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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 %e_"CS  
    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! |/T43ADW  
     
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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 RBGX_v?  
    function z = zernfun(n,m,r,theta,nflag) )qU7`0'8  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. {`"#yl6"  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N vqNsZ 8|`  
    %   and angular frequency M, evaluated at positions (R,THETA) on the -?a<qa?$  
    %   unit circle.  N is a vector of positive integers (including 0), and - u3e5gW  
    %   M is a vector with the same number of elements as N.  Each element csQfic  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) LE=k  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, %[QV,fD'E  
    %   and THETA is a vector of angles.  R and THETA must have the same S h4wqf  
    %   length.  The output Z is a matrix with one column for every (N,M) acW'$@y9?N  
    %   pair, and one row for every (R,THETA) pair. d&(_|xq#  
    % .tXtcf/  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 1np^(['ih  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), #AViM_u  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral TprtE.mP  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, lmCZ8 j(FF  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized XcfKx@l  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. b=[?b+  
    % @QEqB_W  
    %   The Zernike functions are an orthogonal basis on the unit circle. 2+"r~#K*  
    %   They are used in disciplines such as astronomy, optics, and pW?& J>\6  
    %   optometry to describe functions on a circular domain. "ZMkL)'7-  
    % s(2GFc  
    %   The following table lists the first 15 Zernike functions. 5g ;ac~g  
    % Iy7pt~DJ,  
    %       n    m    Zernike function           Normalization MXvXVhCU  
    %       -------------------------------------------------- 'r} fZ  
    %       0    0    1                                 1 O m'(mr  
    %       1    1    r * cos(theta)                    2 k9si| '  
    %       1   -1    r * sin(theta)                    2 K k[`dR;  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) tj1JB%  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Q(@IK&v  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 9'~- U  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) cma*Dc  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) !u;>Wyd W  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) -uR72f  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) GA3sRFZdQ  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) F} DUEDND*  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) TH1B#Y#<J  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 7"v$- Wy  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) u5E]t9~Pq  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) o'V%EQ  
    %       -------------------------------------------------- QLq@u[A  
    % A.%CAGU5w  
    %   Example 1: d^D i*&X  
    % &xS a7FY  
    %       % Display the Zernike function Z(n=5,m=1) 0tz:Wd*<  
    %       x = -1:0.01:1; -8Ti*:  
    %       [X,Y] = meshgrid(x,x); E l&h;N   
    %       [theta,r] = cart2pol(X,Y); e$/B_o7(  
    %       idx = r<=1; 15H6:_+=0  
    %       z = nan(size(X)); Y:QD   
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); mxG]kqi  
    %       figure /.Jb0h[W1  
    %       pcolor(x,x,z), shading interp gUax'^w;V;  
    %       axis square, colorbar d]v+mVAyE  
    %       title('Zernike function Z_5^1(r,\theta)') r0dDHj~F  
    % <,%:   
    %   Example 2: -pb&-@Hul  
    % }ZOFYu0f  
    %       % Display the first 10 Zernike functions ^CT&0  
    %       x = -1:0.01:1; _7)F ?  
    %       [X,Y] = meshgrid(x,x); i8pU|VpA  
    %       [theta,r] = cart2pol(X,Y); h#}YKWL  
    %       idx = r<=1; P&A|PY,P  
    %       z = nan(size(X)); fQLax  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 2 YxTMT  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; `k{& /]  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; x;E2~&E  
    %       y = zernfun(n,m,r(idx),theta(idx)); :os z  
    %       figure('Units','normalized') ]o/|na*  
    %       for k = 1:10 [IBQvL  
    %           z(idx) = y(:,k); !fkep=  
    %           subplot(4,7,Nplot(k)) h5zVGr  
    %           pcolor(x,x,z), shading interp TCVl8)j  
    %           set(gca,'XTick',[],'YTick',[]) jx`QB')kX  
    %           axis square  -7]Xjb5  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) = bt]JRU  
    %       end !Jfs?Hy  
    % # '|'r+  
    %   See also ZERNPOL, ZERNFUN2. 0lk;F  
    b!>\2DlyJ  
    %   Paul Fricker 11/13/2006 Hgc=M  
    !sSQQo2Sv  
    ik,lSTBD  
    % Check and prepare the inputs: }E^S]hdvz  
    % ----------------------------- alFjc.~}  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) |Rzy8j*  
        error('zernfun:NMvectors','N and M must be vectors.') 7 6fIC  
    end I*[tMzE  
    <g2_6C\j  
    if length(n)~=length(m) -`c :}m  
        error('zernfun:NMlength','N and M must be the same length.') B7*}c]^6/  
    end L):qu  
     q" @  
    n = n(:); e_3CSx8Cc  
    m = m(:); w5C*L)l  
    if any(mod(n-m,2)) +FFG#6e  
        error('zernfun:NMmultiplesof2', ... V~{ _3YY  
              'All N and M must differ by multiples of 2 (including 0).') SpTdj^]4>  
    end ni CE\B~  
    -0HkTY  
    if any(m>n) ;DRTQn`m  
        error('zernfun:MlessthanN', ... M~T.n)x2  
              'Each M must be less than or equal to its corresponding N.') cd@.zg'sYn  
    end q`|CrOzO  
    N1EezC'^  
    if any( r>1 | r<0 ) pa .K-e)Mu  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') "kW!{n  
    end -f(/B9}  
    g<*jlM1r  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) zri} h/{  
        error('zernfun:RTHvector','R and THETA must be vectors.') J QKdW  
    end W=}Okq)x9I  
    obClBO)@Y  
    r = r(:); }2>"<)  
    theta = theta(:); tV;% J4E'  
    length_r = length(r); YhKZ|@  
    if length_r~=length(theta) y&T&1o  
        error('zernfun:RTHlength', ... ]n1dp2aH  
              'The number of R- and THETA-values must be equal.') mPZGA\  
    end @ CsV]97`  
    B~WtZ-%%E  
    % Check normalization: ]L_w$ev'  
    % -------------------- &wH:aD  
    if nargin==5 && ischar(nflag) t@zdm y  
        isnorm = strcmpi(nflag,'norm'); ` vk0c  
        if ~isnorm BuQ|~V  
            error('zernfun:normalization','Unrecognized normalization flag.') Jcf"#u-Q/  
        end X!,@ j\L  
    else Q'NmSX)0  
        isnorm = false; ~Vh=5J~  
    end 0OZMlt%z  
    n[+'OU[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4n( E;!s  
    % Compute the Zernike Polynomials 70W"G X&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GUp;AoQ  
    }U 5Y=RYo  
    % Determine the required powers of r: 5a`%)K  
    % ----------------------------------- dz9Y}\2tf  
    m_abs = abs(m); SOh-,c\C  
    rpowers = []; ?s%v0cF  
    for j = 1:length(n) `H%G3M0a  
        rpowers = [rpowers m_abs(j):2:n(j)]; &k>aP0k"  
    end eBr4O i  
    rpowers = unique(rpowers); x!7yU_ls`  
    /="HqBI#i  
    % Pre-compute the values of r raised to the required powers, eb:A1f4L  
    % and compile them in a matrix: mX# "+X|  
    % ----------------------------- y2Bh?>pg  
    if rpowers(1)==0 BNm4k7 ]M  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); {ShgJ ;! Q  
        rpowern = cat(2,rpowern{:}); |^n3{m  
        rpowern = [ones(length_r,1) rpowern]; j+ ::y) $  
    else pK_?}~  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); _2Py\+$  
        rpowern = cat(2,rpowern{:}); d.F)9h]XHO  
    end 'Z!G a.I  
    %CH6lY=lI  
    % Compute the values of the polynomials: /Bv#) -5  
    % -------------------------------------- v"6 \=@  
    y = zeros(length_r,length(n)); 8v_C5d\  
    for j = 1:length(n) F4I6P  
        s = 0:(n(j)-m_abs(j))/2; NlPS#  
        pows = n(j):-2:m_abs(j); Utl t<  
        for k = length(s):-1:1 ?m%h`<wgMc  
            p = (1-2*mod(s(k),2))* ... ISqfU]>[  
                       prod(2:(n(j)-s(k)))/              ... 19u =W(  
                       prod(2:s(k))/                     ... J1F{v)T '?  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... UsW5d]i}Y  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); b{[*N  
            idx = (pows(k)==rpowers); y;`eDS'0.N  
            y(:,j) = y(:,j) + p*rpowern(:,idx); VV3}]GjC  
        end '5.\#=S1  
         E,"&-`/2v  
        if isnorm IM( u<c$  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); zmFws-+A  
        end H oy7RC&  
    end e- 6w8*!i  
    % END: Compute the Zernike Polynomials &w\ I<J`T  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  -;c  
    %vqT#+x  
    % Compute the Zernike functions: C7"HQQ  
    % ------------------------------ .Ao0;:;(2-  
    idx_pos = m>0; !vqC+o>@  
    idx_neg = m<0; LsTffIP  
    R{}qK r  
    z = y; R 1zC.m  
    if any(idx_pos) A|RR]CFJ  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); LJuW${Y  
    end sg?@qc=g  
    if any(idx_neg) lgD]{\O$ip  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ej[Su  
    end &a #GXf  
    qd2xb8r  
    % 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) z^`]7i  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. \r -N(;m  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated |rPAC![=  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Ye |G44z  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, &YX6"S_B  
    %   and THETA is a vector of angles.  R and THETA must have the same lo:~aJ8  
    %   length.  The output Z is a matrix with one column for every P-value, KTmaglgp  
    %   and one row for every (R,THETA) pair. iJnh$jo  
    % TmP8 q  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike i?>Hr|  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) %C *^:\y  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) mK\aI  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 h}6_ybmZ  
    %   for all p. $ KQ,}I  
    % y^s1t2]%  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 > V%Q O>C  
    %   Zernike functions (order N<=7).  In some disciplines it is sR79 K1*j  
    %   traditional to label the first 36 functions using a single mode %zljH"F  
    %   number P instead of separate numbers for the order N and azimuthal 'lQYJ0  
    %   frequency M. Lk nVqZ|k  
    % #v/ry)2Y=  
    %   Example: oyvtZ/@  
    % jT^!J+?6K+  
    %       % Display the first 16 Zernike functions ua#K>su r.  
    %       x = -1:0.01:1; ] 09yy  
    %       [X,Y] = meshgrid(x,x); otnV-7)@  
    %       [theta,r] = cart2pol(X,Y); `ue?Z%p|  
    %       idx = r<=1; ~CFMIQ et  
    %       p = 0:15; 1n3$V:00  
    %       z = nan(size(X)); Xp^$ E6YFy  
    %       y = zernfun2(p,r(idx),theta(idx)); 3+ asP&n  
    %       figure('Units','normalized') !R{em48D  
    %       for k = 1:length(p) }su6izx  
    %           z(idx) = y(:,k); 9[{sEg=C$e  
    %           subplot(4,4,k) Czh8zB+r  
    %           pcolor(x,x,z), shading interp C'<'7g4  
    %           set(gca,'XTick',[],'YTick',[]) e6m1NH4,  
    %           axis square lC{L6&T  
    %           title(['Z_{' num2str(p(k)) '}']) ~XQ$aRl&  
    %       end IUawdB5CB  
    % qw0~ *0}  
    %   See also ZERNPOL, ZERNFUN. Zd XKI{b  
    1 ypjyu  
    %   Paul Fricker 11/13/2006 gMay  
    ua:9`+Dff  
    {X]9^=O"  
    % Check and prepare the inputs: Sj1r s#@1  
    % ----------------------------- ^0eO\wc?O  
    if min(size(p))~=1 j4vB`Gr]  
        error('zernfun2:Pvector','Input P must be vector.') i-"<[*ePd  
    end l#ygb|=x  
    !7Uu]m69n  
    if any(p)>35 +gNX7xuY  
        error('zernfun2:P36', ... %IU4\ZY>  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... `D"1 gD}{A  
               '(P = 0 to 35).']) ](n69XX_  
    end (zEYpTp  
    GZ,j?@  
    % Get the order and frequency corresonding to the function number: X&,N}9>B  
    % ---------------------------------------------------------------- f~iML5lG  
    p = p(:); 2;}xN!8  
    n = ceil((-3+sqrt(9+8*p))/2); .S(^roM;+  
    m = 2*p - n.*(n+2); $~ VcQ  
    D:6N9POB  
    % Pass the inputs to the function ZERNFUN: M;PlSb  
    % ---------------------------------------- 6Ok,_ !  
    switch nargin I*9Gb$]=  
        case 3 Tz2x9b\82  
            z = zernfun(n,m,r,theta); *Ji9%IA  
        case 4 s)Gb!-``  
            z = zernfun(n,m,r,theta,nflag); !8Y3V/)NU  
        otherwise w&9F>`VET  
            error('zernfun2:nargin','Incorrect number of inputs.') qs "s/$  
    end 3U>S]#5}  
    `43vxcMg  
    % 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) Dn: Yi8=  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. fo *!a$)  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of A?\h|u<  
    %   order N and frequency M, evaluated at R.  N is a vector of "3v7gtGG  
    %   positive integers (including 0), and M is a vector with the 0NVG"-Q  
    %   same number of elements as N.  Each element k of M must be a 7/ 4~>D&-b  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) %odw+PhO  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is e1oFnu2R  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix QZWoKGd}+  
    %   with one column for every (N,M) pair, and one row for every l;XUh9RF`A  
    %   element in R. Q4#\{" N!  
    % y+l<vJu  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 1o(+rR<h9  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is |_!PD$i-  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to `Nkx7Z~w:  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 o:h)~[n|  
    %   for all [n,m]. XL=2wh  
    % hcj{%^p  
    %   The radial Zernike polynomials are the radial portion of the twAw01".  
    %   Zernike functions, which are an orthogonal basis on the unit  n})  
    %   circle.  The series representation of the radial Zernike bn5"dxV  
    %   polynomials is FW/6{tm  
    % $4ka +nfU  
    %          (n-m)/2 jBT*~DyN z  
    %            __ >qr=l,Hi  
    %    m      \       s                                          n-2s (q055y  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ;<86P3S  
    %    n      s=0 ,^Ex}Z  
    % mI2|0RWI)l  
    %   The following table shows the first 12 polynomials. g8C+1G8  
    % JiG8jB7%}  
    %       n    m    Zernike polynomial    Normalization .n?5}s+q  
    %       --------------------------------------------- ^Z#<tN;  
    %       0    0    1                        sqrt(2) SZNFE  
    %       1    1    r                           2 >eTf}#s?S  
    %       2    0    2*r^2 - 1                sqrt(6) Z#H@BWN7  
    %       2    2    r^2                      sqrt(6) AEBw#v!,o  
    %       3    1    3*r^3 - 2*r              sqrt(8) #Lu4OSM+  
    %       3    3    r^3                      sqrt(8) e,PQ)1  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) bfcD5:q  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) h}Fu"zK  
    %       4    4    r^4                      sqrt(10) J+-,^8)  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) A{xSbbDk  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) g`y/ _  
    %       5    5    r^5                      sqrt(12) **"zDY*?W  
    %       --------------------------------------------- lsTe*Od  
    % qg/Y;tGSx  
    %   Example: gEX:S(1 QP  
    % 8Xt=eL/P  
    %       % Display three example Zernike radial polynomials W+fkWq7`Xx  
    %       r = 0:0.01:1; }s8*QfK>  
    %       n = [3 2 5]; Z3&XTsq  
    %       m = [1 2 1]; M)bC%(xJ  
    %       z = zernpol(n,m,r); ',v0vyO8  
    %       figure 3/]f4D{MMY  
    %       plot(r,z) X7(rg W8  
    %       grid on So3,Z'z=  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') F 5b]/;|  
    % ^v()iF !  
    %   See also ZERNFUN, ZERNFUN2. aC $h_  
    bYRQI=gW':  
    % A note on the algorithm. 4c493QOd  
    % ------------------------ ,58kjTM  
    % The radial Zernike polynomials are computed using the series wFH(.E0@Q  
    % representation shown in the Help section above. For many special  F<XD^sO  
    % functions, direct evaluation using the series representation can /0'fcjOaQ  
    % produce poor numerical results (floating point errors), because 5cv, >{~5  
    % the summation often involves computing small differences between ~XN]?5GQf  
    % large successive terms in the series. (In such cases, the functions "'LOaf$X  
    % are often evaluated using alternative methods such as recurrence Y D1g]p  
    % relations: see the Legendre functions, for example). For the Zernike <ZN) /,4PS  
    % polynomials, however, this problem does not arise, because the O;.d4pO(tC  
    % polynomials are evaluated over the finite domain r = (0,1), and EV;;N  
    % because the coefficients for a given polynomial are generally all 7ipY*DT8  
    % of similar magnitude. ?L.p9o-S0  
    % ixUiXP  
    % ZERNPOL has been written using a vectorized implementation: multiple >Kqj{/SWK  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] o>!~*b';g,  
    % values can be passed as inputs) for a vector of points R.  To achieve 6r ?cpJV{  
    % this vectorization most efficiently, the algorithm in ZERNPOL e3bAT.P  
    % involves pre-determining all the powers p of R that are required to s`dkEaS  
    % compute the outputs, and then compiling the {R^p} into a single B@: XC&R^  
    % matrix.  This avoids any redundant computation of the R^p, and wZ#~+ }T  
    % minimizes the sizes of certain intermediate variables. *AJezhR  
    % }pU!1GsO  
    %   Paul Fricker 11/13/2006 /-cX(z 7  
    l"V8n BR`  
    Amq8q  
    % Check and prepare the inputs: wHDF TIDI  
    % ----------------------------- UBpM8/U  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Z2Y583D  
        error('zernpol:NMvectors','N and M must be vectors.') <CdG[Ih  
    end o8yEUnqN  
    3]5&&=#  
    if length(n)~=length(m) d%"@#bB  
        error('zernpol:NMlength','N and M must be the same length.') s`7 _J9  
    end pu m9x)y1  
    `dq3=  
    n = n(:); )y [[Se  
    m = m(:); J0Rz.=Y  
    length_n = length(n); HhT8YH  
    hwb(W?*  
    if any(mod(n-m,2)) /R+]}Lt~%*  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') [sh"?  
    end #h|,GvmF<b  
    71,0v`Z<  
    if any(m<0) jL[Is2<@  
        error('zernpol:Mpositive','All M must be positive.') 3C5D~9v  
    end Yk*57&QI  
    u{dN>}{  
    if any(m>n) |<o>$;mZ  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') kA9 X!)2w  
    end D -\'P31  
    8Nl|\3nl-  
    if any( r>1 | r<0 ) c$UpR"+  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') `E1_S  
    end $9u  
    PX>\j&  
    if ~any(size(r)==1) DcvmeGl  
        error('zernpol:Rvector','R must be a vector.') T"0)%k8lJ  
    end '%r@D&*vp  
    1Z{p[\k  
    r = r(:); #j ~FA3O  
    length_r = length(r); QR-R5XNT[  
    mQ `r`DW  
    if nargin==4 R S_lQ{'  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); $5 p'+bE  
        if ~isnorm 38.J:?Q  
            error('zernpol:normalization','Unrecognized normalization flag.') JV*,!5  
        end E)Epr&9S  
    else #K~j9DuR  
        isnorm = false; !-}*jm p<  
    end q\Io6=39x  
    ur quVb  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Xcpm?aTo  
    % Compute the Zernike Polynomials b.u8w2(  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2/o/UfYjgF  
    h],%va[  
    % Determine the required powers of r: WT? U~.U  
    % ----------------------------------- sYW)h$p;D  
    rpowers = []; 4I[FE;^  
    for j = 1:length(n) 2n r UE  
        rpowers = [rpowers m(j):2:n(j)]; 8QgL7  
    end |@9I5Eg)iE  
    rpowers = unique(rpowers); UA u4x 7  
    (6y3"cbe  
    % Pre-compute the values of r raised to the required powers, zNTu j p  
    % and compile them in a matrix: $}c@S0%P"  
    % ----------------------------- \36;csu  
    if rpowers(1)==0 [";5s&)q  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); .F$AmVTN  
        rpowern = cat(2,rpowern{:}); $Lbe5d?\  
        rpowern = [ones(length_r,1) rpowern]; 8`?j*FV7kq  
    else U[ungvU1U  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); $%"}N_M  
        rpowern = cat(2,rpowern{:}); !rqR]nd  
    end JBJ7k19;  
    }EG(!)u  
    % Compute the values of the polynomials: ]6[d-$#^ko  
    % -------------------------------------- #\;w::  
    z = zeros(length_r,length_n); ]|BSX-V.%i  
    for j = 1:length_n (#"s!!b  
        s = 0:(n(j)-m(j))/2; A0k>Nb\c3  
        pows = n(j):-2:m(j); *^5,7}9Qo  
        for k = length(s):-1:1 ~,65/O  
            p = (1-2*mod(s(k),2))* ... \uPTk)oaB  
                       prod(2:(n(j)-s(k)))/          ... D}U<7=\3H  
                       prod(2:s(k))/                 ... e%Xf*64  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... CXFAb1m  
                       prod(2:((n(j)+m(j))/2-s(k)));  ;I@L  
            idx = (pows(k)==rpowers); U: jf9L2  
            z(:,j) = z(:,j) + p*rpowern(:,idx); vj$ 6  
        end N9|.D.#MF  
         w[G_w:$a  
        if isnorm vaZZzv{H  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); {4q:4 i  
        end 0>MI*fnY"  
    end c9@jyq_H?  
    cY]Y8T)  
    % 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)  # HYkzjb  
    %nF\tVP3]  
    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。 y:[]+  
    >nEnX  
    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)