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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 GD -cP5$  
    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! i!i=6m.q7  
     
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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 =eyPo(B  
    function z = zernfun(n,m,r,theta,nflag) JI[{n~bhGD  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. -x VZm8y  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N -A^o5s  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ;Sl%I+?  
    %   unit circle.  N is a vector of positive integers (including 0), and W+I""I*mV  
    %   M is a vector with the same number of elements as N.  Each element @+7CfvM  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) e8 1+as  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, adWH';Q:  
    %   and THETA is a vector of angles.  R and THETA must have the same GDQQ4-|O  
    %   length.  The output Z is a matrix with one column for every (N,M) lFN|)(X  
    %   pair, and one row for every (R,THETA) pair. `d}t?qWS;F  
    % rtdEIk  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike gE9x+g  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), jct'B}@X(  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral t\WU}aKML  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, )4R[C={  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized :?j]W2+kR  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. UCo`l~K)qg  
    % }Ud'j'QMy  
    %   The Zernike functions are an orthogonal basis on the unit circle. .aQ8I1~  
    %   They are used in disciplines such as astronomy, optics, and *Ksk1T+>  
    %   optometry to describe functions on a circular domain. c"diNbm[  
    % v, !`A!{D  
    %   The following table lists the first 15 Zernike functions. ](^FGz  
    % uhU'm@JZ  
    %       n    m    Zernike function           Normalization 73l,PJ  
    %       -------------------------------------------------- AO,^v+ $  
    %       0    0    1                                 1 d*dPi^JjC  
    %       1    1    r * cos(theta)                    2 #y f  
    %       1   -1    r * sin(theta)                    2 Tm2+/qO,  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) uT>"(wnJ|  
    %       2    0    (2*r^2 - 1)                    sqrt(3) D `av9I  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) QYEGiT   
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) E BSjU8  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 7ufTmz#j<  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) bPIo9clq  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) $ I J^  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 40O@a:q*  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7- |N&u  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 6OR)97  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ]:}7-;$V  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) sJMpF8   
    %       -------------------------------------------------- IEe;ygL#  
    % 1'H!S%fS  
    %   Example 1: R5xV_;wD  
    % '$[a-)4  
    %       % Display the Zernike function Z(n=5,m=1) IP^1ca#<  
    %       x = -1:0.01:1; P?@o?  
    %       [X,Y] = meshgrid(x,x); h0C>z2iH  
    %       [theta,r] = cart2pol(X,Y); )<$<9!L4x  
    %       idx = r<=1; Mp(;PbVD  
    %       z = nan(size(X));  +F~B"a  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); !+DhH2;)F  
    %       figure o1k+dJUd  
    %       pcolor(x,x,z), shading interp })j N 8px  
    %       axis square, colorbar >`<qa!9  
    %       title('Zernike function Z_5^1(r,\theta)') 0./Rdf=-1j  
    % 2J (nJT"  
    %   Example 2: c9djBUAk&  
    % bc;?O`I<  
    %       % Display the first 10 Zernike functions 2Z?l,M~  
    %       x = -1:0.01:1; fOdX2{7m  
    %       [X,Y] = meshgrid(x,x); $RYOj{1  
    %       [theta,r] = cart2pol(X,Y); gYloY=.Z$'  
    %       idx = r<=1; qfRrX"  
    %       z = nan(size(X)); g9Ty%|Q7(  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Fzt7@VNxc  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; qC3PKlhv6  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 4ves|pLET  
    %       y = zernfun(n,m,r(idx),theta(idx)); 39d$B'"<1  
    %       figure('Units','normalized') xIH= gK  
    %       for k = 1:10 A p 3B'  
    %           z(idx) = y(:,k); Zy|u5J  
    %           subplot(4,7,Nplot(k)) ND/oKM+?  
    %           pcolor(x,x,z), shading interp -j@IDd7  
    %           set(gca,'XTick',[],'YTick',[]) 3S1{r )[j  
    %           axis square ?X Rl\V  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) J ~KygQ3%  
    %       end pktnX-Slt  
    % )P,pW?h$  
    %   See also ZERNPOL, ZERNFUN2. ce*?crOV  
    $LG.rJ/*  
    %   Paul Fricker 11/13/2006 A-*MH#QUKh  
    $j \jT  
    B5+$ VQ  
    % Check and prepare the inputs: 5=Y(.}6  
    % ----------------------------- aimf,(+  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) "'XYW\bI  
        error('zernfun:NMvectors','N and M must be vectors.') ~qX wQ@  
    end *$3p3-  
    ,c 0]r;u!  
    if length(n)~=length(m) H%Z;Yt8^gt  
        error('zernfun:NMlength','N and M must be the same length.') .EvP%A m  
    end uJ8FzS>[V  
    ;9q$eK%d  
    n = n(:); $.31<@T7  
    m = m(:); x=X&b%09  
    if any(mod(n-m,2)) J(A+mYr{:  
        error('zernfun:NMmultiplesof2', ... l<'}`  
              'All N and M must differ by multiples of 2 (including 0).') FC  
    end L0w2qF  
    PnL?zae  
    if any(m>n) G&`5o*).bb  
        error('zernfun:MlessthanN', ... R^]a<g,  
              'Each M must be less than or equal to its corresponding N.') [{#n?BT  
    end  )\kNufP  
    q^7=/d8  
    if any( r>1 | r<0 ) 19RbIG/X  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 02)Ybp6y  
    end Ga V OMT  
    /||8j.Tm  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 6WoFf  
        error('zernfun:RTHvector','R and THETA must be vectors.') !1@o Z(  
    end ;Wsl 'e/  
    O;T)u4Q&3  
    r = r(:); L(X}37  
    theta = theta(:); e@& 2q{Gi=  
    length_r = length(r); y)TBg8Q  
    if length_r~=length(theta) 6zi Mf  
        error('zernfun:RTHlength', ... ABL5T-*]  
              'The number of R- and THETA-values must be equal.') jpOcug`f  
    end JeAyT48!M  
    3$BO=hI/-  
    % Check normalization: (a~V<v"  
    % -------------------- ;&kZ7%  
    if nargin==5 && ischar(nflag) ]BTISaL-R  
        isnorm = strcmpi(nflag,'norm'); =/\l=*  
        if ~isnorm ~q}]/0-m  
            error('zernfun:normalization','Unrecognized normalization flag.') T+FlN-iy)  
        end l1%*LyD  
    else 5d}bl{  
        isnorm = false; PWyFys  
    end 2P{! n#"  
    o=F!&]+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wy:euKB~   
    % Compute the Zernike Polynomials w(ic$  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fSGaUBiq}  
    Eh[NKgYL  
    % Determine the required powers of r: C\|HN=2eh  
    % ----------------------------------- };*&;GFe  
    m_abs = abs(m); GkKoc v  
    rpowers = []; QqcAmp  
    for j = 1:length(n) W#wC  
        rpowers = [rpowers m_abs(j):2:n(j)]; ): r'IR  
    end +!G)N~o  
    rpowers = unique(rpowers); h(^[WSa  
    Lo" s12fr  
    % Pre-compute the values of r raised to the required powers, U]ZI_[\'U  
    % and compile them in a matrix: W=2]!%3#  
    % ----------------------------- #rp)Gc  
    if rpowers(1)==0 [.;8GMW  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); L_!}R  
        rpowern = cat(2,rpowern{:}); qVd s 2  
        rpowern = [ones(length_r,1) rpowern]; _cJ\A0h^  
    else t3!~=U  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ("=24R=a  
        rpowern = cat(2,rpowern{:}); 18y'#<X!  
    end lvUWs  
    "<"s&ws;k  
    % Compute the values of the polynomials: QR$m i1Vv\  
    % -------------------------------------- } OkK@8?0O  
    y = zeros(length_r,length(n)); V~t; J  
    for j = 1:length(n) ={{q_G\WD  
        s = 0:(n(j)-m_abs(j))/2; =CaSd|   
        pows = n(j):-2:m_abs(j); SWNT}{x]  
        for k = length(s):-1:1 ^n\g,  
            p = (1-2*mod(s(k),2))* ... <V#]3$(S  
                       prod(2:(n(j)-s(k)))/              ... vQ{mEaH  
                       prod(2:s(k))/                     ... 4c.!^EiV  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... +.X3&|@k  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); vnX~OVz2  
            idx = (pows(k)==rpowers); 5g2:o^  
            y(:,j) = y(:,j) + p*rpowern(:,idx); _ n4C~  
        end ]YB,K)WQ  
         *C^TCyBK;  
        if isnorm hr g'Z5n  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); (T",6xBSG  
        end >~T2MlRux  
    end m\K1Ex  
    % END: Compute the Zernike Polynomials >}86#^F  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :/;;|lGw  
    z~;@Mo"*f  
    % Compute the Zernike functions: Angt=q  
    % ------------------------------ Ystd[  
    idx_pos = m>0; KU_""T  
    idx_neg = m<0; {%X[Snv  
    Oq 95zo  
    z = y; a!;K+wL >  
    if any(idx_pos) >< Qp%yT  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); U@)WTH6d  
    end =AeOkie  
    if any(idx_neg) \%.&$z3wz  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); RNX>I,2sh  
    end [ _&z+  
    1xU)nXXb  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) N!13QI H  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. <rNz&;m}  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated #M:Vwn JX  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive }M9I]\  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, sHHu<[psM  
    %   and THETA is a vector of angles.  R and THETA must have the same r6}-EYq=  
    %   length.  The output Z is a matrix with one column for every P-value, E}|IU Pm  
    %   and one row for every (R,THETA) pair. R"e533  
    % R%;dt<Dh  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike M V~3~h8  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) n*N`].r#{=  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) CSMx]jbb  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 \2)~dV:6+  
    %   for all p. _Ns_$_  
    % AJt4I W@  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 .OD{^Kq2  
    %   Zernike functions (order N<=7).  In some disciplines it is =OooTZb:x-  
    %   traditional to label the first 36 functions using a single mode f >\~h,SLL  
    %   number P instead of separate numbers for the order N and azimuthal 1zY" Uxp  
    %   frequency M. 7 9ZYRm2;  
    % _(:bGI'.m  
    %   Example: J|dj`Z ?  
    % );V.le}%(  
    %       % Display the first 16 Zernike functions A)D1 #,0  
    %       x = -1:0.01:1; fb|lWEw5h.  
    %       [X,Y] = meshgrid(x,x); s C?-L  
    %       [theta,r] = cart2pol(X,Y); f_tC:T4a  
    %       idx = r<=1; /QVhT  
    %       p = 0:15; &y:SK)  
    %       z = nan(size(X)); @%$<,$=  
    %       y = zernfun2(p,r(idx),theta(idx)); RMBPm*H  
    %       figure('Units','normalized') 'E#Bz"T  
    %       for k = 1:length(p) zT jk^  
    %           z(idx) = y(:,k); }&IOBYHVDo  
    %           subplot(4,4,k) Np R&`]  
    %           pcolor(x,x,z), shading interp [!b=A:@  
    %           set(gca,'XTick',[],'YTick',[]) hN.{H:skL)  
    %           axis square bF? {  
    %           title(['Z_{' num2str(p(k)) '}']) q!}O+(kt  
    %       end %x|0<@b7-  
    % WB=|Ty ~l  
    %   See also ZERNPOL, ZERNFUN. :<`po4/  
    nSh}1Arp/  
    %   Paul Fricker 11/13/2006 EnXTL]=0S  
    !"N-To-c  
    /}RW~ax  
    % Check and prepare the inputs: #Ue_  
    % ----------------------------- kV+O|9  
    if min(size(p))~=1 4$jb-Aw  
        error('zernfun2:Pvector','Input P must be vector.') kY`L[1G$  
    end >^%TY^7n  
    WhN~R[LE_  
    if any(p)>35 I?%iJ%  
        error('zernfun2:P36', ...  .'^Pg  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... G,h=5y9_J  
               '(P = 0 to 35).']) %P-z3 0FHp  
    end 8zk?:?8%{  
    Kv-4VWh  
    % Get the order and frequency corresonding to the function number: o"@GYc["  
    % ---------------------------------------------------------------- j_HwR9^fd,  
    p = p(:); 3+2cD  
    n = ceil((-3+sqrt(9+8*p))/2); R3gg{hQ  
    m = 2*p - n.*(n+2); h;2n2.Q  
    JO"-"&>  
    % Pass the inputs to the function ZERNFUN: UqaV9  
    % ---------------------------------------- B]|"ePj-  
    switch nargin @EzO bE{  
        case 3 y(0";\V  
            z = zernfun(n,m,r,theta); zQ~8(E]Rf  
        case 4 8.4+4Vxh   
            z = zernfun(n,m,r,theta,nflag); 'J"m`a8no  
        otherwise W4o$J4IX{  
            error('zernfun2:nargin','Incorrect number of inputs.') 8\@&~&(y:  
    end D "9Hv3  
    l|p \8=  
    % 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) *Af:^>mh  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. ^%pM$3ov  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Z:(yX0U,[  
    %   order N and frequency M, evaluated at R.  N is a vector of ,/>hWAx  
    %   positive integers (including 0), and M is a vector with the yC]X&1,:z  
    %   same number of elements as N.  Each element k of M must be a {@8TGHKv  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) %d/Pc4gfc  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 'Bv)UfZ  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix }- P ='AyL  
    %   with one column for every (N,M) pair, and one row for every }^ np  
    %   element in R. kLw07&H  
    % PA(XdT{  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- b;XUv4~V  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is  8DsXw@o  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to D-<9kBZs  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Zw`vPvb!  
    %   for all [n,m]. vhQIkB8  
    % ^ A`@g4!  
    %   The radial Zernike polynomials are the radial portion of the !K~:crUV|S  
    %   Zernike functions, which are an orthogonal basis on the unit d!i#@XZ^  
    %   circle.  The series representation of the radial Zernike _b8?_Zq  
    %   polynomials is <cn{S`  
    % ~\^h;A'3  
    %          (n-m)/2 u'BuZF  
    %            __ &eHhj9  
    %    m      \       s                                          n-2s DcQ[zdEz+  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ZFAi9M  
    %    n      s=0 sm~{fg  
    % <-1(G1v  
    %   The following table shows the first 12 polynomials. "V;5Lp b  
    % ~K-c-Zs#z  
    %       n    m    Zernike polynomial    Normalization NBUSr}8|  
    %       --------------------------------------------- |%@.@c  
    %       0    0    1                        sqrt(2)  '9Hah  
    %       1    1    r                           2 e)WpqaI  
    %       2    0    2*r^2 - 1                sqrt(6) g{}{gBplnl  
    %       2    2    r^2                      sqrt(6) xA-u%Vf7@  
    %       3    1    3*r^3 - 2*r              sqrt(8) e /4{pe+,  
    %       3    3    r^3                      sqrt(8) .%pbKi `  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) qx$-% P  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) nK" XyZ&  
    %       4    4    r^4                      sqrt(10) Vg0$5@  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) EN =oA P  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 5ZRO{rf  
    %       5    5    r^5                      sqrt(12) ;;2Yfn'`9  
    %       --------------------------------------------- J4-64t nZ  
    % x! A.**  
    %   Example: Ie[8Iot?bn  
    % LyRU2A  
    %       % Display three example Zernike radial polynomials d3$&I==;:  
    %       r = 0:0.01:1; i+2fWi6Z+  
    %       n = [3 2 5]; 5jTBPct   
    %       m = [1 2 1]; $:#{Y;d  
    %       z = zernpol(n,m,r); `Eijy3>h  
    %       figure >>ncq$  
    %       plot(r,z) y3]7^+k  
    %       grid on vT#$`M<  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') kXmnLxhS/  
    % l 4zl|6%  
    %   See also ZERNFUN, ZERNFUN2. 1q])"l"<  
    =lzRx%tm  
    % A note on the algorithm. ZZ<uiN$  
    % ------------------------ b#:Pl`n6u  
    % The radial Zernike polynomials are computed using the series =Mb1)^m  
    % representation shown in the Help section above. For many special 1@j0kTJ~m  
    % functions, direct evaluation using the series representation can $\0%"S  
    % produce poor numerical results (floating point errors), because ^=H. .pr  
    % the summation often involves computing small differences between ~JJuM  
    % large successive terms in the series. (In such cases, the functions |hp_<F9.  
    % are often evaluated using alternative methods such as recurrence %V>Ss9;/8  
    % relations: see the Legendre functions, for example). For the Zernike t4a/\{/#9|  
    % polynomials, however, this problem does not arise, because the qH3|x08  
    % polynomials are evaluated over the finite domain r = (0,1), and BrdHTk= Vy  
    % because the coefficients for a given polynomial are generally all bOt6q/f  
    % of similar magnitude. !ys82  
    % GWNLET  
    % ZERNPOL has been written using a vectorized implementation: multiple (8(7:aE $  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] +J^-B}v  
    % values can be passed as inputs) for a vector of points R.  To achieve z%Xz*uu(|  
    % this vectorization most efficiently, the algorithm in ZERNPOL CnJrJ>l  
    % involves pre-determining all the powers p of R that are required to 2{v$GFc/  
    % compute the outputs, and then compiling the {R^p} into a single HAHv^  
    % matrix.  This avoids any redundant computation of the R^p, and U;Iqz1S  
    % minimizes the sizes of certain intermediate variables. c~@Z  
    % YceX)  
    %   Paul Fricker 11/13/2006 g:l5,j.K  
    }=1#ANM1  
    2;Ij~~  
    % Check and prepare the inputs: Svs!C+:le  
    % ----------------------------- ?R7>xrp5  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) mVg$z  
        error('zernpol:NMvectors','N and M must be vectors.') N3D{t\hg  
    end .Ulrv5wJ  
    tgy= .o]  
    if length(n)~=length(m) YEL, TU  
        error('zernpol:NMlength','N and M must be the same length.') CCCd=s.  
    end 0Q81$% @<  
    u!=9.3  
    n = n(:); 7oj ^(R,  
    m = m(:); or?@Ti;  
    length_n = length(n); C@{#OOa  
    <oweLRt  
    if any(mod(n-m,2)) _Eus<c  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') LL|uMe"Jb  
    end K.y2 $b/  
    ,]1oG=`3v  
    if any(m<0) ea"!:cL(g  
        error('zernpol:Mpositive','All M must be positive.') PGaB U3  
    end YVzcV`4w(  
    a J%&Y5L  
    if any(m>n) g_kR5Wxpt  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') n"[VM=YGI  
    end [D8u.8q  
    gnW]5#c@  
    if any( r>1 | r<0 ) 0q|.]:][Eo  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') sFd"VRAV~E  
    end L/2{}l>D  
    T7vSp<i/  
    if ~any(size(r)==1) {[r}&^K15  
        error('zernpol:Rvector','R must be a vector.') |'w_5?|4  
    end aq'd C=y  
    hxIG0d!o  
    r = r(:); >EVlMt27'  
    length_r = length(r); L*;XjacI]  
     .AEOf0t  
    if nargin==4 \E9Hk{V:6  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 9ghZL Q  
        if ~isnorm wv.FL$f[@  
            error('zernpol:normalization','Unrecognized normalization flag.') 80PlbUBb!  
        end ]:lqbg[J  
    else -&4W0JK9  
        isnorm = false;  $D`~X`  
    end [@SLt$9"  
    XkB^.[B  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]zGgx07d  
    % Compute the Zernike Polynomials ")J\} $r  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1b4aY> Z  
    KmuE#Ia  
    % Determine the required powers of r: 8vzjPWu  
    % ----------------------------------- Irk@#,{<  
    rpowers = []; =5NM =K  
    for j = 1:length(n) WM& k  
        rpowers = [rpowers m(j):2:n(j)]; jft%\sY  
    end v&BKl  
    rpowers = unique(rpowers); +UzFHiGy#  
    b`x7%?Qn  
    % Pre-compute the values of r raised to the required powers, rgQ6/3}qc  
    % and compile them in a matrix: \/rK0|2A  
    % ----------------------------- nWTo$*>W  
    if rpowers(1)==0 )&G uZ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ,@+ 7(W  
        rpowern = cat(2,rpowern{:}); ]Lc:M'V#  
        rpowern = [ones(length_r,1) rpowern]; g+5{&YD  
    else ftVA  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); OBGA~E;%  
        rpowern = cat(2,rpowern{:}); em^|E73  
    end _%g}d/v}pO  
    Yg 8AMi  
    % Compute the values of the polynomials: `;[ j`v8O  
    % -------------------------------------- y`"~zq0D  
    z = zeros(length_r,length_n); T[mo PD5  
    for j = 1:length_n 8&15k A  
        s = 0:(n(j)-m(j))/2; !Vtt.j &4  
        pows = n(j):-2:m(j); [ emUyF  
        for k = length(s):-1:1 .#"O VI]#  
            p = (1-2*mod(s(k),2))* ... ^;J@]&[ ~  
                       prod(2:(n(j)-s(k)))/          ... zCrDbGvqF`  
                       prod(2:s(k))/                 ... >KjyxJ7  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ...  &Gp~)%  
                       prod(2:((n(j)+m(j))/2-s(k))); ~#X,)L{y7v  
            idx = (pows(k)==rpowers); |_&Tu#er3  
            z(:,j) = z(:,j) + p*rpowern(:,idx); IUX~dO  
        end mZ;W$y SO  
         "=l<%em  
        if isnorm \;0J6LBc  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); =^. f)  
        end 3kxI'0&T  
    end :t &ib}v  
    __U;fH{c  
    % 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)  )NZ6!3[@  
    $enh>!mU  
    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。 jJl6H~ "q  
    VtF^; f  
    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)