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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 pXa? Q@ 6  
    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! p7er04/}\  
     
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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 #ZA YP  
    function z = zernfun(n,m,r,theta,nflag) Hik[pVK@  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. v &n &i?  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N oWmla*nCKL  
    %   and angular frequency M, evaluated at positions (R,THETA) on the z{\.3G  
    %   unit circle.  N is a vector of positive integers (including 0), and /Ny&;Y  
    %   M is a vector with the same number of elements as N.  Each element N;Bal/kd2  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) %:*HzYf  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, @d\F; o<  
    %   and THETA is a vector of angles.  R and THETA must have the same Bh?;\D'YC  
    %   length.  The output Z is a matrix with one column for every (N,M) K@m^QioMj  
    %   pair, and one row for every (R,THETA) pair. tF|bxXs Z  
    % i7FEjjGtG  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike g5)VV"  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), PBmt.yF  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral mX89^  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ~"k'T9QBY  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized c+JlM1p@  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. !T*izMX}  
    % KVuv%?  
    %   The Zernike functions are an orthogonal basis on the unit circle. Z>l>@wNm  
    %   They are used in disciplines such as astronomy, optics, and ]G:xTv8  
    %   optometry to describe functions on a circular domain. <mN3:G  
    % E' Bt1 u  
    %   The following table lists the first 15 Zernike functions. }1V&(#H2  
    % Nu'rn*Y_  
    %       n    m    Zernike function           Normalization o&]qjFo\m  
    %       -------------------------------------------------- {% P;O ?  
    %       0    0    1                                 1 ~J|0G6H  
    %       1    1    r * cos(theta)                    2 yFSL7`p+  
    %       1   -1    r * sin(theta)                    2 KjadX&JD  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) p.G7Cs  
    %       2    0    (2*r^2 - 1)                    sqrt(3) > L%%B-  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) bm;4NA?Gg  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) cQ`,:t#[  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) <\5{R@A*6  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 3r\QLIr L8  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) g=)@yZ3>v  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 5M*p1^ >  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) [Mi~4b  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) :9<5GF(  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) oW6.c]Vo  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) C.@TX  
    %       -------------------------------------------------- >2a~hW|,  
    % zSu2B6YU}  
    %   Example 1: qVfOf\x.e  
    % T4[eBO  
    %       % Display the Zernike function Z(n=5,m=1) \21!NPXH2  
    %       x = -1:0.01:1; _xJ&p$&  
    %       [X,Y] = meshgrid(x,x); B4kIcHA  
    %       [theta,r] = cart2pol(X,Y); E~B LY{3:  
    %       idx = r<=1; 8L:0Wp  
    %       z = nan(size(X)); [K5afnq`  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); w^~,M3(+)1  
    %       figure z8o Sh t`+  
    %       pcolor(x,x,z), shading interp {S?.bT%&  
    %       axis square, colorbar %lBFj/B  
    %       title('Zernike function Z_5^1(r,\theta)') ek9%Xk8  
    % ' {Q L`L  
    %   Example 2: s SDBl~g  
    % ?IK[]=!  
    %       % Display the first 10 Zernike functions 8 =d9*lm  
    %       x = -1:0.01:1; U-@\V1;C  
    %       [X,Y] = meshgrid(x,x); J? C"be=  
    %       [theta,r] = cart2pol(X,Y); d/MMPge3  
    %       idx = r<=1; k20tn ew  
    %       z = nan(size(X)); avQwbAh[  
    %       n = [0  1  1  2  2  2  3  3  3  3]; LVSJK.B  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ' `S,d[~  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; j:0z/gHp$  
    %       y = zernfun(n,m,r(idx),theta(idx)); |q?A8@\u  
    %       figure('Units','normalized') @ Fu|et  
    %       for k = 1:10 |.YL 2\  
    %           z(idx) = y(:,k); 37VSE@Z+  
    %           subplot(4,7,Nplot(k)) j*GYYEY  
    %           pcolor(x,x,z), shading interp S;Vj5  
    %           set(gca,'XTick',[],'YTick',[]) |g~.]2az  
    %           axis square dI`b AP;\  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) r'&VH]m  
    %       end T!8,R{V]4  
    % ]\{EUx9  
    %   See also ZERNPOL, ZERNFUN2. DUaj]V{_^  
    -0Ps. B  
    %   Paul Fricker 11/13/2006 ?Pa5skqR  
    2vynz,^ET  
    )gZ yW  
    % Check and prepare the inputs: uKK+V6}!kj  
    % ----------------------------- yovC~  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) [j) :2  
        error('zernfun:NMvectors','N and M must be vectors.') _di[PU=Vh  
    end aPRF  
    Ay[6rUO  
    if length(n)~=length(m)  [5H#ay  
        error('zernfun:NMlength','N and M must be the same length.') bO9X;} \6  
    end uT_bA0jK  
    &4LrV+`$V  
    n = n(:); KrB"2e+J  
    m = m(:); " 5=Gu1  
    if any(mod(n-m,2)) nBR4j?':i  
        error('zernfun:NMmultiplesof2', ... MFRM M%`  
              'All N and M must differ by multiples of 2 (including 0).') q.*k J/L  
    end Dc U$sf*  
    L^dF )y?  
    if any(m>n) O.i.<VD7  
        error('zernfun:MlessthanN', ... !Eu}ro.}  
              'Each M must be less than or equal to its corresponding N.') t~3!| @3i  
    end P9BShC5  
    5LR k)@t  
    if any( r>1 | r<0 ) l4RZ!K*X_"  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') O|d"0P  
    end 09/Mg  
    n&Bgpt~  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) |Y4c+6@_  
        error('zernfun:RTHvector','R and THETA must be vectors.') voiWf?X  
    end }Ge$?ZFH  
     (cx Q<5  
    r = r(:); 1 Qln|b8<  
    theta = theta(:); 0tK(:9S  
    length_r = length(r); =A{F&:+a]  
    if length_r~=length(theta) *jM]:GpyoU  
        error('zernfun:RTHlength', ... OQ&l/|{O0?  
              'The number of R- and THETA-values must be equal.') kZ$2Uss  
    end I|(r1.[K  
    Fsz;T;  
    % Check normalization: Qu|H_<8g  
    % -------------------- K|]/BjB/  
    if nargin==5 && ischar(nflag) \8g'v@$wG  
        isnorm = strcmpi(nflag,'norm'); u^, eHO  
        if ~isnorm :<hM@>eFn  
            error('zernfun:normalization','Unrecognized normalization flag.') shKTj5s?  
        end ^VOFkUp)  
    else =bgWUu\F  
        isnorm = false; ]lqLC  
    end Qco8m4n  
    tnE),  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Kl_(4kQE_  
    % Compute the Zernike Polynomials HAwdu1$8  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% H%pD9'q~  
    -!q^/ux  
    % Determine the required powers of r: VkFMr8@|  
    % ----------------------------------- >e>%AMzo[  
    m_abs = abs(m); w{mw?0  
    rpowers = []; Y \Gx|  
    for j = 1:length(n) gWQ(B  
        rpowers = [rpowers m_abs(j):2:n(j)]; tTOBKA89  
    end }k;wSp[3  
    rpowers = unique(rpowers); C cPOK2  
    galzk$D  
    % Pre-compute the values of r raised to the required powers, f*}}Az.4  
    % and compile them in a matrix: 1%ENgb:8  
    % ----------------------------- L>LIN 1A  
    if rpowers(1)==0 # ~Doz7~  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 0CO@@`~4  
        rpowern = cat(2,rpowern{:}); 1J([*)  
        rpowern = [ones(length_r,1) rpowern]; t'1g+g  
    else $Q"D>Qf{G  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); F42<9)I  
        rpowern = cat(2,rpowern{:}); ulE5lG0c  
    end gFqF&t  
    ,?P<=M  
    % Compute the values of the polynomials: 4M#i_.`z  
    % -------------------------------------- C#-HWoSi  
    y = zeros(length_r,length(n)); ^hXm=r4ozR  
    for j = 1:length(n) k3K*{"z  
        s = 0:(n(j)-m_abs(j))/2; oqAO@<dL!  
        pows = n(j):-2:m_abs(j); ]VL} eHZ  
        for k = length(s):-1:1 &(oA/jFQ  
            p = (1-2*mod(s(k),2))* ... u@1 2:U$  
                       prod(2:(n(j)-s(k)))/              ... `Fie'[F5,)  
                       prod(2:s(k))/                     ... C~egF=w  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... @^T~W^+  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); w? >f:2(=[  
            idx = (pows(k)==rpowers); l^Ob60)2  
            y(:,j) = y(:,j) + p*rpowern(:,idx); >$7x]f  
        end XLC9B3Jt  
         ![;={d0  
        if isnorm ,Kl:4 Tv  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); }f;WYz5  
        end /5)*epF+  
    end P0yDL:X[  
    % END: Compute the Zernike Polynomials 6@TU9AZS `  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <o/!M6^:  
    !NH(EWER  
    % Compute the Zernike functions: -'Ay(h   
    % ------------------------------ \_WR:?l  
    idx_pos = m>0; h;,1BpbM  
    idx_neg = m<0; ^R=`<jx   
    $2\ 8Rn6'  
    z = y; 7mq&]4-G  
    if any(idx_pos) i,h30J  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); aX`uF<c9  
    end LD ]-IX&L  
    if any(idx_neg) +N=HI1^54R  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); vo f8bQ{&  
    end @4hzNi+  
    OKAU*}_  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) y QClq{A  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. (/uAn2  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated iP0m1  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive iI{L>  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, K_;vqi^1^&  
    %   and THETA is a vector of angles.  R and THETA must have the same S7)qq  
    %   length.  The output Z is a matrix with one column for every P-value, SK lvZ  
    %   and one row for every (R,THETA) pair. 4d`YZNvZW/  
    % B~w$j/sWU  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike >=[uLY[aK  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) d #1Y^3n  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ;.V/ngaj  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 gom!dB0J  
    %   for all p. R3~,&ab  
    % C< 9x\JY%  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 8@;]@c)m  
    %   Zernike functions (order N<=7).  In some disciplines it is g%&E~V/g$  
    %   traditional to label the first 36 functions using a single mode se\fbe^0  
    %   number P instead of separate numbers for the order N and azimuthal C3}:DIn"w  
    %   frequency M. iX$G($[l(  
    % {1)A"lQu  
    %   Example: F{0\a;U@^  
    % P9/ (f$=  
    %       % Display the first 16 Zernike functions Z^_qXerjP  
    %       x = -1:0.01:1; 6;Z -Y>\c  
    %       [X,Y] = meshgrid(x,x); BM<q;;pO  
    %       [theta,r] = cart2pol(X,Y); _K o#36.S  
    %       idx = r<=1; $D1ha CL  
    %       p = 0:15; B n7uKa{P  
    %       z = nan(size(X)); }T@=I&g;  
    %       y = zernfun2(p,r(idx),theta(idx)); (-gomn  
    %       figure('Units','normalized') KLyRb0V  
    %       for k = 1:length(p) K6kz{R%`  
    %           z(idx) = y(:,k); n9'3~qVZ  
    %           subplot(4,4,k) )i~AXBt}  
    %           pcolor(x,x,z), shading interp S"cTi[9  
    %           set(gca,'XTick',[],'YTick',[]) 4rU/2}. q  
    %           axis square Co1d44Q  
    %           title(['Z_{' num2str(p(k)) '}']) 6Ijt2c'A}  
    %       end (9Zvr4.f7  
    % pR61bl)  
    %   See also ZERNPOL, ZERNFUN. ^ Oh  
    `,qft[1  
    %   Paul Fricker 11/13/2006 BS9VwG <Z  
    (xHmucmwp  
    Cz0FA]-g  
    % Check and prepare the inputs: lL}NiN-)t  
    % ----------------------------- IrMH AM5K  
    if min(size(p))~=1 h[W`P%xZ  
        error('zernfun2:Pvector','Input P must be vector.') 0$*7lQ<a#M  
    end h} `v0E  
    Az&>.*  
    if any(p)>35 )[ V8YiyU  
        error('zernfun2:P36', ... KqK]R6>  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... F\m^slsu7=  
               '(P = 0 to 35).']) .d<K`.O ;  
    end [Fl_R[o  
    .nPOjwEx&Y  
    % Get the order and frequency corresonding to the function number: j'D%eQI,V  
    % ---------------------------------------------------------------- }u_D{bz  
    p = p(:); A NhqS  
    n = ceil((-3+sqrt(9+8*p))/2); 8A#,*@V[  
    m = 2*p - n.*(n+2); W/qXQORv  
    cnu&!>8V  
    % Pass the inputs to the function ZERNFUN: o701RG ~)  
    % ---------------------------------------- ` ,\b_SFg  
    switch nargin 731Lz*IFg  
        case 3 '(.5!7?Qc  
            z = zernfun(n,m,r,theta); yaR>?[h  
        case 4 y98FEG#S}  
            z = zernfun(n,m,r,theta,nflag); |'h (S|  
        otherwise "t0^4=c+7  
            error('zernfun2:nargin','Incorrect number of inputs.') q3x"9i `  
    end tu\XuDk y  
    B4y_{V  
    % 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) U1YqyG8  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. vFUp$[  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of "Hw%@]#  
    %   order N and frequency M, evaluated at R.  N is a vector of 7nB4(A2[S4  
    %   positive integers (including 0), and M is a vector with the ^T&{ORWz  
    %   same number of elements as N.  Each element k of M must be a d:'{h"M6  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) TAYh#T=S  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Ic'D# m  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 5iw\F!op:  
    %   with one column for every (N,M) pair, and one row for every 9C7Npf?~M  
    %   element in R. <Y`(J#  
    % /dCsZA  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- uuM1_nD[  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is _ s 3aaOL  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to OC&BJNOi  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 *$9U/  d  
    %   for all [n,m]. #w;"s*  
    % Tb] 7# v  
    %   The radial Zernike polynomials are the radial portion of the T6/P54S  
    %   Zernike functions, which are an orthogonal basis on the unit cxR.:LD}  
    %   circle.  The series representation of the radial Zernike g[~{iu_$d  
    %   polynomials is "M:ui0YP  
    % a<-aE4wdm  
    %          (n-m)/2 ./I?|ih  
    %            __ (VO'Kd  
    %    m      \       s                                          n-2s =Htt'""DN  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r jG ouwta  
    %    n      s=0 ! VT$U6  
    % ,~3rY,y-  
    %   The following table shows the first 12 polynomials. f}yRTR GJv  
    % ,x\qYz+7|  
    %       n    m    Zernike polynomial    Normalization jTS8 qu  
    %       --------------------------------------------- *C55DO^w  
    %       0    0    1                        sqrt(2) qb;b.P?~D$  
    %       1    1    r                           2 ?$`kT..j,u  
    %       2    0    2*r^2 - 1                sqrt(6) 2|"D\N  
    %       2    2    r^2                      sqrt(6) >,Y+ 1  
    %       3    1    3*r^3 - 2*r              sqrt(8) 53hX%{3  
    %       3    3    r^3                      sqrt(8) r0nnmy]{d  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) *SJ[~  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) o~'p&f  
    %       4    4    r^4                      sqrt(10) A,&711Y  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) jUD^]Qs  
    %       5    3    5*r^5 - 4*r^3            sqrt(12)  3*Q=)}  
    %       5    5    r^5                      sqrt(12) yf*'=q  
    %       --------------------------------------------- &w9*pJR %  
    % aEzf*a|fSV  
    %   Example: ]Sj;\Iz  
    % )@9Eq|jMC  
    %       % Display three example Zernike radial polynomials ZklO9Ox(  
    %       r = 0:0.01:1; Ep(xlHTv  
    %       n = [3 2 5]; ?<F([(  
    %       m = [1 2 1]; )*_G/<N) |  
    %       z = zernpol(n,m,r); g}R#0gkdk}  
    %       figure 'Ev[G6vo  
    %       plot(r,z) 8Vz!zYl  
    %       grid on kxJs4BY0  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') <b'*GBw$  
    % X> 98`  
    %   See also ZERNFUN, ZERNFUN2. +UWv}|  
    +wz1kPRs  
    % A note on the algorithm. Cgln@Rz  
    % ------------------------ Y'000#+  
    % The radial Zernike polynomials are computed using the series 4RctYMz  
    % representation shown in the Help section above. For many special db_Qt'>  
    % functions, direct evaluation using the series representation can ^>%.l'1/(  
    % produce poor numerical results (floating point errors), because %AJ9fs4/  
    % the summation often involves computing small differences between ` Ft-1eE  
    % large successive terms in the series. (In such cases, the functions WI&A+1CK-5  
    % are often evaluated using alternative methods such as recurrence 9:g A0Z  
    % relations: see the Legendre functions, for example). For the Zernike YFu>`w^Y  
    % polynomials, however, this problem does not arise, because the =p5]r:9W  
    % polynomials are evaluated over the finite domain r = (0,1), and ,){#J"W  
    % because the coefficients for a given polynomial are generally all T*@o?U  
    % of similar magnitude. #qk=R7" Q  
    % rRe^7xGe7  
    % ZERNPOL has been written using a vectorized implementation: multiple ?f9M59(l  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Q_p&~PNy5  
    % values can be passed as inputs) for a vector of points R.  To achieve q.R(>ZcV  
    % this vectorization most efficiently, the algorithm in ZERNPOL /tG as  
    % involves pre-determining all the powers p of R that are required to +5I5  
    % compute the outputs, and then compiling the {R^p} into a single p2(ha3PW  
    % matrix.  This avoids any redundant computation of the R^p, and gFuK/]gzI  
    % minimizes the sizes of certain intermediate variables. =\u,4  
    % $Tv~ *|a  
    %   Paul Fricker 11/13/2006 J<H]vs  
    8&HBR #  
    &\ca ? #  
    % Check and prepare the inputs: prt(xr4@  
    % ----------------------------- vN v'%;L  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) FO(QsR=\s  
        error('zernpol:NMvectors','N and M must be vectors.') "5dke^yk0  
    end Uc_ }="  
    Z  #  
    if length(n)~=length(m) 2%fzRXhu%  
        error('zernpol:NMlength','N and M must be the same length.') $bp$[fX(e  
    end zqrqbqK5R  
    WI| -pzg  
    n = n(:); 7bbFUUUG"  
    m = m(:); '/XP4B\(E  
    length_n = length(n); '\d ldg#P  
    a_/4^+  
    if any(mod(n-m,2)) IO&U=-pn&  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 9W(&g)`  
    end ]v5/K  
    "oiN8#Hf  
    if any(m<0) sZ&6g<8#y  
        error('zernpol:Mpositive','All M must be positive.') I)#8}[vK  
    end GK-P6d  
    SJX9oVJeZ  
    if any(m>n) _(?`eWo  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') #%ld~dgz-  
    end ld#x'/  
    "y*3p0E  
    if any( r>1 | r<0 ) 6wu`;>  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') |*w)]2B l  
    end >&e=0@?+G  
    PfU\.[l$  
    if ~any(size(r)==1) KwMt@1Z  
        error('zernpol:Rvector','R must be a vector.') XM+.Hel  
    end >WZbb d-  
    @=AQr4&  
    r = r(:); LKI\(%ba#  
    length_r = length(r); n6,YA2yZO  
    @,= pG  
    if nargin==4 ]!!?gnPd5  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); [O^/"Qk  
        if ~isnorm 451.VI}MR  
            error('zernpol:normalization','Unrecognized normalization flag.') RLL ph  
        end wmVb0~[  
    else ZZ{c  
        isnorm = false; `WCL-OoZc5  
    end 9 4H')(  
    $X-PjQb1Bb  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {E(2.'d  
    % Compute the Zernike Polynomials $ S3b<]B  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% W;R6+@I[  
    -,;woOG  
    % Determine the required powers of r: 3^&`E} r  
    % ----------------------------------- " 1a!]45+  
    rpowers = []; QGOkB  
    for j = 1:length(n) ~.G$0IJY  
        rpowers = [rpowers m(j):2:n(j)]; PHT<]:"`<  
    end aqqo>O3 s  
    rpowers = unique(rpowers); aj|PyX3P:  
    *szs"mQ/  
    % Pre-compute the values of r raised to the required powers, k kD#Bb  
    % and compile them in a matrix: Go:(R {P  
    % ----------------------------- j3%Wrt  
    if rpowers(1)==0 NTZ3Np`  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); i" u|119  
        rpowern = cat(2,rpowern{:}); Bi;a~qE  
        rpowern = [ones(length_r,1) rpowern]; uSI@Cjp  
    else t 3N}):  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 1;kMbl]  
        rpowern = cat(2,rpowern{:}); `)]W~  
    end ,)d`_AD+5  
    `{K-eHlrM9  
    % Compute the values of the polynomials: ns5Dydo{T  
    % -------------------------------------- Z/:yYSq  
    z = zeros(length_r,length_n); ^|vk^`S  
    for j = 1:length_n rq7yNt  
        s = 0:(n(j)-m(j))/2; ]Oo!>iTQi  
        pows = n(j):-2:m(j); t1 9f%d  
        for k = length(s):-1:1 NWiDNK[VE}  
            p = (1-2*mod(s(k),2))* ... q[P>s{"  
                       prod(2:(n(j)-s(k)))/          ... uMG y-c  
                       prod(2:s(k))/                 ... LzLJ6A>;R  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ^Lfwoy7R  
                       prod(2:((n(j)+m(j))/2-s(k))); oF+yh!~mM  
            idx = (pows(k)==rpowers); _(gkYJ+MK  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 6A5.n?B{  
        end !WGQ34R{  
         |zfFB7}v  
        if isnorm mMZrBz7r  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); tAep_GR  
        end ?xMTO  
    end 3l`"(5  
    * Uy>F[%@  
    % 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)  /5f=a  
    "@xL9[d  
    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。 r,GgMk  
    a7z% )i;Z  
    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)