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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 F}[!OYyg  
    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! ^@=4HtA  
     
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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 qF}a7  
    function z = zernfun(n,m,r,theta,nflag) *""W`x  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. .tHc*Eh  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N }?6;;d#  
    %   and angular frequency M, evaluated at positions (R,THETA) on the S fY9PNck\  
    %   unit circle.  N is a vector of positive integers (including 0), and {<}Hut:a  
    %   M is a vector with the same number of elements as N.  Each element } C/+zF6q  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) &~B8~U4%  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ?uJX  
    %   and THETA is a vector of angles.  R and THETA must have the same GA[bo)"  
    %   length.  The output Z is a matrix with one column for every (N,M) Ijz*wq\s;  
    %   pair, and one row for every (R,THETA) pair. >X iT[Ru  
    % ;:R2 P@6f  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .YB/7-%M[  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ~5Mj:{B  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral MwQt/Qv=  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, glROT@  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized }F9#3W&`c  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. cCx{ ")  
    % _.]mES|  
    %   The Zernike functions are an orthogonal basis on the unit circle. {wz_ngQ  
    %   They are used in disciplines such as astronomy, optics, and :.a184ax  
    %   optometry to describe functions on a circular domain. f4d-eXGwx`  
    % (@^ySiU  
    %   The following table lists the first 15 Zernike functions. XUUP#<,s  
    % C v*K.T  
    %       n    m    Zernike function           Normalization :Zob"*T  
    %       -------------------------------------------------- t7V7TL!5'  
    %       0    0    1                                 1 B#5[PX  
    %       1    1    r * cos(theta)                    2 [[N${C  
    %       1   -1    r * sin(theta)                    2 1Q9Hs(s  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) +NvpYz  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Nx*1m BC  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 9OY ao  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) O kT@ _U  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) {%y|A{}c  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) uT<<G)v)  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) D8Mq '$-  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ,PJC FQMR  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) YvP62c \  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ^f"|<r  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Q uw|KL  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) =i;T?*@  
    %       -------------------------------------------------- gnxD'1_  
    % 6zNWDUf  
    %   Example 1: Pq(LW(  
    % ^~bd AO81  
    %       % Display the Zernike function Z(n=5,m=1) $T7 qd  
    %       x = -1:0.01:1; #&L7FBJ"*v  
    %       [X,Y] = meshgrid(x,x); N{@~(>ee^  
    %       [theta,r] = cart2pol(X,Y); @B(E&  
    %       idx = r<=1; Q%J,: J  
    %       z = nan(size(X)); kr |k \  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); t6\--lk_  
    %       figure 9zCuVUcd$.  
    %       pcolor(x,x,z), shading interp 5gC> j(  
    %       axis square, colorbar Lz:FR*  
    %       title('Zernike function Z_5^1(r,\theta)') T:|p[Xbo  
    % ryA+Lli.  
    %   Example 2: xpwy%uo  
    % e:.?T\  
    %       % Display the first 10 Zernike functions Odhr=Hs  
    %       x = -1:0.01:1; 2*Pk1 vrI  
    %       [X,Y] = meshgrid(x,x); lq, ]E/<&  
    %       [theta,r] = cart2pol(X,Y); ,7k1n{C)  
    %       idx = r<=1; ~kDJ-V  
    %       z = nan(size(X)); ,]]IJ;:w  
    %       n = [0  1  1  2  2  2  3  3  3  3]; QF*cdc<  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; A2A_F|f  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 'Yc^9;C(  
    %       y = zernfun(n,m,r(idx),theta(idx)); zM<L_l&  
    %       figure('Units','normalized') 5tLb o  
    %       for k = 1:10 \$ss  
    %           z(idx) = y(:,k); oK4xRv8Hd  
    %           subplot(4,7,Nplot(k)) b^ [ z'  
    %           pcolor(x,x,z), shading interp 72*j6#zS  
    %           set(gca,'XTick',[],'YTick',[]) {{gt>"D,  
    %           axis square 5dD8s-;^T  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) .P?n<n#  
    %       end #)[.Xz:U  
    % Vx> Q  
    %   See also ZERNPOL, ZERNFUN2. [fo#){3K  
    Yw5-:w0f  
    %   Paul Fricker 11/13/2006 N#$]W"U  
    CQrP%}`r  
    ozl!vf# kv  
    % Check and prepare the inputs: NPM2qL9&J  
    % ----------------------------- yaWY>sB  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7-}5 W  
        error('zernfun:NMvectors','N and M must be vectors.') Ld/6{w4ir  
    end S{f,EBE  
    k#8`996P  
    if length(n)~=length(m) |GsMLY:0  
        error('zernfun:NMlength','N and M must be the same length.') G#6Z@|kVw  
    end -!li,&,A1  
    IXR'JZ?fH  
    n = n(:); Em5,Zr_  
    m = m(:); ]+B.=mO_  
    if any(mod(n-m,2)) rX>b R/  
        error('zernfun:NMmultiplesof2', ... a) P r&9I  
              'All N and M must differ by multiples of 2 (including 0).') oGl<i  
    end H=g%>W%3  
    ,&o^}TFkg  
    if any(m>n) z 1^fG)  
        error('zernfun:MlessthanN', ... niW"o-}  
              'Each M must be less than or equal to its corresponding N.') <hTHY E=  
    end ~kSO YvK$'  
    `NEi/jB  
    if any( r>1 | r<0 ) H270)Cwn+  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') o)7Ot\:E  
    end ^yq}>_  
    :M f8q!Q'  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) cs9h\]ZA  
        error('zernfun:RTHvector','R and THETA must be vectors.') .cw)Y#;IG  
    end ,R3TFVV!?  
    {&B_b|g*fW  
    r = r(:); HIvSpO  
    theta = theta(:); la!U  
    length_r = length(r); w%\{4T~  
    if length_r~=length(theta) ^~7Mv^A  
        error('zernfun:RTHlength', ... 8e,F{>N  
              'The number of R- and THETA-values must be equal.') mU?~s7  
    end S_OtY]gF  
    @ F $}/  
    % Check normalization: llWY7u"  
    % -------------------- g7*Uuh#  
    if nargin==5 && ischar(nflag) ]j6K3  
        isnorm = strcmpi(nflag,'norm'); Tcc83_Iq  
        if ~isnorm k`|E&+og  
            error('zernfun:normalization','Unrecognized normalization flag.') vD?D]8.F~Q  
        end "Y&   
    else '-[hy>t  
        isnorm = false; H^@Hco>|  
    end U=69q]  
    qBh@^GxY),  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =s]2?m  
    % Compute the Zernike Polynomials T6=|)UTe1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6DK).|@$r  
    vR2);ywX  
    % Determine the required powers of r: <*dcl2xS  
    % ----------------------------------- cg17e  
    m_abs = abs(m); y%61xA`#  
    rpowers = []; D M+MBK  
    for j = 1:length(n) e!gNd>b {  
        rpowers = [rpowers m_abs(j):2:n(j)]; r^<,f[yH  
    end ~_ZK93o(  
    rpowers = unique(rpowers); SOM? 0.  
    >3KlI  
    % Pre-compute the values of r raised to the required powers, l>pB\<LL  
    % and compile them in a matrix: <HN+pi  
    % ----------------------------- @SiV3k  
    if rpowers(1)==0 rr1'| k "  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 8]`s&d@GY  
        rpowern = cat(2,rpowern{:}); . _|=Btoo  
        rpowern = [ones(length_r,1) rpowern]; pV  u[  
    else ?YZgH>7"  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8Q<Nl=g>'  
        rpowern = cat(2,rpowern{:}); ly0L)L]\  
    end Ax;?~v4Z  
    Zy;jp*Q  
    % Compute the values of the polynomials: mI4GBp  
    % -------------------------------------- )j~{P  
    y = zeros(length_r,length(n)); iQ8{N:58DN  
    for j = 1:length(n) %7aJSuQN%  
        s = 0:(n(j)-m_abs(j))/2; knG:6tQ  
        pows = n(j):-2:m_abs(j); %aK[Yvo6  
        for k = length(s):-1:1 FZ+2{wIV^  
            p = (1-2*mod(s(k),2))* ... 8Nyz{T[  
                       prod(2:(n(j)-s(k)))/              ... 'h'pM#D  
                       prod(2:s(k))/                     ... SM RKEPwp&  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Y,Z$U| U  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); wzd(= *N  
            idx = (pows(k)==rpowers); 0|tyKP|J  
            y(:,j) = y(:,j) + p*rpowern(:,idx); IE996   
        end  ~,&8)1  
         uj.$GAtO)  
        if isnorm (_@5V_U  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ?&eS}skL  
        end JU^Y27  
    end n/Fxjf0W  
    % END: Compute the Zernike Polynomials OEjX(F3=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U2<q dknB  
    6wwbH}*=?  
    % Compute the Zernike functions: Ji9o0YR  
    % ------------------------------ H7&y79mB  
    idx_pos = m>0; E=,5%>C0#%  
    idx_neg = m<0; $poIWJMc  
    ciml:"nQ  
    z = y; R$ +RTG:E  
    if any(idx_pos) [|eIax xR,  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); zc;kNkV#1Y  
    end 36+/MvIT  
    if any(idx_neg) EHn!ZrQgh  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); __$;Z  
    end ^ [m-PS(  
    E2w-b^,5  
    % 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) O]hUOc `k  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. i>h 3UIx\  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated KP0(w(q  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive ZZ^A&%E(a  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, sgK =eBE  
    %   and THETA is a vector of angles.  R and THETA must have the same Mpw]dYM  
    %   length.  The output Z is a matrix with one column for every P-value, W[)HFh(#  
    %   and one row for every (R,THETA) pair. T>| hID  
    % 5cQ]vb  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike }[PwA[k'  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) gE@Pb  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) )hO%W|  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 a-,*iK{_u  
    %   for all p. 3Q62H+MC  
    % H9TeMY  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 !] uB4  
    %   Zernike functions (order N<=7).  In some disciplines it is V(ww F  
    %   traditional to label the first 36 functions using a single mode Y iuV\al  
    %   number P instead of separate numbers for the order N and azimuthal ]Bf1p  
    %   frequency M. $=E4pb4Y  
    % x2)WiO/As  
    %   Example: Gd\/n*j  
    % 8h|}Q_  
    %       % Display the first 16 Zernike functions ^znUf4N1  
    %       x = -1:0.01:1; Wq}6RdY$ZA  
    %       [X,Y] = meshgrid(x,x); }\8-&VoY#X  
    %       [theta,r] = cart2pol(X,Y); Wll0mtv  
    %       idx = r<=1; [olSgq!3  
    %       p = 0:15; {N{eOa<HA  
    %       z = nan(size(X));  }E(w@&  
    %       y = zernfun2(p,r(idx),theta(idx)); cyWb*Wv  
    %       figure('Units','normalized') !+@70|gFF  
    %       for k = 1:length(p) C,> n  
    %           z(idx) = y(:,k); u?SwGXi~8  
    %           subplot(4,4,k) Y9#dAI[Gce  
    %           pcolor(x,x,z), shading interp o!+'< IQ'  
    %           set(gca,'XTick',[],'YTick',[]) |*zgX]-+;  
    %           axis square v ))`U,Gm  
    %           title(['Z_{' num2str(p(k)) '}']) k^K76mB  
    %       end [>p!*%m  
    % z0ufLxq  
    %   See also ZERNPOL, ZERNFUN. \^y~w~g?  
    xh#_K@8  
    %   Paul Fricker 11/13/2006 C "@>NC_  
    OMjPC_  
    b+whZtNk7  
    % Check and prepare the inputs: _IU5HT}2  
    % ----------------------------- ; t9_*)[  
    if min(size(p))~=1 oaq,4FT  
        error('zernfun2:Pvector','Input P must be vector.') [Pp#r&4H  
    end M8 Bp-_  
    hTv*4J&@|  
    if any(p)>35 (HeSL),1  
        error('zernfun2:P36', ... o+}1M  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ak:f4dEd  
               '(P = 0 to 35).']) p</t##]3ks  
    end ~0-)S@  
    0)K~pV0aT  
    % Get the order and frequency corresonding to the function number:  n9&fH  
    % ---------------------------------------------------------------- It/'R-H  
    p = p(:); $/Q\B(X3  
    n = ceil((-3+sqrt(9+8*p))/2); m$_l{|4z  
    m = 2*p - n.*(n+2); .7Qqs=Au  
    2,I]H'}^  
    % Pass the inputs to the function ZERNFUN: 0VJHE~Bgi  
    % ---------------------------------------- v?3xWXX,  
    switch nargin h|'|n/F  
        case 3 @ kv~2m  
            z = zernfun(n,m,r,theta); 9CwtBil<#g  
        case 4 /03 Wst  
            z = zernfun(n,m,r,theta,nflag); j<Pw0?~s6  
        otherwise 3''Kg<k,I  
            error('zernfun2:nargin','Incorrect number of inputs.') U_aI!`WXd  
    end L)7{_s  
    wS7Vo{#@\  
    % 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) 3ErV" R4"$  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. (<OmYnm  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Ya<KMBi3  
    %   order N and frequency M, evaluated at R.  N is a vector of z8D,[`  
    %   positive integers (including 0), and M is a vector with the M,{;xf  
    %   same number of elements as N.  Each element k of M must be a dR,a0+!  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) qOyS8tA.H  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is eo!+UFZbY  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix "J}B lB  
    %   with one column for every (N,M) pair, and one row for every 91a);d  
    %   element in R. TOq xl  
    % ,@N.v?p>  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ~Lu,jLKL=[  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is $'#}f?  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to /d }5R@Oy  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 I(j{D>v  
    %   for all [n,m]. v33[Rk'  
    % q9^.f9-  
    %   The radial Zernike polynomials are the radial portion of the 0,__{?!  
    %   Zernike functions, which are an orthogonal basis on the unit BTA2['  
    %   circle.  The series representation of the radial Zernike vj#m#1\ f  
    %   polynomials is = K`]cEL  
    % #:MoZw`rlw  
    %          (n-m)/2 G[zysxd  
    %            __ xA n|OSe  
    %    m      \       s                                          n-2s `q\v~FT  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r b3GTsX\2|  
    %    n      s=0 9]{Ss$W3x  
    %  1&=2"  
    %   The following table shows the first 12 polynomials. FY+@fy  
    % IL*MB;0>  
    %       n    m    Zernike polynomial    Normalization \naG  
    %       --------------------------------------------- #fyY37-  
    %       0    0    1                        sqrt(2) zRau/1Y0  
    %       1    1    r                           2 t#]VR7]  
    %       2    0    2*r^2 - 1                sqrt(6) A>:31C  
    %       2    2    r^2                      sqrt(6) &(x>J:b  
    %       3    1    3*r^3 - 2*r              sqrt(8) hNp.%XnnZ  
    %       3    3    r^3                      sqrt(8) c Ct5m  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) HE2t0sAYX  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Z\)P|#L$  
    %       4    4    r^4                      sqrt(10) ]HG> Og  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) @$!"}xDR'  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) \/y&l\ k)  
    %       5    5    r^5                      sqrt(12) 8Tc:TaL  
    %       --------------------------------------------- (i@(ZG]/  
    % {N-*eV9#  
    %   Example: bG)6p05Oa  
    % Q6[h;lzGV  
    %       % Display three example Zernike radial polynomials ;q<:iaY9  
    %       r = 0:0.01:1; g87M"kQKA  
    %       n = [3 2 5]; MY&?*pV)  
    %       m = [1 2 1]; LyNur8 Zi  
    %       z = zernpol(n,m,r); `3UvKqe  
    %       figure &v r0{]V^  
    %       plot(r,z) \5) ZI'q  
    %       grid on ia-&?  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 57U;\L;ZmZ  
    % Vf(n  
    %   See also ZERNFUN, ZERNFUN2. 0@#d($'1?Z  
    6 )Qe*S  
    % A note on the algorithm. 3\P/4GK)  
    % ------------------------ /Dk`?  
    % The radial Zernike polynomials are computed using the series QVR-`d/  
    % representation shown in the Help section above. For many special `[f IK,  
    % functions, direct evaluation using the series representation can =Ajw(I[56  
    % produce poor numerical results (floating point errors), because a'\fS7aE0l  
    % the summation often involves computing small differences between Vao3 &#D8  
    % large successive terms in the series. (In such cases, the functions D_I_=0qNd  
    % are often evaluated using alternative methods such as recurrence _3_o/I  
    % relations: see the Legendre functions, for example). For the Zernike IBv9xP]BZ  
    % polynomials, however, this problem does not arise, because the s3gT6  
    % polynomials are evaluated over the finite domain r = (0,1), and xx%*85<  
    % because the coefficients for a given polynomial are generally all }e,*'mCC*  
    % of similar magnitude. M5LqZyY  
    % ;cWFh4_  
    % ZERNPOL has been written using a vectorized implementation: multiple NVo =5  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] N5fMMi(O  
    % values can be passed as inputs) for a vector of points R.  To achieve 2wZyUB;  
    % this vectorization most efficiently, the algorithm in ZERNPOL HG})V PBa  
    % involves pre-determining all the powers p of R that are required to |F>'7JJJ  
    % compute the outputs, and then compiling the {R^p} into a single T(eNK c2  
    % matrix.  This avoids any redundant computation of the R^p, and g*:f#u5  
    % minimizes the sizes of certain intermediate variables. X57\sggK  
    % 8~h.i1L  
    %   Paul Fricker 11/13/2006 )G9,5[  
    :WN*wd  
    Q<Th*t   
    % Check and prepare the inputs: G]fx3=  
    % ----------------------------- ?s{Pp  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) J.npv1F  
        error('zernpol:NMvectors','N and M must be vectors.') QPwUW  
    end l,M?   
    I!,FxOM|$  
    if length(n)~=length(m) Ha/-v?E  
        error('zernpol:NMlength','N and M must be the same length.') T$9tO{  
    end q\\52 :\  
    25`6V>\  
    n = n(:); 09rbu\h  
    m = m(:); &r !*Y&  
    length_n = length(n); `Jon^&^;|  
    Z&0*\.6S~  
    if any(mod(n-m,2)) UPJ3YpK  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).')  |<1  
    end '|l1-yD_  
    GbZqLZ0  
    if any(m<0) HrQft1~N  
        error('zernpol:Mpositive','All M must be positive.') 2=xjgK  
    end Qa=v }d-O  
    BDT1qiC  
    if any(m>n) V@Fj!/  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') AE 2>smp5@  
    end VE5M}kDCZ  
    RI jz7ZG  
    if any( r>1 | r<0 ) }zu?SZH  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') D>Dch0{H,:  
    end |cCrLa2*-  
    Kr|9??`0E  
    if ~any(size(r)==1) &'-ze,k}  
        error('zernpol:Rvector','R must be a vector.') s@vHU4  
    end o72G oUfs  
    =h9&`iwiu  
    r = r(:); ht%:e?@i  
    length_r = length(r); zDO`w0N  
    [1{uK&$e  
    if nargin==4 vEIDf{  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ;y"q uJ'O  
        if ~isnorm 8n^v,s>  
            error('zernpol:normalization','Unrecognized normalization flag.') fB3W} dr  
        end (:]on^|  
    else oO8V0VE\  
        isnorm = false; L<**J\=7M  
    end z tLP {q#  
    K7H` Yt  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Bdd>r# ]  
    % Compute the Zernike Polynomials \-B8`ah  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Wq1%  
    t)Mi,ljY[  
    % Determine the required powers of r: ~ g\GC  
    % ----------------------------------- WM_wkvY l  
    rpowers = []; IMDGinHAy  
    for j = 1:length(n) _Hn-bp[?>  
        rpowers = [rpowers m(j):2:n(j)]; m:,S1V_jl  
    end q'%-8t  
    rpowers = unique(rpowers); d)&}% 2ku  
    s<t*g]0`/  
    % Pre-compute the values of r raised to the required powers, 2Po e-=  
    % and compile them in a matrix: N>S_Vgk}  
    % ----------------------------- Z;6v`;[  
    if rpowers(1)==0 tGcp48R-:+  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); :NJ(QkTZv  
        rpowern = cat(2,rpowern{:}); P<@V  
        rpowern = [ones(length_r,1) rpowern]; Lgh. 1foK  
    else -5~&A6+ILn  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); `|\z#Et  
        rpowern = cat(2,rpowern{:}); *0GR }k  
    end R7 )2@;i  
    GDgq 4vfj  
    % Compute the values of the polynomials: WML%yO\.;  
    % -------------------------------------- VgHVj)ir  
    z = zeros(length_r,length_n); V9tG2m Lf>  
    for j = 1:length_n J~3+j6?%  
        s = 0:(n(j)-m(j))/2; D.hj9  
        pows = n(j):-2:m(j); %,G&By&,  
        for k = length(s):-1:1 ;-:Nw6 E  
            p = (1-2*mod(s(k),2))* ... 0T{Z'3^=  
                       prod(2:(n(j)-s(k)))/          ... #G;0yB:76  
                       prod(2:s(k))/                 ... > :s#MwIwm  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... jU~ ! *]  
                       prod(2:((n(j)+m(j))/2-s(k))); M~Tx 4_t  
            idx = (pows(k)==rpowers); 1/O7K R`K  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 2aef[TY  
        end gi|j ! m  
         fZQC'Z>EX  
        if isnorm ^Gc#D:zU  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); mlsM;A d2  
        end |]tIE{d  
    end Gf(|?" H  
    "w#jC ~J<W  
    % 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)  bR\Oyd~e  
    rEoMj)~\4&  
    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。 #YiphR&  
    ro %Jg  
    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)