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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 bojx:g  
    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! at@B>Rb  
     
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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 :^+ aJ]  
    function z = zernfun(n,m,r,theta,nflag) tkBp?Wl  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. **L. !/  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N U$j*{`$4  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Hn%n>Bnl  
    %   unit circle.  N is a vector of positive integers (including 0), and 9IgozYj  
    %   M is a vector with the same number of elements as N.  Each element PSX-b)wb  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ;Ub;AqY  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, -AUdBG  
    %   and THETA is a vector of angles.  R and THETA must have the same ?Xscc mN  
    %   length.  The output Z is a matrix with one column for every (N,M) #F\}PCBe'  
    %   pair, and one row for every (R,THETA) pair. Iy\{)+}aS  
    % oR'8|~U@B  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike %/17K2g  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), H tIl;E  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 6$TE-l  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, m&xyw9a  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized U$R+&@;  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. kYwk'\s  
    % %xE\IRlR  
    %   The Zernike functions are an orthogonal basis on the unit circle. Ur`Ri?  
    %   They are used in disciplines such as astronomy, optics, and gbOd(ugH  
    %   optometry to describe functions on a circular domain. $+eDoI'f  
    % }Wf\\  
    %   The following table lists the first 15 Zernike functions. 0;,4.hsh  
    % DN)Ehd.  
    %       n    m    Zernike function           Normalization ek~bXy{O`  
    %       -------------------------------------------------- Fw!CssW  
    %       0    0    1                                 1 (J(JB}[X,  
    %       1    1    r * cos(theta)                    2 V QE *B  
    %       1   -1    r * sin(theta)                    2 >'3J. FY  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) &KC^Vn3Nj  
    %       2    0    (2*r^2 - 1)                    sqrt(3) LyM"  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) qP<wf=wY  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) wehZ7eqm  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ^v. ~FFK  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) #gbJ$1s  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) f6x}M9xS%  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) p!<Y 'G  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) kIS_ 6!  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ,"!t[4p=f  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |,c\R"8xS  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) vy?Zz<c;  
    %       -------------------------------------------------- B`,4M&  
    % w8M,35b  
    %   Example 1: AyZL(  
    % zoYw[YP9  
    %       % Display the Zernike function Z(n=5,m=1) V=}AFGC85  
    %       x = -1:0.01:1; |IL..C  
    %       [X,Y] = meshgrid(x,x); Iuk!A?XV  
    %       [theta,r] = cart2pol(X,Y); IHCEuK  
    %       idx = r<=1; {f;]  
    %       z = nan(size(X)); MM8r*T4g/  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); AW;"` ].  
    %       figure 1Ao YG_  
    %       pcolor(x,x,z), shading interp W$=MuF7R  
    %       axis square, colorbar O(BAw  
    %       title('Zernike function Z_5^1(r,\theta)') x}I'W?g  
    % =H&@9=D*  
    %   Example 2: &Pu}"M$[MH  
    % dLQV>oF  
    %       % Display the first 10 Zernike functions  S^;D\6(r  
    %       x = -1:0.01:1; S<"T:Y &  
    %       [X,Y] = meshgrid(x,x); A<esMDX  
    %       [theta,r] = cart2pol(X,Y); Q%6Lc.i  
    %       idx = r<=1; s,Uc cA@  
    %       z = nan(size(X)); kWs"v6B  
    %       n = [0  1  1  2  2  2  3  3  3  3]; z7X[$T$V  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 0#f;/ c0i  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; r:u,  
    %       y = zernfun(n,m,r(idx),theta(idx)); `4E6&&E+S  
    %       figure('Units','normalized') nzI}w7>VU  
    %       for k = 1:10 __jFSa`at  
    %           z(idx) = y(:,k); |,k,X}gP  
    %           subplot(4,7,Nplot(k)) NsYeg&>`  
    %           pcolor(x,x,z), shading interp jFYv4!\ju  
    %           set(gca,'XTick',[],'YTick',[]) -z%| Jk  
    %           axis square NWCJ|  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) vr#_pu)f4  
    %       end lTOO`g  
    % ts rcX  
    %   See also ZERNPOL, ZERNFUN2. FL -yt  
    rdd%"u+  
    %   Paul Fricker 11/13/2006 oW]~\vp^0  
    h\GlyH~  
    bN-ljw0&  
    % Check and prepare the inputs: W ~sP7&sp  
    % ----------------------------- &y-(UOqbkP  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) B=K& +  
        error('zernfun:NMvectors','N and M must be vectors.') (vHB`@x  
    end ZsjDe{TH  
    F.:B_t  
    if length(n)~=length(m) ;  ntq%  
        error('zernfun:NMlength','N and M must be the same length.') X.V6v4  
    end Aa^%_5  
    @ %LrpD  
    n = n(:);  )L}6to  
    m = m(:); &_cMbFLBP  
    if any(mod(n-m,2)) Ys|n9pW  
        error('zernfun:NMmultiplesof2', ... Ms8& $  
              'All N and M must differ by multiples of 2 (including 0).') (h;4irfX  
    end -A}U^-'a}  
    -:w+`x?XaB  
    if any(m>n) }lZfZ?oAz  
        error('zernfun:MlessthanN', ... d\Q~L 3x  
              'Each M must be less than or equal to its corresponding N.') Qp9)Rc5  
    end RGrra<  
    Cnp\2Fu/  
    if any( r>1 | r<0 ) NEInro<  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') U#3Y3EdF<  
    end k.b->U  
    ]+RBykr  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) hiKgV|ZD  
        error('zernfun:RTHvector','R and THETA must be vectors.') @SA:64 9  
    end 4Eq$f (QJ  
    md8r"  
    r = r(:); Kts#e:k@  
    theta = theta(:); -X#Zn>#  
    length_r = length(r); Kfho:e,  
    if length_r~=length(theta) E3X6-J|  
        error('zernfun:RTHlength', ... 4,D$% .  
              'The number of R- and THETA-values must be equal.') 24u;'i-y5  
    end @SH%l]  
    P{qi>FJqe  
    % Check normalization:  "5\<.  
    % -------------------- d;GF<bz  
    if nargin==5 && ischar(nflag) y^"[^+F3 .  
        isnorm = strcmpi(nflag,'norm'); n_}=G RR  
        if ~isnorm ;{xk[f m=  
            error('zernfun:normalization','Unrecognized normalization flag.') @k_xA-a  
        end "o+E9'Dm  
    else px!lJtvgo  
        isnorm = false; &gdtI  
    end hrsMAh!  
    D,FX&{TYU  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G,+-}~$_  
    % Compute the Zernike Polynomials SF?Ublc!   
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :{za[,  
    yQ5F'.m9e  
    % Determine the required powers of r: iwJeV J  
    % ----------------------------------- f|eUpf%)  
    m_abs = abs(m); di^E8egR$  
    rpowers = []; H^UuT  
    for j = 1:length(n) e !_+TyI  
        rpowers = [rpowers m_abs(j):2:n(j)]; B&J;yla6`d  
    end DIx!Sw7EC  
    rpowers = unique(rpowers); l ;TWs_N  
    <pAN{:  
    % Pre-compute the values of r raised to the required powers, xO2e>[W  
    % and compile them in a matrix: F' eV%g  
    % ----------------------------- &PJ&XTR  
    if rpowers(1)==0 W( O)J$j  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Uy8r !9O  
        rpowern = cat(2,rpowern{:}); Ko6>h  
        rpowern = [ones(length_r,1) rpowern]; *;(wtMg  
    else S.,om;`  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); M'Ec:p=X"  
        rpowern = cat(2,rpowern{:}); _ ^5w f  
    end 0Q\6GCzN\  
    Tk(ciwB  
    % Compute the values of the polynomials: t[L0kF9en  
    % -------------------------------------- \UKr|[P  
    y = zeros(length_r,length(n)); GEJEhwO;H  
    for j = 1:length(n) >lZ9Y{Y4v  
        s = 0:(n(j)-m_abs(j))/2; @9yY`\"ed  
        pows = n(j):-2:m_abs(j); @m*^v\q<u  
        for k = length(s):-1:1 R*m=V{iu`  
            p = (1-2*mod(s(k),2))* ... Yxe%:  
                       prod(2:(n(j)-s(k)))/              ... N@Ie VF  
                       prod(2:s(k))/                     ... D]NfA2B7  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... >]DnEF&  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); & ,KxE(C  
            idx = (pows(k)==rpowers); (_2;}eg  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Yo`#G-]  
        end mGf@J6wGz  
         3vs;ZBM  
        if isnorm p-p]dV  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 0(+3w\_!  
        end rlQ4+~  
    end VK7lm|J+  
    % END: Compute the Zernike Polynomials #dcfQ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +mc0:e{WF  
    (`z`ni  
    % Compute the Zernike functions: lIs<&-0  
    % ------------------------------ $:v!*0/  
    idx_pos = m>0; 7 (}gs?&w  
    idx_neg = m<0; 4d\1W?i-  
    3zV{cm0  
    z = y; *|Cmm>z"7  
    if any(idx_pos) _FG?zE  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); i,77F!  
    end (QARle(i  
    if any(idx_neg) EX]LH({?+L  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); y81B3`@  
    end EfTuHg$pe  
    $Tc"7nYu  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) Tc{n]TV  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. h5vvizruy  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ]z^*1^u^ig  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive ukZ>_ke`+  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, U{^~X_?  
    %   and THETA is a vector of angles.  R and THETA must have the same x)+3SdH  
    %   length.  The output Z is a matrix with one column for every P-value, Wmm'j&hI  
    %   and one row for every (R,THETA) pair. 3k5C;5  
    % 4P1<Zi+<  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ?b}d"QsmU  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) WyO7,Qr\   
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) s>A!Egmo  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 W,\LdQ  
    %   for all p. Pz=x$aY  
    % O@[jNs)].  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 -d|Q|zF^x  
    %   Zernike functions (order N<=7).  In some disciplines it is X4- _l$j  
    %   traditional to label the first 36 functions using a single mode d[cqs9=\  
    %   number P instead of separate numbers for the order N and azimuthal v>e%5[F  
    %   frequency M. >?pWbL  
    % g$z9 (i+  
    %   Example: PNjZbOmzS  
    % 8$c_M   
    %       % Display the first 16 Zernike functions zvzS$Gpe  
    %       x = -1:0.01:1; k7R8Q~4  
    %       [X,Y] = meshgrid(x,x); dtXA EL\q  
    %       [theta,r] = cart2pol(X,Y); qUZm6)p6[a  
    %       idx = r<=1; 2;82*0Y%  
    %       p = 0:15; 'dkKBLsx  
    %       z = nan(size(X)); k^x[(gw  
    %       y = zernfun2(p,r(idx),theta(idx)); "kYzgi  
    %       figure('Units','normalized') l6YToYzE2  
    %       for k = 1:length(p) ??4#)n k  
    %           z(idx) = y(:,k); R{GT? wl  
    %           subplot(4,4,k) uQ}0hs  
    %           pcolor(x,x,z), shading interp 3 &aBU [  
    %           set(gca,'XTick',[],'YTick',[]) KGVAP  
    %           axis square ucVWvXCr  
    %           title(['Z_{' num2str(p(k)) '}']) m'L7K K-Y)  
    %       end ?PMF]ah  
    % l'~~hQ{h/  
    %   See also ZERNPOL, ZERNFUN. u$3wdZ2&m  
    *@EItj`  
    %   Paul Fricker 11/13/2006 ? iX1;c9  
    <c,/+ lQ^  
    H 3e(-  
    % Check and prepare the inputs: T)!$-qdz/  
    % ----------------------------- yMJY6$Ct  
    if min(size(p))~=1 c@+;4Iz  
        error('zernfun2:Pvector','Input P must be vector.') ^KKU@ab9  
    end )_MIUQ%  
    ftL>oOz[  
    if any(p)>35 X2 Z E9b  
        error('zernfun2:P36', ... -T s8y  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... (c'=jJX  
               '(P = 0 to 35).']) `u./2]n  
    end #[4MwM3  
    fs43\m4= m  
    % Get the order and frequency corresonding to the function number: v7V.,^6+  
    % ---------------------------------------------------------------- Mp8FYPjZ  
    p = p(:); FXAP]iqo  
    n = ceil((-3+sqrt(9+8*p))/2); SP&Y|I$:  
    m = 2*p - n.*(n+2); nJdO~0}3  
    j:JM v  
    % Pass the inputs to the function ZERNFUN: :X?bWxOJ  
    % ---------------------------------------- `I\)Kk@*b9  
    switch nargin \Y EV 5  
        case 3 <@Lw '  
            z = zernfun(n,m,r,theta); "Yk3K^`1T.  
        case 4 !hBzT7CO  
            z = zernfun(n,m,r,theta,nflag); .g|D  
        otherwise oX2J2O  
            error('zernfun2:nargin','Incorrect number of inputs.') }G:5P3f  
    end 75O-%9lFF  
    |o:[*2-   
    % 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) .szs?  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. w|"cf{$^x  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of A[,[j?wC  
    %   order N and frequency M, evaluated at R.  N is a vector of }]qx "  
    %   positive integers (including 0), and M is a vector with the 5y}kI  
    %   same number of elements as N.  Each element k of M must be a m &U $V  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) vd4}b>  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is g{`rWKj  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix `kx+Kc  
    %   with one column for every (N,M) pair, and one row for every q{rc[ s?  
    %   element in R. UE3#(:x A  
    % &"90pBGK  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- C ?^si  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ,oW8im   
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to uq}>5  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 \Z +O9T%  
    %   for all [n,m]. 9$9Pv%F:j  
    % c'9-SY1'~  
    %   The radial Zernike polynomials are the radial portion of the -&#H@Gyw  
    %   Zernike functions, which are an orthogonal basis on the unit 7qyv.{+  
    %   circle.  The series representation of the radial Zernike Qi_De '@  
    %   polynomials is Wcgy:4K3  
    % H:~41f[  
    %          (n-m)/2 (IbT5  
    %            __ uW.)(l  
    %    m      \       s                                          n-2s ^,Sl^ 9K  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r  c`'2  
    %    n      s=0 a;Nj'M~U  
    % ,{@,dw`lUz  
    %   The following table shows the first 12 polynomials. K22'XrN  
    % l!B)1  
    %       n    m    Zernike polynomial    Normalization Q k`yK|(0=  
    %       --------------------------------------------- cVzOW|NVx  
    %       0    0    1                        sqrt(2) hRn[ 9B  
    %       1    1    r                           2 hM!D6: t  
    %       2    0    2*r^2 - 1                sqrt(6) EDm,Y  
    %       2    2    r^2                      sqrt(6) F5CV<-jB  
    %       3    1    3*r^3 - 2*r              sqrt(8) htn"rY(  
    %       3    3    r^3                      sqrt(8) G/F0 )M  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 6%mF iX  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) AZmABl  
    %       4    4    r^4                      sqrt(10) W_zv"c  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) !MOgM  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ZMSP8(V  
    %       5    5    r^5                      sqrt(12) ToUeXU [  
    %       --------------------------------------------- e ;4y5i  
    % +4kBd<0Y  
    %   Example: y;N[#hY#CD  
    % !aSu;Ln  
    %       % Display three example Zernike radial polynomials &yKUf  
    %       r = 0:0.01:1; O k-*xd  
    %       n = [3 2 5]; C|kZT<,]  
    %       m = [1 2 1]; /f!CX|U  
    %       z = zernpol(n,m,r); Td&w  
    %       figure ~-+lZ4}  
    %       plot(r,z) OzFA>FK0f;  
    %       grid on f IUz%YFn  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') rPV\ F  
    % JrF\7*rh9  
    %   See also ZERNFUN, ZERNFUN2. :*wnO;eN  
    Z/ "jLfP  
    % A note on the algorithm. Qrt[MJ+#  
    % ------------------------ p]d3F^*i  
    % The radial Zernike polynomials are computed using the series R3]Ra&h6N)  
    % representation shown in the Help section above. For many special LoHL}1BG-  
    % functions, direct evaluation using the series representation can M1Jnn4w*d  
    % produce poor numerical results (floating point errors), because q%u;+/|l  
    % the summation often involves computing small differences between iJg3`1@j  
    % large successive terms in the series. (In such cases, the functions tUXq!r<'dT  
    % are often evaluated using alternative methods such as recurrence ~!c~jcq]lZ  
    % relations: see the Legendre functions, for example). For the Zernike d%$'Y|  
    % polynomials, however, this problem does not arise, because the 6?U2Et  
    % polynomials are evaluated over the finite domain r = (0,1), and nw3CI&Y`  
    % because the coefficients for a given polynomial are generally all Z5/g\G[  
    % of similar magnitude. pKrol]cth8  
    % DkP%1Crdr  
    % ZERNPOL has been written using a vectorized implementation: multiple z}'*zB>  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ,:!X]F#d$  
    % values can be passed as inputs) for a vector of points R.  To achieve ?)9mHo^  
    % this vectorization most efficiently, the algorithm in ZERNPOL J4 yT|  
    % involves pre-determining all the powers p of R that are required to zWxKp;.  
    % compute the outputs, and then compiling the {R^p} into a single 1uTbN  
    % matrix.  This avoids any redundant computation of the R^p, and ?XVJ$nzW  
    % minimizes the sizes of certain intermediate variables. ;Ry )^5Q  
    % Ly?yW S-x  
    %   Paul Fricker 11/13/2006 :^mfTj$  
    m/"\+Hv  
    !BHIp7p  
    % Check and prepare the inputs: hB#z8D  
    % ----------------------------- .7-Yu1{2  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) EM+_c)d}  
        error('zernpol:NMvectors','N and M must be vectors.') ~Tv %6iaeE  
    end Az2HlKF"L  
    %(`4wo},  
    if length(n)~=length(m) |C9qM  
        error('zernpol:NMlength','N and M must be the same length.') |qS<{WZ!h  
    end iVM{ L  
    iP1u u  
    n = n(:); bdiyS.a-  
    m = m(:); <$s G]l!\  
    length_n = length(n); 1]r+$L3  
    B9;-Blh  
    if any(mod(n-m,2)) /8baJ+D"4\  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).')  }SHF  
    end hS4Ljyeg  
    rIz"_r  
    if any(m<0) Qc2_B\K^  
        error('zernpol:Mpositive','All M must be positive.') z<~gv"  
    end ?U]/4]  
    dq?q(_9  
    if any(m>n) 7kM_Ijd$  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') vVF#]t b|  
    end { U a19~'>  
    IAbK]kA  
    if any( r>1 | r<0 ) FJ3Xeo s4|  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') EJYfk?(B  
    end {9KG06%+  
    jp2AU,Cl  
    if ~any(size(r)==1) )J+vmY~&  
        error('zernpol:Rvector','R must be a vector.') 4 Yq|Z  
    end O&93QN0  
    Fl GKy9k  
    r = r(:); '\dau>  
    length_r = length(r); *ms?UFV[r  
    Dqu1!f  
    if nargin==4 LQSno)OZ  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); >S5:zz\  
        if ~isnorm ?\:ysTVu  
            error('zernpol:normalization','Unrecognized normalization flag.') RRy3N )HR  
        end G0$ 1"9u\w  
    else +x$;T*0  
        isnorm = false; Y3jb 'S4(  
    end F7^d@hSV  
    Qh-k[w0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CqDMq!  
    % Compute the Zernike Polynomials v"Bv\5f,Ys  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% H@er"boi  
    Y'kD_T`f,  
    % Determine the required powers of r: aX6.XHWbDf  
    % ----------------------------------- _T^ip.o  
    rpowers = []; li\hHd5  
    for j = 1:length(n) u2'xM0nQ  
        rpowers = [rpowers m(j):2:n(j)]; 0q"&AxNsP  
    end Kd CPt!  
    rpowers = unique(rpowers); N4"%!.Y  
    6l IFxc  
    % Pre-compute the values of r raised to the required powers, eFvw9B+  
    % and compile them in a matrix: .EGZv (rz&  
    % ----------------------------- &O(z|-&| x  
    if rpowers(1)==0 :h1itn  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); GOHRBV  
        rpowern = cat(2,rpowern{:}); =x}27f%-Mg  
        rpowern = [ones(length_r,1) rpowern]; >:5/V0;,  
    else _ I+#K M  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ;ej;<7+  
        rpowern = cat(2,rpowern{:}); nn$,|/  
    end zLue j'  
    )DuOo83n["  
    % Compute the values of the polynomials: t)XNS!6#]?  
    % -------------------------------------- NvXds;EC  
    z = zeros(length_r,length_n); eu ~WFI  
    for j = 1:length_n OUn,URI  
        s = 0:(n(j)-m(j))/2; GWRKiTu9  
        pows = n(j):-2:m(j); N5[QQtQ  
        for k = length(s):-1:1 <LQwH23@  
            p = (1-2*mod(s(k),2))* ... RUm1;MWs  
                       prod(2:(n(j)-s(k)))/          ... Z<z(;)?c  
                       prod(2:s(k))/                 ... & :x_  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... d_Ll,*J9  
                       prod(2:((n(j)+m(j))/2-s(k))); %1a\"F![  
            idx = (pows(k)==rpowers); CD%wi:C%|  
            z(:,j) = z(:,j) + p*rpowern(:,idx); QNzI  
        end ~j",ePl  
         ^,'!j/w5  
        if isnorm FVsVY1  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 8vK Z;  
        end 95>(NwST4  
    end )#Ea~>v  
    pUZe.S>G  
    % 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)  M|8vP53=q  
    8*o*?1.  
    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。 Rr#Zcs!G  
    0x BO5[w,Y  
    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)