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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 0%vXPlfnY  
    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! #bnFR  
     
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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 J7$5<  
    function z = zernfun(n,m,r,theta,nflag) Es1Yx\/:  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. cvi+AZ=  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N anHP5gD  
    %   and angular frequency M, evaluated at positions (R,THETA) on the pz~AsF  
    %   unit circle.  N is a vector of positive integers (including 0), and Qr$ uFh/y  
    %   M is a vector with the same number of elements as N.  Each element %=%jy  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 3%>"|Ye}A  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 76(&O  
    %   and THETA is a vector of angles.  R and THETA must have the same yin"+&<T  
    %   length.  The output Z is a matrix with one column for every (N,M) Fod2KS;g  
    %   pair, and one row for every (R,THETA) pair.  ]Ocf %(  
    % <5G*#0gw  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike @zW'!Ol  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), =DUsQN!  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ,@8>=rT  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, f?[IwA`  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Ju Kj  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. PKty'}KF  
    % -(@dMY  
    %   The Zernike functions are an orthogonal basis on the unit circle. K'7i$bl%  
    %   They are used in disciplines such as astronomy, optics, and Kmk<  
    %   optometry to describe functions on a circular domain. o0_RU<bWN  
    % #P<v[O/rA  
    %   The following table lists the first 15 Zernike functions.  Hi|'  
    % esWgYAc3{  
    %       n    m    Zernike function           Normalization FX4](oM  
    %       -------------------------------------------------- +(QGlRd  
    %       0    0    1                                 1 bw ' yX  
    %       1    1    r * cos(theta)                    2 -aXV}ZY"  
    %       1   -1    r * sin(theta)                    2 !zVuO*+  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 48Z{wV,  
    %       2    0    (2*r^2 - 1)                    sqrt(3) [ wi "  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ;XRLp:y  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) fOF02WP^  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) T1LtO O  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ;a[56W  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) (Rve<n6{A  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10)   9Ld3  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) &Dgho  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) "n=`{~F  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Da0E)  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ]+{Cy\*kR  
    %       -------------------------------------------------- H_3S#.  
    % 1BmevE a)  
    %   Example 1: {;=I69 X  
    % +MIDq{B  
    %       % Display the Zernike function Z(n=5,m=1) &NL=Bd  
    %       x = -1:0.01:1;  +,gI|  
    %       [X,Y] = meshgrid(x,x); @q}.BcSg  
    %       [theta,r] = cart2pol(X,Y); %F` c Nw]  
    %       idx = r<=1; !FX;QD@"  
    %       z = nan(size(X)); "W?k~.uw  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Y7zg  
    %       figure eo24I0 `N  
    %       pcolor(x,x,z), shading interp x~?,Wv|cm  
    %       axis square, colorbar "[q/2vC  
    %       title('Zernike function Z_5^1(r,\theta)') "@;q! B.qo  
    % )0 .gW  
    %   Example 2: MMN2X xS  
    % @(,k%84z  
    %       % Display the first 10 Zernike functions hCD0Zel  
    %       x = -1:0.01:1; ;$wS<zp6  
    %       [X,Y] = meshgrid(x,x); \f}S Hh  
    %       [theta,r] = cart2pol(X,Y); &jT>)MXPu  
    %       idx = r<=1; R#"kh/M  
    %       z = nan(size(X)); A|,\}9)4X[  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ,2qJXMg"=$  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ;O}%_ef@  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ?Lbw o<E  
    %       y = zernfun(n,m,r(idx),theta(idx)); b'pbf  
    %       figure('Units','normalized') :_~UO^*h  
    %       for k = 1:10 Ou"QUn|  
    %           z(idx) = y(:,k); eu@-v"=w  
    %           subplot(4,7,Nplot(k)) #I'W[\l~+  
    %           pcolor(x,x,z), shading interp i/2OE&*O[  
    %           set(gca,'XTick',[],'YTick',[]) $W<H[k&(B  
    %           axis square [rC-3sGar  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 5?r#6:(yI  
    %       end >_!pg<{,  
    % ClCb.Ozj4  
    %   See also ZERNPOL, ZERNFUN2. @NWjYHM[`  
    E ~<SEA  
    %   Paul Fricker 11/13/2006 ;nyV)+t+a  
    9<I@}w  
    +AhR7R!  
    % Check and prepare the inputs: v`A^6)U#M  
    % ----------------------------- W$O^IC  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) PspH[db  
        error('zernfun:NMvectors','N and M must be vectors.') ,%w_E[2  
    end 1&\_|2  
    }QU9+<Z[r  
    if length(n)~=length(m) G(~d1%(  
        error('zernfun:NMlength','N and M must be the same length.') `r e]Q0IO  
    end .+t{o [  
    {mY<R`Ee  
    n = n(:); _iLXs  
    m = m(:); kSv?p1\@&P  
    if any(mod(n-m,2)) LzB)o\a  
        error('zernfun:NMmultiplesof2', ... Tw/kD)u{  
              'All N and M must differ by multiples of 2 (including 0).') 'g$~ij ;x  
    end JR|yg=E  
    oUIa/}}w5  
    if any(m>n) :{pvA;f  
        error('zernfun:MlessthanN', ... *[*LtyCQt4  
              'Each M must be less than or equal to its corresponding N.') BQ{Gp 2N  
    end 3Bee6N>  
    }jBr[S5  
    if any( r>1 | r<0 ) lEIX,amwa  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') &Y%Kr`.h  
    end pN6!IxN$  
    /tM<ois*  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) v|t_kNX;v*  
        error('zernfun:RTHvector','R and THETA must be vectors.') #F@53N  
    end 8+{WH/}y8  
    ^)<>5.%1''  
    r = r(:); [X0Wfb}{  
    theta = theta(:); {>tgNW>)  
    length_r = length(r); ]|18tVXc  
    if length_r~=length(theta) _m;0%]+  
        error('zernfun:RTHlength', ... %Js3Y9AL C  
              'The number of R- and THETA-values must be equal.') ; 29q  
    end ;ZPAnd:pb  
    o\vIYQ   
    % Check normalization: G,9osTt/  
    % --------------------  Z+`mla  
    if nargin==5 && ischar(nflag) YNA %/  
        isnorm = strcmpi(nflag,'norm'); hV#+joT8i  
        if ~isnorm #~*fZ|sq+3  
            error('zernfun:normalization','Unrecognized normalization flag.') uy)iB'st&  
        end {LYA?w^GT  
    else ,uqSq  
        isnorm = false; ?1?D[7$  
    end !^cQPX2<  
    ugcWFB5|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !31v@v:)  
    % Compute the Zernike Polynomials lTW5> %  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8.HqQ:?&2t  
    v/=O:SM}  
    % Determine the required powers of r: a9 7A{7I&  
    % ----------------------------------- vT"T*FKh:  
    m_abs = abs(m); ?&EPZqI  
    rpowers = []; B;9X{"  
    for j = 1:length(n) KGd L1~  
        rpowers = [rpowers m_abs(j):2:n(j)]; :$."x '  
    end Ug*:o d  
    rpowers = unique(rpowers); GV#"2{t j  
    "gjy+eosY  
    % Pre-compute the values of r raised to the required powers, z1wy@1o'  
    % and compile them in a matrix: ,2q LiE>  
    % ----------------------------- b/cc\d<  
    if rpowers(1)==0 ~f0Bu:A)  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); [U@#whEO  
        rpowern = cat(2,rpowern{:}); 0][PL%3Z  
        rpowern = [ones(length_r,1) rpowern]; m-S4"!bl  
    else :\9E%/aAD  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); -8;U1^#  
        rpowern = cat(2,rpowern{:}); e8 4[B.  
    end 0FD#9r  
    u!?cKZw  
    % Compute the values of the polynomials: hz{=@jX  
    % -------------------------------------- uq~$HXdc  
    y = zeros(length_r,length(n)); &+;z`A'|8  
    for j = 1:length(n) wZ/Zc} .  
        s = 0:(n(j)-m_abs(j))/2; *t.L` G  
        pows = n(j):-2:m_abs(j); Jj4!O3\I  
        for k = length(s):-1:1 ' _Ij9{M  
            p = (1-2*mod(s(k),2))* ... ,0O9!^  
                       prod(2:(n(j)-s(k)))/              ... (b%&DyOt  
                       prod(2:s(k))/                     ... p9rnhqH6  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... h'YC!hjp   
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Wcd;B7OH  
            idx = (pows(k)==rpowers); T(zE RWo  
            y(:,j) = y(:,j) + p*rpowern(:,idx);  vp7J';  
        end B'"(qzE-kM  
         hi4#8W  
        if isnorm !PJD+SrG  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); >utm\!Gac  
        end *-"DZ  
    end k2DT+}u7G  
    % END: Compute the Zernike Polynomials }bIbMEMn  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0J7)UqMf.  
    XM o#LS  
    % Compute the Zernike functions: ':YFm  
    % ------------------------------ It>8XKS  
    idx_pos = m>0; 0m k-o  
    idx_neg = m<0; ovJwo r  
    0V6gNEAUg  
    z = y; N9@@n:JT  
    if any(idx_pos) dnt: U!TW@  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); $?RxmWsP  
    end &?C% -"|c  
    if any(idx_neg) e<o{3*%p)  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ?EQ]f34  
    end VsEMF i=  
    <nDuN*|  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) g2Pa-}{  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ^W!w~g+  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated \yDr  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive a&oz<4oT  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, }i,LP1R  
    %   and THETA is a vector of angles.  R and THETA must have the same  <sdC#j  
    %   length.  The output Z is a matrix with one column for every P-value, d +0(H   
    %   and one row for every (R,THETA) pair. b {fZU?o  
    % 7;0$UYDU*  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike UC&f  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2)  %T9'dcM  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) IJX75hE0g  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 G-FeDP  
    %   for all p. MP"Pqt  
    % {[B^~Y>Lr  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ?+6w8j%\  
    %   Zernike functions (order N<=7).  In some disciplines it is iIrH&}2  
    %   traditional to label the first 36 functions using a single mode ,EhQTVJ  
    %   number P instead of separate numbers for the order N and azimuthal l6o?(!:!%  
    %   frequency M. mx\b6w7  
    % 7O)j]eeoL  
    %   Example: NlLgXn!  
    % OJX* :Q  
    %       % Display the first 16 Zernike functions PeCU V6  
    %       x = -1:0.01:1; bWp40&vx  
    %       [X,Y] = meshgrid(x,x); 4-ijuqjN  
    %       [theta,r] = cart2pol(X,Y); m]}%Ag^x  
    %       idx = r<=1; C|"BMam  
    %       p = 0:15; uh,~Cv XU]  
    %       z = nan(size(X)); 6k14xPj  
    %       y = zernfun2(p,r(idx),theta(idx)); dt -EY  
    %       figure('Units','normalized') c;RB!`9"  
    %       for k = 1:length(p) u~ %xU~v  
    %           z(idx) = y(:,k); g$=y#<2?  
    %           subplot(4,4,k) eAkC-Fm  
    %           pcolor(x,x,z), shading interp tzl`|UwF  
    %           set(gca,'XTick',[],'YTick',[]) `bXP )$  
    %           axis square Fh)`A5#  
    %           title(['Z_{' num2str(p(k)) '}']) #p<1@,  
    %       end 4(2iR0N  
    % [}p/pj=  
    %   See also ZERNPOL, ZERNFUN. X MkyX&y  
    i;+]Y   
    %   Paul Fricker 11/13/2006 R.ZC|bPiD  
    =lyP &u  
    {~cG'S Y%  
    % Check and prepare the inputs: # MpW\yX  
    % ----------------------------- 'j6)5WL$  
    if min(size(p))~=1 "l83O8 L  
        error('zernfun2:Pvector','Input P must be vector.') , Oqd4NS  
    end o XKH,r  
    =^nb-9.  
    if any(p)>35 ]U9f4ODt  
        error('zernfun2:P36', ... vsFRWpq  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ,gdf7&r  
               '(P = 0 to 35).']) (t^&L  
    end f[S$ Gu4-  
    6x6xv:\  
    % Get the order and frequency corresonding to the function number: 4 VPJv>^  
    % ---------------------------------------------------------------- j?eWh#[K"  
    p = p(:); CuS"Wj  
    n = ceil((-3+sqrt(9+8*p))/2); hu=b ,  
    m = 2*p - n.*(n+2); h~\bJ*Zp  
    L\O}q  
    % Pass the inputs to the function ZERNFUN: -;VKtBXP</  
    % ---------------------------------------- \W^+aNbv=8  
    switch nargin d5b \kRr  
        case 3 *ud"?{)Z  
            z = zernfun(n,m,r,theta); ~c;D@.e\  
        case 4 u0 & aw  
            z = zernfun(n,m,r,theta,nflag); `#v(MK{9+V  
        otherwise $s[DT!8N  
            error('zernfun2:nargin','Incorrect number of inputs.') Muhq,>!U  
    end SfHs,y6  
    n aQ0TN,  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) jjs1Vj1@<  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. h3B s  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ~-1!?t/%  
    %   order N and frequency M, evaluated at R.  N is a vector of 81(.{Y839_  
    %   positive integers (including 0), and M is a vector with the 9PpPAF  
    %   same number of elements as N.  Each element k of M must be a $U{ \T4  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) > x$eKN  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ^p'iX4M  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix upD 2vtU  
    %   with one column for every (N,M) pair, and one row for every 9}\{0;9  
    %   element in R. 2N,<~L`FX'  
    % #D{jNSB  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- aN87^[  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 01 UEd8  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to *2$I, ~(P  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 "F,d}3}  
    %   for all [n,m]. d"7l<y5  
    % C2 ~t  
    %   The radial Zernike polynomials are the radial portion of the h^u 9W7.  
    %   Zernike functions, which are an orthogonal basis on the unit dHE\+{K%-  
    %   circle.  The series representation of the radial Zernike }s0?RH  
    %   polynomials is -ZmccT"8  
    % FG?69b>  
    %          (n-m)/2 L(a){<c  
    %            __ #i#4h<R  
    %    m      \       s                                          n-2s e``X6=rcG  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ~|LlT^C  
    %    n      s=0 =bVaB<!  
    % "`sr#  
    %   The following table shows the first 12 polynomials. z[7j`J|Kk  
    % F]#rH   
    %       n    m    Zernike polynomial    Normalization **zh>Y}6  
    %       --------------------------------------------- ?5%0zMC  
    %       0    0    1                        sqrt(2) O Oa}+^-j  
    %       1    1    r                           2 4 Ar\`{c>  
    %       2    0    2*r^2 - 1                sqrt(6) B/*`u  
    %       2    2    r^2                      sqrt(6) dy4! >zxF  
    %       3    1    3*r^3 - 2*r              sqrt(8) ~"<AYJlO  
    %       3    3    r^3                      sqrt(8) fvW7a8k3  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) QQ+?J~  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) -^y$RJC  
    %       4    4    r^4                      sqrt(10) oGK 1D  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) *AO^oBeY  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 8x`?Yc  
    %       5    5    r^5                      sqrt(12) RJ#xq#l  
    %       --------------------------------------------- 1:.0^?Gz  
    % l.DC20bs  
    %   Example: $oefG}h2  
    % \'rh7!v-u  
    %       % Display three example Zernike radial polynomials IX!Q X  
    %       r = 0:0.01:1; EF7Y4lp  
    %       n = [3 2 5]; (6xrs_ea  
    %       m = [1 2 1]; U!GG8;4  
    %       z = zernpol(n,m,r); jIZpv|t)  
    %       figure JN{.-k4Ha  
    %       plot(r,z) eiL  ;  
    %       grid on /$'R!d5r  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') yQ33JQr  
    % ``~7z;E%@  
    %   See also ZERNFUN, ZERNFUN2. =:T pH>f*  
    $6BD6\@  
    % A note on the algorithm. .A: #l?  
    % ------------------------ y4aW8J#  
    % The radial Zernike polynomials are computed using the series !nQ!J+ g  
    % representation shown in the Help section above. For many special x Z 3b)j2D  
    % functions, direct evaluation using the series representation can cx]&ae*  
    % produce poor numerical results (floating point errors), because Et\z^y  
    % the summation often involves computing small differences between ZH9sf~7  
    % large successive terms in the series. (In such cases, the functions \r_-gn'1b  
    % are often evaluated using alternative methods such as recurrence #6])\  
    % relations: see the Legendre functions, for example). For the Zernike &ukYTDM  
    % polynomials, however, this problem does not arise, because the H /E.R[\+x  
    % polynomials are evaluated over the finite domain r = (0,1), and L-7?:  
    % because the coefficients for a given polynomial are generally all =.@{ uu;  
    % of similar magnitude. ogt<vng  
    % 8pc=Oor2Tv  
    % ZERNPOL has been written using a vectorized implementation: multiple ;&|MNN^  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Ax D&_GT  
    % values can be passed as inputs) for a vector of points R.  To achieve 6 ufF34tA  
    % this vectorization most efficiently, the algorithm in ZERNPOL LY}9$1G]  
    % involves pre-determining all the powers p of R that are required to YT)jBS~&  
    % compute the outputs, and then compiling the {R^p} into a single 5*.JXx E;U  
    % matrix.  This avoids any redundant computation of the R^p, and DKd:tL24&  
    % minimizes the sizes of certain intermediate variables. (Rqn)<<2  
    % 3"ALohlL  
    %   Paul Fricker 11/13/2006 Ae&470  
    S4/CL4=  
    mnK SO  
    % Check and prepare the inputs: +{Qk9Z  
    % ----------------------------- )aW;w|#n  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 0dv# [  
        error('zernpol:NMvectors','N and M must be vectors.') PU0Ha  
    end {rQ SB;3  
    MfJ;":]O!  
    if length(n)~=length(m) 3)jFv7LAU  
        error('zernpol:NMlength','N and M must be the same length.') ;T!w$({V0z  
    end u n{LwZH  
    MP5 vc5[  
    n = n(:); 9M3XHj  
    m = m(:); _K(w &Kr  
    length_n = length(n); >w,o|  
    p^QEk~qw  
    if any(mod(n-m,2)) rJ2yi6TB\  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') [If%+mHdU  
    end QnsD,F; /  
    huj 6Ysr  
    if any(m<0) fq-zgqF<  
        error('zernpol:Mpositive','All M must be positive.') B d#D*"gx  
    end vrr&Ve  
    "bI'XaSv  
    if any(m>n) > /,7j:X  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') z8HOig?  
    end zGtWyXP  
    QU4/hS;Ux  
    if any( r>1 | r<0 ) wc&%icF*cr  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') &L&6 y()G  
    end oD_n+95B  
    El3Y1g3+3  
    if ~any(size(r)==1) &e2|]C4  
        error('zernpol:Rvector','R must be a vector.') C%hMh/Li;  
    end [1 pWg^  
    tO0MYEx"  
    r = r(:); 1C,=1bY  
    length_r = length(r); Q8] lz}  
    y~,mIM$[@  
    if nargin==4 O=2"t%Gc  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 6Vr:?TI7  
        if ~isnorm Lye^G% {  
            error('zernpol:normalization','Unrecognized normalization flag.') [sxJ<  
        end uc>u=kEue  
    else f{s}[p~  
        isnorm = false;  y5"b(nb  
    end azEN_oUV  
    >_R,^iH"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [^oTC;  
    % Compute the Zernike Polynomials teRK#: .P  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }vp pn=[Y  
    *{_WM}G  
    % Determine the required powers of r: 3aD\J_  
    % ----------------------------------- Y9YE:s  
    rpowers = []; nT(Lh/  
    for j = 1:length(n) *@2+$fgz  
        rpowers = [rpowers m(j):2:n(j)]; BZ2frG\0&I  
    end KwEyMR!  
    rpowers = unique(rpowers); nSMw5  
    2[^p6s[  
    % Pre-compute the values of r raised to the required powers, Y~FN` =O  
    % and compile them in a matrix: L[Z SgRTu  
    % ----------------------------- NeG$;z7  
    if rpowers(1)==0 #0;H'GO?c  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); PWavq?SR  
        rpowern = cat(2,rpowern{:}); bSK> p3  
        rpowern = [ones(length_r,1) rpowern]; -w>2!@8  
    else vvWje:H  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9E@}@ZV(  
        rpowern = cat(2,rpowern{:}); Z@Tb3N/[  
    end \=3fO(  
    )GbVgYkk  
    % Compute the values of the polynomials: hv]}b'M$  
    % -------------------------------------- lWiC$  
    z = zeros(length_r,length_n); @ V_@r@A  
    for j = 1:length_n 0!Zp4>l\Z  
        s = 0:(n(j)-m(j))/2; U};~ff+  
        pows = n(j):-2:m(j); 2q4dCbJ!  
        for k = length(s):-1:1 71g\fGG\  
            p = (1-2*mod(s(k),2))* ... 8y9`xRy  
                       prod(2:(n(j)-s(k)))/          ... =N,ahq  
                       prod(2:s(k))/                 ... 5V"Fy&}:  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... zB/)_AW  
                       prod(2:((n(j)+m(j))/2-s(k))); k&<cFZU  
            idx = (pows(k)==rpowers); 3U.?Jbm-8  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ~s$ jiA1  
        end 0 It[Pa qG  
         <KBzZ !n5  
        if isnorm .u<i<S  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ,_G((oS40  
        end 4`KQ@m  
    end ?3=D-Xrb  
    z_gjC%(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)  _T{ "F  
    soTmKqj E  
    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。 %:DH _0  
    %+{[%?xh  
    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)