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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 vR<fdV  
    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! O7! fI'R  
     
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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  "@UU[o  
    function z = zernfun(n,m,r,theta,nflag) 9RCB$Ka6X  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. , }xpYq_/  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N XL"v21X  
    %   and angular frequency M, evaluated at positions (R,THETA) on the A?6{  
    %   unit circle.  N is a vector of positive integers (including 0), and [[.&,6  
    %   M is a vector with the same number of elements as N.  Each element ~T;a jvJ  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) #*ZnA,  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, b.w(x*a  
    %   and THETA is a vector of angles.  R and THETA must have the same pw(U< )  
    %   length.  The output Z is a matrix with one column for every (N,M) Vsm%h^]d  
    %   pair, and one row for every (R,THETA) pair. 5 b#" G"  
    % sqMNon`5  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Gdc ~Lh  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), SevfxR  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral )Rm 'YmO  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, .:r2BgL  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 0NuL9  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ]HZa:aPY  
    % F$sF 'cw  
    %   The Zernike functions are an orthogonal basis on the unit circle. %~8](]p  
    %   They are used in disciplines such as astronomy, optics, and >M8^ Jgh  
    %   optometry to describe functions on a circular domain. h[[/p {z  
    % `o^;fcnG  
    %   The following table lists the first 15 Zernike functions. +r#=n7 t  
    % "p6:ekw  
    %       n    m    Zernike function           Normalization mPw56>  
    %       -------------------------------------------------- ba:mO$  
    %       0    0    1                                 1 TS~Y\Cp  
    %       1    1    r * cos(theta)                    2 J?qcRg`1E  
    %       1   -1    r * sin(theta)                    2 Hc_hO  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) m_PrasZ>  
    %       2    0    (2*r^2 - 1)                    sqrt(3) tc49Ty9$[  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) |=h)efo}  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) dg'CHxU  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) }77=<N br  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 2gC&R1 H  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 4LKs'$:A=  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) F~d7;x =g  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4 L~;>]7  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) DbNi;m  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) J:TI>*tn  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) w7*b}D@65\  
    %       -------------------------------------------------- Z%HEn$t  
    % ^&Rxui  
    %   Example 1: )2^/?jK  
    % Oa_o"p<Lr  
    %       % Display the Zernike function Z(n=5,m=1) 2*7s 9g  
    %       x = -1:0.01:1; #QyK?i*  
    %       [X,Y] = meshgrid(x,x); 61Iy{-/ZV  
    %       [theta,r] = cart2pol(X,Y); ym,Ot1  
    %       idx = r<=1; UV *tO15i  
    %       z = nan(size(X)); ZjI/zqBm  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx));  _.J[w6  
    %       figure Ow .)h(y/  
    %       pcolor(x,x,z), shading interp >I66R;  
    %       axis square, colorbar [Yahxw}  
    %       title('Zernike function Z_5^1(r,\theta)') g]PLW3  
    % $M3A+6["H  
    %   Example 2: w]5f3CIm  
    % 39a]B`y  
    %       % Display the first 10 Zernike functions T~ q'y~9o  
    %       x = -1:0.01:1; glKs8^W  
    %       [X,Y] = meshgrid(x,x); O^="T^J  
    %       [theta,r] = cart2pol(X,Y); y\f8Ird  
    %       idx = r<=1; )hZ}$P1  
    %       z = nan(size(X)); _ry En  
    %       n = [0  1  1  2  2  2  3  3  3  3]; vdFQf ^l  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; B+q+)O+  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; (.nJT"&  
    %       y = zernfun(n,m,r(idx),theta(idx)); a ~iEps  
    %       figure('Units','normalized') [ sO<6?LY  
    %       for k = 1:10 l<MCmKuYp  
    %           z(idx) = y(:,k); d(B;vL@R2V  
    %           subplot(4,7,Nplot(k)) *,XJN_DKj  
    %           pcolor(x,x,z), shading interp H1ui#5n2  
    %           set(gca,'XTick',[],'YTick',[]) O@(.ei*HJ!  
    %           axis square u1|Y;*  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ZWe$(?  
    %       end $O</akn;  
    % Ckl]fy@D}  
    %   See also ZERNPOL, ZERNFUN2. =smY/q^3  
    uY%3X/^j  
    %   Paul Fricker 11/13/2006 ]O(HZD%  
    }d*sWSPu(  
    rJ~(Xu>,s  
    % Check and prepare the inputs: Kmf-l*7}  
    % ----------------------------- _<~Vxz9  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) )Jjw}}$}Y  
        error('zernfun:NMvectors','N and M must be vectors.') >_% g8T'  
    end P}u<NPy3Q  
    Ex&RR< 5  
    if length(n)~=length(m) 0c;"bA0>Sx  
        error('zernfun:NMlength','N and M must be the same length.') n\)f.}YD8d  
    end 2iINQK$  
    5lA 8e  
    n = n(:); |0 pBBDw  
    m = m(:); u H;^>`DT  
    if any(mod(n-m,2)) }sNZQ89V*v  
        error('zernfun:NMmultiplesof2', ... X1~A "sW[  
              'All N and M must differ by multiples of 2 (including 0).')  D)eKq!_  
    end }8KL]11b  
    S gsR;)2  
    if any(m>n) dz.MH  
        error('zernfun:MlessthanN', ... kK6>>lD'  
              'Each M must be less than or equal to its corresponding N.') +fR`@HI  
    end v+2q R0,LM  
    ba1QFzN  
    if any( r>1 | r<0 ) rG%_O$_dO  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 2&f=4b`Z  
    end V1V4 <Zj  
    IIEU{},}z  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 2Yf;b9-k  
        error('zernfun:RTHvector','R and THETA must be vectors.') !Y i<h/:  
    end 5DBd [u3  
    _4#psxl[M  
    r = r(:); |,~A9  
    theta = theta(:); t`3T_t Y  
    length_r = length(r); )8>f  
    if length_r~=length(theta) `\uv+^x{  
        error('zernfun:RTHlength', ... /9# jv]C:  
              'The number of R- and THETA-values must be equal.') _C#( )#  
    end KT?s\w  
    QlXF:Gx"=  
    % Check normalization: m1Z8SM+  
    % -------------------- i58CA?  
    if nargin==5 && ischar(nflag) +~AI(h  
        isnorm = strcmpi(nflag,'norm'); qUg4-Z4  
        if ~isnorm *\+ 'tFT6  
            error('zernfun:normalization','Unrecognized normalization flag.') AUpC HG7  
        end VDN]P3   
    else 3CRBu:)m  
        isnorm = false; tzN;;h4C  
    end #iU/Yg!  
    e;3 (,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?m^7O_1  
    % Compute the Zernike Polynomials N4NH)x  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h--!pE+  
    #ms98pw%5  
    % Determine the required powers of r: -"L6^IH7  
    % ----------------------------------- 1 niTkop  
    m_abs = abs(m); ~q>ilnL"h  
    rpowers = []; m 1;jS|  
    for j = 1:length(n) uV:;y}T^Z  
        rpowers = [rpowers m_abs(j):2:n(j)]; #8|NZ6x,  
    end a5&j=3)|  
    rpowers = unique(rpowers); AVZ@?aJgF  
    g?M69~G$:x  
    % Pre-compute the values of r raised to the required powers, FZ/&[;E!  
    % and compile them in a matrix: DF =. G1  
    % ----------------------------- S5!2%-;<k  
    if rpowers(1)==0 y70gNPuTOD  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); |7fBiVo  
        rpowern = cat(2,rpowern{:}); o(qmI/h  
        rpowern = [ones(length_r,1) rpowern]; SQk!o{  
    else t,6=EK*3T  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); nQ6'yd"  
        rpowern = cat(2,rpowern{:}); VG^-aR_F  
    end _m-r}9au   
    n-_w0Y  
    % Compute the values of the polynomials: \_'pUp22  
    % -------------------------------------- `lzH:B  
    y = zeros(length_r,length(n)); vt,X:3  
    for j = 1:length(n) DdgFBO  
        s = 0:(n(j)-m_abs(j))/2; t|lv6-Hy9  
        pows = n(j):-2:m_abs(j); j>23QPG`6U  
        for k = length(s):-1:1 Q0-~&e_'  
            p = (1-2*mod(s(k),2))* ... N h%8;  
                       prod(2:(n(j)-s(k)))/              ... >MH@FnUL  
                       prod(2:s(k))/                     ... "k/@tX1:R  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Hua8/:![+  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 1[ Pbsb  
            idx = (pows(k)==rpowers); Ek0.r)Nw  
            y(:,j) = y(:,j) + p*rpowern(:,idx); z_TK (;j  
        end Rz]bCiD3 B  
         )M~5F,)  
        if isnorm F\;1:y~1  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); FTe#@\I  
        end "'L SLp  
    end 7Jk.U=vY  
    % END: Compute the Zernike Polynomials ^D)C|T  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /_8V+@im  
    #s%$kYp 1  
    % Compute the Zernike functions: x uF_^  
    % ------------------------------ .v{ty  
    idx_pos = m>0; XJ+sm^`vOf  
    idx_neg = m<0; teb(\% ,  
    8:MYeE5  
    z = y; T5)?6i -N  
    if any(idx_pos) uwJkqlUOz  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); <U*d   
    end Y/gCtSF  
    if any(idx_neg) )U` c9*.  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); UpbzH(?#  
    end #]2u!a ma  
    uJizR F  
    % EOF zernfun
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag)  2B#WWb  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. <[Vr(.A  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated UOyP6ej  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive h!.(7qdd  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, kI]1J  
    %   and THETA is a vector of angles.  R and THETA must have the same p\ASf  
    %   length.  The output Z is a matrix with one column for every P-value, #AHIlUH"m  
    %   and one row for every (R,THETA) pair. Y+E@afsKs  
    % *T3"U|0_y  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike lWR  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ;8!D8o(+  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) <mxUgU  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 _Uq'eZol  
    %   for all p. ^U1;5+2G+~  
    % m~v Ie c  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 *UTk. :G5  
    %   Zernike functions (order N<=7).  In some disciplines it is *m7e>]-  
    %   traditional to label the first 36 functions using a single mode UCQL~  
    %   number P instead of separate numbers for the order N and azimuthal wFe</U-';  
    %   frequency M. wG B'c's*  
    % eWFlJ;=  
    %   Example: *oF{ R^  
    % 8/=2N  
    %       % Display the first 16 Zernike functions =LC5o2bLy  
    %       x = -1:0.01:1; '{|87kI  
    %       [X,Y] = meshgrid(x,x); ?h5Y^}8Qg  
    %       [theta,r] = cart2pol(X,Y); ]2<g"zo0  
    %       idx = r<=1; ,{%[/#~6  
    %       p = 0:15; ?lTQjw{  
    %       z = nan(size(X)); hX^XtIC=  
    %       y = zernfun2(p,r(idx),theta(idx)); Ruf*aF(  
    %       figure('Units','normalized') EV}%D9:  
    %       for k = 1:length(p) {uw]s< 6  
    %           z(idx) = y(:,k); )TLDNpH?J  
    %           subplot(4,4,k) ALG +  
    %           pcolor(x,x,z), shading interp DP?gozm  
    %           set(gca,'XTick',[],'YTick',[]) U_:/>8})d  
    %           axis square </fzBaTo  
    %           title(['Z_{' num2str(p(k)) '}']) zUOYH4+  
    %       end b_B4  
    % MT3UJ6~P  
    %   See also ZERNPOL, ZERNFUN. {5,CW  
    -v]7}[ .[  
    %   Paul Fricker 11/13/2006 y(%6?a @  
    -1@kt<Es  
    R_-.:n%.z  
    % Check and prepare the inputs: ,Rf<6/A  
    % ----------------------------- u+-}|  
    if min(size(p))~=1 J^u{7K,  
        error('zernfun2:Pvector','Input P must be vector.') RW3&]l=  
    end U+\\#5$  
    J~~WV<6  
    if any(p)>35 a{y ;Ub  
        error('zernfun2:P36', ... lwV#j}G  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... \E n^Vf  
               '(P = 0 to 35).']) |'Jz(dv[  
    end z 6p.{M  
    rx%lL  
    % Get the order and frequency corresonding to the function number: (*#S%4(YX  
    % ---------------------------------------------------------------- J"|o g|Tz  
    p = p(:); V ] Z{0  
    n = ceil((-3+sqrt(9+8*p))/2); &Y\`FY\   
    m = 2*p - n.*(n+2); -&+[/  
    ?8`b  
    % Pass the inputs to the function ZERNFUN: -cMqq$  
    % ---------------------------------------- D&.+Dx^G  
    switch nargin y3d`$'7H>  
        case 3 At"@`1n_u'  
            z = zernfun(n,m,r,theta); gx3arVa  
        case 4 BYRf MtT@+  
            z = zernfun(n,m,r,theta,nflag); P#iBwmwN+.  
        otherwise v&|o5om  
            error('zernfun2:nargin','Incorrect number of inputs.') aCQAh[T  
    end {>90d(j  
    j2V^1  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) R e-4y5f  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. X$)<>e]!>  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of T=%,^  
    %   order N and frequency M, evaluated at R.  N is a vector of 2{(_{9<>z  
    %   positive integers (including 0), and M is a vector with the h<JV6h:8  
    %   same number of elements as N.  Each element k of M must be a &Yb!j  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) cJ=0zEv  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 4;=+qb  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix jhg0H2C8  
    %   with one column for every (N,M) pair, and one row for every ,Tjc\;~%  
    %   element in R. lG6P+ Z/nf  
    % ?`8jn$W^  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- HW"@~-\  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is @#rF8;  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to "dQ02y  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 @p"m{  
    %   for all [n,m]. /?VwoSgV^  
    % BS!VAHO"V  
    %   The radial Zernike polynomials are the radial portion of the EZypqe):/C  
    %   Zernike functions, which are an orthogonal basis on the unit *> LA30R*v  
    %   circle.  The series representation of the radial Zernike OlI|.~  
    %   polynomials is n3 y`='D  
    % vq/3a  
    %          (n-m)/2 b1\.hi  
    %            __ W"$sN8K>)  
    %    m      \       s                                          n-2s \SKobO?qI  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r grrM[Y7#~b  
    %    n      s=0 F=EG#<@u  
    % *MC+i$  
    %   The following table shows the first 12 polynomials. x4v@o?zW  
    % wwaw|$  
    %       n    m    Zernike polynomial    Normalization &L`^\B]k|  
    %       --------------------------------------------- =Z}$X: $  
    %       0    0    1                        sqrt(2) i24t$7q  
    %       1    1    r                           2 x,L<{A`z  
    %       2    0    2*r^2 - 1                sqrt(6) -?z#  
    %       2    2    r^2                      sqrt(6) ;S0Kf{DN2  
    %       3    1    3*r^3 - 2*r              sqrt(8) gxPu/VD4  
    %       3    3    r^3                      sqrt(8) JCO+_d#x  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ur\<NApT;  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) %>Q[j`9y  
    %       4    4    r^4                      sqrt(10) O pavno%&  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) z;iNfs0i$  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Gn&=<q :H  
    %       5    5    r^5                      sqrt(12) Uhs/F:E[A  
    %       --------------------------------------------- [eLMb)n  
    % 6({TG&`!]  
    %   Example: '2XIeR  
    % @k+ K_gR  
    %       % Display three example Zernike radial polynomials /Vdu|k=  
    %       r = 0:0.01:1; ` {/"?s|  
    %       n = [3 2 5]; Y#[xX2z9  
    %       m = [1 2 1]; +9exap27  
    %       z = zernpol(n,m,r); WYJH+"@%j  
    %       figure )sN}ClgJ  
    %       plot(r,z) q{Ao j  
    %       grid on 9$f%  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ij5|P4Eka  
    % 4ibOVBG:*,  
    %   See also ZERNFUN, ZERNFUN2. *[Im].  
    \r1nMw3&  
    % A note on the algorithm. ;vG%[f`K  
    % ------------------------ ?k`UQi]Q  
    % The radial Zernike polynomials are computed using the series .fAHP 5-  
    % representation shown in the Help section above. For many special T].Xx`  
    % functions, direct evaluation using the series representation can dk/f_m  
    % produce poor numerical results (floating point errors), because >=1Aa,_tc  
    % the summation often involves computing small differences between m`BE{%  
    % large successive terms in the series. (In such cases, the functions uA4x xY  
    % are often evaluated using alternative methods such as recurrence S-5O$EnD  
    % relations: see the Legendre functions, for example). For the Zernike R0~w F>  
    % polynomials, however, this problem does not arise, because the FQBE1h@k0u  
    % polynomials are evaluated over the finite domain r = (0,1), and s}qtM.^W  
    % because the coefficients for a given polynomial are generally all TXT!Ae  
    % of similar magnitude.  qC6@  
    % lk*w M?Z  
    % ZERNPOL has been written using a vectorized implementation: multiple s~06%QEG  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] m*|G 2  
    % values can be passed as inputs) for a vector of points R.  To achieve !&},h=  
    % this vectorization most efficiently, the algorithm in ZERNPOL b$q~(Z}  
    % involves pre-determining all the powers p of R that are required to &'k:?@J[  
    % compute the outputs, and then compiling the {R^p} into a single < &kl:|  
    % matrix.  This avoids any redundant computation of the R^p, and [}I|tb>Pg  
    % minimizes the sizes of certain intermediate variables. +#L'g c  
    % bgeJVI  
    %   Paul Fricker 11/13/2006 v]\T&w%9  
    |G)P I`BH  
    ~p?D[]h  
    % Check and prepare the inputs: 3/y"kl:< -  
    % ----------------------------- !Qq~lAJO;  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ;#L]7ZY9:-  
        error('zernpol:NMvectors','N and M must be vectors.') =6a=`3r!I  
    end T 9FGuit9  
    .oM;D~(=9  
    if length(n)~=length(m) ?)gc;K  
        error('zernpol:NMlength','N and M must be the same length.') [Lcy &+  
    end 2 ?F?C  
    [9d\WPLC  
    n = n(:); RdB,;Um9f  
    m = m(:); z+KZ6h  
    length_n = length(n); O03F@v  
    d*x&Uh[K  
    if any(mod(n-m,2)) [e>2HIS,  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') @1~cPt   
    end @^%YOorr  
    FqZD'Uu7  
    if any(m<0) OaKr_m  
        error('zernpol:Mpositive','All M must be positive.') s<;{q+1#  
    end U8{^-#(Uz  
    tr58J% Mu  
    if any(m>n) 7)RRCsn  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ]E.\ |I(  
    end .l,]yWwfK  
    XqGa]/;}  
    if any( r>1 | r<0 ) *^KEb")$  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ]@m`bs_6  
    end 'Z|Czd8E  
    )Y`ybADd3  
    if ~any(size(r)==1) eM]>"  
        error('zernpol:Rvector','R must be a vector.') y Ni3@f  
    end v|dt[>G  
    *TrpW?]Y&  
    r = r(:); >U.7>K V&  
    length_r = length(r); 9rIv-&7'm  
    #7"";"{ z|  
    if nargin==4 N/[!$B0H@  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); WHqw=! G  
        if ~isnorm |n;5D,r0C  
            error('zernpol:normalization','Unrecognized normalization flag.') l3+G]C&<  
        end )=cJW(nfP  
    else {P3gMv;  
        isnorm = false; ;X:Bh8tEV  
    end Vh^ :.y   
    W.59Al'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G5lBCm   
    % Compute the Zernike Polynomials y4VO\N!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _bRgr  
    w]fVELU  
    % Determine the required powers of r: ]z2x`P^oI  
    % ----------------------------------- %0({ MU  
    rpowers = []; ^)o]hE|  
    for j = 1:length(n) 7%&e4'SZO  
        rpowers = [rpowers m(j):2:n(j)]; [+ : zlA  
    end ;Ah eeq746  
    rpowers = unique(rpowers); qW /&.  
    w4R~0jXy  
    % Pre-compute the values of r raised to the required powers, b>9?gmR{  
    % and compile them in a matrix: UGvUU<N|N  
    % ----------------------------- =>JA; ft  
    if rpowers(1)==0 p}JGx^X ~  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); -X3CrW  
        rpowern = cat(2,rpowern{:}); %zR5q  Lb  
        rpowern = [ones(length_r,1) rpowern]; WqS$C;]%  
    else sGh TP/  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); \tA@A  
        rpowern = cat(2,rpowern{:}); VA`VDUG,  
    end 4;(W0RQa  
    2@Q5Ta #h  
    % Compute the values of the polynomials: C>F5=&  
    % -------------------------------------- 6G(K8Q{>  
    z = zeros(length_r,length_n); F6\4[B  
    for j = 1:length_n %$bhg&}  
        s = 0:(n(j)-m(j))/2; tv2k&\1  
        pows = n(j):-2:m(j); TH55@1W,[  
        for k = length(s):-1:1 CYsLyk  
            p = (1-2*mod(s(k),2))* ... =`2jnvx  
                       prod(2:(n(j)-s(k)))/          ... b2RW=m-  
                       prod(2:s(k))/                 ... ||fCY+x*8  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... q"WfKz!U  
                       prod(2:((n(j)+m(j))/2-s(k))); x_<,GE@  
            idx = (pows(k)==rpowers); YgtW(j[  
            z(:,j) = z(:,j) + p*rpowern(:,idx); }9N-2]  
        end Hn/V*RzQ  
         !Q,Dzv"7  
        if isnorm eT?vZH[N  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ^)'D eP/  
        end $7-S\sDr  
    end e&K7n@  
    9JeT1\VvHY  
    % 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)  ?1H>k<Jp  
    _9z+xl  
    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。 {InW%qSn_  
    o H]FT{  
    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)