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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  O@skd2  
    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! =>O{hT ^F  
     
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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 8mO_dQ  
    function z = zernfun(n,m,r,theta,nflag) }$ a *XY1  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. x@|10GC#:  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 8/~@3-9EK  
    %   and angular frequency M, evaluated at positions (R,THETA) on the T ^/\Rr  
    %   unit circle.  N is a vector of positive integers (including 0), and Wq<H sJd/  
    %   M is a vector with the same number of elements as N.  Each element +hs:W'`%  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Ia:M+20n  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, V Y@`)  
    %   and THETA is a vector of angles.  R and THETA must have the same D"{%[;J  
    %   length.  The output Z is a matrix with one column for every (N,M) #.LI `nYA  
    %   pair, and one row for every (R,THETA) pair. j ~I_by  
    % NYR:dH]N~d  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike "DRiJ.|APs  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Bo0T}P~  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral kAW2vh  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Ze?H  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized xg;F};}5$  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. m5W':vM  
    % 0bu!(Tpg7  
    %   The Zernike functions are an orthogonal basis on the unit circle. Q=epUHFs  
    %   They are used in disciplines such as astronomy, optics, and lEw!H^O4  
    %   optometry to describe functions on a circular domain. *QoQ$alHH  
    % 9q2x}  
    %   The following table lists the first 15 Zernike functions. raM{!T:  
    % mw83pU6  
    %       n    m    Zernike function           Normalization 1([?EfC  
    %       -------------------------------------------------- _znpzr9H  
    %       0    0    1                                 1 unr`.}A2>  
    %       1    1    r * cos(theta)                    2 QO4eDSW  
    %       1   -1    r * sin(theta)                    2 8w~X4A,  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ]hbrzv o  
    %       2    0    (2*r^2 - 1)                    sqrt(3) T|5uywA|  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) hS^8/]E={  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) tHFUV\D;,  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) }'uV{$  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) V}h)e3X  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) l_ LH!Tu  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) P R_| 8H|  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0.B'Bvn=s2  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) {$C"yksr  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) T5_rPz  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) \WZSY||C|_  
    %       -------------------------------------------------- ] B>.}  
    % LE g#W  
    %   Example 1: c3O&sa V!  
    % o\nFSG kn  
    %       % Display the Zernike function Z(n=5,m=1) Qo80u? *  
    %       x = -1:0.01:1; F*y7 4j,  
    %       [X,Y] = meshgrid(x,x); z/pxZ B ~"  
    %       [theta,r] = cart2pol(X,Y); ^fbzlu?G4-  
    %       idx = r<=1; Xzqx8Kd  
    %       z = nan(size(X)); fhro"5/4  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 9Wdx"g52_D  
    %       figure <"7Wb"+  
    %       pcolor(x,x,z), shading interp W}WDj:  
    %       axis square, colorbar w1+ %+x  
    %       title('Zernike function Z_5^1(r,\theta)') 2>xEE  
    % 2hb>6Z;r]K  
    %   Example 2: nDz.61$[  
    % ~vMJ?P@  
    %       % Display the first 10 Zernike functions ,fhK  
    %       x = -1:0.01:1; 1gX$U00:  
    %       [X,Y] = meshgrid(x,x); =@d->d  
    %       [theta,r] = cart2pol(X,Y); tjcsT>  
    %       idx = r<=1; "lB%"}  
    %       z = nan(size(X)); u_Xp\RJ  
    %       n = [0  1  1  2  2  2  3  3  3  3]; @$;I%  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; .Z@iz5  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; $/Llzpvny  
    %       y = zernfun(n,m,r(idx),theta(idx)); QF$s([  
    %       figure('Units','normalized') |zy` ]p9  
    %       for k = 1:10 dfXBgsc6i  
    %           z(idx) = y(:,k); <#)Q.P  
    %           subplot(4,7,Nplot(k))  wKbU}29c  
    %           pcolor(x,x,z), shading interp Bsj^R\  
    %           set(gca,'XTick',[],'YTick',[]) >|1-o;UU  
    %           axis square Y 9BKd78Y  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) F1% ^,;  
    %       end pzT`.#N:M  
    % L^Fb;sJYI  
    %   See also ZERNPOL, ZERNFUN2. k: z)Sw  
    }RUK?:lEA  
    %   Paul Fricker 11/13/2006 R 3*{"!O  
    =RHIB1  
    ZSLvr-,D  
    % Check and prepare the inputs: {3``B#}  
    % ----------------------------- JcC2Zn6  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) I.U=%{.  
        error('zernfun:NMvectors','N and M must be vectors.') )c<[@ ::i  
    end &_'3(xIO  
    ,2mq}u>WU  
    if length(n)~=length(m) E=ObfN"ge  
        error('zernfun:NMlength','N and M must be the same length.') nHQWO   
    end oKPG0iM:  
    %kuUQ%W1  
    n = n(:); ;Ao`yC2(v  
    m = m(:); 2|${2u`$&y  
    if any(mod(n-m,2)) 5 axt\  
        error('zernfun:NMmultiplesof2', ... }wC=p>zA  
              'All N and M must differ by multiples of 2 (including 0).') ~NIqO4 D  
    end af&P;#U  
    7s 0pH+  
    if any(m>n) 'T]Ok\  
        error('zernfun:MlessthanN', ...  ">q?(i\  
              'Each M must be less than or equal to its corresponding N.') UryHte  
    end lN*"?%<x>  
    -`PLewvX  
    if any( r>1 | r<0 ) CJ6vS  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') R+9 hog  
    end 8o466m6/  
    A" IaFXB  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) !#S"[q  
        error('zernfun:RTHvector','R and THETA must be vectors.') it->)?"(6  
    end -~ Dn^B1^  
    e]V7 7oc  
    r = r(:); S~^]ib0  
    theta = theta(:); $v=(`=  
    length_r = length(r); '2SZ]   
    if length_r~=length(theta) Sre:l'.  
        error('zernfun:RTHlength', ... Li\b ,_C  
              'The number of R- and THETA-values must be equal.') l=47#zbpZ]  
    end w=thaF.  
    VI)hA ^ S  
    % Check normalization: 1{G@'# (  
    % -------------------- &H2j3De  
    if nargin==5 && ischar(nflag) Us3zvpy)o  
        isnorm = strcmpi(nflag,'norm'); WPPD vB  
        if ~isnorm jm[f|4\  
            error('zernfun:normalization','Unrecognized normalization flag.') Pxgal4{6  
        end @QDpw1;V'  
    else y_T%xWK5  
        isnorm = false; CH$* =3M  
    end kE1k@h#/  
    |oR#j `  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Vr #o]v  
    % Compute the Zernike Polynomials u\)q.`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [!4V_yOb  
    PrF('PH7i  
    % Determine the required powers of r: x #t?`  
    % ----------------------------------- (mx}6A  
    m_abs = abs(m); )9+H[  
    rpowers = []; +B4i,]lCx  
    for j = 1:length(n) T^Hq 5Oy  
        rpowers = [rpowers m_abs(j):2:n(j)]; 0kaMYV?  
    end 3vEwui-5  
    rpowers = unique(rpowers); 4r9AUmJqw  
    E/_n}$Z  
    % Pre-compute the values of r raised to the required powers, 7+rroCr"  
    % and compile them in a matrix: 'i 8`LPQ  
    % ----------------------------- x/%/MFK)>8  
    if rpowers(1)==0 $Y%,?>AL<  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); |j4;XaG)  
        rpowern = cat(2,rpowern{:}); cK'}+  
        rpowern = [ones(length_r,1) rpowern]; R%Xz3Z&|  
    else o>I,$=  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); yg"FF:^T  
        rpowern = cat(2,rpowern{:}); K\5/||gi  
    end Q1x=@lXR  
    la`f@~Bbr1  
    % Compute the values of the polynomials: XKvH^Z4h{l  
    % -------------------------------------- kM3#[#6$!  
    y = zeros(length_r,length(n)); L"( {6H  
    for j = 1:length(n) /=KEM gI?  
        s = 0:(n(j)-m_abs(j))/2; 4"Mq]_D  
        pows = n(j):-2:m_abs(j); t5EYu*  
        for k = length(s):-1:1 mA5sK?W  
            p = (1-2*mod(s(k),2))* ... COA>y?  
                       prod(2:(n(j)-s(k)))/              ... c`7dNx  
                       prod(2:s(k))/                     ... qrc ir-+  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 8B6 -f:  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ?qbq\t  
            idx = (pows(k)==rpowers); Om2w+yU  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 2QKt.a  
        end l2kUa'O-  
         | zOwC9-6  
        if isnorm tQ`|MO&o  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 2FEi-m}  
        end MK <\:g  
    end 5]2 p>%G  
    % END: Compute the Zernike Polynomials "FLiSz%ME  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ccy q~  
    TmJXkR.5  
    % Compute the Zernike functions: >&Y\g?Z6G  
    % ------------------------------ "MyMByomQ  
    idx_pos = m>0; ME*A6/h  
    idx_neg = m<0; -6# _t  
    Sea6xGdq  
    z = y; jH37{S-  
    if any(idx_pos) zEw~t&:e  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); (dHjf;  
    end +(h{ 3Y|  
    if any(idx_neg) 5e&;f  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); A&X XL~yH  
    end 2j$~lI  
    -a7BVEFts  
    % 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) chM-YuN|  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. o Va[  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated (t,mtdD#1  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive LVdtI  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, G^#? ~  
    %   and THETA is a vector of angles.  R and THETA must have the same 8tzL.P^  
    %   length.  The output Z is a matrix with one column for every P-value, {a(<E8-^  
    %   and one row for every (R,THETA) pair. {,= hIXo>  
    % ruy?#rk  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 7?\r9bD  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) <}F(G-kV6  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) gl!ht@;>ak  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 3;R`_#t+  
    %   for all p.  &%T*sR  
    % Uh'W d_?  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 BRv#`  
    %   Zernike functions (order N<=7).  In some disciplines it is UU ' 9  
    %   traditional to label the first 36 functions using a single mode c)c_Qv  
    %   number P instead of separate numbers for the order N and azimuthal V>ZDJW"G!  
    %   frequency M. >]2^5C;  
    % hPUZ{#;n  
    %   Example: |*L/ m0'L  
    % ^Zlbs goZ  
    %       % Display the first 16 Zernike functions wV,=hMTd&\  
    %       x = -1:0.01:1;  JY_!G  
    %       [X,Y] = meshgrid(x,x); MPLeqk$;  
    %       [theta,r] = cart2pol(X,Y); PmT<S,}L  
    %       idx = r<=1; 1 /SB[[g  
    %       p = 0:15; a&[>kO  
    %       z = nan(size(X)); <80M$a g  
    %       y = zernfun2(p,r(idx),theta(idx)); {'#7b# DB>  
    %       figure('Units','normalized') etk|%%J  
    %       for k = 1:length(p) P#"_H}qC*  
    %           z(idx) = y(:,k); K] &GSro  
    %           subplot(4,4,k) ,? Q1JZPy@  
    %           pcolor(x,x,z), shading interp {fMo#`9=  
    %           set(gca,'XTick',[],'YTick',[]) ,/[1hhP@  
    %           axis square Gi&/`vm  
    %           title(['Z_{' num2str(p(k)) '}']) =q[ynZ8O\w  
    %       end M->#WGl\B  
    % VYrs4IFT$  
    %   See also ZERNPOL, ZERNFUN. ;`Z>^.CB  
    r$%,k*X^ k  
    %   Paul Fricker 11/13/2006 \3)U~[O>:  
    ~L.5;8a3Pe  
    {6F]w_\  
    % Check and prepare the inputs: H-8_&E?6m  
    % ----------------------------- iu{QHjZK(  
    if min(size(p))~=1 <XzRRCYQ  
        error('zernfun2:Pvector','Input P must be vector.') )7Oj  
    end ?l`|j*  
    u3HaWf3  
    if any(p)>35 $[b1_Db  
        error('zernfun2:P36', ... yaAg!mW  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... !@> :k3DC&  
               '(P = 0 to 35).']) X$ A ]7t  
    end #vTF:r  
    g5 y*-t  
    % Get the order and frequency corresonding to the function number: *k0;R[IAV  
    % ---------------------------------------------------------------- Wr,pm#gl6  
    p = p(:); mHNqzdaa  
    n = ceil((-3+sqrt(9+8*p))/2); =BzBM`-o  
    m = 2*p - n.*(n+2); +@<@x4yt  
    &CfzhIi*!  
    % Pass the inputs to the function ZERNFUN: &pAmFe  
    % ---------------------------------------- UBx0Z0Y  
    switch nargin z@2nre  
        case 3 ;\iu*1>Z,&  
            z = zernfun(n,m,r,theta);  8vUq8[[  
        case 4 &p=(0$0&-  
            z = zernfun(n,m,r,theta,nflag); :oJ!9\5  
        otherwise bW zUWLa  
            error('zernfun2:nargin','Incorrect number of inputs.') `[tYe<  
    end [LSs|f  
    ^!SwY_>  
    % 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) S#|5&SR  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. m_UzmWF  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of xGX U7w:X  
    %   order N and frequency M, evaluated at R.  N is a vector of ))uki*UNK  
    %   positive integers (including 0), and M is a vector with the %UCuI9  
    %   same number of elements as N.  Each element k of M must be a !r+SE  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) *w1R>  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is @FuX^Q.[  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix |exjrsmM*  
    %   with one column for every (N,M) pair, and one row for every BR0P :h  
    %   element in R. - [7S.  
    % ]ov"&,J  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- R<ZyP~  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is "C]_pWk  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to S&~;l/  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 *l'5z)]  
    %   for all [n,m]. {c=H#- A  
    % |A:+[35  
    %   The radial Zernike polynomials are the radial portion of the m[&pR2T  
    %   Zernike functions, which are an orthogonal basis on the unit N#vV;  
    %   circle.  The series representation of the radial Zernike .T7S1C $HP  
    %   polynomials is MT.D#jv&  
    % /Y*6mQ:  
    %          (n-m)/2 Ga $EM  
    %            __ %<'PSri  
    %    m      \       s                                          n-2s q]z%<`.9*  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r MJ`BlE,Fmb  
    %    n      s=0 1.q a//'RW  
    % ~H`(zzk  
    %   The following table shows the first 12 polynomials. I#](mRJ6  
    % +q)B4A'J!  
    %       n    m    Zernike polynomial    Normalization _,E! <  
    %       --------------------------------------------- bOdyrynh  
    %       0    0    1                        sqrt(2) O`='8'6zW\  
    %       1    1    r                           2 #jX%nqMxW  
    %       2    0    2*r^2 - 1                sqrt(6) yyu f  
    %       2    2    r^2                      sqrt(6) *Duxabo?  
    %       3    1    3*r^3 - 2*r              sqrt(8) PH]ui=  
    %       3    3    r^3                      sqrt(8) nV?e(}D  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) YX6[m6L U  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) REKv&^FLN  
    %       4    4    r^4                      sqrt(10) )Xd2qbi  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) FLOSdMYdw  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ,2^zX]dgM  
    %       5    5    r^5                      sqrt(12) C-L["O0[  
    %       --------------------------------------------- HXRK<6k$  
    % c^BeT;  
    %   Example: "%{J$o  
    % -nC!kpo  
    %       % Display three example Zernike radial polynomials :X ~{,J  
    %       r = 0:0.01:1; rTLo6wI  
    %       n = [3 2 5]; ~0XV[$`L  
    %       m = [1 2 1]; FR1se  
    %       z = zernpol(n,m,r); agxR V  
    %       figure Rac4a@hZ  
    %       plot(r,z) s4Y7x.-  
    %       grid on 'Lv>!s 7  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') =;@?bTmqD  
    % {,b:f  
    %   See also ZERNFUN, ZERNFUN2. ys$X!Ep  
    B3iU#   
    % A note on the algorithm. L# NW<T  
    % ------------------------ 1r;.r|  
    % The radial Zernike polynomials are computed using the series #u6ZCv7u  
    % representation shown in the Help section above. For many special z9[BQ(9t  
    % functions, direct evaluation using the series representation can 9<S};I;  
    % produce poor numerical results (floating point errors), because NHq*&xy  
    % the summation often involves computing small differences between (.54`[2+L  
    % large successive terms in the series. (In such cases, the functions h"5!puN+  
    % are often evaluated using alternative methods such as recurrence {`CmE/`{  
    % relations: see the Legendre functions, for example). For the Zernike \3v}:E+3  
    % polynomials, however, this problem does not arise, because the 6N.+  
    % polynomials are evaluated over the finite domain r = (0,1), and 60&4?<lR4  
    % because the coefficients for a given polynomial are generally all w0N8a%  
    % of similar magnitude. SRf .8j  
    % q2KWSh5  
    % ZERNPOL has been written using a vectorized implementation: multiple #9) D.d|5  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 9Wi+7_)  
    % values can be passed as inputs) for a vector of points R.  To achieve vx8-~Oq{|;  
    % this vectorization most efficiently, the algorithm in ZERNPOL a)GT\1q  
    % involves pre-determining all the powers p of R that are required to gXI8$W>  
    % compute the outputs, and then compiling the {R^p} into a single BSib/)p   
    % matrix.  This avoids any redundant computation of the R^p, and bp?4)C*R  
    % minimizes the sizes of certain intermediate variables. |=9=a@l]P  
    % R-2V C  
    %   Paul Fricker 11/13/2006 >X!A/; $  
    -%#F5br%  
    ~}fQ.F*7R  
    % Check and prepare the inputs: =ADdfuKN  
    % ----------------------------- JHZ`LWq  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) P_f^gB7  
        error('zernpol:NMvectors','N and M must be vectors.') Ue22,Pp6  
    end El)WjcmH  
    h16i]V  
    if length(n)~=length(m) ($ l t@j  
        error('zernpol:NMlength','N and M must be the same length.') N N;'QiE  
    end ,>Lj>g{~  
    4BgrG[l)  
    n = n(:); ZhpbbS  
    m = m(:); 9:tn! <^=I  
    length_n = length(n); }yW*vy6`  
    2XN];,{  
    if any(mod(n-m,2)) ,:1_I`d>#X  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') O~WT$  
    end sjISVJ?  
    ?Q XS?  
    if any(m<0) T@ecWRro  
        error('zernpol:Mpositive','All M must be positive.') }JKK"d}U  
    end @3:oo /;  
    {tlt5p!4  
    if any(m>n) [Q=NGHB1/  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') .%rB-vO:g  
    end )ZA3m _w]  
    f4TNy^-  
    if any( r>1 | r<0 ) dLs40 -R  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') Bco_\cpt]z  
    end eZ^-gk?  
    *75?%l  
    if ~any(size(r)==1) e+t2F |xDh  
        error('zernpol:Rvector','R must be a vector.') z` :uvEX0  
    end j5yxdjx9  
    9. 7XRxR^  
    r = r(:); 'kz[Gh*8  
    length_r = length(r); C {GSf`D!T  
    !h "6h  
    if nargin==4 Jn>6y:s  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 0WjPo  
        if ~isnorm %j3 *j  
            error('zernpol:normalization','Unrecognized normalization flag.') UF)4K3X  
        end x3JX}yCX  
    else ]^63n/Twj  
        isnorm = false; +4,v. B@  
    end )OAd[u<  
    nz=X/J6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z,~EH  
    % Compute the Zernike Polynomials -R[ *S "  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BWbM$@'x  
    `n# {}%  
    % Determine the required powers of r: '01ifA^  
    % ----------------------------------- ;c-J)Ky  
    rpowers = []; _;Q1P gT  
    for j = 1:length(n) JDyP..Dt  
        rpowers = [rpowers m(j):2:n(j)]; ,c%>M^d  
    end WzC_M>_  
    rpowers = unique(rpowers); V _&>0P{q  
    `nxm<~-\  
    % Pre-compute the values of r raised to the required powers, KKz{a{ePY%  
    % and compile them in a matrix: jo.Sg:7&  
    % ----------------------------- U2DE"  
    if rpowers(1)==0 CCp8,  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); J8T?=%?=  
        rpowern = cat(2,rpowern{:}); c:/ H}2/C  
        rpowern = [ones(length_r,1) rpowern]; Wq+6`o  
    else m{/?6h 1  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); [HKTXF{n  
        rpowern = cat(2,rpowern{:}); (qDu|S3P  
    end V'";u?h#S  
    ;BsPms@U  
    % Compute the values of the polynomials: c({V[eGY  
    % -------------------------------------- <23oyMR0  
    z = zeros(length_r,length_n); 5/:BtlFx  
    for j = 1:length_n a]<y*N?qu  
        s = 0:(n(j)-m(j))/2; pV>M, f  
        pows = n(j):-2:m(j); h|_E>6d)  
        for k = length(s):-1:1 /mb?C/CI  
            p = (1-2*mod(s(k),2))* ... <Ct b^4$  
                       prod(2:(n(j)-s(k)))/          ... GgoPwl#{  
                       prod(2:s(k))/                 ... 0i[,`>-Av  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 'nOc_b0  
                       prod(2:((n(j)+m(j))/2-s(k))); C0xj M0  
            idx = (pows(k)==rpowers); Q1fJ`A=  
            z(:,j) = z(:,j) + p*rpowern(:,idx); *Txt`z[|  
        end !+;'kI2  
         ~ >af"<  
        if isnorm bp06xHMu  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); q1Ja*=r  
        end IpX>G]"-C  
    end >Cb[  
    CH6^;.  
    % 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)  4a'GWzUtS  
    (E'f'g  
    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。  4wLp  
    5v51:g>c  
    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)