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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 9G&l qfX:  
    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! 7ml0  
     
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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 Q}Ze-JIL$  
    function z = zernfun(n,m,r,theta,nflag) V"7<[u]K|  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. I^M#[xA  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 11B{gUv.]  
    %   and angular frequency M, evaluated at positions (R,THETA) on the {wp Mg  
    %   unit circle.  N is a vector of positive integers (including 0), and V8nz-DL{  
    %   M is a vector with the same number of elements as N.  Each element Y*kh$E%<#  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) %%as>}.  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 2%5^Fi  
    %   and THETA is a vector of angles.  R and THETA must have the same 4h;f>BG  
    %   length.  The output Z is a matrix with one column for every (N,M) =MJ-s;raq  
    %   pair, and one row for every (R,THETA) pair. 8sR  
    % Pu$kj"|q*[  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike co<2e#p;  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Zr.\`mG4f  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral +(z_"[l"  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, L,L ~ .E  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized (RDa,&  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. o5=)~D{/G3  
    % FtFv<UV  
    %   The Zernike functions are an orthogonal basis on the unit circle. "$~}'`(]  
    %   They are used in disciplines such as astronomy, optics, and _OJ0 < {E  
    %   optometry to describe functions on a circular domain. qXrt0s[  
    % N"YK@)*Q  
    %   The following table lists the first 15 Zernike functions. ot@|blVC8  
    % l$k]O  
    %       n    m    Zernike function           Normalization ;L G %s  
    %       -------------------------------------------------- GhG%>U#&a  
    %       0    0    1                                 1 24H^ hN9  
    %       1    1    r * cos(theta)                    2 J.bF v/R  
    %       1   -1    r * sin(theta)                    2 P\q<d  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) < v|%K.yd  
    %       2    0    (2*r^2 - 1)                    sqrt(3) }[>RxHd  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) X+dR<GN+YX  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ]5} -y3  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) _l24Ba$F6  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) L _vblUDq  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) <CZI7]PM7  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Mvy6"Q:  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) jw/'*e  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) '[>\N4WD  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) FF%\g J  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ~/j$TT"  
    %       -------------------------------------------------- /*u#Ba<<  
    % .mvB99P{<  
    %   Example 1: {E3xI2  
    % z>cIiprX  
    %       % Display the Zernike function Z(n=5,m=1) t{/:(Nu  
    %       x = -1:0.01:1; Rro?q  
    %       [X,Y] = meshgrid(x,x); .abyYVrN4?  
    %       [theta,r] = cart2pol(X,Y); Y brx%  
    %       idx = r<=1; =d go!k  
    %       z = nan(size(X)); [kPD`be2#  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); |7svA<<[  
    %       figure 5~@?>)TBv  
    %       pcolor(x,x,z), shading interp o2;(VSKhS  
    %       axis square, colorbar p//T7r s  
    %       title('Zernike function Z_5^1(r,\theta)') lo cW_/  
    % ! 9d _Gf-  
    %   Example 2: ~gu=x&{  
    % wVx,JL5Jr  
    %       % Display the first 10 Zernike functions XOu+&wOu  
    %       x = -1:0.01:1; J?._/RL8-  
    %       [X,Y] = meshgrid(x,x); 1pd 9s8CA  
    %       [theta,r] = cart2pol(X,Y); _R EqT  
    %       idx = r<=1; yJDeX1+,  
    %       z = nan(size(X)); EfFz7j&X  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Gx.P ]O3  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; {I4%   
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; v2Dt3$@H6  
    %       y = zernfun(n,m,r(idx),theta(idx)); 4cott^K.  
    %       figure('Units','normalized') )HEfU31IC  
    %       for k = 1:10 MVeF e\r  
    %           z(idx) = y(:,k); 7sXy`+TZ->  
    %           subplot(4,7,Nplot(k)) D,c!#(v cK  
    %           pcolor(x,x,z), shading interp sB?2*S"X)<  
    %           set(gca,'XTick',[],'YTick',[]) *R5`.j =  
    %           axis square "Owct(9  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) k) "ao2iXL  
    %       end cb +l"FI7  
    % >eQbipn  
    %   See also ZERNPOL, ZERNFUN2. Rb)|66&3&  
    `&7mHa61  
    %   Paul Fricker 11/13/2006 yC W*fIaq  
    F7\BF  
    VLiIO"u;  
    % Check and prepare the inputs: G;/Q>V  
    % ----------------------------- 1hR (N  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) & B}Lo  
        error('zernfun:NMvectors','N and M must be vectors.') 5B1G?`]?  
    end cC' ~  
     #;`Oj  
    if length(n)~=length(m) :-)GNf yGz  
        error('zernfun:NMlength','N and M must be the same length.') ,"B?_d6  
    end 4AQ[igTDP  
    7skljw(  
    n = n(:); C)OG62  
    m = m(:); b6|Z"{TI _  
    if any(mod(n-m,2)) ~F;CE"3A  
        error('zernfun:NMmultiplesof2', ... cQX:%Ix=  
              'All N and M must differ by multiples of 2 (including 0).') :V-k'hm &  
    end "# 2pT H~  
    qYK4)JP  
    if any(m>n) [9OSpq  
        error('zernfun:MlessthanN', ... h}h^L+4  
              'Each M must be less than or equal to its corresponding N.') BBxc*alG0  
    end #: #Dz.$L  
     r@k"4ce-  
    if any( r>1 | r<0 ) cJ. 7Mt  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') \ZMP_UU(  
    end -j&Vtr  
    qbb6,DL7J  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ll%G!VR  
        error('zernfun:RTHvector','R and THETA must be vectors.') ^2E hlK^)  
    end ?KB@Zm+#~  
    +i.u< T  
    r = r(:); )7dEi+v52  
    theta = theta(:); 9*\g`fWc}{  
    length_r = length(r); }#6xFTH  
    if length_r~=length(theta) \,R!S/R#  
        error('zernfun:RTHlength', ... F;P5D<  
              'The number of R- and THETA-values must be equal.') o\4CoeG  
    end zJY']8ah  
    Qs l80~n_7  
    % Check normalization: Ux}W&K/?'  
    % -------------------- rLzW`  
    if nargin==5 && ischar(nflag) \0?$wIH?  
        isnorm = strcmpi(nflag,'norm'); 2JZdw  
        if ~isnorm qnJ50 VVW  
            error('zernfun:normalization','Unrecognized normalization flag.') |@RpWp>2  
        end tuLH}tkNY  
    else ^I`a;  
        isnorm = false; 1k[GuG%/K  
    end *~2cG;B"e  
    jXp. qK\"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Nqc p1J"  
    % Compute the Zernike Polynomials mb1Vu  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m[:K"lZ ]2  
    a9l8{ 3  
    % Determine the required powers of r: ^m%52Tm h  
    % ----------------------------------- OCNPi4  
    m_abs = abs(m); 9x?'}  
    rpowers = []; &94W-zh  
    for j = 1:length(n) &RO7{,`  
        rpowers = [rpowers m_abs(j):2:n(j)]; V_"f|[1  
    end {DwIjy31T  
    rpowers = unique(rpowers); T SjI z5  
    ,mKObMu  
    % Pre-compute the values of r raised to the required powers, 9S>g6}[E#0  
    % and compile them in a matrix: f%XJ;y\,9H  
    % ----------------------------- "^Rv#  
    if rpowers(1)==0 z vO:"w}  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 6kR\xP]Kr  
        rpowern = cat(2,rpowern{:}); bd== +   
        rpowern = [ones(length_r,1) rpowern]; ^DB{qU  
    else 0<.R A%dj  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,T|x)"uA`  
        rpowern = cat(2,rpowern{:}); cWa)#:JOV  
    end zzIr2so  
    kOjf #@c  
    % Compute the values of the polynomials: UyiJU~r1  
    % -------------------------------------- %3o`j<  
    y = zeros(length_r,length(n)); <)U4Xz?  
    for j = 1:length(n) U|5-0u5  
        s = 0:(n(j)-m_abs(j))/2; 6 BAW  
        pows = n(j):-2:m_abs(j); fS=hpL6]@  
        for k = length(s):-1:1 (Rd$VYuf  
            p = (1-2*mod(s(k),2))* ... qP1FJ89H  
                       prod(2:(n(j)-s(k)))/              ... ;Vu5p#,O<M  
                       prod(2:s(k))/                     ... 41V e}%  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 2SG$LIV 9Y  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); sKL:p3r  
            idx = (pows(k)==rpowers); )9L/sKz  
            y(:,j) = y(:,j) + p*rpowern(:,idx); %j+xgX/&  
        end rv7{Ow_Y  
         3BQ!qO17^d  
        if isnorm _}gtcyx  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); )uheV,ZnY  
        end d@ J a}`  
    end N#ioJ^}n:  
    % END: Compute the Zernike Polynomials c#cx>wq9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'V&Y[7Aeq  
    M;.ZM<Ga  
    % Compute the Zernike functions: V diJ>d[  
    % ------------------------------ GTl xq%?b  
    idx_pos = m>0; dl~|Izm  
    idx_neg = m<0; -e]7n*}H$  
    ",Cr,;]  
    z = y; n<7q`tM#  
    if any(idx_pos) bt/ =Kq#  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); r ?m6$  
    end @n+=vC.xO  
    if any(idx_neg) _NZ@4+aW  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 3n;K!L%zMT  
    end z=Cr7-  
    l.+yn91%>  
    % 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) Zu*K-ep"  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. |CFRJN-J"  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 9i q""  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive p{$p $/A  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, wq!iV |  
    %   and THETA is a vector of angles.  R and THETA must have the same X6e/g{S)  
    %   length.  The output Z is a matrix with one column for every P-value, cmwPuK$  
    %   and one row for every (R,THETA) pair. f58?5(Dc|  
    % 5\fCd|  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike rf &M!d}!  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2)  ;q>9W,jy  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) s%4M$ e  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ;3eKqr0  
    %   for all p. TI|/u$SJ<Z  
    % 9LC&6Q5O&  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ),}AI/j;zY  
    %   Zernike functions (order N<=7).  In some disciplines it is ]e?x# <S  
    %   traditional to label the first 36 functions using a single mode Y"g.IK`V  
    %   number P instead of separate numbers for the order N and azimuthal r=.A'"Kf  
    %   frequency M. +j14Q$  
    % ^Q<mV*~  
    %   Example: m"eteA,"k_  
    % !U%T&?E l  
    %       % Display the first 16 Zernike functions KJn!Ap  
    %       x = -1:0.01:1; O`1!  
    %       [X,Y] = meshgrid(x,x); ,MPB/j^o5!  
    %       [theta,r] = cart2pol(X,Y); (.Y/  
    %       idx = r<=1; 26?W nu60  
    %       p = 0:15; I{'f|+1  
    %       z = nan(size(X)); k;W@LfP  
    %       y = zernfun2(p,r(idx),theta(idx)); ZD/jX_!t  
    %       figure('Units','normalized') & WOiik  
    %       for k = 1:length(p) jhgX{xc  
    %           z(idx) = y(:,k); T4/fdORS  
    %           subplot(4,4,k) [(kB 5 a  
    %           pcolor(x,x,z), shading interp >r@.F%  
    %           set(gca,'XTick',[],'YTick',[]) =<@2#E)  
    %           axis square {=2DqkTD  
    %           title(['Z_{' num2str(p(k)) '}']) ;h=*!7:  
    %       end <yA}i"-1W  
    % :'L2J  
    %   See also ZERNPOL, ZERNFUN. F'}'(t+oAm  
    q<W=#Sx  
    %   Paul Fricker 11/13/2006 uE/T2BX*  
    O)|P,?  
    ~5 N)f UI\  
    % Check and prepare the inputs: ,QIF &  
    % ----------------------------- `A$!]&[~|  
    if min(size(p))~=1 lT&wOm3  
        error('zernfun2:Pvector','Input P must be vector.') 8F(h*e_?  
    end }kHdK vZ  
    Jq.lT(E8D  
    if any(p)>35 w>fdQ!RdP  
        error('zernfun2:P36', ... -Y#sI3o*R8  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... n"PJ,ao  
               '(P = 0 to 35).']) `N//A}9  
    end TcTM]ixr  
    Cb t{ H}I3  
    % Get the order and frequency corresonding to the function number: )4U> !KrY  
    % ---------------------------------------------------------------- WF&[HKOy/  
    p = p(:); gbeghLP[?  
    n = ceil((-3+sqrt(9+8*p))/2); l- pe4x  
    m = 2*p - n.*(n+2); b+-f.!j  
    W W2Ob*  
    % Pass the inputs to the function ZERNFUN: mP38T{  
    % ---------------------------------------- jxa D&4Fs8  
    switch nargin yq-=],h  
        case 3 %=AxJp!a  
            z = zernfun(n,m,r,theta); qW:)!z3\  
        case 4 % }|cb7l  
            z = zernfun(n,m,r,theta,nflag); nMfFH[I4  
        otherwise -4rDbDsr  
            error('zernfun2:nargin','Incorrect number of inputs.') 9//+Bh  
    end `!:q;i]}  
    aJL^AG  
    % 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) V@7KsB  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Xtz-\v#0o'  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 5(W"-A}  
    %   order N and frequency M, evaluated at R.  N is a vector of JXG"M#{  
    %   positive integers (including 0), and M is a vector with the zf4Ec-)  
    %   same number of elements as N.  Each element k of M must be a ""Zp:8o  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) +')f6P;t>=  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ~U(,TjJb  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix BR^7_q4q  
    %   with one column for every (N,M) pair, and one row for every _LAS~x7,  
    %   element in R. W"{v2xi  
    % Q9d`zR]  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ms($9Lv/  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is =.]l*6W V  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to %p^.\ch9  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 i,V;xB2  
    %   for all [n,m]. <R%;~){  
    % :+{ ?  
    %   The radial Zernike polynomials are the radial portion of the %N;!+ ;F_g  
    %   Zernike functions, which are an orthogonal basis on the unit V._6=ZJ  
    %   circle.  The series representation of the radial Zernike T5Q{{@Q  
    %   polynomials is fP3_d  
    % <9=9b_z  
    %          (n-m)/2 8ul&x~2;X  
    %            __ BR'I+lQ  
    %    m      \       s                                          n-2s j-CnT)W<  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r \;VhYvEH  
    %    n      s=0 $M_x!f'{>  
    % @)kO=E d  
    %   The following table shows the first 12 polynomials. K.G$]H  
    % 1Z[/KJ  
    %       n    m    Zernike polynomial    Normalization  hjO*~  
    %       --------------------------------------------- {k4CEt;  
    %       0    0    1                        sqrt(2) rC:?l(8ng3  
    %       1    1    r                           2 cPgfTT  
    %       2    0    2*r^2 - 1                sqrt(6) k>dsw:  
    %       2    2    r^2                      sqrt(6) =n^!VXaL]]  
    %       3    1    3*r^3 - 2*r              sqrt(8) \MxoZ  
    %       3    3    r^3                      sqrt(8) Qn ^bVhG+  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) < Dx]b*H  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) _#$ *y  
    %       4    4    r^4                      sqrt(10) iX'rU@C  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Tirux ;  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) v+jsC`m  
    %       5    5    r^5                      sqrt(12) Ladsw  
    %       --------------------------------------------- tb :L\A^:  
    % 5XuT={o  
    %   Example: ]$U xCu  
    % ?ER-25S  
    %       % Display three example Zernike radial polynomials Ku&!?m@C  
    %       r = 0:0.01:1; V\V)<BARe  
    %       n = [3 2 5]; K1V#cB WO  
    %       m = [1 2 1]; A2}Rl%+X]6  
    %       z = zernpol(n,m,r); 2+Px'U\  
    %       figure #fj/~[Ajv  
    %       plot(r,z) qQ!1t>j+H  
    %       grid on 0y&I/2  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') b':|uu*/  
    % ZoKcJA  
    %   See also ZERNFUN, ZERNFUN2. xEuN   
    7PR#(ftz  
    % A note on the algorithm. m/NdJMoN=  
    % ------------------------ {JV@"t-X3"  
    % The radial Zernike polynomials are computed using the series pZ#ap<|>I  
    % representation shown in the Help section above. For many special IV lf=k  
    % functions, direct evaluation using the series representation can %4 \OPw&  
    % produce poor numerical results (floating point errors), because [m+iQVk'  
    % the summation often involves computing small differences between zI~owK)%Z  
    % large successive terms in the series. (In such cases, the functions +GsWTEz   
    % are often evaluated using alternative methods such as recurrence 3~e8bcb  
    % relations: see the Legendre functions, for example). For the Zernike nC {K$  
    % polynomials, however, this problem does not arise, because the $+}+zZX5  
    % polynomials are evaluated over the finite domain r = (0,1), and ]|_\xO(  
    % because the coefficients for a given polynomial are generally all *&Z7m^`FQ  
    % of similar magnitude. UD~p'^.m_  
    % fw oQ' &  
    % ZERNPOL has been written using a vectorized implementation: multiple :';L/x>  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] .qk]$LJF7  
    % values can be passed as inputs) for a vector of points R.  To achieve s:00yQ  
    % this vectorization most efficiently, the algorithm in ZERNPOL smG>sEp2  
    % involves pre-determining all the powers p of R that are required to %+ZJhHT  
    % compute the outputs, and then compiling the {R^p} into a single +i\&6HGK;-  
    % matrix.  This avoids any redundant computation of the R^p, and +.y .Mp  
    % minimizes the sizes of certain intermediate variables. r%DFve:%  
    % Z ,^9 Z  
    %   Paul Fricker 11/13/2006 %!ebO*8q  
    ~j#~ \Ir  
    E,n}HiAz7V  
    % Check and prepare the inputs: K/ &?VIi`z  
    % ----------------------------- H A}f,),G  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~.%K/=wK@  
        error('zernpol:NMvectors','N and M must be vectors.') =66Nw(E.  
    end Vtppuu$  
    gn5)SP8  
    if length(n)~=length(m) 4/ X/>Y1  
        error('zernpol:NMlength','N and M must be the same length.') Nr2C@FU:0  
    end :V)lbn\  
    XW JwJ  
    n = n(:); ( 6(x'ByT  
    m = m(:); @DW[Z`X  
    length_n = length(n); ?=GXqbS"  
    5 ,0d  
    if any(mod(n-m,2)) +.RKi !  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') @`FCiHM  
    end 3Rm#-T s  
    9;F bnp'  
    if any(m<0) b]E|*  
        error('zernpol:Mpositive','All M must be positive.') +7Kyyu)y@  
    end Hn,:`mj4-6  
    )pw&c_x  
    if any(m>n) 0'&X T^"  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') LtT\z<bAI  
    end co _oMc  
    W~_t~Vg5  
    if any( r>1 | r<0 ) ~f|Z%&l|  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 7j5f ;O^+  
    end E2GGEKrW  
    X &2oPo  
    if ~any(size(r)==1) hzI *{  
        error('zernpol:Rvector','R must be a vector.') 0oy-os  
    end RkF D*E$  
    &iN--~}!$  
    r = r(:); @1zQce>  
    length_r = length(r); 2?@j~I=s2h  
    GFSt<k)  
    if nargin==4 9iN.3/T8  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); %tA57Pn>  
        if ~isnorm S{',QO*D6  
            error('zernpol:normalization','Unrecognized normalization flag.') G ;?qWB,  
        end Y}6n]n;uR  
    else N__H*yP  
        isnorm = false; NGYyn`Lx  
    end 7dihVvL $  
    \EbbkN:D  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F2(q>#<_  
    % Compute the Zernike Polynomials ^s\3/z>b4!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x0a.!  
    r%Rs0)$yj  
    % Determine the required powers of r: {|9}+ @5Q1  
    % ----------------------------------- elJ)4Em  
    rpowers = []; ~Lfcg*  
    for j = 1:length(n) dsK&U\ej}  
        rpowers = [rpowers m(j):2:n(j)];  Z:2I/  
    end R)!`JKeO/  
    rpowers = unique(rpowers); ')+0nPV  
    '%v#v3'  
    % Pre-compute the values of r raised to the required powers, ,]R8(bD)  
    % and compile them in a matrix: X fz`^x>M  
    % ----------------------------- 9?+9UlJ7K  
    if rpowers(1)==0 4n 3Tp{Y}  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); mxrG)n6Y  
        rpowern = cat(2,rpowern{:}); ;D ~L|  
        rpowern = [ones(length_r,1) rpowern]; 4{9d#[KW  
    else l#3($QV,  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); [n,?WwC  
        rpowern = cat(2,rpowern{:}); NTs;FX~g[  
    end 8U~.\`H-PT  
    9T2xU3UyY  
    % Compute the values of the polynomials: 0Flu\w/+P  
    % -------------------------------------- pw>m.=9|y  
    z = zeros(length_r,length_n); Hr;h4J  
    for j = 1:length_n S_J :&9L  
        s = 0:(n(j)-m(j))/2; z?8~[h{i%  
        pows = n(j):-2:m(j); uMXc0fs!$  
        for k = length(s):-1:1 &!7+Yb(1  
            p = (1-2*mod(s(k),2))* ... eN0P9.eqM  
                       prod(2:(n(j)-s(k)))/          ... bv?0.{Z  
                       prod(2:s(k))/                 ... OKuD"   
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... U` R;P-  
                       prod(2:((n(j)+m(j))/2-s(k))); ~M ?|Vn  
            idx = (pows(k)==rpowers); g=]&A  
            z(:,j) = z(:,j) + p*rpowern(:,idx); E|Bd>G  
        end M\/XP| 7  
         #S QXTR  
        if isnorm ~JZ3a0$^  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); -Q$nA>trKA  
        end fhp)S",  
    end )&NAs  
    7-iIay1h"  
    % 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)  <tU :U<ea]  
    {Ot[WF  
    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。 (V^QQ !:  
    Nq  U9/  
    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)