETA) pair. tTub W=H  
    % OQKc_z'"  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ^|hVFM2  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), >LH}A6dUC  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral f|F=)tJO  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, =*zde0T?l  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized JR&yaOws  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. -XK;B--c  
    % p&)d]oV>  
    %   The Zernike functions are an orthogonal basis on the unit circle. R?tjobk!  
    %   They are used in disciplines such as astronomy, optics, and yx*<c#Uf  
    %   optometry to describe functions on a circular domain. Of$R+n.  
    % \IudS{ .?;  
    %   The following table lists the first 15 Zernike functions. \j BA4?(S  
    % >El]5M7h7  
    %       n    m    Zernike function           Normalization gSj0+|  
    %       -------------------------------------------------- &@BAVc z  
    %       0    0    1                                 1  ylS6D  
    %       1    1    r * cos(theta)                    2 Q"c/]Sk)  
    %       1   -1    r * sin(theta)                    2 UWK|_RT6SA  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) +9pock  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 0M&~;`W}  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) W2zG"Q  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ^Oeixi@f  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) kUT^o  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) C@zG(?X  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ._<, Eodv  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) sX3qrRY  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;_|4c7  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Uq{$j5p8  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :xbj& l  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) |-S+x]9  
    %       -------------------------------------------------- :*DWL!a  
    % lFSvHs5  
    %   Example 1: _' X  
    % b?lRada{I  
    %       % Display the Zernike function Z(n=5,m=1) E`hR(UL ?  
    %       x = -1:0.01:1; XZ3fWcw[  
    %       [X,Y] = meshgrid(x,x); jAv3qMQA  
    %       [theta,r] = cart2pol(X,Y); bhbTloCR  
    %       idx = r<=1; 2mMi=pv9  
    %       z = nan(size(X)); ?~.:C'  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 0E,QOF{o  
    %       figure {.[EXMX  
    %       pcolor(x,x,z), shading interp JRZp 'Ln  
    %       axis square, colorbar gu~R4 @3  
    %       title('Zernike function Z_5^1(r,\theta)') zxH<~2  
    % 4sRg+mMI  
    %   Example 2: "USzk7=&.  
    % R$A%Zh6  
    %       % Display the first 10 Zernike functions 4<)*a]\c5M  
    %       x = -1:0.01:1; z 0zB&}  
    %       [X,Y] = meshgrid(x,x); suW|hh1/Ya  
    %       [theta,r] = cart2pol(X,Y); .X"&k O>G  
    %       idx = r<=1; v6[VdWOx5  
    %       z = nan(size(X)); 8O60pB;4  
    %       n = [0  1  1  2  2  2  3  3  3  3]; h(J$-SUs  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; e>.^RtDF  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ],~[^0  
    %       y = zernfun(n,m,r(idx),theta(idx)); J=(i0A  
    %       figure('Units','normalized') zxD=q5in  
    %       for k = 1:10 2Ub-ufkU  
    %           z(idx) = y(:,k); 5} ur,0{  
    %           subplot(4,7,Nplot(k)) #CAZ}];Qx  
    %           pcolor(x,x,z), shading interp j6$@vA)  
    %           set(gca,'XTick',[],'YTick',[]) }$qrNbLJ  
    %           axis square JKO*bbj  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) yJO Jw o^  
    %       end *O@Zn  
    % j!oX\Y-:&  
    %   See also ZERNPOL, ZERNFUN2. S')DAx  
    D^P0X:T]  
    %   Paul Fricker 11/13/2006 YWDgRb  
    5L~lF8  
    (: k n)  
    % Check and prepare the inputs: 0dS(g&ZR  
    % ----------------------------- N#)Klq87z  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) S1@r.z2L  
        error('zernfun:NMvectors','N and M must be vectors.') Nq\)o{<1  
    end <SOG?Lh~  
    IR:{{ (  
    if length(n)~=length(m) 2@pEiq3  
        error('zernfun:NMlength','N and M must be the same length.') P$N5j~*  
    end Mqk|H~l5c  
    * a1q M?  
    n = n(:); "lC>_A  
    m = m(:); F2_'U' a  
    if any(mod(n-m,2)) ~)>.%`v&  
        error('zernfun:NMmultiplesof2', ... s|c}9/Xe)  
              'All N and M must differ by multiples of 2 (including 0).') cXf/  
    end tlg}"lY  
    nhC8Tq[m  
    if any(m>n) MZcvr9y  
        error('zernfun:MlessthanN', ... i O?f&u  
              'Each M must be less than or equal to its corresponding N.') PNo:vRtsq  
    end [q_62[-X  
    qdKqc,R1{  
    if any( r>1 | r<0 ) Ie=gI+2  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') x%goyXK  
    end %hZX XpuO  
    vdB2T2F  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) (JnEso-V  
        error('zernfun:RTHvector','R and THETA must be vectors.') }Y!s:w#  
    end .m>Qlh  
    O'#;Ge/,  
    r = r(:); w'$>E4\   
    theta = theta(:); n+Conp/  
    length_r = length(r); "$K]+0ryG<  
    if length_r~=length(theta) $FX$nY  
        error('zernfun:RTHlength', ... a_{'I6a*,  
              'The number of R- and THETA-values must be equal.') *b 0z/ 6  
    end v,ni9DIu  
    @|">j#0  
    % Check normalization: 5rCJIl.  
    % -------------------- &(Hw:W 9  
    if nargin==5 && ischar(nflag) |wQ3+WN|  
        isnorm = strcmpi(nflag,'norm'); ~]?EV?T  
        if ~isnorm u8|CeA  
            error('zernfun:normalization','Unrecognized normalization flag.') !Y7$cU &  
        end Cc`-34/%  
    else r2i]9>w  
        isnorm = false; \+Y=}P>  
    end *&_cp]3-WF  
    cq gCcO ,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4oryTckS  
    % Compute the Zernike Polynomials ePv`R'#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% T\6,@7  
    1{d;Ngx  
    % Determine the required powers of r: N=T}  
    % ----------------------------------- q*Hg-J}  
    m_abs = abs(m); 5[)#3vY  
    rpowers = []; fz|_c*&64  
    for j = 1:length(n) $dK430_B  
        rpowers = [rpowers m_abs(j):2:n(j)]; T3SFG]H  
    end GVn'p Wg  
    rpowers = unique(rpowers); #8M^;4N >[  
    %{:pBt:Z  
    % Pre-compute the values of r raised to the required powers, 7 H:y=?X6  
    % and compile them in a matrix: 0YfmAF$/B  
    % ----------------------------- 0o6o<ggi  
    if rpowers(1)==0 +\&6Zbn  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); rLmc(-q  
        rpowern = cat(2,rpowern{:}); ~Jsu"kr  
        rpowern = [ones(length_r,1) rpowern]; 'o0o.&/=  
    else 6|3 X*Orn  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); '|5o(6u'  
        rpowern = cat(2,rpowern{:}); `ZM$\Q=:  
    end 6w m-uu  
    $""k Z  
    % Compute the values of the polynomials: ;XjXv'  
    % -------------------------------------- #;@I.  
    y = zeros(length_r,length(n)); bXXX-Xc  
    for j = 1:length(n) 'X6Y!VDd  
        s = 0:(n(j)-m_abs(j))/2; S'ms>ZENC  
        pows = n(j):-2:m_abs(j); \{~CO{II  
        for k = length(s):-1:1 d=uGB"  
            p = (1-2*mod(s(k),2))* ... H`URJ8k$Q  
                       prod(2:(n(j)-s(k)))/              ... VGxab;#,:3  
                       prod(2:s(k))/                     ... :~srl)|)  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... whP5 u/857  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); W _Hoa*~  
            idx = (pows(k)==rpowers); 1x\k:2U  
            y(:,j) = y(:,j) + p*rpowern(:,idx); DS7L}]  
        end 1MnC5[Q  
         =Bm|9A1  
        if isnorm \*b  .f  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 9b,0_IMHH  
        end 59W~bWHCP  
    end ~$j;@ 4  
    % END: Compute the Zernike Polynomials l`:u5\ rM  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $G }9iV7  
    Y{#*;p*I  
    % Compute the Zernike functions: R)*l)bpZ#  
    % ------------------------------ M3F1O6=4j  
    idx_pos = m>0; dw5"}-D  
    idx_neg = m<0; zF{~Md1  
    /]-yZ0hX0O  
    z = y; ~!g2+^G7+P  
    if any(idx_pos) f/IQ2yT-:D  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); +Ig%h[1a  
    end z#P`m,~t0  
    if any(idx_neg) .7LQ l ?  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); c|aX4=Z  
    end WQiRbbX  
    L+ XAbL)  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) cS:O|R#%t  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. >G%oWRk  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated v'Ehr**]+  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 2QAP$f0Ln  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, CnZEBAU  
    %   and THETA is a vector of angles.  R and THETA must have the same FkuD Gg~a  
    %   length.  The output Z is a matrix with one column for every P-value, CxJ3u  
    %   and one row for every (R,THETA) pair. lA1  
    % d6JW"  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike / j%~#@  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ,c?( |tF  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) aBG^Xhx  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 w%X@os}E  
    %   for all p. tK/,U =+  
    % Ml{ ]{n  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 oaPWeM+  
    %   Zernike functions (order N<=7).  In some disciplines it is 4KR`  
    %   traditional to label the first 36 functions using a single mode ISK 8t  
    %   number P instead of separate numbers for the order N and azimuthal l:JVt`A4?  
    %   frequency M. v7KBYN  
    % + WMXd.iN,  
    %   Example: \f(zMP  
    % -LUZ7,!/>o  
    %       % Display the first 16 Zernike functions 8Yf*vp>T/x  
    %       x = -1:0.01:1; oA7DhU5n  
    %       [X,Y] = meshgrid(x,x); ZS(%!+M  
    %       [theta,r] = cart2pol(X,Y); /d:hW4}<}.  
    %       idx = r<=1; c(2?./\|  
    %       p = 0:15; <rK[&JlJ  
    %       z = nan(size(X)); }+pwSjsno  
    %       y = zernfun2(p,r(idx),theta(idx)); %*npLDi  
    %       figure('Units','normalized') K?! W9lUq  
    %       for k = 1:length(p) GK1nGdT]  
    %           z(idx) = y(:,k); 8IxIW0  
    %           subplot(4,4,k) 0!ZaR 6  
    %           pcolor(x,x,z), shading interp %Y=r5'6l  
    %           set(gca,'XTick',[],'YTick',[]) w{xa@Q]t-  
    %           axis square 8M,@Mb n  
    %           title(['Z_{' num2str(p(k)) '}']) 0,0Z!-Y  
    %       end T|D^kL%m!  
    % JA9NTu(  
    %   See also ZERNPOL, ZERNFUN. PlS)Zv3  
    00dY?d{[D  
    %   Paul Fricker 11/13/2006 3F!)7  
    h%W,O,K/  
    D]}~`SO  
    % Check and prepare the inputs: \< T7EV.  
    % ----------------------------- )8#-IXxp  
    if min(size(p))~=1 _a& Z$2O  
        error('zernfun2:Pvector','Input P must be vector.') ]a&riPh"  
    end c*E7nc)u  
    >4kQ9lXL  
    if any(p)>35 Wex2Fd?DO  
        error('zernfun2:P36', ... U\qbr.<  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... $|J+  
               '(P = 0 to 35).']) oi,KA  
    end u pUJF`3  
    0uW)&>W  
    % Get the order and frequency corresonding to the function number: '/ Hoq  
    % ---------------------------------------------------------------- Fv %@k{  
    p = p(:); =>3,]hnep  
    n = ceil((-3+sqrt(9+8*p))/2); I(7iD. ^:  
    m = 2*p - n.*(n+2); >]gB@tn[  
    er-0i L@  
    % Pass the inputs to the function ZERNFUN: @ %L  
    % ---------------------------------------- *aWh]x9TlU  
    switch nargin HpIi-Es7C  
        case 3 9e>2kd  
            z = zernfun(n,m,r,theta); /<$|tp\Rc  
        case 4 w42{)S"  
            z = zernfun(n,m,r,theta,nflag);  + @f  
        otherwise U; <{P  
            error('zernfun2:nargin','Incorrect number of inputs.') 7_qsVhh]$E  
    end oPaoQbR(A  
    XP}5i!}}7=  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) W1 xPK*  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. bDh4p]lm  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of )z z{~Cf  
    %   order N and frequency M, evaluated at R.  N is a vector of v*JKLA  
    %   positive integers (including 0), and M is a vector with the Rc4=zimr+  
    %   same number of elements as N.  Each element k of M must be a d#bg(y\G|  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) A=|a!N/  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is G"u4]!$/  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix mSu$1m8  
    %   with one column for every (N,M) pair, and one row for every *)^ ZUk  
    %   element in R. g +gcH  
    % 3PRU  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- <im}R9eJ1  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 7zE1>.  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to !J<0.nO/:  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 = B;qy7?  
    %   for all [n,m]. :KG=3un]  
    % $J)`Ru6.  
    %   The radial Zernike polynomials are the radial portion of the Xg_l4!T_l  
    %   Zernike functions, which are an orthogonal basis on the unit IV':sNV  
    %   circle.  The series representation of the radial Zernike &{a#8sbf#c  
    %   polynomials is G@jx&#v  
    % 06.8m;{N  
    %          (n-m)/2 OT|0_d?bD  
    %            __ pMViq0  
    %    m      \       s                                          n-2s [U^Cz{G  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r _G<Wq`0w)  
    %    n      s=0 l"X,[  
    % knV*,   
    %   The following table shows the first 12 polynomials. Ic!x y  
    % \?8q&o1=]  
    %       n    m    Zernike polynomial    Normalization a x;<idC}  
    %       --------------------------------------------- 8JR&s  
    %       0    0    1                        sqrt(2) 5{1=BZftZ  
    %       1    1    r                           2 DyZ6&*s$  
    %       2    0    2*r^2 - 1                sqrt(6) \21Gg%W5AE  
    %       2    2    r^2                      sqrt(6) .Sa=VC?EZ  
    %       3    1    3*r^3 - 2*r              sqrt(8) 7AGUi+!ICl  
    %       3    3    r^3                      sqrt(8) =c&.I}^1L  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) &%/T4$'+Y+  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) e F}KOOfC  
    %       4    4    r^4                      sqrt(10) DXO'MZon3  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) eUR+j?5I  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) :2vuc!Pu  
    %       5    5    r^5                      sqrt(12) wCv9VvF`  
    %       --------------------------------------------- /#eS3`48  
    % E|_8#xvb  
    %   Example: gy|o#&e]%  
    % /6y{ ?0S  
    %       % Display three example Zernike radial polynomials !a!4^zqp  
    %       r = 0:0.01:1; 3N2d@R  
    %       n = [3 2 5]; xy&*s\=:  
    %       m = [1 2 1]; 7aPA+gA/  
    %       z = zernpol(n,m,r); ZzR0k  
    %       figure L|-|DOgw  
    %       plot(r,z) `}$bJCSF.n  
    %       grid on aAn p7\7  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') Z:x`][vg  
    % mC0Dj O  
    %   See also ZERNFUN, ZERNFUN2. 368 g> /#'  
    hV) `e"r\s  
    % A note on the algorithm. tBl (E  
    % ------------------------ z?|bs?HKS  
    % The radial Zernike polynomials are computed using the series 4,Uqcw?!F'  
    % representation shown in the Help section above. For many special #a tL2(wJ  
    % functions, direct evaluation using the series representation can -*J!Ws(9  
    % produce poor numerical results (floating point errors), because /"~UGn]R  
    % the summation often involves computing small differences between z" ?WT$  
    % large successive terms in the series. (In such cases, the functions j:2 F97  
    % are often evaluated using alternative methods such as recurrence m&(qr5>b  
    % relations: see the Legendre functions, for example). For the Zernike 8.IenU9  
    % polynomials, however, this problem does not arise, because the a gL@A  
    % polynomials are evaluated over the finite domain r = (0,1), and mC(YO y  
    % because the coefficients for a given polynomial are generally all EaL>~: j  
    % of similar magnitude. {/aHZ<I&^h  
    % N Mx:Jh-YN  
    % ZERNPOL has been written using a vectorized implementation: multiple {9XNh[NbP  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] I};*O6D`  
    % values can be passed as inputs) for a vector of points R.  To achieve +/M%%:>mY  
    % this vectorization most efficiently, the algorithm in ZERNPOL #@1(  
    % involves pre-determining all the powers p of R that are required to em- <V5fb  
    % compute the outputs, and then compiling the {R^p} into a single :LdPqFXj  
    % matrix.  This avoids any redundant computation of the R^p, and S>j.i  
    % minimizes the sizes of certain intermediate variables. 0XqxW\8_l  
    % N:+d=G`x  
    %   Paul Fricker 11/13/2006 Wfc~"GQq4  
    ?FR-a Xx  
    H(M{hfa|  
    % Check and prepare the inputs: j}(m$j'  
    % ----------------------------- AA=eWg  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) $ye>;Ek  
        error('zernpol:NMvectors','N and M must be vectors.') ub`zS-vb  
    end WNKP';(a@G  
    P6'Se'f8  
    if length(n)~=length(m) ;mwnAO  
        error('zernpol:NMlength','N and M must be the same length.') ga S}>?qk  
    end _.BT%4  
    KU]o=\ak%  
    n = n(:); Unb3 Gv#O  
    m = m(:); "Y- WY,H  
    length_n = length(n); *8)va  
    M#m;jJqON  
    if any(mod(n-m,2)) )1 HWD]>4  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') L*vKIP<EMM  
    end Qj(ppep\U"  
    `c-omNu  
    if any(m<0) n"Bc2}{  
        error('zernpol:Mpositive','All M must be positive.') ]bpgsW:Xu  
    end Q[|*P ] w  
    HTvUt*U1  
    if any(m>n) +PKsiUJ|  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') m&'!^{av  
    end 1Ax;|.KQH  
    $7i[7S4  
    if any( r>1 | r<0 ) /m 7~-~$V  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') be5N{lPT@;  
    end Vry_X2  
    ;_E|I=%'E  
    if ~any(size(r)==1) e6jA4X+a  
        error('zernpol:Rvector','R must be a vector.') 6CW5ay_,  
    end ~vM99hW  
    d~{$,"!-f  
    r = r(:); urCTP.F  
    length_r = length(r); 4i+%~X@p  
    d2-oy5cEB  
    if nargin==4 n5^57[(  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); # h4FLF_w  
        if ~isnorm BpYxH#4  
            error('zernpol:normalization','Unrecognized normalization flag.') TOS'|xQ  
        end ;YW@ 3F-h  
    else 6DgdS5GhT_  
        isnorm = false; (+/d*4  
    end V{A`?Jl6{  
    SO[ u4b_"h  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RgQs`aI  
    % Compute the Zernike Polynomials mdEl CC0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^w<aS w  
    uLPBl~Y  
    % Determine the required powers of r: Fkq^2o ]  
    % ----------------------------------- cF8X  
    rpowers = []; ,u)jZ7  
    for j = 1:length(n) $AE5n>ZD$  
        rpowers = [rpowers m(j):2:n(j)]; 1+XM1(|c`  
    end 0 _MtmmL.  
    rpowers = unique(rpowers); 5$cjCjY  
    '\xE56v)F  
    % Pre-compute the values of r raised to the required powers, RwOOe7mv  
    % and compile them in a matrix: VO=Ibu&X  
    % ----------------------------- 5$N#=i`V  
    if rpowers(1)==0 h,]VWG  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); jC{KI!kPt  
        rpowern = cat(2,rpowern{:}); 7C,giCYU  
        rpowern = [ones(length_r,1) rpowern]; 6yMZ2%  
    else GdVF;  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); >8=lX`9f{  
        rpowern = cat(2,rpowern{:}); g=@d!]Z~[  
    end 4nh=Dq[  
    /sT?p=[.  
    % Compute the values of the polynomials: b.@4yW  
    % --------------------------------------  ms&1P  
    z = zeros(length_r,length_n); 5\#I4\  
    for j = 1:length_n : MjDcI~  
        s = 0:(n(j)-m(j))/2; ci$J?a  
        pows = n(j):-2:m(j); K%x]:|,>M  
        for k = length(s):-1:1 Ro"'f7(v.  
            p = (1-2*mod(s(k),2))* ... m|c [C\)By  
                       prod(2:(n(j)-s(k)))/          ... 6l;2kztGp  
                       prod(2:s(k))/                 ... =W:=}ODD  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... $[,4Ib_|  
                       prod(2:((n(j)+m(j))/2-s(k))); 6L$KMYHE  
            idx = (pows(k)==rpowers); s8>y&b.  
            z(:,j) = z(:,j) + p*rpowern(:,idx); (=Kv1 HaD  
        end -eyF9++`  
         *q k7e[IP  
        if isnorm T]-MrnO  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 9 i/ (  
        end <<A#4!f  
    end !U m9ceK  
    6uFw+Ya#  
    % EOF zernpol
    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)  @\y{q;  
    _,L_H[FN  
    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。 @FN|=?8%  
    (2# Xa,pb  
    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)