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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 *Zt#U#  
    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! NT e5  
     
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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 n `&/ D  
    function z = zernfun(n,m,r,theta,nflag) Xejo_SV&?  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 9Uj $K>:  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N N G "C&v  
    %   and angular frequency M, evaluated at positions (R,THETA) on the rH_\ d?b  
    %   unit circle.  N is a vector of positive integers (including 0), and (tIo:j  
    %   M is a vector with the same number of elements as N.  Each element &cxRD  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) gW>uR3Ca4  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Fl kcU `j  
    %   and THETA is a vector of angles.  R and THETA must have the same tzZ`2pSh  
    %   length.  The output Z is a matrix with one column for every (N,M) :S<f?* }:  
    %   pair, and one row for every (R,THETA) pair. 8u6:=fxb  
    % 6-z%633DL  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike %?}33yV  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 95 ;x=ju  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 9$cWU_q{  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, WY?[,_4U  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized QZ6D7t Uc8  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. e+~\+:[?  
    % }+.}J  
    %   The Zernike functions are an orthogonal basis on the unit circle. `|{-+m  
    %   They are used in disciplines such as astronomy, optics, and QEz? w}b*  
    %   optometry to describe functions on a circular domain. cAY:AtD  
    % fI&t]   
    %   The following table lists the first 15 Zernike functions. 06O2:5zF  
    % oB}BU`-l  
    %       n    m    Zernike function           Normalization yE:+Lo`>  
    %       -------------------------------------------------- c3 jx+Q  
    %       0    0    1                                 1 OGK}EI  
    %       1    1    r * cos(theta)                    2 kD=WO4}  
    %       1   -1    r * sin(theta)                    2 lAb*fafQy  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) w,#>G07D  
    %       2    0    (2*r^2 - 1)                    sqrt(3) /N=b\-]  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) \-h%O jf4  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 8(pp2rlR  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) d,+Hd2o^X  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) }>>1<P<8-  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) T|nDTezr  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) U' H$`$Ov  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) PVe xa|aaX  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) (}Z@R#njH  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) I'A_x$ib6  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) e?0l"  
    %       -------------------------------------------------- [tlI!~Z  
    % \pPY37l  
    %   Example 1: >0/i[k-dk  
    % C _'%N lJ'  
    %       % Display the Zernike function Z(n=5,m=1) l4F%VR4KT  
    %       x = -1:0.01:1; +"rDT1^V  
    %       [X,Y] = meshgrid(x,x); tr<Nm6!  
    %       [theta,r] = cart2pol(X,Y); SIBtmm1W  
    %       idx = r<=1; J\+0[~~  
    %       z = nan(size(X)); u(@$a4z  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); uaT!(Y6  
    %       figure Bmr>n6|  
    %       pcolor(x,x,z), shading interp xN5)   
    %       axis square, colorbar *=8JIs A>!  
    %       title('Zernike function Z_5^1(r,\theta)') u_@f$  
    % CDsSrKhx  
    %   Example 2: J"!vu.[  
    % ")SFi^]  
    %       % Display the first 10 Zernike functions &5\iM^  
    %       x = -1:0.01:1; VEWi_;=J1  
    %       [X,Y] = meshgrid(x,x); Fq0i`~L~  
    %       [theta,r] = cart2pol(X,Y); ?9!6%]2D  
    %       idx = r<=1; Nv#t:J9f  
    %       z = nan(size(X)); /5S30 |K  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 9]k @Q_  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; v[ . cd*b  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; i+A3~w5c  
    %       y = zernfun(n,m,r(idx),theta(idx)); =$u! 59_dE  
    %       figure('Units','normalized') 8[a=OP  
    %       for k = 1:10 qB5j;@ r  
    %           z(idx) = y(:,k); IdzxS  
    %           subplot(4,7,Nplot(k)) D9<!mH  
    %           pcolor(x,x,z), shading interp B^1>PE  
    %           set(gca,'XTick',[],'YTick',[]) p)`{Sos  
    %           axis square 2<i!{;u$qL  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) :0Bq^G"ge  
    %       end PY{ G [  
    % m4**~xfC  
    %   See also ZERNPOL, ZERNFUN2. tI`Q/a5@  
    #jkf1"8C  
    %   Paul Fricker 11/13/2006 [A~y%bI"  
    U_M$#i{_  
    m,VOx7%n  
    % Check and prepare the inputs: {&cJDqz5=  
    % ----------------------------- =b%MXT  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Yrb{ByO&  
        error('zernfun:NMvectors','N and M must be vectors.')  DGRXd#  
    end *QpMF/<?  
    b,5~b&<h  
    if length(n)~=length(m) y`VyQWW  
        error('zernfun:NMlength','N and M must be the same length.') vq0Vq(V=  
    end bfFeBBi  
    SzAJ2:qhl  
    n = n(:); @ju@WY45$^  
    m = m(:); r A`V}>Xj  
    if any(mod(n-m,2)) 8*W#DH!  
        error('zernfun:NMmultiplesof2', ... pM+ AjPr  
              'All N and M must differ by multiples of 2 (including 0).')  ]3x?  
    end @'w"R/,n-@  
     w^?>e;/\  
    if any(m>n) ~Y`ldL  
        error('zernfun:MlessthanN', ... )mg:_K  
              'Each M must be less than or equal to its corresponding N.') sQ=]NF)\  
    end Z~AO0zUKY  
    S ^"y4- 2  
    if any( r>1 | r<0 ) >W%EmnLK  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Q! o'}nA  
    end oL!EYbFD'Z  
    .~)q};Z  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ],>@";9u"  
        error('zernfun:RTHvector','R and THETA must be vectors.') 4qO+_!x{)  
    end 1 8|m)(W  
    Tre]"2l  
    r = r(:); EOIN^4V"  
    theta = theta(:); :WjpzgPuN  
    length_r = length(r); wu7Lk3  
    if length_r~=length(theta) {'N Z.  
        error('zernfun:RTHlength', ... US+Q~GTA  
              'The number of R- and THETA-values must be equal.') 68NYIyTW9  
    end (lXGmx8  
    S{Kiy#ltWc  
    % Check normalization: FTH|9OP  
    % -------------------- ZXu>,Jy  
    if nargin==5 && ischar(nflag) [^R^8k  
        isnorm = strcmpi(nflag,'norm'); i{Uc6 R6  
        if ~isnorm QHDXW1+|^  
            error('zernfun:normalization','Unrecognized normalization flag.') &x=.$76  
        end v6[!o<@"a  
    else .sxcCrQE  
        isnorm = false; uX"H4l O~  
    end )s)I2Z+  
    T]R|qlZ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% szb_*)k  
    % Compute the Zernike Polynomials S(o#K|)>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x5|I  
    O#n8=B4  
    % Determine the required powers of r: Bz_^~b7  
    % ----------------------------------- 45=bGf#  
    m_abs = abs(m); aFc1|.Nm  
    rpowers = []; 6 +Sxr  
    for j = 1:length(n) }^4Xv^dW>g  
        rpowers = [rpowers m_abs(j):2:n(j)]; %OtFHhb  
    end Eav[/cU  
    rpowers = unique(rpowers); H ;7(}:.  
    0v6)t.]s  
    % Pre-compute the values of r raised to the required powers, u~r=)His  
    % and compile them in a matrix: 00<cYy  
    % ----------------------------- )}''L{k-  
    if rpowers(1)==0 W$QcDp]#p}  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); O ~(pg  
        rpowern = cat(2,rpowern{:}); 7WZrSC  
        rpowern = [ones(length_r,1) rpowern]; LZ\q3 7UV  
    else )r';lGh2#  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); V GL aN%|  
        rpowern = cat(2,rpowern{:}); < z+t,<3D  
    end Okgv!Nt8)A  
    cO-7ke  
    % Compute the values of the polynomials: 68bQ;Dv  
    % -------------------------------------- Q0$8j-1I  
    y = zeros(length_r,length(n)); Om\o#{D  
    for j = 1:length(n) #:%&x@@c3P  
        s = 0:(n(j)-m_abs(j))/2; ,4 hJT  
        pows = n(j):-2:m_abs(j); @(l^]9(V\  
        for k = length(s):-1:1 y9_V  
            p = (1-2*mod(s(k),2))* ... -Bt k 3  
                       prod(2:(n(j)-s(k)))/              ... Z<U6<{b  
                       prod(2:s(k))/                     ... OHv[#xGuV?  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... `{4i)n%e&  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 3NZK*!@ '  
            idx = (pows(k)==rpowers); M])ZK  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 3sc+3-TF  
        end c@YI;HS_g  
         "-y-iJ  
        if isnorm wWgWWXGT}  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); k2E0/ @f{k  
        end JgG$?n\  
    end $v,dz_O*\  
    % END: Compute the Zernike Polynomials &6DMk-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M-\Y"]sW  
    XV!6dh!  
    % Compute the Zernike functions: X"MB|N y  
    % ------------------------------ dCb`xR}  
    idx_pos = m>0; TP VVck-T8  
    idx_neg = m<0; w'L\?pI  
    \ Fl+\?~D  
    z = y; M=.:,wRm  
    if any(idx_pos) <wZQc  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); !P ~_Dl2d  
    end PEc,l>u9  
    if any(idx_neg) Qg^cf<X{i  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); k- Q%.o  
    end z+ s6)Ad  
    KuWWUjCE  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) ]e >RK'  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 3Uqr,0$p  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated L{:9Cx!F  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive qNI, 62  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 0- ><q  
    %   and THETA is a vector of angles.  R and THETA must have the same "@evXql3`  
    %   length.  The output Z is a matrix with one column for every P-value, |4 v0:ETb$  
    %   and one row for every (R,THETA) pair. FSUttg"  
    % y'FS/=u>0  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 1<+2kBuY  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ?in|qevL  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) .R)PJc5^  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 m1n.g4Z&*  
    %   for all p. s?zAP O8Sz  
    % ktK_e  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 (&&4J{`W9  
    %   Zernike functions (order N<=7).  In some disciplines it is *o-.6OxZ$  
    %   traditional to label the first 36 functions using a single mode |nbf'  
    %   number P instead of separate numbers for the order N and azimuthal qZ]pq2G  
    %   frequency M. F&])P- !3  
    % ii%+jdi.  
    %   Example: KQcs3F@t  
    % df*5,NV'-*  
    %       % Display the first 16 Zernike functions uqM yoIc  
    %       x = -1:0.01:1; 7uT:b!^f[  
    %       [X,Y] = meshgrid(x,x); GFfq+=se  
    %       [theta,r] = cart2pol(X,Y); D6CS8 ~"  
    %       idx = r<=1; p>vn7;s2#  
    %       p = 0:15; !icT/5  
    %       z = nan(size(X)); Kk(9O06j  
    %       y = zernfun2(p,r(idx),theta(idx)); 'D6T8B4  
    %       figure('Units','normalized') 9m|kgY# 4  
    %       for k = 1:length(p) \I["2C]3M  
    %           z(idx) = y(:,k); xBUya4w  
    %           subplot(4,4,k) oHMo>*?  
    %           pcolor(x,x,z), shading interp jAy^J(+  
    %           set(gca,'XTick',[],'YTick',[]) Jo$G,Q  
    %           axis square \=+b}mKV m  
    %           title(['Z_{' num2str(p(k)) '}']) }K1JU`Lz  
    %       end 3= DNb+D!  
    % 9Rn? :B~W:  
    %   See also ZERNPOL, ZERNFUN. N;Dni#tQ`  
    AgOp.~*Z~V  
    %   Paul Fricker 11/13/2006 v$(lZa1  
    \ {qI4=  
    u\LiSGePN  
    % Check and prepare the inputs: )'/|)  
    % ----------------------------- pAb.c  
    if min(size(p))~=1 !Gu%U$d  
        error('zernfun2:Pvector','Input P must be vector.') S;~g3DC d  
    end (T>nPbv)  
    -od!J\ KCy  
    if any(p)>35 AtU v71D:  
        error('zernfun2:P36', ... u$*56y   
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... @[#$J0q q  
               '(P = 0 to 35).']) cCo`~7rE  
    end chV9_(8  
    I8+~ &V}  
    % Get the order and frequency corresonding to the function number: SrFS#  
    % ---------------------------------------------------------------- CNr/U*+  
    p = p(:); 2-=\~<)  
    n = ceil((-3+sqrt(9+8*p))/2); :ml2.vP  
    m = 2*p - n.*(n+2); b !%hH  
    `W dD8E  
    % Pass the inputs to the function ZERNFUN: %FS$zOsgGK  
    % ---------------------------------------- >w-;Z>3Q@  
    switch nargin Ma'_e=+A  
        case 3 V$"ujRp  
            z = zernfun(n,m,r,theta); VOc8q-hK  
        case 4 Ypeiy `.  
            z = zernfun(n,m,r,theta,nflag); @%tRhG  
        otherwise txZ?=8j_Y  
            error('zernfun2:nargin','Incorrect number of inputs.') PqyA1  
    end 6ZKsz5:=  
    k% sO 0  
    % 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) =/u% c!  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. a@@M+9Q  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of o]ag"Q  
    %   order N and frequency M, evaluated at R.  N is a vector of "]m+z)lWd  
    %   positive integers (including 0), and M is a vector with the -pU|hSW*b  
    %   same number of elements as N.  Each element k of M must be a n:0}utU4  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ?;wpd';c  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is $`8Ar,Xz`  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ik"sq}u_]E  
    %   with one column for every (N,M) pair, and one row for every  ],ZzI  
    %   element in R. A%Xt|=^_  
    % ?E9DXg  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- s-Aw<Q)d  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is \":?xh_H  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to WpS1a440  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 4EXB;[ ]  
    %   for all [n,m]. n4R2^gXAw  
    % b1gaj"]  
    %   The radial Zernike polynomials are the radial portion of the Z<#hS=eY  
    %   Zernike functions, which are an orthogonal basis on the unit >Jw6l0z  
    %   circle.  The series representation of the radial Zernike T"p(]@Ng  
    %   polynomials is zOHypazOTq  
    % `/"nTB  
    %          (n-m)/2 l{:a1^[>y  
    %            __  cO\-  
    %    m      \       s                                          n-2s fQ'.8'>T  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r &(z fa&j|  
    %    n      s=0 R7s|`\  
    % x$wd O  
    %   The following table shows the first 12 polynomials. xq %{}  
    % q8Nn%o=5V  
    %       n    m    Zernike polynomial    Normalization rVtw-[p  
    %       --------------------------------------------- MlJVeod  
    %       0    0    1                        sqrt(2) '~ 4pl0TWc  
    %       1    1    r                           2 E15vq6DKF  
    %       2    0    2*r^2 - 1                sqrt(6) Vvt  ;  
    %       2    2    r^2                      sqrt(6) W%e_~$H0  
    %       3    1    3*r^3 - 2*r              sqrt(8) K9[e>  
    %       3    3    r^3                      sqrt(8) 6*nAo8gl  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) .fzu"XAPu  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 'r} zY-FM`  
    %       4    4    r^4                      sqrt(10) mmEp'E  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) *U=]@I}J  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) pilh@#_h  
    %       5    5    r^5                      sqrt(12) H]\Zn%.#  
    %       --------------------------------------------- ' )-M\'S$E  
    % 8ga_pNe  
    %   Example: _P,^_%}V06  
    % [4+q+  
    %       % Display three example Zernike radial polynomials F?u^"}%Fc  
    %       r = 0:0.01:1; z *9FlV  
    %       n = [3 2 5]; S2C]?6cTq  
    %       m = [1 2 1]; W3&tJ8*3  
    %       z = zernpol(n,m,r); I\Glc=T*  
    %       figure )9=(|Lp  
    %       plot(r,z) J$9:jE-4  
    %       grid on h?UVDzI!O  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') hzY[ G :  
    % G",.,Px  
    %   See also ZERNFUN, ZERNFUN2. K{cbn1\,H  
    rS*$rQCr=  
    % A note on the algorithm. :XV} c(+d  
    % ------------------------ Rt(J/%;  
    % The radial Zernike polynomials are computed using the series +VU4s$w6  
    % representation shown in the Help section above. For many special K(T\9J.  
    % functions, direct evaluation using the series representation can f+Dn9t  
    % produce poor numerical results (floating point errors), because 7Bz*r0 9S  
    % the summation often involves computing small differences between U5/qf8)yO  
    % large successive terms in the series. (In such cases, the functions JmOW~W  
    % are often evaluated using alternative methods such as recurrence GZ }/leR  
    % relations: see the Legendre functions, for example). For the Zernike %G?K@5?j?  
    % polynomials, however, this problem does not arise, because the W9J1=  
    % polynomials are evaluated over the finite domain r = (0,1), and {hqAnZ@]vr  
    % because the coefficients for a given polynomial are generally all V+Xl9v4O  
    % of similar magnitude. C:\(~D *GS  
    % Wv K(G3  
    % ZERNPOL has been written using a vectorized implementation: multiple {UH9i'y:t  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ujgLJ77  
    % values can be passed as inputs) for a vector of points R.  To achieve v })Q  
    % this vectorization most efficiently, the algorithm in ZERNPOL {<w +3Va  
    % involves pre-determining all the powers p of R that are required to `_ (~ Ud  
    % compute the outputs, and then compiling the {R^p} into a single ivrXwZ7jT  
    % matrix.  This avoids any redundant computation of the R^p, and %85Icg  
    % minimizes the sizes of certain intermediate variables. dEp/dd~(&  
    % (:\LWJX0=  
    %   Paul Fricker 11/13/2006 FfMnul  
    We`'>'W0  
    8SnS~._9  
    % Check and prepare the inputs: Fa3gJ[ZAqf  
    % -----------------------------  s&*yk p  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) H3 m8  
        error('zernpol:NMvectors','N and M must be vectors.') `1p 8C%  
    end FWPW/oC  
    A%ywj'|z  
    if length(n)~=length(m) zTCP )x  
        error('zernpol:NMlength','N and M must be the same length.') "S(X[Y'  
    end C|z%P}u#p  
    r(uP!n1+  
    n = n(:); R Td^ImV  
    m = m(:); "D> ]ES%5  
    length_n = length(n); R]b! $6Lt  
    p/|(,)'+jx  
    if any(mod(n-m,2)) : d'65KMi  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') x3p9GAd#  
    end T$b\Q  
    ;;LuU<,$  
    if any(m<0) Etmo7 8e  
        error('zernpol:Mpositive','All M must be positive.') gOE_ ]  
    end W\ULUK  
    o+)A'S  
    if any(m>n) wz@FrRP=  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') e)pTC97^L  
    end Uu2N9.5  
    mC(u2  
    if any( r>1 | r<0 ) l]vohLz 3!  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') %yw=[]Vjze  
    end bf98B4<  
    rycJyiw<-  
    if ~any(size(r)==1) _Y YP4lEL  
        error('zernpol:Rvector','R must be a vector.') ag]*DsBt  
    end Pc4R!Tc  
    nGZ \<-  
    r = r(:); =49o U  
    length_r = length(r); \|{*arS  
    >Pkdu}xP3  
    if nargin==4 !V( `ZH  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); *98Ti|  
        if ~isnorm @f=RL)$|  
            error('zernpol:normalization','Unrecognized normalization flag.') `Yogq)G}  
        end BD1K H;  
    else X\G)81Q.S  
        isnorm = false; wG:$6  
    end -><QFJ  
    LD@7(?mlU  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~j}J<4&OvC  
    % Compute the Zernike Polynomials Gmgeve  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;E2kT GT  
    }B*,mn2N  
    % Determine the required powers of r: nL\ZId  
    % ----------------------------------- r03I*b  
    rpowers = []; M 5rwoyn  
    for j = 1:length(n) v|y<_Ya  
        rpowers = [rpowers m(j):2:n(j)]; {fElto   
    end 0Q*-g}wXfS  
    rpowers = unique(rpowers); NV(fN-L  
    (.oaMA"B  
    % Pre-compute the values of r raised to the required powers, Z]e4pR6!  
    % and compile them in a matrix: Uk"Y/Ddm  
    % ----------------------------- E~69^ cd  
    if rpowers(1)==0 _3KZME  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); =u[rOU{X"W  
        rpowern = cat(2,rpowern{:}); ^OjvL6 A/p  
        rpowern = [ones(length_r,1) rpowern]; .='3bQ(UZ4  
    else g(aNyn  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9M0d+:YJ  
        rpowern = cat(2,rpowern{:}); #_,uE9  
    end $Gd5wmb!  
    B^Bbso'{1  
    % Compute the values of the polynomials: ?V6 %>RU  
    % -------------------------------------- j$%yw4dsj  
    z = zeros(length_r,length_n); Yg|l?d"  
    for j = 1:length_n Vy&f"4~  
        s = 0:(n(j)-m(j))/2; 1 O- E],  
        pows = n(j):-2:m(j); ^h"n03VFA  
        for k = length(s):-1:1 t0I>5#*WU  
            p = (1-2*mod(s(k),2))* ... JYl\<Z' {  
                       prod(2:(n(j)-s(k)))/          ... `J]<_0kX}%  
                       prod(2:s(k))/                 ... wFn@\3%l`  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... o9~h%&  
                       prod(2:((n(j)+m(j))/2-s(k))); &s2#1  
            idx = (pows(k)==rpowers); &l?N:(r  
            z(:,j) = z(:,j) + p*rpowern(:,idx); % Mw'e/?  
        end K]%N-F>r  
         nh?9R&  
        if isnorm L;4[ k;5  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); /bu'6/!`  
        end p9k4w% ~:  
    end % i %ew4  
    ,uKs>T^  
    % 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)  !ho5VA t  
    .%J?T5D  
    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。 U'Ja\Ek/f  
    (A]m=  
    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)