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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 dI!x Ai  
    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! FY%v \`@1*  
     
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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 dsP|j (y  
    function z = zernfun(n,m,r,theta,nflag) GF5WR e(E  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 6U;pYWht  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 2SU G/-P#  
    %   and angular frequency M, evaluated at positions (R,THETA) on the BQVpp,]  
    %   unit circle.  N is a vector of positive integers (including 0), and lTe}[@(  
    %   M is a vector with the same number of elements as N.  Each element \KfngYD]W  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) |D)NP N&  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, .{D[!Dp#h  
    %   and THETA is a vector of angles.  R and THETA must have the same bHcb+TR3  
    %   length.  The output Z is a matrix with one column for every (N,M) <tK 6+isc  
    %   pair, and one row for every (R,THETA) pair. (gBP`*2  
    % nt\6o?W  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike e#Jx|Ej=  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), }h<\qvCcU  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral lbrob' '+  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, v'*  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized kvs^*X''Ep  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. QytqO {B^  
    % :P`sK&b_  
    %   The Zernike functions are an orthogonal basis on the unit circle. Hno@  
    %   They are used in disciplines such as astronomy, optics, and f XxdOn.  
    %   optometry to describe functions on a circular domain. tr"iluwGc  
    % T/G1v;]  
    %   The following table lists the first 15 Zernike functions. E :*!an  
    % X5U.8qI3  
    %       n    m    Zernike function           Normalization " |RP_v2  
    %       -------------------------------------------------- /b,M492  
    %       0    0    1                                 1 z<c^<hE:l  
    %       1    1    r * cos(theta)                    2 [P)'LY6F  
    %       1   -1    r * sin(theta)                    2 eKn&`\j6  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) \1D~4Gz6}  
    %       2    0    (2*r^2 - 1)                    sqrt(3) x|*v(,7b]!  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) STu!v5XY}-  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) {55{ YDqx  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) /_v5B>  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) {U(-cdU{e`  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) _Hi;Y  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) T8>:@EL-k  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .6SdSB ^M  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) W3* BdpTw  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6PT ,m  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) leyhiL<  
    %       -------------------------------------------------- t3u"2B7oG  
    % O_0|Q@  
    %   Example 1: z=<T[Uy  
    % owZj Q  
    %       % Display the Zernike function Z(n=5,m=1) 1B= vrGq  
    %       x = -1:0.01:1; ElhTB  
    %       [X,Y] = meshgrid(x,x); *MW)APw=  
    %       [theta,r] = cart2pol(X,Y); .g DWv  
    %       idx = r<=1; xc Wr hg  
    %       z = nan(size(X)); ,e( |,u  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 3v;o`Em&  
    %       figure <`5>;Xn=  
    %       pcolor(x,x,z), shading interp eS fT +UL  
    %       axis square, colorbar AuUT 'E@E  
    %       title('Zernike function Z_5^1(r,\theta)') k:s}`h _n  
    % ,kuJWaUC@  
    %   Example 2: } IFZ$Y  
    % 7}-.U=tnP  
    %       % Display the first 10 Zernike functions 2\l7=9 ]\3  
    %       x = -1:0.01:1; [VIdw 92  
    %       [X,Y] = meshgrid(x,x); 5tHv'@  
    %       [theta,r] = cart2pol(X,Y); (G<"nnjK  
    %       idx = r<=1; 85$MHod}[,  
    %       z = nan(size(X)); W+BM|'%}|  
    %       n = [0  1  1  2  2  2  3  3  3  3]; #9}KC 9f  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; *CA|}l  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; \lCr~D5  
    %       y = zernfun(n,m,r(idx),theta(idx)); *$/7;CLq  
    %       figure('Units','normalized') |;'V":yDs  
    %       for k = 1:10 rtv\Pf|  
    %           z(idx) = y(:,k); Bk a\0+  
    %           subplot(4,7,Nplot(k)) 1c&/&6 #5  
    %           pcolor(x,x,z), shading interp K6~N{:.s  
    %           set(gca,'XTick',[],'YTick',[]) w_@N T}  
    %           axis square (ZQ{%-i?qR  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) O^G/(  
    %       end -'BJhi\Y]~  
    % <8Nh dCO6  
    %   See also ZERNPOL, ZERNFUN2. ;j=/2vU~@  
    ySP1WK  
    %   Paul Fricker 11/13/2006 M|?qSFv:  
    U4%d #  
    VF!?B>  
    % Check and prepare the inputs: s#;|8_L M  
    % ----------------------------- c.m ' %4  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 5!}xl9D  
        error('zernfun:NMvectors','N and M must be vectors.') [%YA42_`LD  
    end DF%\ 1C>  
    ,d)!&y  
    if length(n)~=length(m) H:#b(&qw2  
        error('zernfun:NMlength','N and M must be the same length.') .a:"B\B`  
    end Xw`vf7z*  
    );}k@w fw)  
    n = n(:); '?E^\\"*  
    m = m(:); ~-GgVi*I  
    if any(mod(n-m,2)) r^ S 4 I&  
        error('zernfun:NMmultiplesof2', ... Vi23pDZ5  
              'All N and M must differ by multiples of 2 (including 0).') /tc*jXB  
    end _zI9 5  
    ->6 /L)  
    if any(m>n) vi4u `  
        error('zernfun:MlessthanN', ... ;TF(opW:  
              'Each M must be less than or equal to its corresponding N.') 24Z7;'  
    end ylLQKdcL  
    a-4'jT:  
    if any( r>1 | r<0 ) r|:|\"Yk  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') T2Z;)e$m_  
    end i]Lt8DiRq  
    <?&GBCe  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 9'o!9_j  
        error('zernfun:RTHvector','R and THETA must be vectors.') :1q+[T/ @  
    end 49nZWv48"_  
    @y;N u   
    r = r(:); ~1`.iA  
    theta = theta(:); *;Dd:D9  
    length_r = length(r); dI5Z*"`R9  
    if length_r~=length(theta) Pr{?A]dQ  
        error('zernfun:RTHlength', ... &1%W-&bc6  
              'The number of R- and THETA-values must be equal.') 6%JKY+n^  
    end f*Xonb  
    @z#;O2  
    % Check normalization: .S`Ue,H  
    % -------------------- Op,Ce4A  
    if nargin==5 && ischar(nflag) $/}*HWVZ  
        isnorm = strcmpi(nflag,'norm'); Ht5 %fcD  
        if ~isnorm ~&CaC  
            error('zernfun:normalization','Unrecognized normalization flag.') J<p<5):R;  
        end }el. qZ  
    else 00Tm0rY  
        isnorm = false; :J@q Xa  
    end Vpt)?];P  
    Z 7t0=U  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <P3r+ 1|R  
    % Compute the Zernike Polynomials <t,uj.9_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% K%Mm'$fTw  
    FviLlly6  
    % Determine the required powers of r: ;j-@ $j  
    % ----------------------------------- Cmm"K[>Rx  
    m_abs = abs(m); X, <&#l  
    rpowers = []; +hpXMO%?  
    for j = 1:length(n) Dn) =V.  
        rpowers = [rpowers m_abs(j):2:n(j)]; Kf76./  
    end {q~Bss{z  
    rpowers = unique(rpowers); F:%= u =  
    30*^ERO  
    % Pre-compute the values of r raised to the required powers, "FT5]h  
    % and compile them in a matrix: b=9(gZ 9  
    % ----------------------------- ?1xBhKq  
    if rpowers(1)==0 n&i WYECz  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 6dV@.(][a  
        rpowern = cat(2,rpowern{:}); o{4ya jt  
        rpowern = [ones(length_r,1) rpowern]; GxLoNVr  
    else 1.o-2:]E  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); brb8C%j}9  
        rpowern = cat(2,rpowern{:}); QUaz;kNC7  
    end _{8f^@I"+  
    JS642T  
    % Compute the values of the polynomials: = PqQJE}  
    % -------------------------------------- Hig=PG5I  
    y = zeros(length_r,length(n)); y%|nE((  
    for j = 1:length(n) >a%NC'~rc  
        s = 0:(n(j)-m_abs(j))/2; UG s <<  
        pows = n(j):-2:m_abs(j); ~=Z&l  
        for k = length(s):-1:1 0Tp?ED_  
            p = (1-2*mod(s(k),2))* ... HPCzh  
                       prod(2:(n(j)-s(k)))/              ... U)zd~ug?m  
                       prod(2:s(k))/                     ... oIt.Pc~;'#  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... q+x4Od3  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); {DI_i +2  
            idx = (pows(k)==rpowers); y+(<Is0w  
            y(:,j) = y(:,j) + p*rpowern(:,idx); h.%VWsAO7  
        end 7 ZET@  
         0'Tq W9P  
        if isnorm nR6~oB{-  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); H.7gSB1  
        end ::dLOf8o  
    end {dg3 qg~  
    % END: Compute the Zernike Polynomials K-)*S\<}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >L$y|8 O  
    %aaOws  
    % Compute the Zernike functions: +#6WORH0S  
    % ------------------------------ Q0gO1 T  
    idx_pos = m>0; r4gkSwy  
    idx_neg = m<0; %Wom]/&,'  
    o2 W pi  
    z = y; A%"XNk  
    if any(idx_pos) !Wvzum@5D  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 8doT`rI1  
    end 7t\kof  
    if any(idx_neg) )"m FlS<I  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); y`E2IE2o  
    end Z%`} `(  
    ?fXg_?+{'g  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) vO]gj/SaT  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. UldKlQ8  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated kC01s  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 56>Zqtp*  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, l2 gI2Cioa  
    %   and THETA is a vector of angles.  R and THETA must have the same pA4/ '7nCl  
    %   length.  The output Z is a matrix with one column for every P-value, *W(b=u  
    %   and one row for every (R,THETA) pair. bLCrh(<  
    % S^I38gJd  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 5;`([oX|_  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) klT6?'S  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ^P"t "  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 CS50wY  
    %   for all p. ^[&*B#(  
    % ;y=w :r\A  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 `WWf?g  
    %   Zernike functions (order N<=7).  In some disciplines it is Bpo~x2p  
    %   traditional to label the first 36 functions using a single mode { zlq6z  
    %   number P instead of separate numbers for the order N and azimuthal \Vm{5[:SA  
    %   frequency M. m;K Mr6sO  
    % /W:}p(>4a  
    %   Example: S]<G|mn,  
    % dXe763~<  
    %       % Display the first 16 Zernike functions ,m3AVHa*G  
    %       x = -1:0.01:1; g|)e3q{M  
    %       [X,Y] = meshgrid(x,x); {EW}Wd  
    %       [theta,r] = cart2pol(X,Y); \M:,Vg  
    %       idx = r<=1; u+(e,t  
    %       p = 0:15; +(PtOo.  
    %       z = nan(size(X)); LFen!FnM  
    %       y = zernfun2(p,r(idx),theta(idx)); YX^{lD1Jj  
    %       figure('Units','normalized') *'d5~dz=  
    %       for k = 1:length(p) wKk  
    %           z(idx) = y(:,k); lba*&j]w=  
    %           subplot(4,4,k) :".!6~:2  
    %           pcolor(x,x,z), shading interp vk{4:^6.TV  
    %           set(gca,'XTick',[],'YTick',[]) oJ6 d:  
    %           axis square HeSnj-mtr}  
    %           title(['Z_{' num2str(p(k)) '}']) "q#(}1Zd  
    %       end iW* 0V3  
    % C$ZY=UXz!T  
    %   See also ZERNPOL, ZERNFUN. BOt\"N  
    `q$DNOrS  
    %   Paul Fricker 11/13/2006 AuO%F YKY  
    8P5xRUkV  
    }pTw$B  
    % Check and prepare the inputs: gMHH3^\VH)  
    % ----------------------------- 7-^df0  
    if min(size(p))~=1 ,wRrx&  
        error('zernfun2:Pvector','Input P must be vector.') JXPn <  
    end `kekc.*-[@  
    qn+mlduU  
    if any(p)>35 ;GH(A=}/Y  
        error('zernfun2:P36', ... zoUW}O  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... t{?_]2vl  
               '(P = 0 to 35).']) WY*}|R2R  
    end _r^&.'q  
    /Zm@.%.  
    % Get the order and frequency corresonding to the function number: >vP^l {SD  
    % ---------------------------------------------------------------- &1w,;45  
    p = p(:); 1$Eiv8xd  
    n = ceil((-3+sqrt(9+8*p))/2); b[*d i{?-  
    m = 2*p - n.*(n+2); JmY"Ja,&  
     ISq^V  
    % Pass the inputs to the function ZERNFUN: _10#rucr  
    % ---------------------------------------- YI\^hP#  
    switch nargin fl)Oto7  
        case 3 -P.) 0d(  
            z = zernfun(n,m,r,theta); Ef?_d]  
        case 4 gO$!_!@LM  
            z = zernfun(n,m,r,theta,nflag); Z.h`yRhO  
        otherwise @]2cL  
            error('zernfun2:nargin','Incorrect number of inputs.') ipU"|{NK  
    end 8|J%IE  
    9k$uo_i'  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) c\RDa|B,  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. o9Agx{'oV  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 2QBq  
    %   order N and frequency M, evaluated at R.  N is a vector of v"bOv"!al  
    %   positive integers (including 0), and M is a vector with the Llg[YBJ7>  
    %   same number of elements as N.  Each element k of M must be a 7v=Nh  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) { >4exyu6  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is zIm$S/Qe*  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 6BMRl%3>Z  
    %   with one column for every (N,M) pair, and one row for every `Th!bk  
    %   element in R. ' ;$2j~  
    % %tpt+N?  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- FTk!Mn88  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is }l_) d  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 5[;p<GqGN  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Nf3Kz#!B  
    %   for all [n,m]. 2^Y1S?g.  
    % ybB/sShGM  
    %   The radial Zernike polynomials are the radial portion of the RL>[t  
    %   Zernike functions, which are an orthogonal basis on the unit n:;2Z  
    %   circle.  The series representation of the radial Zernike > Y[{m $-  
    %   polynomials is ZpWG  
    % }_Jai4O  
    %          (n-m)/2 D^f;dT;-  
    %            __ `Y$LXF~,Om  
    %    m      \       s                                          n-2s  LqU]&AAh  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r rK"$@ tc  
    %    n      s=0 L$Ss]Ar=  
    % Jzfz y0$  
    %   The following table shows the first 12 polynomials. FIW*N r  
    % 44b'40  
    %       n    m    Zernike polynomial    Normalization #&Biu }4D  
    %       --------------------------------------------- x{IOn;>R  
    %       0    0    1                        sqrt(2) -a/5   
    %       1    1    r                           2 (bZ)pW/iw  
    %       2    0    2*r^2 - 1                sqrt(6) wUl}x)xo  
    %       2    2    r^2                      sqrt(6) qzmY]N+w|  
    %       3    1    3*r^3 - 2*r              sqrt(8) R3.tkFZq]  
    %       3    3    r^3                      sqrt(8) {n |Ra[9_  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) jJg 'Y:K9q  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) <A!v'Y  
    %       4    4    r^4                      sqrt(10) ssoe$Gr7>  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) _DH,$evS%  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) + PGfQN  
    %       5    5    r^5                      sqrt(12) J6&;pCAi  
    %       --------------------------------------------- 1C'lT,twl  
    % QT?fp >'  
    %   Example: 1Te: &d  
    % {TlS)i`  
    %       % Display three example Zernike radial polynomials $yhQ)@#1  
    %       r = 0:0.01:1; &OWiA;e?f  
    %       n = [3 2 5]; \e( h6,@  
    %       m = [1 2 1]; b9`MUkGGd  
    %       z = zernpol(n,m,r); y{5ZC~Z<!  
    %       figure \6nWt6M  
    %       plot(r,z) #;2Ju'e#z  
    %       grid on HJoPk'p%  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') =yvyd0|35  
    % }1Q]C"hY  
    %   See also ZERNFUN, ZERNFUN2. V,{ydxfB  
    @P[Tu; 4  
    % A note on the algorithm. uFG]8pj2V1  
    % ------------------------ 3Pkzzyk_|D  
    % The radial Zernike polynomials are computed using the series obS|wTG~  
    % representation shown in the Help section above. For many special S}ZM;M  
    % functions, direct evaluation using the series representation can "-(yZigQ  
    % produce poor numerical results (floating point errors), because v8_HaA$5Y  
    % the summation often involves computing small differences between 9C3q4.$D  
    % large successive terms in the series. (In such cases, the functions RLecKw&1{3  
    % are often evaluated using alternative methods such as recurrence zziujs:  
    % relations: see the Legendre functions, for example). For the Zernike :] {+ 3A  
    % polynomials, however, this problem does not arise, because the MS><7lk-  
    % polynomials are evaluated over the finite domain r = (0,1), and [2.pZB  
    % because the coefficients for a given polynomial are generally all ~kw[Aw3?D\  
    % of similar magnitude. v lOMB  
    % o#H"tYP  
    % ZERNPOL has been written using a vectorized implementation: multiple sf7'8+wj>  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] )\'U$  
    % values can be passed as inputs) for a vector of points R.  To achieve zeNvg/LI^  
    % this vectorization most efficiently, the algorithm in ZERNPOL Y0aO/6  
    % involves pre-determining all the powers p of R that are required to Z!7#"wO9+V  
    % compute the outputs, and then compiling the {R^p} into a single vvFXdHP  
    % matrix.  This avoids any redundant computation of the R^p, and ^|-xmUC  
    % minimizes the sizes of certain intermediate variables. .B_) w:oF  
    % Tld %NE  
    %   Paul Fricker 11/13/2006 ;40!2P8t  
    Z/6B[,V  
    O0';j!?X  
    % Check and prepare the inputs: _U%!&_m6  
    % ----------------------------- b78'yM&  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ??ah  
        error('zernpol:NMvectors','N and M must be vectors.') 9bq#&~+  
    end ,8cw jS2E  
    2*F["E  
    if length(n)~=length(m) j*rra  
        error('zernpol:NMlength','N and M must be the same length.') Tg)Fr)  
    end 4(|x@: wxm  
    `lqMifD  
    n = n(:); Qte=<Z)  
    m = m(:); %AXa(C\1  
    length_n = length(n); $xa#+  
    -Q<3Q_  
    if any(mod(n-m,2)) ?)'j;1_=E3  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Vq -!1.v3  
    end 2.Th29]  
    wVP{R3  
    if any(m<0) P`cEu6:  
        error('zernpol:Mpositive','All M must be positive.') _t 'Kj \  
    end VfAIx]Fa  
    n*{e0,gp`  
    if any(m>n) <RKh%4#~  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') i/NY86A  
    end l;b5v]~  
    =YR/X@&  
    if any( r>1 | r<0 ) 2_ <  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') oJvF)d@gU  
    end #3f\,4K5  
     # G0jMQ  
    if ~any(size(r)==1) r> eOq[z  
        error('zernpol:Rvector','R must be a vector.') :%{MMhb x  
    end F'|D  
    GX4# IRq  
    r = r(:); TWK(vEDM  
    length_r = length(r); "|gNNmr  
    +D& W!m  
    if nargin==4 1X.5cl?V  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); UW8b(b[-6b  
        if ~isnorm b6 &`]O;%  
            error('zernpol:normalization','Unrecognized normalization flag.') ;Bd0 =C  
        end #f5-f  
    else ,/BBG\mJ  
        isnorm = false; fpwge/w  
    end / ~'ZtxA  
    5#p [Q _  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6! \a8q'z  
    % Compute the Zernike Polynomials )wQR2$x~  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% qh Rs5QXL  
    <VS\z(K  
    % Determine the required powers of r: MS%xOB*6  
    % ----------------------------------- S^]i  
    rpowers = []; B<T wTv  
    for j = 1:length(n) fT!n*;h  
        rpowers = [rpowers m(j):2:n(j)]; osB[KRT>("  
    end M\BLuD  
    rpowers = unique(rpowers); v[T5D:  
    iEux`CcJ.  
    % Pre-compute the values of r raised to the required powers, we9R4 *j  
    % and compile them in a matrix: 2_6x2Ia4  
    % ----------------------------- @@9#od O  
    if rpowers(1)==0 f*04=R?w7>  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); <v&>&;>3  
        rpowern = cat(2,rpowern{:}); yHT8I  
        rpowern = [ones(length_r,1) rpowern]; ,YX[6eZr  
    else [~,~ e   
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7Q7z6p/\v  
        rpowern = cat(2,rpowern{:}); #>g]CRN  
    end r4D*$H-rR  
    *q{/`Z{wy  
    % Compute the values of the polynomials: @[D-2s  
    % -------------------------------------- ~rN~Ql%S  
    z = zeros(length_r,length_n); a*o#,T5A  
    for j = 1:length_n `{s:lf  
        s = 0:(n(j)-m(j))/2; BX;Z t9"*  
        pows = n(j):-2:m(j); W! v8'T  
        for k = length(s):-1:1 dU+28  
            p = (1-2*mod(s(k),2))* ... w&:"x@ -|  
                       prod(2:(n(j)-s(k)))/          ... T52A}vf4  
                       prod(2:s(k))/                 ... YQxVeS(  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... _[zO?Div[  
                       prod(2:((n(j)+m(j))/2-s(k))); ' \Z54$  
            idx = (pows(k)==rpowers); fn>MOD!l  
            z(:,j) = z(:,j) + p*rpowern(:,idx); zFmoo4P/  
        end SxcNr5F   
         [H$rdh[+  
        if isnorm %?1k}(qUeY  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); "l6v[yv  
        end f5<qF ]Y/  
    end 8Ih+^Y a  
    v@TP_Ka  
    % 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)   m[>pv1o  
    Q L 1e  
    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。 frh!dN  
    Xh5&J9pw   
    07年就写过这方面的计算程序了。
    让光学不再神秘,让光学变得容易,快速实现客户关于光学的设想与愿望。
    www.rivolens.com