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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 4#xDgxg\f  
    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! @[<><uTH  
     
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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 X; \+<LE  
    function z = zernfun(n,m,r,theta,nflag) |}s*E_/[  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. NqazpB*  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N u^ +7hkk  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 58tARLDr  
    %   unit circle.  N is a vector of positive integers (including 0), and Ha0M)0Anv  
    %   M is a vector with the same number of elements as N.  Each element 9iIhte.  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) m<T%Rb4?@  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Db}j?ik/  
    %   and THETA is a vector of angles.  R and THETA must have the same n`B:;2X,  
    %   length.  The output Z is a matrix with one column for every (N,M) 17%,7P9pg  
    %   pair, and one row for every (R,THETA) pair. Pe_W;q.  
    %  lHY+}v0  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ,*TmIPNK  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), [[Ls_ZL!=  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral TVtvuvQ2K  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, J@HtoTDO3  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized hc(#{]].  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. b5dD/-Vj  
    % hP%M?MKC  
    %   The Zernike functions are an orthogonal basis on the unit circle. r4b 6 c  
    %   They are used in disciplines such as astronomy, optics, and O-0x8O^B  
    %   optometry to describe functions on a circular domain. #_ ;lf1x!  
    % .]Y$o^mf  
    %   The following table lists the first 15 Zernike functions. B?gOHG*vd>  
    % x*\Y)9Vgy  
    %       n    m    Zernike function           Normalization k<nZ+! M  
    %       -------------------------------------------------- ~|D Ut   
    %       0    0    1                                 1 A7Cm5>Y_S  
    %       1    1    r * cos(theta)                    2 `iFmrC<  
    %       1   -1    r * sin(theta)                    2 Fh&G;aEq  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) y4 #>X  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 9rA0lqr]5  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) FJ GlP&v<  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 1APe=tJ  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) $D~0~gn~  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) #'nr Er <  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) DZ 3wCLQtK  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 13$%,q)  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) hlvK5Z   
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ^,lIK+#Elz  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Q",t3i4  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) T$)^gHS  
    %       -------------------------------------------------- ,a{P4Bq  
    % RtkEGxw*^  
    %   Example 1: DD+7V@  
    % ?um;s-x)  
    %       % Display the Zernike function Z(n=5,m=1) rQ{7j!Im  
    %       x = -1:0.01:1; .FP$m?  
    %       [X,Y] = meshgrid(x,x); ^&9zw\x;z  
    %       [theta,r] = cart2pol(X,Y); /e5O"@  
    %       idx = r<=1; T8?Ghbn  
    %       z = nan(size(X)); p;`>e>$  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 7~G9'P<  
    %       figure pTth}JM>  
    %       pcolor(x,x,z), shading interp hIYNhZv  
    %       axis square, colorbar v|)4ocFK  
    %       title('Zernike function Z_5^1(r,\theta)') "=HA Y  
    % @(EAq<5{  
    %   Example 2: ,i ^9 |Oeq  
    % =g7x' kN  
    %       % Display the first 10 Zernike functions W]$w@.oW[  
    %       x = -1:0.01:1; k>Is:P  
    %       [X,Y] = meshgrid(x,x); ]\-A;}\e  
    %       [theta,r] = cart2pol(X,Y); W 8<&gh+  
    %       idx = r<=1; t5^{D>S1  
    %       z = nan(size(X)); T= 80,  
    %       n = [0  1  1  2  2  2  3  3  3  3]; @o].He@L<j  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ol\Utq,  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; W<h)HhyG  
    %       y = zernfun(n,m,r(idx),theta(idx)); hk;5w{t}}  
    %       figure('Units','normalized') M><yGaaX/  
    %       for k = 1:10 Ye%~I`@?  
    %           z(idx) = y(:,k); ^ox=HNV  
    %           subplot(4,7,Nplot(k)) rET\n(AJ  
    %           pcolor(x,x,z), shading interp aL\PGdgO  
    %           set(gca,'XTick',[],'YTick',[]) &N$<e(K  
    %           axis square lf`{zc r:  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) do hA0  
    %       end u4cnE"  
    % >%_\;svZG  
    %   See also ZERNPOL, ZERNFUN2.  \{_q.;}  
    7uqzm  
    %   Paul Fricker 11/13/2006 O0x,lq  
    Qab>|eSm  
    Y sC>i`n9  
    % Check and prepare the inputs: TIqtF&@o4  
    % ----------------------------- df8k7D;~e  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) .fqN|[>  
        error('zernfun:NMvectors','N and M must be vectors.') OU\~::  
    end {f_={k  
    G{~J|{t\yz  
    if length(n)~=length(m) |w~nVRb  
        error('zernfun:NMlength','N and M must be the same length.') /obfw^  
    end oi7@s0@  
    |u% )gk  
    n = n(:); *gb*LhgO  
    m = m(:); b<[Or^X ]  
    if any(mod(n-m,2)) e-/&$Qq  
        error('zernfun:NMmultiplesof2', ... LtF,kAIt7v  
              'All N and M must differ by multiples of 2 (including 0).') 2 0h} [Q(  
    end 4/~E4"8  
    AEI>\Y  
    if any(m>n) H064BM  
        error('zernfun:MlessthanN', ... _IHV7*u{;  
              'Each M must be less than or equal to its corresponding N.') aH(J,XY  
    end h]&GLb&<?  
    {GT*ZU*  
    if any( r>1 | r<0 ) bn&TF3b  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') #<"~~2?  
    end %bn jgy  
    PCee<W_%YE  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) dh\'<|\K  
        error('zernfun:RTHvector','R and THETA must be vectors.') edq4D53  
    end CT <7mi!  
    wc@X.Q[  
    r = r(:); V*;(kEqj  
    theta = theta(:); St9?RD{4;  
    length_r = length(r); #powub  
    if length_r~=length(theta) 9Q^r O26+  
        error('zernfun:RTHlength', ... B2vh-%63  
              'The number of R- and THETA-values must be equal.') |Pax=oJ\M  
    end \A#41  
    WM$ MPs  
    % Check normalization: 2DDtu[}  
    % -------------------- T@B/xAq5!  
    if nargin==5 && ischar(nflag) OX0%C.K)hZ  
        isnorm = strcmpi(nflag,'norm'); vzAaxk%  
        if ~isnorm oG?Xk%7&\  
            error('zernfun:normalization','Unrecognized normalization flag.') &vMb_;~B  
        end Y;M|D'y+  
    else !;v|'I  
        isnorm = false; hp X9[3  
    end ^ig' bw+WS  
    `UyG_;  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x.6:<y  
    % Compute the Zernike Polynomials M#6W(|V/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wH&!W~M  
    2 c{34:  
    % Determine the required powers of r: %3-y[f  
    % ----------------------------------- g}{aZ$sta  
    m_abs = abs(m); (NU NHxi5B  
    rpowers = []; R4cM%l_#W  
    for j = 1:length(n) ]y '>=a|T  
        rpowers = [rpowers m_abs(j):2:n(j)]; ql{ OETn#  
    end %)W2H^  
    rpowers = unique(rpowers); OX!tsARC@  
    D2 eckLT  
    % Pre-compute the values of r raised to the required powers, D_*WYV  
    % and compile them in a matrix: _S1>j7RQo  
    % ----------------------------- 5coyr`7mP  
    if rpowers(1)==0 Y eo]]i{  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); dn+KH+v  
        rpowern = cat(2,rpowern{:}); \'D0'\:vz  
        rpowern = [ones(length_r,1) rpowern]; 5L%'@`mX  
    else t\,PB{P:J  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); =s2*H8]  
        rpowern = cat(2,rpowern{:}); ,!y$qVg'\f  
    end Y"aJur=`  
    S`0(*A[W*  
    % Compute the values of the polynomials: WPMSm<[  
    % -------------------------------------- oW*16>IN9l  
    y = zeros(length_r,length(n)); $|@@Qk/T  
    for j = 1:length(n) +gtbcF@rx  
        s = 0:(n(j)-m_abs(j))/2; Id .nu/  
        pows = n(j):-2:m_abs(j); zKJ#`OhT  
        for k = length(s):-1:1 ]Ie 0S~  
            p = (1-2*mod(s(k),2))* ... vMH  
                       prod(2:(n(j)-s(k)))/              ... "7F?@D$e  
                       prod(2:s(k))/                     ... 7' V@+5  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 3$>1FoSk  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); U$ElV]N  
            idx = (pows(k)==rpowers); ;))+>%SGCt  
            y(:,j) = y(:,j) + p*rpowern(:,idx); h2]P]@nW;W  
        end u?(d gJ  
         Vaw+.sG`AP  
        if isnorm 9vc2VB$  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); k9 I%PH  
        end G@X% +$I  
    end K;H&n1  
    % END: Compute the Zernike Polynomials +.FEq*V  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L48_96  
    D8?Vn"  
    % Compute the Zernike functions: !``,gExH  
    % ------------------------------  {Gk1vcq  
    idx_pos = m>0; {]@= ijjf  
    idx_neg = m<0; '4Bm;&6M  
    KBc1{adDx@  
    z = y; >jLY"  
    if any(idx_pos) /%1ON9o>  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Vv=. -&'  
    end sBg.u  
    if any(idx_neg) xdt- ;w|  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); :{l_FY436  
    end z,p~z*4  
    G<J?"oQbRT  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) N S[l/0F&  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. }|NCboM^_  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated uHzU-FZ|B  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive ;[OH(!  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ?%[@Qb=2  
    %   and THETA is a vector of angles.  R and THETA must have the same ]GkfEh7/J  
    %   length.  The output Z is a matrix with one column for every P-value, }WXi$(@v  
    %   and one row for every (R,THETA) pair. Eo]xNn/g  
    % t-bB>q#3>  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike -x`@6  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) +',S]Edx  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Dp-z[]})1  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 Fywv  
    %   for all p. /@TF5]Ri  
    % BUXpC xQ  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 lzVq1@B  
    %   Zernike functions (order N<=7).  In some disciplines it is yl+gL?IES  
    %   traditional to label the first 36 functions using a single mode j'"J%e]  
    %   number P instead of separate numbers for the order N and azimuthal >!1-lfa8  
    %   frequency M. E{P|)`,V  
    % 6mxfLlZ  
    %   Example: \\;jw[P0  
    % 1K50Z.o&@  
    %       % Display the first 16 Zernike functions ` 7V]y -  
    %       x = -1:0.01:1; .Vvx,>>D  
    %       [X,Y] = meshgrid(x,x); Ean5b>\  
    %       [theta,r] = cart2pol(X,Y); ],Do6 @M-  
    %       idx = r<=1; 4O!ikmY:t  
    %       p = 0:15; z5*'{t)  
    %       z = nan(size(X)); M&9+6e'-F  
    %       y = zernfun2(p,r(idx),theta(idx)); 6q.Uhe_B  
    %       figure('Units','normalized') _ *Pf  
    %       for k = 1:length(p) i2SR{e8:GF  
    %           z(idx) = y(:,k); dJNe+ MB`  
    %           subplot(4,4,k) &Hs!:43E-<  
    %           pcolor(x,x,z), shading interp Yufc{M00  
    %           set(gca,'XTick',[],'YTick',[]) 59;KQ  
    %           axis square V/9!K%y  
    %           title(['Z_{' num2str(p(k)) '}']) d)Y}>@:W  
    %       end \bvfEP  
    % |[b{)s?x  
    %   See also ZERNPOL, ZERNFUN. |z^^.d~a0  
    p947w,1![  
    %   Paul Fricker 11/13/2006 )|# sfHv7  
    LG#t<5y~  
    #5o(h+w)  
    % Check and prepare the inputs: bq0zxg%  
    % ----------------------------- f x+/C8GK  
    if min(size(p))~=1 -r]W  
        error('zernfun2:Pvector','Input P must be vector.') 3eQ&F~S  
    end q9s=~d7  
    G2: agqL/  
    if any(p)>35 NyNXP_8  
        error('zernfun2:P36', ... p9{mS7R9T  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... <x>M o   
               '(P = 0 to 35).']) =fFP5e ['  
    end d5:c^`  
    FXkM#}RgNm  
    % Get the order and frequency corresonding to the function number: BR;D@R``}  
    % ---------------------------------------------------------------- xMG~N`r  
    p = p(:); FJ)$f?=Qd  
    n = ceil((-3+sqrt(9+8*p))/2); $*=<Yw4  
    m = 2*p - n.*(n+2); <frutU16\  
    k~1?VQ+?M  
    % Pass the inputs to the function ZERNFUN: Qh\60f>0  
    % ---------------------------------------- 6i3$CW  
    switch nargin \z(gqkc 6  
        case 3 'm kLCS  
            z = zernfun(n,m,r,theta); 1#+S+g@#  
        case 4 v):Or'$~M  
            z = zernfun(n,m,r,theta,nflag); c'yxWZEv  
        otherwise {F.[&/A  
            error('zernfun2:nargin','Incorrect number of inputs.') E+;7>ja  
    end ^^D0^k!R  
    KJ4.4Zq{c  
    % 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) 7zG_(83)K  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. qY!Zt_Be6  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of {R6ZKB  
    %   order N and frequency M, evaluated at R.  N is a vector of +52{-a,>  
    %   positive integers (including 0), and M is a vector with the U # qK.  
    %   same number of elements as N.  Each element k of M must be a * 0=j?~&  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) -(H0>Ap  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 1iF1GkLEq  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ~Z' ?LV<t  
    %   with one column for every (N,M) pair, and one row for every 3h`f  6  
    %   element in R. P~X2^bw  
    % $6poFo)U+  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- nAdf=D'P  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is l,5+@i`5i  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to aQ@oH#  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 _X x/(.O  
    %   for all [n,m]. \,0oX!<YY  
    % L.JT[zOfb  
    %   The radial Zernike polynomials are the radial portion of the b4N[)%@  
    %   Zernike functions, which are an orthogonal basis on the unit IW] rb/H  
    %   circle.  The series representation of the radial Zernike CRy|kkT  
    %   polynomials is R0*|Lo$6  
    % 6.yu-xm  
    %          (n-m)/2 ]:J$w]\  
    %            __ "VMz]ybi^  
    %    m      \       s                                          n-2s @f3E`8  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 63IM]J  
    %    n      s=0 Pa: |_IXA  
    % @`9]F7h5W  
    %   The following table shows the first 12 polynomials. '!B&:X)  
    % am6L8N  
    %       n    m    Zernike polynomial    Normalization "E4a=YH_  
    %       --------------------------------------------- {]4LULq  
    %       0    0    1                        sqrt(2) ]3Sp W{=^(  
    %       1    1    r                           2 inp7K41  
    %       2    0    2*r^2 - 1                sqrt(6) ,Ma^&ypH  
    %       2    2    r^2                      sqrt(6) +9sQZB# (  
    %       3    1    3*r^3 - 2*r              sqrt(8) dioGAai'  
    %       3    3    r^3                      sqrt(8) e~"U @8xk~  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 1 [Bk%G@D&  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) xr^LFn)  
    %       4    4    r^4                      sqrt(10)  _;\_l  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) j8:\%|  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) F#5~M<`.o  
    %       5    5    r^5                      sqrt(12) IO<6  
    %       --------------------------------------------- nKj7.,>;:<  
    % tw)mepwB  
    %   Example: }3WxZv]I}  
    % Ar#(psU  
    %       % Display three example Zernike radial polynomials $U-0)4yf  
    %       r = 0:0.01:1; "qy,*{~  
    %       n = [3 2 5]; S~G ]~gt  
    %       m = [1 2 1]; t\O16O7S  
    %       z = zernpol(n,m,r);  &q*Aj17  
    %       figure QIFgQ0{  
    %       plot(r,z) rEz^  
    %       grid on k$:|-_(w  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') p!AAFmc  
    % &_8 947  
    %   See also ZERNFUN, ZERNFUN2. h 'nY3GrU  
    KD.]i' d<  
    % A note on the algorithm. |CbikE}kL  
    % ------------------------ 0jWVp- y  
    % The radial Zernike polynomials are computed using the series < I``&>  
    % representation shown in the Help section above. For many special lr&a;aZp  
    % functions, direct evaluation using the series representation can lPAQ3t!,  
    % produce poor numerical results (floating point errors), because w_VP J  
    % the summation often involves computing small differences between _7y[B&g[r  
    % large successive terms in the series. (In such cases, the functions ;8 lfOMf  
    % are often evaluated using alternative methods such as recurrence 9} .z;prz  
    % relations: see the Legendre functions, for example). For the Zernike V>3X\)qu  
    % polynomials, however, this problem does not arise, because the hOK8(U0  
    % polynomials are evaluated over the finite domain r = (0,1), and 4s oJ.j8  
    % because the coefficients for a given polynomial are generally all 3Tm+g2w2V8  
    % of similar magnitude. z`b,h\  
    % uCB=u[]y4  
    % ZERNPOL has been written using a vectorized implementation: multiple 'dc#F3  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] j_j]"ew)  
    % values can be passed as inputs) for a vector of points R.  To achieve >y+B  
    % this vectorization most efficiently, the algorithm in ZERNPOL tfWS)y7  
    % involves pre-determining all the powers p of R that are required to dlnX_+((KC  
    % compute the outputs, and then compiling the {R^p} into a single b| (: [nB  
    % matrix.  This avoids any redundant computation of the R^p, and 8H`[*|{'  
    % minimizes the sizes of certain intermediate variables. llDkJ)\  
    % `XDl_E+>l  
    %   Paul Fricker 11/13/2006 uhq8   
    w&.a QGR#  
    7a}k  
    % Check and prepare the inputs: F((4U"   
    % ----------------------------- #B w0,\  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) y8xE 6i  
        error('zernpol:NMvectors','N and M must be vectors.') cm+P]8o%{  
    end \z)%$#I  
    m(P]k'ZH?  
    if length(n)~=length(m) 62NsJ<#>  
        error('zernpol:NMlength','N and M must be the same length.') N6TH}~62}  
    end JlJ a #  
    PZzMHK?hP  
    n = n(:); f%8C!W]Dm  
    m = m(:); K@%].:  
    length_n = length(n); V28M lP  
    bW:!5"_{H  
    if any(mod(n-m,2)) y<.5xq5_3  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 1B\WA8  
    end -tU'yKhn  
    lk=<A"^S  
    if any(m<0) o#N+Y?O  
        error('zernpol:Mpositive','All M must be positive.') dQG=G%W  
    end qxJ\ye+'*  
    c[s4EUG  
    if any(m>n) u]G\H!Wk Q  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') {\\T gs  
    end - ! S_ryL  
    0IWf!Sk ]  
    if any( r>1 | r<0 ) _WbxH  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') c4zR*  
    end Y|/ 8up  
    UL9n-M =  
    if ~any(size(r)==1) :fJN->wY^s  
        error('zernpol:Rvector','R must be a vector.') HOJV,9v N  
    end . 'yCw#f  
    P+HXn8@  
    r = r(:); EBmt9S  
    length_r = length(r); d0 /#nz  
    aQI(Y^&%3  
    if nargin==4 -%4,@ x`  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); t3^&; &[  
        if ~isnorm <\S:'g"(  
            error('zernpol:normalization','Unrecognized normalization flag.') HLi%%"'  
        end q75s#[<ap  
    else FE;x8(;W8  
        isnorm = false; h FBe,'3M  
    end xe$_aBU  
    ,v}k{( 16{  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]J]h#ZHx  
    % Compute the Zernike Polynomials HkVB80hv  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SZCze"`[  
    rQ snhv  
    % Determine the required powers of r: @=f\<"$vt  
    % ----------------------------------- j*m%*_kO  
    rpowers = []; ;xn0;V'=  
    for j = 1:length(n) k~z Iy;AZ  
        rpowers = [rpowers m(j):2:n(j)]; M rb)  
    end ku M$UYTTX  
    rpowers = unique(rpowers); 1m0c|ckb  
    3HK\BS  
    % Pre-compute the values of r raised to the required powers, ] @fk] ]R  
    % and compile them in a matrix: )Xyn q(  
    % ----------------------------- I1&aM}y{G  
    if rpowers(1)==0 IO:G1;[/2L  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); f(7GX3?  
        rpowern = cat(2,rpowern{:}); *}W_+qo"  
        rpowern = [ones(length_r,1) rpowern]; bi;1s'Y<D  
    else "tpSg  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); L9#g)tf 8T  
        rpowern = cat(2,rpowern{:}); o#1 $q`Z  
    end B4 }bVjs  
    "@8li^  
    % Compute the values of the polynomials: 18:%~>.!  
    % -------------------------------------- lU8Hd|@-  
    z = zeros(length_r,length_n); }\k"n{!"  
    for j = 1:length_n .(2ik5A%9  
        s = 0:(n(j)-m(j))/2; ;i+#fQO7Q  
        pows = n(j):-2:m(j); FJ?IUy 6  
        for k = length(s):-1:1 \Y}8S/]  
            p = (1-2*mod(s(k),2))* ... 8, >P  
                       prod(2:(n(j)-s(k)))/          ... u\nh[1)a)  
                       prod(2:s(k))/                 ... ^_mj  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... q'MZ R'<@  
                       prod(2:((n(j)+m(j))/2-s(k))); "g8M0[7e3  
            idx = (pows(k)==rpowers); b>JDH1)  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ^K@C"j?M/  
        end [}]Q?*_  
         $L]lHji  
        if isnorm P7/X|M z  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ,s;Uf F  
        end jrh43 \$*  
    end `*KHS A  
    v|2T%y_ u  
    % 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)  0rs"o-s<  
    Sp]0c[37R  
    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。 rlSeu5X6  
    L2i_X@/  
    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)