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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 |J $A%27  
    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! U/>I! 7oe  
     
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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 Z~:)hwF  
    function z = zernfun(n,m,r,theta,nflag) B:>:$LIL  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 2)EqqX[D  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N VgPlIIHh5  
    %   and angular frequency M, evaluated at positions (R,THETA) on the /&6{}n  
    %   unit circle.  N is a vector of positive integers (including 0), and jV% VN  
    %   M is a vector with the same number of elements as N.  Each element :k9T`Aa]  
    %   k of M must be a positive integer, with possible values M(k) = -N(k)  l!1_~!{y  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, `.@udfog^0  
    %   and THETA is a vector of angles.  R and THETA must have the same yp~z-aRa  
    %   length.  The output Z is a matrix with one column for every (N,M) ^"Bhp:o2  
    %   pair, and one row for every (R,THETA) pair. S @[]znH  
    % gj|5"'g%  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike $YJ 1P  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ?0)K[Kd'Y  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral gY+d[3N  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1,  $H*8H`  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 6+=_p$crMx  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. >4g!ic~O  
    % %XR(K@V  
    %   The Zernike functions are an orthogonal basis on the unit circle. h<L_ =)lH  
    %   They are used in disciplines such as astronomy, optics, and S6bW r0XR  
    %   optometry to describe functions on a circular domain. hUpour |b  
    % "]3o93 3 D  
    %   The following table lists the first 15 Zernike functions. qt:B]#j@  
    % we}xGb.u  
    %       n    m    Zernike function           Normalization .QY>@b\  
    %       -------------------------------------------------- H~*N:$C  
    %       0    0    1                                 1 M|nLD+d~8  
    %       1    1    r * cos(theta)                    2 X$xf@|<a  
    %       1   -1    r * sin(theta)                    2 o^@#pU <  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) pZ Uy (  
    %       2    0    (2*r^2 - 1)                    sqrt(3) #ChTel  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) IFW(nB(  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Zl[EpXlZ  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) PU%Zay  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) I484c R2.  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) <\nM5-wR  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 42e[OG-  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^/%o I;O{  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ]prw=rD  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) rHk(@T.]  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) e'~Qe_  
    %       -------------------------------------------------- H/t0#  
    % H-t$A, [  
    %   Example 1: YdV.+v(30  
    % I!b"Rv=Nf-  
    %       % Display the Zernike function Z(n=5,m=1) TFldYKd/l  
    %       x = -1:0.01:1; {^ BZ#)m|  
    %       [X,Y] = meshgrid(x,x); R;,5LS&*a  
    %       [theta,r] = cart2pol(X,Y); gHgqElr(  
    %       idx = r<=1; N9ipwr'P  
    %       z = nan(size(X)); b+Sj\3fX  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); &pY$\  
    %       figure <IU   
    %       pcolor(x,x,z), shading interp (]k Q9}8  
    %       axis square, colorbar @Y,t]  
    %       title('Zernike function Z_5^1(r,\theta)') [cFD\"gJAr  
    % ((?"2 }1r  
    %   Example 2: A|Ft:_Y  
    % 0rX%z$D+@  
    %       % Display the first 10 Zernike functions ;=0-B&+v  
    %       x = -1:0.01:1; l_2Xao$  
    %       [X,Y] = meshgrid(x,x); m, +E5^  
    %       [theta,r] = cart2pol(X,Y); t .&JPTK-H  
    %       idx = r<=1; Cm5L99Y  
    %       z = nan(size(X)); Ww~C[8q  
    %       n = [0  1  1  2  2  2  3  3  3  3]; W rT_7  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; @@a#DjE%/  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; "4KyJ;RA*  
    %       y = zernfun(n,m,r(idx),theta(idx)); E Id>%0s5  
    %       figure('Units','normalized') 1A93ol=  
    %       for k = 1:10 p  Dg!Cs  
    %           z(idx) = y(:,k); X'.l h#&  
    %           subplot(4,7,Nplot(k)) DZ`,QWuA  
    %           pcolor(x,x,z), shading interp  Z a,o  
    %           set(gca,'XTick',[],'YTick',[]) Ur[ai6LNG  
    %           axis square  vWW Q/^  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) I+Y Z+  
    %       end ; p+C0!B2  
    % \7UeV:3Ojn  
    %   See also ZERNPOL, ZERNFUN2. @Nm{H  
    j0F& WKk  
    %   Paul Fricker 11/13/2006 J;V#a=I  
    K7$Q .  
    @6[aLF]F  
    % Check and prepare the inputs: 7u1o>a %9  
    % ----------------------------- 'e>'J ZR  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 8u*Q^-fpo0  
        error('zernfun:NMvectors','N and M must be vectors.') sj+ )   
    end 3]NKAPY  
    :3se/4y}  
    if length(n)~=length(m) <,*w$  
        error('zernfun:NMlength','N and M must be the same length.') ~urk Uz  
    end "<L9-vb  
    uI)z4Z  
    n = n(:); !!6@r|.  
    m = m(:); ?r$& O*;  
    if any(mod(n-m,2)) ?<OE|nb&  
        error('zernfun:NMmultiplesof2', ... Nog{w  
              'All N and M must differ by multiples of 2 (including 0).') AHa]=ka>  
    end AgDXpaq  
    C:!&g~{cKi  
    if any(m>n) Q>z (!'dw  
        error('zernfun:MlessthanN', ... .<K9Zyi  
              'Each M must be less than or equal to its corresponding N.') fTy{`}>  
    end V+u0J"/8  
    qP/McH?  
    if any( r>1 | r<0 ) qe uc^+P;  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ?Rh[S  
    end m9'bDyyK  
    3!KyO)8  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) HT_nxe`E  
        error('zernfun:RTHvector','R and THETA must be vectors.') r-hb]!t  
    end JFRbW Q0  
    4{$ L]toP  
    r = r(:); uE#"wm'J  
    theta = theta(:); kCZ'p  
    length_r = length(r); #E/|W T  
    if length_r~=length(theta) Q9g^'a  
        error('zernfun:RTHlength', ... efyGjfoO  
              'The number of R- and THETA-values must be equal.') 9:!V":8q  
    end w!UIz[ajI  
    *Xu?(Jd  
    % Check normalization: _bCIVf`  
    % -------------------- V4*/t#L/  
    if nargin==5 && ischar(nflag) o~x49%X<c  
        isnorm = strcmpi(nflag,'norm'); :9|CpC`.  
        if ~isnorm `:gXQmt  
            error('zernfun:normalization','Unrecognized normalization flag.') LD;! s  
        end X-yS9E  
    else @Bsvk9}  
        isnorm = false; GS GaYq  
    end 5N#Sic M  
    ;3 =RM\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ad<ZdO*h  
    % Compute the Zernike Polynomials +W|VCz  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "6WE6zq   
    F@xKL;'N74  
    % Determine the required powers of r: 1?y QjW,  
    % ----------------------------------- #!j wn^yq  
    m_abs = abs(m); iT~ gt/K  
    rpowers = []; %G, d&%f  
    for j = 1:length(n) uF@DJX}>  
        rpowers = [rpowers m_abs(j):2:n(j)]; d`xDv$QZ  
    end Zu ![v0  
    rpowers = unique(rpowers); |zp}u(N  
    70A* !v  
    % Pre-compute the values of r raised to the required powers, Cyp%E5b7  
    % and compile them in a matrix: gGbJk&E  
    % ----------------------------- [58qC:  
    if rpowers(1)==0 P7 qzZ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Mgux (5`;  
        rpowern = cat(2,rpowern{:}); Z"9D1Uk  
        rpowern = [ones(length_r,1) rpowern]; qc/)l~]?g{  
    else <xD6}h/  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); $btk48a7  
        rpowern = cat(2,rpowern{:}); rVb61$  
    end xtd1>|  
    Wl{}>F`W[  
    % Compute the values of the polynomials: r4pR[G._  
    % -------------------------------------- CuYSvW  
    y = zeros(length_r,length(n)); ?,UO$#Xm  
    for j = 1:length(n) NY%=6><t!  
        s = 0:(n(j)-m_abs(j))/2; <)$JA  
        pows = n(j):-2:m_abs(j); Nj}-"R\u  
        for k = length(s):-1:1 pq*4yaTT'  
            p = (1-2*mod(s(k),2))* ... LE+#%>z>  
                       prod(2:(n(j)-s(k)))/              ... }\.Z{h:t ?  
                       prod(2:s(k))/                     ... 'dd[= vzK  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... a_Z[@W  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); NU%W9jQYS  
            idx = (pows(k)==rpowers); +{&++^(}a  
            y(:,j) = y(:,j) + p*rpowern(:,idx); .10$n*  
        end O.'\GM  
         x|A{|oFC  
        if isnorm 6$\'dkufQ  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); j<-YK4.t  
        end &&|c-mD+*  
    end @<=<?T> 1  
    % END: Compute the Zernike Polynomials )uH#+IU  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F)uS2  
    =.6JvX<d1*  
    % Compute the Zernike functions: hdy N   
    % ------------------------------ j%Z%_{6Ds*  
    idx_pos = m>0; !WQS.&  
    idx_neg = m<0; 8i?:aN[.1b  
    +IbQVU~/  
    z = y; mI3 \n  
    if any(idx_pos) 7\Wq:<JL  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); @x/D8HK2  
    end kTS #>uS  
    if any(idx_neg) 3W"l}.&ZJ"  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); *ta?7uSiT  
    end P~OD d(  
    f]]UNS$AYQ  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) geSH3I   
    %ZERNFUN2 Single-index Zernike functions on the unit circle. )CUB7D)=  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated hScC< =W  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive s|o+ Im  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 2H2Yxe7?-  
    %   and THETA is a vector of angles.  R and THETA must have the same oTLpq:9J  
    %   length.  The output Z is a matrix with one column for every P-value, Xi81?F?[  
    %   and one row for every (R,THETA) pair. y6N }R  
    % KVZ-T1K  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike fFJu]  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) oS Apa  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) l"1at eM3  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 zJX _EO  
    %   for all p. DNy 6Kw  
    % Eao^/MKx-  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 K%RjWX=H  
    %   Zernike functions (order N<=7).  In some disciplines it is Vs@H>97,G  
    %   traditional to label the first 36 functions using a single mode "=3bL>\<  
    %   number P instead of separate numbers for the order N and azimuthal ud:5_*  
    %   frequency M. Rr6}$]1  
    % ylwh_&>2  
    %   Example: k;xIo(:  
    % K *xca(6  
    %       % Display the first 16 Zernike functions XX[CTh?O%  
    %       x = -1:0.01:1; U)iq  
    %       [X,Y] = meshgrid(x,x); 6Dz N.fz  
    %       [theta,r] = cart2pol(X,Y);  Va3/#is'  
    %       idx = r<=1; Y]])Tq;h5  
    %       p = 0:15; t?;\'  
    %       z = nan(size(X)); 8m"jd+  
    %       y = zernfun2(p,r(idx),theta(idx)); u;3wg`e  
    %       figure('Units','normalized') .LA?2N  
    %       for k = 1:length(p) >N8*O3  
    %           z(idx) = y(:,k); y';"tDFb  
    %           subplot(4,4,k) }\3jcnn  
    %           pcolor(x,x,z), shading interp tiQeON-Q_  
    %           set(gca,'XTick',[],'YTick',[]) =Cg1I\  
    %           axis square O#72h]  
    %           title(['Z_{' num2str(p(k)) '}']) qEajT"?  
    %       end 1Zt>andBF  
    % EUjA-L(  
    %   See also ZERNPOL, ZERNFUN. ?{rpzrc!*  
    wjc&S'[  
    %   Paul Fricker 11/13/2006 M~,N~ N1  
    gUHx(Fi[4  
    iWp 6^g  
    % Check and prepare the inputs: :hFKmoy#  
    % ----------------------------- @M&qH[tK-A  
    if min(size(p))~=1 p4^&G/'  
        error('zernfun2:Pvector','Input P must be vector.') +Hk r\  
    end r}i}4K[1  
    S:8 WBY]M  
    if any(p)>35 fOJTy0jX8  
        error('zernfun2:P36', ... )zK@@E  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... lFZl}x  
               '(P = 0 to 35).'])  rBUWzpE"  
    end }BI|M_q.1~  
     *"Uf|  
    % Get the order and frequency corresonding to the function number: l?)!^}Qc  
    % ---------------------------------------------------------------- OAo;vC:^  
    p = p(:); L25%KGg' o  
    n = ceil((-3+sqrt(9+8*p))/2); d3"QCl  
    m = 2*p - n.*(n+2); 7(l>Ck3B#  
    Y1R?, 5  
    % Pass the inputs to the function ZERNFUN: C2C 1 @=w  
    % ---------------------------------------- kJK*wq]U6  
    switch nargin \[&&4CN{  
        case 3 s`gfz}/  
            z = zernfun(n,m,r,theta); 8F9x2CM-[C  
        case 4 G.3yuok9  
            z = zernfun(n,m,r,theta,nflag); ],s{%a5wC  
        otherwise apZPHau6h  
            error('zernfun2:nargin','Incorrect number of inputs.') -CLBf'a  
    end y;<F|zIm  
    DIw9ov>k  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) Xo$SQ0K  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. P^AI*tH"m  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of RT|1M"?$  
    %   order N and frequency M, evaluated at R.  N is a vector of ;Z); k`j  
    %   positive integers (including 0), and M is a vector with the #>6Jsnv1  
    %   same number of elements as N.  Each element k of M must be a +kN,OK~  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) %n 6NVi_[  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is WK0:3q(P  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix zx5#eMD  
    %   with one column for every (N,M) pair, and one row for every (67byO{  
    %   element in R. X;n09 L`CB  
    % +)LCYDRV7  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- l <p(zLR  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is -^3uQa<zN^  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to !jvl"+_FV  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ST2:&xH(  
    %   for all [n,m]. ^a<kp69qS  
    % g(9kc<`3'D  
    %   The radial Zernike polynomials are the radial portion of the 8@Bm2?$}g  
    %   Zernike functions, which are an orthogonal basis on the unit /24}>oAH  
    %   circle.  The series representation of the radial Zernike v*OV\h.  
    %   polynomials is @]!9;?so  
    % {Fqwr>e  
    %          (n-m)/2 /b\c<'3NY  
    %            __ [(eX\kL  
    %    m      \       s                                          n-2s N0%q 66]1  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r "j&'R#$&d  
    %    n      s=0 <<b]v I  
    % 2Z5_@Y  
    %   The following table shows the first 12 polynomials. \~g,;>%7Y  
    % mFeR~Bi>!  
    %       n    m    Zernike polynomial    Normalization YmpaLZJ  
    %       --------------------------------------------- !9.FI{W  
    %       0    0    1                        sqrt(2) ':3[?d1Es  
    %       1    1    r                           2 % m"Qg<  
    %       2    0    2*r^2 - 1                sqrt(6) OE}FZCX F  
    %       2    2    r^2                      sqrt(6) >bd@2au9!  
    %       3    1    3*r^3 - 2*r              sqrt(8) s/.P/g%tA>  
    %       3    3    r^3                      sqrt(8) @v#,SF{  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 684|Uuf7  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 8Ihl}aguW  
    %       4    4    r^4                      sqrt(10) DJ*mWi.  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) vw4b@v-XQ3  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ^`B;SSV  
    %       5    5    r^5                      sqrt(12) ``eam8Az_U  
    %       --------------------------------------------- ;>L8&m)R5  
    % ;rF[y7\  
    %   Example: H>W8F2VT  
    % C fM[<w   
    %       % Display three example Zernike radial polynomials 1= 7ASS9  
    %       r = 0:0.01:1; ;b:'i& r  
    %       n = [3 2 5]; D6H?*4f]  
    %       m = [1 2 1]; R7U%v"F>`  
    %       z = zernpol(n,m,r); 9K#3JyW*  
    %       figure -cijLlz%+  
    %       plot(r,z) reNf?7G+m  
    %       grid on V[uSo$k+>  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') vS)>g4  
    % s2<[@@@q  
    %   See also ZERNFUN, ZERNFUN2. L~^5Ez6U  
    Dk>6PBl  
    % A note on the algorithm. "l9aBBiu  
    % ------------------------ +wJ!zab`  
    % The radial Zernike polynomials are computed using the series JSi0-S[Y{  
    % representation shown in the Help section above. For many special ;-XfbqZ\  
    % functions, direct evaluation using the series representation can @"MQ6u G>  
    % produce poor numerical results (floating point errors), because ]ly" K!1,  
    % the summation often involves computing small differences between tv.<pP9-C  
    % large successive terms in the series. (In such cases, the functions G<Eb~]. 1'  
    % are often evaluated using alternative methods such as recurrence WubV?NX;EF  
    % relations: see the Legendre functions, for example). For the Zernike A=5Ebu!z  
    % polynomials, however, this problem does not arise, because the ,oh;(|=  
    % polynomials are evaluated over the finite domain r = (0,1), and 8I *N  
    % because the coefficients for a given polynomial are generally all $xbW*w  
    % of similar magnitude. \Dy|}LE  
    % b0YEIV<$  
    % ZERNPOL has been written using a vectorized implementation: multiple `d$@1  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] (S?Y3l|  
    % values can be passed as inputs) for a vector of points R.  To achieve rv(?%h`  
    % this vectorization most efficiently, the algorithm in ZERNPOL 9$Z0mzk  
    % involves pre-determining all the powers p of R that are required to F-,chp  
    % compute the outputs, and then compiling the {R^p} into a single  u\L}B!  
    % matrix.  This avoids any redundant computation of the R^p, and RG""/x ;  
    % minimizes the sizes of certain intermediate variables. IlB8~{p_  
    % %^HE^ &  
    %   Paul Fricker 11/13/2006 ~^V&n`*7D  
    ?KOw~-u  
    uBa<5YDF  
    % Check and prepare the inputs: R-j*fO}  
    % ----------------------------- Jp_#pV*}:  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) >>,G3/Zd*  
        error('zernpol:NMvectors','N and M must be vectors.') GaG>0 x   
    end 4minzrKM\  
    8ZVQM7O  
    if length(n)~=length(m) * l1*zaE  
        error('zernpol:NMlength','N and M must be the same length.') (X,i,qK/  
    end j}eb _K+I  
    ESIP+  
    n = n(:); *H/3xPh,*  
    m = m(:); twq~.:<o  
    length_n = length(n); MCmb/.&wu  
    !43 !JfD  
    if any(mod(n-m,2)) %g}d}5s  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') KDq="=q  
    end ^]nLE]M  
    _E`+0;O  
    if any(m<0) v/q-{ 1   
        error('zernpol:Mpositive','All M must be positive.') )ZpI%M?)  
    end c\1X NPGG  
    |cf-S8pwY  
    if any(m>n) PY.K_(D  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 'UIFP#GtFO  
    end ovTL'j!  
    B5;%R01A  
    if any( r>1 | r<0 ) ,UMr_ e{|  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') w3(|A> s3  
    end hr fF1 >A  
    %-540V{q  
    if ~any(size(r)==1) #f2k*8"eAF  
        error('zernpol:Rvector','R must be a vector.') j)mU`b_  
    end )N&SrzqTK  
    lD0-S0i  
    r = r(:); #5@(^N5p`  
    length_r = length(r); FOsd{Fw  
    i D IY|  
    if nargin==4 1@}F8&EZ  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); M?eP1v:<+G  
        if ~isnorm v'@gUgC  
            error('zernpol:normalization','Unrecognized normalization flag.') WzN c=@[W  
        end {Ya$Q#l  
    else +Y sGH~jX  
        isnorm = false; 9j>2C  
    end &-yRa45?  
    bE !SW2:M  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fvl\.  
    % Compute the Zernike Polynomials z4:!*:.Asu  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  j%Au0k  
    yS W$zA,  
    % Determine the required powers of r: Q"XDxa'7"  
    % ----------------------------------- a|7C6#iz$  
    rpowers = []; j#5a&Z  
    for j = 1:length(n) NZB*;U~t  
        rpowers = [rpowers m(j):2:n(j)]; .KA-=$~J1  
    end 3U@jw,K!{A  
    rpowers = unique(rpowers); )[Tm[o?Y.  
    Dt: Q$  
    % Pre-compute the values of r raised to the required powers, 6VGY4j}:(  
    % and compile them in a matrix: nHdQe  
    % ----------------------------- h+Co:pr  
    if rpowers(1)==0 2?t@<M]  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); oe|#!SM(  
        rpowern = cat(2,rpowern{:}); Z!"-LQJ  
        rpowern = [ones(length_r,1) rpowern]; 7 m{lOR  
    else 3CoZ2  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ]->"4,}  
        rpowern = cat(2,rpowern{:}); *4U_MM#rX  
    end I`V<Sh^Qd  
    9p W~Gz  
    % Compute the values of the polynomials: = j1Jl^[  
    % -------------------------------------- og}Ri!^  
    z = zeros(length_r,length_n); gXdMGO>  
    for j = 1:length_n $gUlM+sK  
        s = 0:(n(j)-m(j))/2; S0^a)#D &  
        pows = n(j):-2:m(j); t eY@) F  
        for k = length(s):-1:1 ,UY1.tR(  
            p = (1-2*mod(s(k),2))* ... {cXr!N^K  
                       prod(2:(n(j)-s(k)))/          ... ( eTrqI`  
                       prod(2:s(k))/                 ... -#|;qFD]  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... lmeTW0U@9(  
                       prod(2:((n(j)+m(j))/2-s(k))); }(nT(9|  
            idx = (pows(k)==rpowers); ..)J6L5l  
            z(:,j) = z(:,j) + p*rpowern(:,idx); >@2<^&K`  
        end WO qDW~  
         A(FnU:  
        if isnorm A & iv  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); .4+R ac  
        end U]cXE1c>F  
    end  k[r^@|  
    &b} \).5E  
    % 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)  2 }xePX9?  
    tJ(xeb  
    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。 ^6W}ZLp  
    ASXGM0t  
    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)