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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 HNL;s5gq  
    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!  .Aa(  
     
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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 T*sB Wn'am  
    function z = zernfun(n,m,r,theta,nflag) d|(@#*{T]  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. J+6bp0RIh  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 2OJ=Xb1  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 7IH^5r  
    %   unit circle.  N is a vector of positive integers (including 0), and } h pTS_  
    %   M is a vector with the same number of elements as N.  Each element j?rq%rQd  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) XT 'v7  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, {:r8X  
    %   and THETA is a vector of angles.  R and THETA must have the same 9&uWj'%ia  
    %   length.  The output Z is a matrix with one column for every (N,M) n9Xssl0  
    %   pair, and one row for every (R,THETA) pair. v"dj%75O?e  
    % 92HxZ*t7km  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike _~b$6Nf!83  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 27!9LU  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral OCVF+D :  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, /_G^d1T1?L  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized }TS4D={1  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. m;"i4!  
    % d,9YrwbD  
    %   The Zernike functions are an orthogonal basis on the unit circle. QjlwT2o'  
    %   They are used in disciplines such as astronomy, optics, and " )87GQ(R  
    %   optometry to describe functions on a circular domain. " %)zTH  
    % d;D8$q)8Q  
    %   The following table lists the first 15 Zernike functions. /"M7YPX;  
    % Gf{FFIe(  
    %       n    m    Zernike function           Normalization s!d"(K9E  
    %       -------------------------------------------------- S4?N_"m9  
    %       0    0    1                                 1 TZ,kmk#  
    %       1    1    r * cos(theta)                    2 ~~_!&  
    %       1   -1    r * sin(theta)                    2 ;w_f^R #  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) IT u6m<V  
    %       2    0    (2*r^2 - 1)                    sqrt(3) K;wd2/jmJ  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) _DK%-,Spu  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) okO^ /"  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) $m;rOKVU  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 8[|RsM   
    %       3    3    r^3 * sin(3*theta)             sqrt(8) L[Dr[  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Ox` +Z0)a  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =A,6KY=E  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) MWS=$N)v*  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0{P Rv./`  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ;(0E#hGN  
    %       -------------------------------------------------- 3d2|vQx,K  
    % |oSx*Gh  
    %   Example 1: j<LDJi>O  
    % t(|\3$z  
    %       % Display the Zernike function Z(n=5,m=1) kR0d]"dr  
    %       x = -1:0.01:1; ]~SOGAFW  
    %       [X,Y] = meshgrid(x,x); S"Dw8_y7}  
    %       [theta,r] = cart2pol(X,Y); ?{ "_9g9  
    %       idx = r<=1; d+Vx:`tT  
    %       z = nan(size(X)); tp,e:4\ 8Q  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); xJ|3}o:,  
    %       figure W7a aL  
    %       pcolor(x,x,z), shading interp Ifm|_  
    %       axis square, colorbar gt9{u"o  
    %       title('Zernike function Z_5^1(r,\theta)') i$Q$y hT{  
    % P-?ya!@"  
    %   Example 2: 52$7vYMto  
    % +a%Vp!y  
    %       % Display the first 10 Zernike functions qd9CKd  
    %       x = -1:0.01:1; fJ3*'(  
    %       [X,Y] = meshgrid(x,x);  ;Q;u^T`  
    %       [theta,r] = cart2pol(X,Y);  lqO"  
    %       idx = r<=1; S?bG U8R5  
    %       z = nan(size(X)); CV~\xYY  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 0{/P1  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; l;I)$=={=  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; U`D.cEMfH  
    %       y = zernfun(n,m,r(idx),theta(idx)); 7[wHNJ7)r  
    %       figure('Units','normalized') `3Gjj&c  
    %       for k = 1:10 6]%79?'A  
    %           z(idx) = y(:,k); B*+3A!{s  
    %           subplot(4,7,Nplot(k)) l@8UL</W  
    %           pcolor(x,x,z), shading interp f((pRP   
    %           set(gca,'XTick',[],'YTick',[]) a sDq(J`sQ  
    %           axis square K +oFu%  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) *uAsKU  
    %       end BTXS+mvl  
    % q][{?  
    %   See also ZERNPOL, ZERNFUN2. =|lKB;  
    g.v)qB  
    %   Paul Fricker 11/13/2006 Hz}6XS@  
    k\T,CZ<  
    P< +5So0  
    % Check and prepare the inputs: *^XfEO  
    % ----------------------------- JfmNI~%  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) GbC-6.~  
        error('zernfun:NMvectors','N and M must be vectors.') L~yu  
    end !$"DD[~\  
    SCClD6k=V  
    if length(n)~=length(m) gWo`i  
        error('zernfun:NMlength','N and M must be the same length.') W|K"0ab  
    end h 7feZ_  
    aI$D qnF4  
    n = n(:); yv:8=.r}M  
    m = m(:); biCX: m+_?  
    if any(mod(n-m,2)) qc}r.'p  
        error('zernfun:NMmultiplesof2', ...  =#N;ZG  
              'All N and M must differ by multiples of 2 (including 0).') <_HK@E<_HO  
    end \bze-|C  
    W ?;kMGW-  
    if any(m>n) -e"~UDq`  
        error('zernfun:MlessthanN', ... x.rOP_rs  
              'Each M must be less than or equal to its corresponding N.') 8Z TN  
    end 8SvPDGu `]  
    &UhI1mi]h  
    if any( r>1 | r<0 ) 3:Aw.-,i\  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') =9UR~-`d\  
    end J`U\3:b`SP  
    D];%Ey  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) (U"Ub;[7  
        error('zernfun:RTHvector','R and THETA must be vectors.') -c-#1_X5  
    end EG<YxNX,  
    \atztC{-L>  
    r = r(:); \ltA&}!  
    theta = theta(:); s)#8>s-  
    length_r = length(r); `6KTQk'  
    if length_r~=length(theta) i5  x[1  
        error('zernfun:RTHlength', ... {EKzPr/  
              'The number of R- and THETA-values must be equal.') d\Xi1&&  
    end 60KhwD1  
    j9zK=eG  
    % Check normalization: H6ff b)&  
    % -------------------- K1rF;7Y6  
    if nargin==5 && ischar(nflag) 'J R2@W`]]  
        isnorm = strcmpi(nflag,'norm'); @1#QbNp#  
        if ~isnorm .\kcWeC\  
            error('zernfun:normalization','Unrecognized normalization flag.') FNpMu3Q  
        end :3k&[W*  
    else q=bW!.#?  
        isnorm = false; VvuwgJX  
    end )3_I-Ia  
    6Q<^,`/T  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% aa8xo5tIp  
    % Compute the Zernike Polynomials jk-hIl&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d\aarhD8*  
    Si,[7um  
    % Determine the required powers of r: 6LUO  
    % ----------------------------------- w ^ v*1KA&  
    m_abs = abs(m); OhmKjY/}  
    rpowers = []; W2L:  
    for j = 1:length(n) t^HQ=*c  
        rpowers = [rpowers m_abs(j):2:n(j)]; 7XKPC+)1ya  
    end c\i`=>%b@  
    rpowers = unique(rpowers); e0O2 >w  
    1O bxQ_x  
    % Pre-compute the values of r raised to the required powers, Txkmt$h  
    % and compile them in a matrix: & 2MI(9v  
    % ----------------------------- K~"J<798{  
    if rpowers(1)==0 `UFRv   
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); (0s7<&Iu  
        rpowern = cat(2,rpowern{:}); l4+!H\2  
        rpowern = [ones(length_r,1) rpowern]; QJc3@  
    else 70p1&Y7or  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )ndcBwQc"  
        rpowern = cat(2,rpowern{:}); =5NrkCk#V  
    end ^6!C":f  
    +g_+JLQ  
    % Compute the values of the polynomials: BZy&;P  
    % -------------------------------------- [%(}e1T(  
    y = zeros(length_r,length(n)); p<1z!`!P  
    for j = 1:length(n) &fJ92v?%^S  
        s = 0:(n(j)-m_abs(j))/2; {9sA'5  
        pows = n(j):-2:m_abs(j); ta]B9&c  
        for k = length(s):-1:1 {6%vmMbJ  
            p = (1-2*mod(s(k),2))* ... rj qX|  
                       prod(2:(n(j)-s(k)))/              ... 9] /xAsD  
                       prod(2:s(k))/                     ... Bq~!_6fB  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... l2uh"!  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); P( >*gp  
            idx = (pows(k)==rpowers); cjzhuH/y  
            y(:,j) = y(:,j) + p*rpowern(:,idx); EL!V\J`S_  
        end &jCT-dj  
         dR"H,$UH  
        if isnorm E~?0Yrm F  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); o -tc}Aa  
        end Zw+VcZz3  
    end :USN`"  
    % END: Compute the Zernike Polynomials KK; 3<kX  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% su]CaHU  
    j .Ro(0%  
    % Compute the Zernike functions: =;DmD?nZ  
    % ------------------------------ BrYU*aPW;  
    idx_pos = m>0; SH>L3@Za  
    idx_neg = m<0; Rd6? ,  
    1qWIku  
    z = y; &7* |rshZ  
    if any(idx_pos) USz |Rh  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); VU+`yQp  
    end Va^Y3/  
    if any(idx_neg) j-wSsjLk  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); RAMkTS  
    end nR)/k,3W  
    Ed[ tmaEuV  
    % 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) AR9D;YfR~  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Ay7I_" %  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ,ra!O=d~0  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive , ~^0AtLv  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, `"CIy_m  
    %   and THETA is a vector of angles.  R and THETA must have the same O~t]:p9_  
    %   length.  The output Z is a matrix with one column for every P-value, 6 &~8TH  
    %   and one row for every (R,THETA) pair. W}&[p=PAS  
    % Yud]s~N  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike JCoDe.  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) *_G(*yAe(  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ]IbX<  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 Oax*3TD  
    %   for all p. [J0f:&7\  
    % Ewjzm,2  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ,&>LBdG`  
    %   Zernike functions (order N<=7).  In some disciplines it is GE;S5 X]X  
    %   traditional to label the first 36 functions using a single mode 3IXai)6U  
    %   number P instead of separate numbers for the order N and azimuthal +"8 [E~Bih  
    %   frequency M. kev|AU (WX  
    % ^0A}iJL  
    %   Example: 4.7 YIM  
    % (S1c6~  
    %       % Display the first 16 Zernike functions 0vN<0  
    %       x = -1:0.01:1; (Fc\*Vn  
    %       [X,Y] = meshgrid(x,x); RbPD3& .  
    %       [theta,r] = cart2pol(X,Y); x*}41;j}C  
    %       idx = r<=1; ~>C@n'\lv  
    %       p = 0:15; YbaaX{7^  
    %       z = nan(size(X)); ~)!yl. H  
    %       y = zernfun2(p,r(idx),theta(idx)); ? yL3XB>  
    %       figure('Units','normalized') }DH3_M!  
    %       for k = 1:length(p) T: zO9C/  
    %           z(idx) = y(:,k); 5`su^  
    %           subplot(4,4,k) )8`7i{F  
    %           pcolor(x,x,z), shading interp HgH\2QL3&  
    %           set(gca,'XTick',[],'YTick',[]) 0#\K9|.  
    %           axis square K%NNw7\A  
    %           title(['Z_{' num2str(p(k)) '}']) gc~nT/lfK  
    %       end Y0nnn  
    % +tsF.Is!t  
    %   See also ZERNPOL, ZERNFUN. [ .] x y  
    *nb `DR  
    %   Paul Fricker 11/13/2006 tC;L A 4  
    {wUbr^  
    iBCZx>![;  
    % Check and prepare the inputs: MA}~bfB  
    % ----------------------------- #G</RYM~m  
    if min(size(p))~=1 _=,\uIrk  
        error('zernfun2:Pvector','Input P must be vector.') F"p7&e\W|l  
    end )OjTn"  
    ?D 8<}~Do  
    if any(p)>35 kV >[$6  
        error('zernfun2:P36', ... @9,=|kxK  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... rGa@!^hk  
               '(P = 0 to 35).']) NC]]`O2r@  
    end &j}08aK%  
    N, `q1B  
    % Get the order and frequency corresonding to the function number: WBb@\|V|  
    % ---------------------------------------------------------------- ?vA)F)MS   
    p = p(:); x/BtB"e*5  
    n = ceil((-3+sqrt(9+8*p))/2); zL8Z8eh">  
    m = 2*p - n.*(n+2); .bdp=vbA  
    VO"/cG;]*  
    % Pass the inputs to the function ZERNFUN: e"=/zZH3  
    % ---------------------------------------- gMXs&`7P  
    switch nargin &xhwx>C`K  
        case 3 o!6~tO=%  
            z = zernfun(n,m,r,theta); 5OHg% ^  
        case 4 *}F>c3x]  
            z = zernfun(n,m,r,theta,nflag); @`Fv}RY{  
        otherwise b#uNdq3  
            error('zernfun2:nargin','Incorrect number of inputs.') |}^me7C,[  
    end B#Q` !B4v  
    fL xGaOT  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) k kZ2Jxvx  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. ^_4e^D]P"  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of G_m$?0\  
    %   order N and frequency M, evaluated at R.  N is a vector of (6}[y\a+  
    %   positive integers (including 0), and M is a vector with the `p!&>,lrk  
    %   same number of elements as N.  Each element k of M must be a N>TmaUk  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) hqXp>.W  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is x/fhlf}a}=  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix jj,CBNo(  
    %   with one column for every (N,M) pair, and one row for every =l)D$l  
    %   element in R. -(dtAo6  
    % ^l:~r2  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- Ir\P[A  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ^ABt g#  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to AX%N:)_$|  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 40e(p/Qka  
    %   for all [n,m]. 'iM#iA8  
    % %nS(>X<B  
    %   The radial Zernike polynomials are the radial portion of the Z bW!c1s{  
    %   Zernike functions, which are an orthogonal basis on the unit @Ojbu@A  
    %   circle.  The series representation of the radial Zernike AZ9;6Df  
    %   polynomials is QkFB \v  
    % 0&~ JC>S  
    %          (n-m)/2 ~xa yGk  
    %            __ 7z2Q!0Sz  
    %    m      \       s                                          n-2s '^n,)oA/G  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r pqCp>BO?O  
    %    n      s=0 sck.2-f"  
    % HUFm@?  
    %   The following table shows the first 12 polynomials. :[:*kbWN-  
    % n #l~B@  
    %       n    m    Zernike polynomial    Normalization HJi FlL3  
    %       --------------------------------------------- 6`h}#@ (  
    %       0    0    1                        sqrt(2) l\/uXP?  
    %       1    1    r                           2 a'%eyN  
    %       2    0    2*r^2 - 1                sqrt(6) @tX8M[.eA  
    %       2    2    r^2                      sqrt(6) 7;I;(iY  
    %       3    1    3*r^3 - 2*r              sqrt(8) >"q?P^f/  
    %       3    3    r^3                      sqrt(8) >vR7l&"  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) | |u  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) }Ug O$1  
    %       4    4    r^4                      sqrt(10) 43}uW, P  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ,/ YTW@N  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 1`sTGNo  
    %       5    5    r^5                      sqrt(12) O[|_~v:^  
    %       --------------------------------------------- >1qum'  
    % #AR$'TE#  
    %   Example: U>i}C_7g  
    % -kP$S qR~  
    %       % Display three example Zernike radial polynomials Rb_%vOM  
    %       r = 0:0.01:1; aKFY&zN?  
    %       n = [3 2 5]; tZ.hSDH  
    %       m = [1 2 1]; h+!@`c>)Y  
    %       z = zernpol(n,m,r); >g;995tG  
    %       figure V"|`Z}XW  
    %       plot(r,z) YO9ofT  
    %       grid on aH&Efz^  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ;04< 9i  
    % [#SiwhF|  
    %   See also ZERNFUN, ZERNFUN2. m++=FsiX=  
    >|pN4FS  
    % A note on the algorithm. hRNnj  
    % ------------------------ `)tIXMn  
    % The radial Zernike polynomials are computed using the series Zg4kO;r08  
    % representation shown in the Help section above. For many special sE])EwZ  
    % functions, direct evaluation using the series representation can O'{g{  
    % produce poor numerical results (floating point errors), because d}2(G2z^  
    % the summation often involves computing small differences between (j-_iOQ]i+  
    % large successive terms in the series. (In such cases, the functions eUKl(  
    % are often evaluated using alternative methods such as recurrence 489xoP  
    % relations: see the Legendre functions, for example). For the Zernike 48,uO !  
    % polynomials, however, this problem does not arise, because the Mt*V-`+\  
    % polynomials are evaluated over the finite domain r = (0,1), and 9DmFa5E  
    % because the coefficients for a given polynomial are generally all ]m :Y|,:6  
    % of similar magnitude. h-]c   
    % XFv^j SF  
    % ZERNPOL has been written using a vectorized implementation: multiple uE}$ZBi q  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] q $=[v  
    % values can be passed as inputs) for a vector of points R.  To achieve wg4Ol*y'  
    % this vectorization most efficiently, the algorithm in ZERNPOL C^fn[plL  
    % involves pre-determining all the powers p of R that are required to &bigLe  
    % compute the outputs, and then compiling the {R^p} into a single y7Nd3\v [\  
    % matrix.  This avoids any redundant computation of the R^p, and 61}eB/;7  
    % minimizes the sizes of certain intermediate variables. i!*8@:VI  
    % c5Hyja=  
    %   Paul Fricker 11/13/2006 +*IRI/KUD  
    3 TRG] 5  
    9 /=+2SZ  
    % Check and prepare the inputs: WIN3*z7oW  
    % ----------------------------- A*{CT>  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2;x+#D8  
        error('zernpol:NMvectors','N and M must be vectors.') Nj.;mr<  
    end w8bvqTQ  
    *#1J  
    if length(n)~=length(m) HLCI  
        error('zernpol:NMlength','N and M must be the same length.') {(l,Uhxl""  
    end ovZ!}  
    ,hWuAu6.L  
    n = n(:); "TVmxE%(  
    m = m(:); 8v)iOPmDC  
    length_n = length(n); K,,'{j2#f  
    q7pe\~q  
    if any(mod(n-m,2)) ;?v&=Z't.  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') V}ls|B$Y  
    end ~sdM~9@ '  
    /i{V21(%  
    if any(m<0) [ @2$W?0i  
        error('zernpol:Mpositive','All M must be positive.') ;u=%Vn"2a  
    end BZ(DP_}&D  
    f VJWW):  
    if any(m>n) rl x6a@MiD  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') Fd<eh(g9P  
    end &(m01  
    +~sqv?8  
    if any( r>1 | r<0 ) 6m@B.+1  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') XRTiC #6  
    end 7~SnY\B|  
    _}cD_$D  
    if ~any(size(r)==1) j^/=.cD|  
        error('zernpol:Rvector','R must be a vector.') M^lP`=sSv  
    end agq4Zy  
    0 Bk-)z|V  
    r = r(:); 7nsovWp  
    length_r = length(r); /6Kx249Dw  
    m ?*h\NaB  
    if nargin==4 !Sl_qL  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); &}t8O?!  
        if ~isnorm ,YJn=9pTl  
            error('zernpol:normalization','Unrecognized normalization flag.') lq mr`\@)  
        end .#Z"Sj  
    else  ?<T=g  
        isnorm = false; vOq N=bp  
    end SXC 7LJm<g  
    x<7?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q7I!wD9Cff  
    % Compute the Zernike Polynomials /.~zk(-&h  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nb ?(zDJ8  
    v57<b&p26  
    % Determine the required powers of r: Xc4zUEO9  
    % ----------------------------------- < FY%QB)h  
    rpowers = []; K & %8w  
    for j = 1:length(n) zN=s]b=/  
        rpowers = [rpowers m(j):2:n(j)]; ]^Xj!01~  
    end C;%dZ  
    rpowers = unique(rpowers); zZP/C   
    E^0a; |B[  
    % Pre-compute the values of r raised to the required powers, xZY7X&C4  
    % and compile them in a matrix: x&r f]R  
    % ----------------------------- SE(c_ sX  
    if rpowers(1)==0 [7NO !^  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 4aW[`  
        rpowern = cat(2,rpowern{:}); :, 3S5!(y  
        rpowern = [ones(length_r,1) rpowern]; Z:^ S-h  
    else aMVq%{U  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); s< Fp17  
        rpowern = cat(2,rpowern{:}); C Sz+cS  
    end p 16+(m  
    7u rD  
    % Compute the values of the polynomials: Xoha.6$l5  
    % -------------------------------------- ] g8z@r"b  
    z = zeros(length_r,length_n); Z{/GT7 /  
    for j = 1:length_n KzJJ@D*4M]  
        s = 0:(n(j)-m(j))/2; JcMl*k  
        pows = n(j):-2:m(j); KUpj.[5 qo  
        for k = length(s):-1:1 m#-&<=  
            p = (1-2*mod(s(k),2))* ... A~^x*#q{4  
                       prod(2:(n(j)-s(k)))/          ... $sUn'62JlU  
                       prod(2:s(k))/                 ... ))nTd=  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Wc,`L$Jx  
                       prod(2:((n(j)+m(j))/2-s(k))); "d /uyS$6  
            idx = (pows(k)==rpowers); bAy\Sr #/  
            z(:,j) = z(:,j) + p*rpowern(:,idx); s%8,'3&  
        end A-J#$B  
         i29a1nD4Hm  
        if isnorm  ;]bW  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 4Xww(5?3  
        end TQPrOs?  
    end o,S(;6pDJ  
    6{0MprY  
    % 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)  [p}~M-$V8Y  
    Wm5[+z|2?9  
    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。 Q~k|lTf  
    Ggst s  
    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)