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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 hsz$S:am  
    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! epYj+T  
     
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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=0iPy,m>  
    function z = zernfun(n,m,r,theta,nflag) )x y9X0  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. UzXDi#Ky  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N HW^{;'kH~  
    %   and angular frequency M, evaluated at positions (R,THETA) on the oC5gME"2  
    %   unit circle.  N is a vector of positive integers (including 0), and t!NrB X  
    %   M is a vector with the same number of elements as N.  Each element r#ks>s  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) }o~Tw?z-|  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, L!`*R)I45  
    %   and THETA is a vector of angles.  R and THETA must have the same _.u~)Q`6  
    %   length.  The output Z is a matrix with one column for every (N,M) RJQ/y3  
    %   pair, and one row for every (R,THETA) pair. (L]T*03#  
    % w;@`Yi.WQ  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 4&#vU(-H  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 77)OW $G  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral mm3zQ!2j.  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, :k Rv  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized A;K{&x  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. FA5k45w L  
    % QSO5 z2|  
    %   The Zernike functions are an orthogonal basis on the unit circle. KB$ vQ@N  
    %   They are used in disciplines such as astronomy, optics, and LPtx|Sx![  
    %   optometry to describe functions on a circular domain. u0<d2Y  
    % <6~/sa4GN  
    %   The following table lists the first 15 Zernike functions. {6REfY c  
    % w;yar=n  
    %       n    m    Zernike function           Normalization rCV$N&rK  
    %       -------------------------------------------------- GA({ri  
    %       0    0    1                                 1 J$o[$G_Z  
    %       1    1    r * cos(theta)                    2 ,Gf+U7'K  
    %       1   -1    r * sin(theta)                    2 !&W"f#_Z  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) uOy\{5s8  
    %       2    0    (2*r^2 - 1)                    sqrt(3) "Wzij&WkQ  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) pP=_@ 3 D  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) U`},)$  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) C`=`Ce~|d  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) (cbB %  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) O% j,:t'"  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ;tZ}i4Ud  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) BbXmT"@  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) $\=6."R5<  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) &q kl*#]  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) dA3`b*nC  
    %       -------------------------------------------------- iX&Z  
    % Br?++\  
    %   Example 1: ZVCv(J  
    % 5k!(#@a_T  
    %       % Display the Zernike function Z(n=5,m=1) kr &:;  
    %       x = -1:0.01:1; @DjG? yLK$  
    %       [X,Y] = meshgrid(x,x); 7]0\[9DyJ  
    %       [theta,r] = cart2pol(X,Y); 5Lo==jHif  
    %       idx = r<=1; -0[>}!l=G  
    %       z = nan(size(X)); ^+.e5roBKj  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); EV;;N  
    %       figure 7ipY*DT8  
    %       pcolor(x,x,z), shading interp ?L.p9o-S0  
    %       axis square, colorbar ixUiXP  
    %       title('Zernike function Z_5^1(r,\theta)') >Kqj{/SWK  
    % o>!~*b';g,  
    %   Example 2: 6r ?cpJV{  
    % e3bAT.P  
    %       % Display the first 10 Zernike functions s`dkEaS  
    %       x = -1:0.01:1; B@: XC&R^  
    %       [X,Y] = meshgrid(x,x); wZ#~+ }T  
    %       [theta,r] = cart2pol(X,Y); TO8\4p*tE  
    %       idx = r<=1; J^e|"0d  
    %       z = nan(size(X)); ,& {5,=  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 4%Wn}@  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; *PA1iNdKS  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; =h1 QN  
    %       y = zernfun(n,m,r(idx),theta(idx)); 2T{-J!k  
    %       figure('Units','normalized') ^Ypb"Wx8  
    %       for k = 1:10 Rg!aKdDl$  
    %           z(idx) = y(:,k); a|^-z|.  
    %           subplot(4,7,Nplot(k)) %[31ZFYB  
    %           pcolor(x,x,z), shading interp y0Q/B|&[  
    %           set(gca,'XTick',[],'YTick',[]) Yqj.z|}Nb  
    %           axis square }@t'rK[  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) F'T= Alf  
    %       end N*c?Er@8U  
    % {mq$W  
    %   See also ZERNPOL, ZERNFUN2. blQzVp-  
    88X*:Kf?:  
    %   Paul Fricker 11/13/2006 fuwpp  
    XzTH,7[n  
    0 Ci"tA3"  
    % Check and prepare the inputs: GqF.T#|  
    % ----------------------------- wr6xuoH  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) `'{%szmD  
        error('zernfun:NMvectors','N and M must be vectors.') 5d>YE  
    end .$T:n[@  
    "$wPq@  
    if length(n)~=length(m) w[n>4?"{  
        error('zernfun:NMlength','N and M must be the same length.') 1Tk\n  
    end )}g4Rvr  
    %W|Zj QI^  
    n = n(:); mk3e^,[A  
    m = m(:); Z6 |'k:R8  
    if any(mod(n-m,2)) qCFXaj   
        error('zernfun:NMmultiplesof2', ... d$C|hT  
              'All N and M must differ by multiples of 2 (including 0).') ;),O*Z|"v  
    end 0jx~_zq-j  
    OrqJo!FEg{  
    if any(m>n) 8f`b=r(a>  
        error('zernfun:MlessthanN', ... %l$&_xV-  
              'Each M must be less than or equal to its corresponding N.') "u> sS  
    end r:\5/0(  
    Wy-quq03"&  
    if any( r>1 | r<0 ) b "3T(#2<*  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 7XI4=O};&%  
    end X9BBnZ  
    i{x0#6_Y  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 9tW.}5V  
        error('zernfun:RTHvector','R and THETA must be vectors.')  B*~Bm.  
    end _WkcJe`  
    NCh(-E  
    r = r(:); 9;WOqBD  
    theta = theta(:); \:)o'-   
    length_r = length(r); }\qdow-  
    if length_r~=length(theta) g|*eN{g]uE  
        error('zernfun:RTHlength', ... f0'Wq^^  
              'The number of R- and THETA-values must be equal.') H\>I&gC'  
    end 4Xho0lO&  
    #YMp,i  
    % Check normalization: GP k Cgb(  
    % -------------------- vCe<-k  
    if nargin==5 && ischar(nflag) <("w'd}  
        isnorm = strcmpi(nflag,'norm'); L5P}%1 _  
        if ~isnorm mZJzBYM)  
            error('zernfun:normalization','Unrecognized normalization flag.') B*?PB]  
        end 2A;[Ek6{q  
    else u z2s-,  
        isnorm = false; 7%x+7  
    end uM6!RR!~  
     V# %spW  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'ah0IYe  
    % Compute the Zernike Polynomials 2g8P$+;  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Yt<PKs#E  
    a9_KQ=&CI  
    % Determine the required powers of r: Tsp-]-)  
    % ----------------------------------- ~O7(0RsCN  
    m_abs = abs(m); %H~gN9Vn#@  
    rpowers = []; )'CEWc%  
    for j = 1:length(n) zjZTar1Re  
        rpowers = [rpowers m_abs(j):2:n(j)]; :NyEd<'  
    end ]<?)(xz  
    rpowers = unique(rpowers); ZvKMRW  
    4gNRln-  
    % Pre-compute the values of r raised to the required powers, ~0{Kga  
    % and compile them in a matrix: )GKgK;=~  
    % ----------------------------- n^)9QQ  
    if rpowers(1)==0 _Cs}&Bic_  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); -Dm.z16  
        rpowern = cat(2,rpowern{:}); ">&:(<  
        rpowern = [ones(length_r,1) rpowern]; 1@dx(_  
    else ~ J{{n_G{  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); TRok4uc  
        rpowern = cat(2,rpowern{:}); :P1c>:j[  
    end #iDFGkK/  
    A>2p/iMc  
    % Compute the values of the polynomials: E,:pIw  
    % -------------------------------------- @O @yJ{(I  
    y = zeros(length_r,length(n)); sYP@>tHC  
    for j = 1:length(n) Xkm2C)  
        s = 0:(n(j)-m_abs(j))/2; kw}1CXD  
        pows = n(j):-2:m_abs(j); <\EfG:e  
        for k = length(s):-1:1 (:x"p{  
            p = (1-2*mod(s(k),2))* ... i)3\jO0&GU  
                       prod(2:(n(j)-s(k)))/              ... oA%[x  
                       prod(2:s(k))/                     ... i?=.; 0[|  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... x7@HPf  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); * v]UgPk  
            idx = (pows(k)==rpowers); Y\|J1I,Z4  
            y(:,j) = y(:,j) + p*rpowern(:,idx); "A+F&C>  
        end w8ld* z  
         -y.AJ~T  
        if isnorm k4rB S  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ,e_#   
        end wO%:WL$5  
    end /CE d 14.  
    % END: Compute the Zernike Polynomials =lD]sk  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O3: dOL/C  
    <]^D({`  
    % Compute the Zernike functions: BAHx7x#(  
    % ------------------------------ S$WM&9U   
    idx_pos = m>0; c10).zZ  
    idx_neg = m<0; lHqx}n@e  
    A$6b=2hc>  
    z = y; 9-6_:N>  
    if any(idx_pos) "6QMa,)D  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); V,5}hQJ F  
    end ~Xw?>&  
    if any(idx_neg) .&xNJdsY  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); f|0QN#$  
    end #Q7$I.O]  
    sdD[`#  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) T%/w^27E  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. p> S/6 [X  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated J3;KQ}F.I  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive e`F|sz]k"H  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, FG:BRS<m~  
    %   and THETA is a vector of angles.  R and THETA must have the same W<o0Z OO  
    %   length.  The output Z is a matrix with one column for every P-value, m)}MkC-  
    %   and one row for every (R,THETA) pair. d2sq]Q  
    % BH a>2N  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike yw!`1#3.  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) q2vz#\A?  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) GGtrH~zx  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 e]?S-J'z  
    %   for all p. IOl"Xgn5  
    % U$uO%:4%  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 f34&:xz2U  
    %   Zernike functions (order N<=7).  In some disciplines it is gQ#T7  
    %   traditional to label the first 36 functions using a single mode wP?q5r5  
    %   number P instead of separate numbers for the order N and azimuthal "@$STptkc  
    %   frequency M. *pp1Wa7O  
    % 89mre;v`  
    %   Example: eCD,[At/  
    % ]5/U}Um  
    %       % Display the first 16 Zernike functions Ms)zEy>[Ql  
    %       x = -1:0.01:1; Ya 4$7|(  
    %       [X,Y] = meshgrid(x,x); ^MV%\0o  
    %       [theta,r] = cart2pol(X,Y); V.:A'!$#  
    %       idx = r<=1; dC#\ut%l  
    %       p = 0:15; ;(6lN<i U  
    %       z = nan(size(X)); 4'&BpFDUb  
    %       y = zernfun2(p,r(idx),theta(idx)); ZRGZ'+hw  
    %       figure('Units','normalized') y/eX(l<{  
    %       for k = 1:length(p) kH -b!  
    %           z(idx) = y(:,k); 3HR]TQ%r  
    %           subplot(4,4,k) !Jl0Eu  
    %           pcolor(x,x,z), shading interp |LH*)GrD*t  
    %           set(gca,'XTick',[],'YTick',[]) s;$TX304  
    %           axis square >+8I =S  
    %           title(['Z_{' num2str(p(k)) '}']) P@`"MNS  
    %       end ygt)7f5  
    % u6T?oK9j  
    %   See also ZERNPOL, ZERNFUN. REBDr;tv  
    j],.`Y  
    %   Paul Fricker 11/13/2006 olxP`iK  
     o f  
    $VIq)s2az|  
    % Check and prepare the inputs: #!# X3j  
    % ----------------------------- vK`h;  
    if min(size(p))~=1 J5 ( D7rp#  
        error('zernfun2:Pvector','Input P must be vector.') z}8L}:  
    end @ibPL+~-_  
     WPKTX,k  
    if any(p)>35 u?Mu*r?  
        error('zernfun2:P36', ... PGl-2Cr  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ER1mA:8>E  
               '(P = 0 to 35).']) [;YBX] t  
    end BM~niW;k  
    pu*u[n  
    % Get the order and frequency corresonding to the function number: kA=~ 8N  
    % ---------------------------------------------------------------- L b;vrh;A  
    p = p(:); E9 q;>)}  
    n = ceil((-3+sqrt(9+8*p))/2); 8lSn*;S,  
    m = 2*p - n.*(n+2); aZGDtzNG5h  
    q%Jy>IXt  
    % Pass the inputs to the function ZERNFUN: 4,ynt&  
    % ---------------------------------------- Al=? j#J6p  
    switch nargin |ZlT>u  
        case 3 YKOO(?lv  
            z = zernfun(n,m,r,theta); ?$4R <  
        case 4 .|`=mx  
            z = zernfun(n,m,r,theta,nflag); (ul-J4E\O  
        otherwise qpqz. {\  
            error('zernfun2:nargin','Incorrect number of inputs.') 9Ru%E>el-  
    end 8'WMspX  
    q)xl$*g  
    % 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) OK?3,<x  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. [^}>AC*im  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Bx : So6:  
    %   order N and frequency M, evaluated at R.  N is a vector of pkN:D+g S  
    %   positive integers (including 0), and M is a vector with the u$=ogp =0  
    %   same number of elements as N.  Each element k of M must be a Y!1^@;)^  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) UtBlP+bE?y  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is OG^WZ.YU  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix /\a]S:V-j  
    %   with one column for every (N,M) pair, and one row for every ENx@Ex  
    %   element in R. % X ,B-h^  
    % p@7i=hyt`p  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- fqk Dk  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is h$7Fe +#I#  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to H"q`k5R  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 hp]ng!I{\u  
    %   for all [n,m]. {.3  
    % =Q8H]F  
    %   The radial Zernike polynomials are the radial portion of the `\F%l?aY  
    %   Zernike functions, which are an orthogonal basis on the unit '0_j{ig  
    %   circle.  The series representation of the radial Zernike $,e?X}4  
    %   polynomials is [b i3%yWh  
    % hi3sOK*r;<  
    %          (n-m)/2 sE%<"h\_0  
    %            __ gAr`hXO  
    %    m      \       s                                          n-2s &Ky u@Tt  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r yw* mA1v  
    %    n      s=0 NB W%.z  
    % =yTa,PY  
    %   The following table shows the first 12 polynomials. X=p3KzzX  
    % *L;pcg8{  
    %       n    m    Zernike polynomial    Normalization ( ztim  
    %       --------------------------------------------- L;--d`[  
    %       0    0    1                        sqrt(2) aq0iNbv@  
    %       1    1    r                           2 Dz8:; $/  
    %       2    0    2*r^2 - 1                sqrt(6) TXJY2J*24  
    %       2    2    r^2                      sqrt(6) m/<F 5R  
    %       3    1    3*r^3 - 2*r              sqrt(8) u JQaHL!  
    %       3    3    r^3                      sqrt(8) iJZ|[jEDV  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) Kl aZZJ  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ne"?90~  
    %       4    4    r^4                      sqrt(10) zD)IU_GWa  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ckf<N9  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) eg2U+g4  
    %       5    5    r^5                      sqrt(12) 2 ]V>J  
    %       --------------------------------------------- i[2bmd!H  
    % k'@7ZH  
    %   Example: 0;FqX*  
    % pM&]&Nk  
    %       % Display three example Zernike radial polynomials # cN_y  
    %       r = 0:0.01:1; H}sS4[z  
    %       n = [3 2 5]; c/<Sa|'  
    %       m = [1 2 1]; bB:r]*_ s]  
    %       z = zernpol(n,m,r); -Wlp=#9  
    %       figure crJ7pe9  
    %       plot(r,z) #*Yi4Cn<  
    %       grid on U/X|i /  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') O}K_l1  
    % \K?(  
    %   See also ZERNFUN, ZERNFUN2. Qe>i{:N  
    xb9Pc.A[  
    % A note on the algorithm. =% q?Cr  
    % ------------------------ IpWy)B>Fl3  
    % The radial Zernike polynomials are computed using the series UCn*UX  
    % representation shown in the Help section above. For many special MX!u$ei  
    % functions, direct evaluation using the series representation can ;-KA UgL2  
    % produce poor numerical results (floating point errors), because k_D4'(V:b  
    % the summation often involves computing small differences between GOy=p3mQ  
    % large successive terms in the series. (In such cases, the functions j3x^<a\gJ  
    % are often evaluated using alternative methods such as recurrence (C`FicY  
    % relations: see the Legendre functions, for example). For the Zernike pg~zUOY  
    % polynomials, however, this problem does not arise, because the aO.\Qe+j  
    % polynomials are evaluated over the finite domain r = (0,1), and bp]^EVx  
    % because the coefficients for a given polynomial are generally all U1,~bO9  
    % of similar magnitude. bQ-Gp;]  
    % CM%|pB/z  
    % ZERNPOL has been written using a vectorized implementation: multiple jWH{;V&ZV  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] A1T<  
    % values can be passed as inputs) for a vector of points R.  To achieve Sl3KpZ  
    % this vectorization most efficiently, the algorithm in ZERNPOL =\~E n5  
    % involves pre-determining all the powers p of R that are required to P%zH>K  
    % compute the outputs, and then compiling the {R^p} into a single cGgM8  
    % matrix.  This avoids any redundant computation of the R^p, and {$EH@$./  
    % minimizes the sizes of certain intermediate variables. Sa3I?+  
    % 0a"igH}  
    %   Paul Fricker 11/13/2006 UL86-R!  
    C#MF pT  
    KX?o nsZ  
    % Check and prepare the inputs: 3iE-6udCS  
    % ----------------------------- $ A-+E\vQ@  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) I jZ]_*^!  
        error('zernpol:NMvectors','N and M must be vectors.') Lw*;tL<,  
    end H>60D|v[  
    .6>  hD1'  
    if length(n)~=length(m) C%giv9a  
        error('zernpol:NMlength','N and M must be the same length.') _& 8O~8tW  
    end 0:Ar| to$m  
    2R^O,Vu*W  
    n = n(:); x|]\1sb"  
    m = m(:); B\Xh 3l]+j  
    length_n = length(n); CF]i}xpWV  
    yGU .AM  
    if any(mod(n-m,2)) vB[~pQ;Z  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') pv$mZi4i  
    end LmP qLH'(Q  
    6}ce1|mkg/  
    if any(m<0) ;W]D ~X&  
        error('zernpol:Mpositive','All M must be positive.') 4L8z>9D  
    end Lp_$?MCD.  
    Ls&+XlrX8  
    if any(m>n) G+0><,S  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ,eR8 ~(`=  
    end rkkU"l$v  
    94\t1fE  
    if any( r>1 | r<0 ) &~RR&MdZ2  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') BR+nL6sU  
    end z9[[C^C  
    l :/&E 6 9  
    if ~any(size(r)==1) ~A6"sb=  
        error('zernpol:Rvector','R must be a vector.') fX_#S|DlSG  
    end [`d$X^<y;  
    Jlp<koy  
    r = r(:); !<&m]K  
    length_r = length(r); nSS>\$  
    c! @F  
    if nargin==4 gw"~RV0  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); _5mc('  
        if ~isnorm P''X_1oMC  
            error('zernpol:normalization','Unrecognized normalization flag.') 'l~6ErBSg  
        end r!7Y'|  
    else cB#nsu>  
        isnorm = false; \#CM <%  
    end  ?>af'o:  
    Br}h/!NU/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -D^L}b  
    % Compute the Zernike Polynomials =VNSi K>F  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% petW M@  
    '0:i<`qv#g  
    % Determine the required powers of r: Ow3P-UzU3  
    % ----------------------------------- #Z\ O}<  
    rpowers = [];  B$^7h!  
    for j = 1:length(n) .-0%6] cFD  
        rpowers = [rpowers m(j):2:n(j)]; k@V#HC{t  
    end } VEq:^o.  
    rpowers = unique(rpowers); ZsZcQj6G,  
    %K(0W8&  
    % Pre-compute the values of r raised to the required powers, X eoJ$PfT  
    % and compile them in a matrix: q_ %cbAcD  
    % ----------------------------- [|[>}z:  
    if rpowers(1)==0 k6!4Zz_8  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); *:_P8G;  
        rpowern = cat(2,rpowern{:}); B<7/,d'  
        rpowern = [ones(length_r,1) rpowern]; EATu KLP\  
    else y:d{jG^  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); @m~RtC-Q  
        rpowern = cat(2,rpowern{:}); B6] <G-  
    end o%[U  
    &.Q8Mi aT  
    % Compute the values of the polynomials: [3N[i(Wlk  
    % -------------------------------------- w5KPB5/zu  
    z = zeros(length_r,length_n); u=r`t(Z1H  
    for j = 1:length_n #`;/KNp 9  
        s = 0:(n(j)-m(j))/2; 2 -Xdoxw  
        pows = n(j):-2:m(j);  )zq.4  
        for k = length(s):-1:1 K=?VDN  
            p = (1-2*mod(s(k),2))* ... ar.AL'  
                       prod(2:(n(j)-s(k)))/          ... W2Luz;(U  
                       prod(2:s(k))/                 ... |.P/:e9  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Jq ]:<TQ  
                       prod(2:((n(j)+m(j))/2-s(k))); 9b;A1gu  
            idx = (pows(k)==rpowers); Q7gY3flg  
            z(:,j) = z(:,j) + p*rpowern(:,idx); @]HXP_lyD/  
        end \*0yaSQF  
          U47}QDh  
        if isnorm la <npX  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); }P9Ap3?  
        end `zpbnxOL$T  
    end ]"~51HQZ  
    Vp}^NNYf  
    % 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)  _k8A$s<d  
    ;nC.fBu  
    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。 |#6QThK  
    MlLb|!,)T  
    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)