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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 BDB-OJ  
    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! gb@!Co3  
     
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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 kSI,Q!e\  
    function z = zernfun(n,m,r,theta,nflag) EoOrA@N  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. KNK0w5  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N hcN$p2-  
    %   and angular frequency M, evaluated at positions (R,THETA) on the  gu"Agct4  
    %   unit circle.  N is a vector of positive integers (including 0), and xt3IR0  
    %   M is a vector with the same number of elements as N.  Each element u6%56 %^f  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) y XS/3_A{  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, { ! FrI@  
    %   and THETA is a vector of angles.  R and THETA must have the same ]-ZD;kOr  
    %   length.  The output Z is a matrix with one column for every (N,M) Dnd  
    %   pair, and one row for every (R,THETA) pair. ZZeqOu7^  
    % Gt 2rJ<>  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike M8g=t[\  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), HVk3F| ]V  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral n P69W  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ^U`[P@T  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 8:0l5cZE  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. AE<AEq  
    % YJ:CqTy  
    %   The Zernike functions are an orthogonal basis on the unit circle. xDVzHgbf  
    %   They are used in disciplines such as astronomy, optics, and IWMqmCbv  
    %   optometry to describe functions on a circular domain. E^|b3G6T  
    % IAtc^'l#  
    %   The following table lists the first 15 Zernike functions. 7p~@S4  
    % =-vk}O0C  
    %       n    m    Zernike function           Normalization ^ 0TJys%  
    %       -------------------------------------------------- j.m-6  
    %       0    0    1                                 1 !Ug J^v  
    %       1    1    r * cos(theta)                    2 rW1 > t+  
    %       1   -1    r * sin(theta)                    2 ls/:/x(5d  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) R)<>} y  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 2 3>lE}^G  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) [F6=JZ  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) jo"[$%0`  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) vKI,|UD&-  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) g0ug:- R  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ^:DlrI$  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) GLk7# Y  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -R:1-0I$  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) D6v0n6w  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .oSKSld  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) CBO8^M<K  
    %       -------------------------------------------------- JBg",2w |C  
    % MiRMjQ2  
    %   Example 1: -@i2]o  
    % 6?hv ,^  
    %       % Display the Zernike function Z(n=5,m=1) 8LkC/  
    %       x = -1:0.01:1; m&; t;&#  
    %       [X,Y] = meshgrid(x,x); K` U\+AE  
    %       [theta,r] = cart2pol(X,Y); ~v<r\8`OI2  
    %       idx = r<=1; 6o{anHBB  
    %       z = nan(size(X)); l "d&Sgnj  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); M g;;o  
    %       figure )6!SFj>.O  
    %       pcolor(x,x,z), shading interp N;ssO,  
    %       axis square, colorbar /n:s9eq  
    %       title('Zernike function Z_5^1(r,\theta)') Gz6FwU8L  
    % ~_h4|vG  
    %   Example 2: D0-C:gz  
    % Que)kjp  
    %       % Display the first 10 Zernike functions op}x}Ioz  
    %       x = -1:0.01:1; }3vB_0[r  
    %       [X,Y] = meshgrid(x,x); aY"qEH7]  
    %       [theta,r] = cart2pol(X,Y); . vYGJ8(P  
    %       idx = r<=1; M,mj{OY~x  
    %       z = nan(size(X)); b z<wihZj  
    %       n = [0  1  1  2  2  2  3  3  3  3]; W_M]fjL.  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; k*^.-v  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; qWr`cO~hc  
    %       y = zernfun(n,m,r(idx),theta(idx)); 5oORwOP  
    %       figure('Units','normalized') SHh g&~B  
    %       for k = 1:10 }*? e w  
    %           z(idx) = y(:,k); 5*4P_q(AxD  
    %           subplot(4,7,Nplot(k)) m ;[z)-&"  
    %           pcolor(x,x,z), shading interp ~L4"t_-  
    %           set(gca,'XTick',[],'YTick',[]) bt~-=\  
    %           axis square 3>?ip;  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) }3N8EmS  
    %       end Hm4lR{A  
    % q9!5J2P  
    %   See also ZERNPOL, ZERNFUN2. EB>laZy>  
    5#:tL&q  
    %   Paul Fricker 11/13/2006 NUm3E4  
    W.H_G.C%  
    ts)0+x  
    % Check and prepare the inputs: ?#]c{Tlpz  
    % ----------------------------- MR8-xO'w  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ,g^Bu {?  
        error('zernfun:NMvectors','N and M must be vectors.') EStHl(DUPq  
    end /&ph-4\i  
    @tp/0E?  
    if length(n)~=length(m) pY-iz M L  
        error('zernfun:NMlength','N and M must be the same length.') Ry/NfF=  
    end 8/=[mYn`-  
    ^3*gf}  
    n = n(:); h=)Im )  
    m = m(:); V ;>{-p  
    if any(mod(n-m,2)) {J|P2a[  
        error('zernfun:NMmultiplesof2', ... 1 w\Y ._jK  
              'All N and M must differ by multiples of 2 (including 0).') kv)LH{  
    end x6F\|nb  
    z RsA[F#  
    if any(m>n) IK}T. *[  
        error('zernfun:MlessthanN', ... G::6?+S  
              'Each M must be less than or equal to its corresponding N.') 9E (>mN  
    end R?X9U.AcW  
    V+D "_  
    if any( r>1 | r<0 ) b8QW^Z  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Jbs:}]2  
    end Qaagi `  
    tD>m%1'&  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) L{(r@Vu  
        error('zernfun:RTHvector','R and THETA must be vectors.') Sw(%j1uL  
    end ydlH6>  
    z<@$$Z=0UF  
    r = r(:); *TMg.  
    theta = theta(:); $ar:5kif  
    length_r = length(r); sW=@G'}3  
    if length_r~=length(theta) R HF;AX n  
        error('zernfun:RTHlength', ... :  l]>nF4  
              'The number of R- and THETA-values must be equal.') ;z%& 3u/  
    end 0BE%~W  
    G+5G,|}  
    % Check normalization: xD_jfAH'  
    % -------------------- "~FXmKcX  
    if nargin==5 && ischar(nflag) YQ?|Vb U  
        isnorm = strcmpi(nflag,'norm'); yy #Xs:/  
        if ~isnorm vtvr{Uqo@  
            error('zernfun:normalization','Unrecognized normalization flag.') }Efp{E  
        end 5^%^8o  
    else -"a])- j  
        isnorm = false; bfa5X<8  
    end \HH|{   
    JWxPH5L  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4.VEE~sH$  
    % Compute the Zernike Polynomials blp)a  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6+LX oR'  
    SX F F  
    % Determine the required powers of r: EA8(_}  
    % ----------------------------------- =`/X Wem  
    m_abs = abs(m); I5 2wTl0  
    rpowers = []; 89 SsSb  
    for j = 1:length(n) U&B~GJT+  
        rpowers = [rpowers m_abs(j):2:n(j)]; B,gQeW&  
    end @MN>ye'T  
    rpowers = unique(rpowers); j*6!7u.,K  
    Q'\jm=k  
    % Pre-compute the values of r raised to the required powers, !`aodz*PO  
    % and compile them in a matrix: `|PxEif+J  
    % ----------------------------- K1eoZ8=!  
    if rpowers(1)==0 `zep`j&8^  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); i.fDH57  
        rpowern = cat(2,rpowern{:}); q].C>R*ux8  
        rpowern = [ones(length_r,1) rpowern]; QZwRg&d<o  
    else xw?G?(WO  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ~" $9auQtC  
        rpowern = cat(2,rpowern{:}); Kfj*#) SZ  
    end -7+Fb^"L  
    -<<!eH  
    % Compute the values of the polynomials: B3yn:=80  
    % -------------------------------------- :F<a~_k  
    y = zeros(length_r,length(n)); E8-p ,e,  
    for j = 1:length(n) r[\47cG  
        s = 0:(n(j)-m_abs(j))/2; Pb~S{):  
        pows = n(j):-2:m_abs(j); Riw>cVi~  
        for k = length(s):-1:1  ! $d:k|b  
            p = (1-2*mod(s(k),2))* ... MM5#B!BB  
                       prod(2:(n(j)-s(k)))/              ... gjs-j{*  
                       prod(2:s(k))/                     ... As>po +T*  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... oVsl,V  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 95(VY)_6#A  
            idx = (pows(k)==rpowers); &7<~Q\XZbI  
            y(:,j) = y(:,j) + p*rpowern(:,idx); XRNL;X%}7  
        end |L+GM"hg  
         pF8'S{y  
        if isnorm $iF7hyZ  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1w5p*U0 ;  
        end 8[y7(Xw  
    end _c #P  
    % END: Compute the Zernike Polynomials F,EHZ,<V  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i>w>UA*t  
    lX7#3ti:  
    % Compute the Zernike functions: UbuxD})  
    % ------------------------------ (8>k_  
    idx_pos = m>0; V5A7w V3~  
    idx_neg = m<0; 9GQTe1[t4  
    S@*@*>s^  
    z = y; ,f1+jC  
    if any(idx_pos) "n05y}  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); o-(jSaH :;  
    end o@>5[2b4  
    if any(idx_neg) %R_8`4IQ  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); @{$SjR8Q $  
    end ',CcLN  
    F 'h[g.\}  
    % 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) QR,i b  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. g|T' oK  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated (d~'H{q  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive KT|$vw2b  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ] #J ]f  
    %   and THETA is a vector of angles.  R and THETA must have the same *.K}`89T  
    %   length.  The output Z is a matrix with one column for every P-value, c(eu[vj:  
    %   and one row for every (R,THETA) pair. GEvif4  
    % a-kU?&* y  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike <vj&e(D^  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) bGSgph  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) QP qa\87  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 7aU*7!U  
    %   for all p.  M,6AD]  
    % 4e5Ka{# <  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ]V9\4#I4  
    %   Zernike functions (order N<=7).  In some disciplines it is 1f~D Uku=  
    %   traditional to label the first 36 functions using a single mode c8u&ev.U  
    %   number P instead of separate numbers for the order N and azimuthal SWmdU]  
    %   frequency M. *D9QwQ _|  
    % )X7ZX#ttH  
    %   Example: E]e6a^J#  
    % v8WoV*  
    %       % Display the first 16 Zernike functions TQ>1u  
    %       x = -1:0.01:1; @ 8SYV}0H  
    %       [X,Y] = meshgrid(x,x); a(<nk5  
    %       [theta,r] = cart2pol(X,Y); irL ehPX9  
    %       idx = r<=1; a u#IA  
    %       p = 0:15; fa6L+wt4O  
    %       z = nan(size(X)); sNNt0q(  
    %       y = zernfun2(p,r(idx),theta(idx)); 6ZF5f^M^  
    %       figure('Units','normalized') #q=?Zu^Da  
    %       for k = 1:length(p) 3 =S.-  
    %           z(idx) = y(:,k); T{ojla(  
    %           subplot(4,4,k) 19lx;^b  
    %           pcolor(x,x,z), shading interp a{{([uZ  
    %           set(gca,'XTick',[],'YTick',[]) .E@yB`AR  
    %           axis square uEk$Y=p7!  
    %           title(['Z_{' num2str(p(k)) '}']) Kj}}O2  
    %       end i|2Q}$3t2  
    % /FQumqbnt  
    %   See also ZERNPOL, ZERNFUN. "V^(i%E;  
    6T)D6;@L  
    %   Paul Fricker 11/13/2006 jF'S"_/?  
    [jY_e`S  
    $ A ( #^&  
    % Check and prepare the inputs: ^6obxwVG  
    % ----------------------------- v/(< fI^  
    if min(size(p))~=1 _w'4f )7  
        error('zernfun2:Pvector','Input P must be vector.') |KkVt]ZQe9  
    end d F),  
    "Z,'NL>&  
    if any(p)>35 Hm*n ,8_  
        error('zernfun2:P36', ... l3.HL> o  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... \.}* s]6  
               '(P = 0 to 35).']) :r!nz\%WW  
    end _I75[W!  
    2vK{Yw   
    % Get the order and frequency corresonding to the function number: I*'QD)  
    % ---------------------------------------------------------------- 6Si z9  
    p = p(:); !K3 #4   
    n = ceil((-3+sqrt(9+8*p))/2); ,%D \  
    m = 2*p - n.*(n+2); TqzkF7;k4  
    W4X=.vr  
    % Pass the inputs to the function ZERNFUN: <@JK;qm>S  
    % ---------------------------------------- ]G&d`DNV  
    switch nargin <w 8*Ly:L  
        case 3 %e=BC^VW  
            z = zernfun(n,m,r,theta); eB5; wH  
        case 4 mKn:EqA  
            z = zernfun(n,m,r,theta,nflag); Un7jzAvQ  
        otherwise -3F|)qwK  
            error('zernfun2:nargin','Incorrect number of inputs.') ix6j=5{  
    end # bP1rQ0  
    sJMT _yt;  
    % 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) "0pu_  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. eM{,B  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ~X)Aw 3}F  
    %   order N and frequency M, evaluated at R.  N is a vector of 'z>|N{-xG  
    %   positive integers (including 0), and M is a vector with the e@w-4G(;  
    %   same number of elements as N.  Each element k of M must be a !S$LRm\ '  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) Jvgx+{Xu  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is DTH;d-Z  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 7CWz)LT  
    %   with one column for every (N,M) pair, and one row for every <$qe2Ft Uq  
    %   element in R. ?45bvkCT  
    % H0LEK(K  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ,l1A]Wx  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is }f?$QSF  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to sZxf.  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 h3[^uY e  
    %   for all [n,m]. F4`ud;1H  
    % fX^ <H_1$G  
    %   The radial Zernike polynomials are the radial portion of the >;:235'(M  
    %   Zernike functions, which are an orthogonal basis on the unit 4$~eG"wu  
    %   circle.  The series representation of the radial Zernike x2%xrlv<J/  
    %   polynomials is a9}7K/Y=d  
    % CD]"Q1 t}  
    %          (n-m)/2 )O;6S$z9Y  
    %            __ 8Qd*OO  
    %    m      \       s                                          n-2s R6v~Sy&n!  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Bi:%}8STH  
    %    n      s=0 #Ie/|  
    % "}fJ 2G3  
    %   The following table shows the first 12 polynomials. EhB0w;c  
    % `n)e] dn  
    %       n    m    Zernike polynomial    Normalization | KY6IGcqV  
    %       --------------------------------------------- (_1(<Jw  
    %       0    0    1                        sqrt(2) v P;  
    %       1    1    r                           2 AAuH}W>n  
    %       2    0    2*r^2 - 1                sqrt(6) rvfS[@>v  
    %       2    2    r^2                      sqrt(6) R2,Z`I  
    %       3    1    3*r^3 - 2*r              sqrt(8) VC~1QPC9  
    %       3    3    r^3                      sqrt(8) ,6@s N'c  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ?V&# nA  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) g6aIS^mU  
    %       4    4    r^4                      sqrt(10) UZvF5Hoe+O  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) eO%w i.Q  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) @:s (L]  
    %       5    5    r^5                      sqrt(12) = j)5kY`  
    %       --------------------------------------------- ZP-^10  
    % u]0{#wu;g  
    %   Example: wB'GV1|jL  
    % Y2$wL9">  
    %       % Display three example Zernike radial polynomials H. o=4[  
    %       r = 0:0.01:1; `O,^oD4  
    %       n = [3 2 5]; 8/gA]I 6=#  
    %       m = [1 2 1]; U-+o6XX  
    %       z = zernpol(n,m,r); )?y${T   
    %       figure 1egq:bh  
    %       plot(r,z) E>_N|j)9  
    %       grid on -<0xS.^  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') <DR$WsDG  
    % {3Y R_^>?  
    %   See also ZERNFUN, ZERNFUN2. qzk!'J3*r<  
    QzLE9   
    % A note on the algorithm. Nhf@Y}Cu  
    % ------------------------ {HO,d{{  
    % The radial Zernike polynomials are computed using the series +,c]FAx4  
    % representation shown in the Help section above. For many special /0m0""  
    % functions, direct evaluation using the series representation can OV2/?  
    % produce poor numerical results (floating point errors), because +khVi}  
    % the summation often involves computing small differences between {q!GTO  
    % large successive terms in the series. (In such cases, the functions zu_bno!  
    % are often evaluated using alternative methods such as recurrence ~~r7TPq  
    % relations: see the Legendre functions, for example). For the Zernike kY?w] lS)t  
    % polynomials, however, this problem does not arise, because the 3-Bz5sj9  
    % polynomials are evaluated over the finite domain r = (0,1), and ]621Z1  
    % because the coefficients for a given polynomial are generally all 7?@ -|{  
    % of similar magnitude. n:"0mWnL$y  
    % PRal>s&f  
    % ZERNPOL has been written using a vectorized implementation: multiple nPW=m`jG  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] `eEiSf  
    % values can be passed as inputs) for a vector of points R.  To achieve q:cCk#ra  
    % this vectorization most efficiently, the algorithm in ZERNPOL 8hV>Q  
    % involves pre-determining all the powers p of R that are required to <s=i5t My5  
    % compute the outputs, and then compiling the {R^p} into a single W:VX^8</  
    % matrix.  This avoids any redundant computation of the R^p, and Xl,707  
    % minimizes the sizes of certain intermediate variables. PiIP%$72O  
    % Og-v][  
    %   Paul Fricker 11/13/2006 O $ARk+  
    719lfI&s  
    XwZR Kh\>=  
    % Check and prepare the inputs: * ,L e--t  
    % ----------------------------- k 1l K`p  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) qm/#kPlM  
        error('zernpol:NMvectors','N and M must be vectors.') dv cLZK  
    end M 4E|^p=5  
     CH$K_\  
    if length(n)~=length(m) ujWC!*W(Q  
        error('zernpol:NMlength','N and M must be the same length.') bfq%.<W  
    end Z&|Dp*Z  
    BU<Qp$ &  
    n = n(:); ]T=o>%  
    m = m(:); .I Io   
    length_n = length(n); G:!3X)b  
    R$x(3eyx  
    if any(mod(n-m,2)) 0nPg`@e.  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') weMufT  
    end 4axuE]  
    P>EG;u@.  
    if any(m<0) ;w;+<Rd  
        error('zernpol:Mpositive','All M must be positive.') =4uO"o  
    end |RH^|2:x9Q  
    *7{{z%5Pu  
    if any(m>n) N C3XJ 4  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 8/@*6J  
    end F?Fxm*Wa/  
    FI@kE19  
    if any( r>1 | r<0 ) iU|X/>k?  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') p^C$(}Yh  
    end yu jv^2/  
    MKh}2B#S  
    if ~any(size(r)==1) by$S#e f  
        error('zernpol:Rvector','R must be a vector.') `j4OKZ  
    end [U,hb1Wi3  
    N97WI+`  
    r = r(:); Bxf&gDwjgr  
    length_r = length(r); RgD:"zeM  
    *|,ye5"  
    if nargin==4 WtlLqD!_D  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); sWq@E6,I  
        if ~isnorm 7g4IAsoD  
            error('zernpol:normalization','Unrecognized normalization flag.') NftR2  
        end U9XOs)^  
    else Ny$N5/b!!  
        isnorm = false; qgxGq(6K  
    end cS>xT cj  
    ybcCq]cgt  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @=?#nB&  
    % Compute the Zernike Polynomials RijFN.s  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  n[7=  
    (Bss%\  
    % Determine the required powers of r: n],"!>=+  
    % ----------------------------------- ${tBu#$-d  
    rpowers = []; /BrbP7  
    for j = 1:length(n) UAds$ 9  
        rpowers = [rpowers m(j):2:n(j)]; o;v_vCLO  
    end 2 U3WH.o  
    rpowers = unique(rpowers); #;\tgUQ  
    Me-H'Mp~  
    % Pre-compute the values of r raised to the required powers, &g!yRvM!;Q  
    % and compile them in a matrix: -e.ygiK.`S  
    % ----------------------------- ,[u.5vC  
    if rpowers(1)==0 &ZJ$V  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ]eI|_O^u  
        rpowern = cat(2,rpowern{:}); Gdr7d  
        rpowern = [ones(length_r,1) rpowern]; [a k[ZXC,  
    else  s-S|#5  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); V7?Pv Q  
        rpowern = cat(2,rpowern{:}); mW#p&{  
    end (kJ"M4*<F'  
    ~g&Gi)je  
    % Compute the values of the polynomials: -V52?Hq  
    % -------------------------------------- \; zix(N[5  
    z = zeros(length_r,length_n); Gu%}B@4^  
    for j = 1:length_n AE4>pzBe  
        s = 0:(n(j)-m(j))/2; Zv8G[(  
        pows = n(j):-2:m(j); b\+9#)Up@  
        for k = length(s):-1:1 F"a31`L>H  
            p = (1-2*mod(s(k),2))* ... ~GjM:*  
                       prod(2:(n(j)-s(k)))/          ... 9]|G-cyt  
                       prod(2:s(k))/                 ... 2w:cdAv$  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ETaLE[T%1  
                       prod(2:((n(j)+m(j))/2-s(k))); A w)P%r  
            idx = (pows(k)==rpowers); %loe8yt  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 1y.!x~Pi,  
        end (C hL$!x  
         =mh)b]].4\  
        if isnorm !{4bC  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); M9wj };vy  
        end WC 5v#*Jd  
    end )%9 P ;/  
    3qq 6X?y*  
    % 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)  tndtwM*B'  
    =yf LqU  
    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。 }7Si2S  
    wPDA_ns~  
    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)