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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 XL?A w  
    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! 7n [12:  
     
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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 \}U[}5Pk&  
    function z = zernfun(n,m,r,theta,nflag) <[/PyNYK  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 'E@2I9Kj  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N >~.Zr3P6kC  
    %   and angular frequency M, evaluated at positions (R,THETA) on the (QA-"9v#i,  
    %   unit circle.  N is a vector of positive integers (including 0), and D9e+  
    %   M is a vector with the same number of elements as N.  Each element ],H1  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) y*y`t6D  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, &NlS  =  
    %   and THETA is a vector of angles.  R and THETA must have the same rsd2v9  
    %   length.  The output Z is a matrix with one column for every (N,M) FGV}5L  
    %   pair, and one row for every (R,THETA) pair. >cBGw'S  
    % TEH*@~P"  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 4!NfQk>X  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9k714bnMLX  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral E_ o{c5N  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, i#CaKS  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized j` [#Ij  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. L"Qh_+   
    % E1$Hu{  
    %   The Zernike functions are an orthogonal basis on the unit circle. ; ,Of\Efc|  
    %   They are used in disciplines such as astronomy, optics, and ~ >&I^4  
    %   optometry to describe functions on a circular domain. ({D}QEP  
    % iSSc5ek4  
    %   The following table lists the first 15 Zernike functions. j;1~=j])  
    % N*_/@qM> a  
    %       n    m    Zernike function           Normalization N1D6D$s0  
    %       -------------------------------------------------- ws*~$x?7  
    %       0    0    1                                 1 *#9VC)Q  
    %       1    1    r * cos(theta)                    2 'd|Q4RE+W  
    %       1   -1    r * sin(theta)                    2 GI0x>Z+  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ^8o_Iz)r,  
    %       2    0    (2*r^2 - 1)                    sqrt(3) pDLu+ }@  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) hj[+d%YZY"  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) kX ~-g  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) [ HC8-N^.}  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) *" |VNnB  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) lWu9/r 1  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) hLDch5J5~  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) KdBq@  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) LUe>)eqw  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1YF+(fk  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 8`L#1ybMO  
    %       -------------------------------------------------- _IQU<Za  
    % 4yJ*85e]  
    %   Example 1: Q1O_CC}  
    % Gvt;Q,hH  
    %       % Display the Zernike function Z(n=5,m=1) EI?d(K  
    %       x = -1:0.01:1; 1Pw(.8P  
    %       [X,Y] = meshgrid(x,x); :Y}Y&mA4  
    %       [theta,r] = cart2pol(X,Y); Rye ~w6  
    %       idx = r<=1; rL!_&|  
    %       z = nan(size(X)); Mp^OL7p^^  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Zq\RNZ}  
    %       figure :_{{PY0PK  
    %       pcolor(x,x,z), shading interp v&[X&Hu[  
    %       axis square, colorbar &;~2sEo,  
    %       title('Zernike function Z_5^1(r,\theta)') LK   
    % w (vE2Y ?  
    %   Example 2: d'lr:=GQ  
    % QoT3;<r}  
    %       % Display the first 10 Zernike functions IF36K^K  
    %       x = -1:0.01:1; yL.PGF1(  
    %       [X,Y] = meshgrid(x,x); 0gwm gc/#  
    %       [theta,r] = cart2pol(X,Y); g~ppPAH  
    %       idx = r<=1; xzMeKC `  
    %       z = nan(size(X)); ]2aYi9)  
    %       n = [0  1  1  2  2  2  3  3  3  3]; oPBg+Bh*  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; dIBKE0`  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; azR;*j8Q'  
    %       y = zernfun(n,m,r(idx),theta(idx)); DJD]aI  
    %       figure('Units','normalized') 4BduUH  
    %       for k = 1:10 O$<%z[  
    %           z(idx) = y(:,k); [G'!`^V,  
    %           subplot(4,7,Nplot(k)) 6`s%%v  
    %           pcolor(x,x,z), shading interp /IrR,bvA  
    %           set(gca,'XTick',[],'YTick',[]) U'Ja\Ek/f  
    %           axis square {LB }v;?l  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) HP4'8#3o  
    %       end 3gV&`>@  
    % z 1#0  
    %   See also ZERNPOL, ZERNFUN2. r:WgjjA%  
    IQk#  
    %   Paul Fricker 11/13/2006 t=E|RYC(k  
    c:@OX[##  
    >^a"Z[s[  
    % Check and prepare the inputs: R+kZLOE  
    % ----------------------------- |=^#d\?]j  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) mNnw G);$  
        error('zernfun:NMvectors','N and M must be vectors.') guUr1Ij  
    end U Qi^udGFD  
    Vk N[=0a,  
    if length(n)~=length(m) 2 l[A=Z  
        error('zernfun:NMlength','N and M must be the same length.') WFeMr%Zqh>  
    end |W~V@n8"6  
    'wB Huq  
    n = n(:); $!l2=^\3  
    m = m(:); '4^V4i  
    if any(mod(n-m,2)) U$/Hp#~X  
        error('zernfun:NMmultiplesof2', ... OnPy8mC  
              'All N and M must differ by multiples of 2 (including 0).') _/sf@R  
    end {YKMQI^O/  
    PgG |7='  
    if any(m>n) T956L'.+G  
        error('zernfun:MlessthanN', ... &x0TnW"g  
              'Each M must be less than or equal to its corresponding N.') }N#>q.M  
    end OJ_2z|f<  
    X!+Mgh6  
    if any( r>1 | r<0 ) Y?vm%t`K  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') CI,`R&=xO  
    end 6JFDRsX>)?  
    EYx2IJ  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) .e _D3Xp<  
        error('zernfun:RTHvector','R and THETA must be vectors.') J6["j   
    end 5#9Wd9LP  
    ndCS<ojcBP  
    r = r(:); 4 _U,-%/  
    theta = theta(:); MZP><Je&  
    length_r = length(r); pv m'pu78  
    if length_r~=length(theta) 't]EkH]BC  
        error('zernfun:RTHlength', ... |YGiATD4DG  
              'The number of R- and THETA-values must be equal.') 0)`lx9&h  
    end d Xo'#.  
    SJ[@fUxO)  
    % Check normalization: @aD~YtL"n  
    % -------------------- hPeKQwzC0  
    if nargin==5 && ischar(nflag) |nH0~P#!  
        isnorm = strcmpi(nflag,'norm'); _6-/S!7Y\  
        if ~isnorm mQA<t)1  
            error('zernfun:normalization','Unrecognized normalization flag.') ^n45N&916  
        end kzVI:  
    else 9hs{uxwuEE  
        isnorm = false; W];6u  
    end 5G ]#yb74  
    {O&liU4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  ISnS;  
    % Compute the Zernike Polynomials vBn=bb'W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3D09P5$W  
    =ci5&B?  
    % Determine the required powers of r: vS t=Ax3]  
    % ----------------------------------- np\Q&  
    m_abs = abs(m); gAUQQ  
    rpowers = []; <K[Zl/7I  
    for j = 1:length(n) ' bw,K*  
        rpowers = [rpowers m_abs(j):2:n(j)]; (Nlm4*{h  
    end PKM$*_LcGI  
    rpowers = unique(rpowers); ?a0}^:6  
    yzNX2u1  
    % Pre-compute the values of r raised to the required powers, 4%v+ark8  
    % and compile them in a matrix: |p4OlUq  
    % ----------------------------- a=B0ytNm  
    if rpowers(1)==0 Dw ;vDK  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 4e#K.HU_  
        rpowern = cat(2,rpowern{:}); WfbNar[  
        rpowern = [ones(length_r,1) rpowern]; re7\nZ<\|  
    else B*iz+"H  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3N%Ev o  
        rpowern = cat(2,rpowern{:}); 5GFnfc}  
    end !BikF4Y1L&  
    .x$T a l  
    % Compute the values of the polynomials: ~m|?! ]n  
    % -------------------------------------- ?ZV0   
    y = zeros(length_r,length(n)); BG8)bh k;/  
    for j = 1:length(n) qf=[*ZY  
        s = 0:(n(j)-m_abs(j))/2; f>+}U;)EF  
        pows = n(j):-2:m_abs(j); RHAr[$  
        for k = length(s):-1:1 x-#9i  
            p = (1-2*mod(s(k),2))* ... kJeOlO[  
                       prod(2:(n(j)-s(k)))/              ... 5)v^ cR?&  
                       prod(2:s(k))/                     ... bfI -!,  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... tWOze, N  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); =+=|{l?F  
            idx = (pows(k)==rpowers); nJ#@W b@  
            y(:,j) = y(:,j) + p*rpowern(:,idx); >(ww6vk2  
        end ,$qs9b~  
         (l_de)N7  
        if isnorm 8=o(nFJw  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); %1 ^jd\  
        end o4f9EJY   
    end EF=D}"E6pO  
    % END: Compute the Zernike Polynomials ,k!f`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ![!b^:f  
    ~+nSI-L  
    % Compute the Zernike functions: Iw |[*Nu-  
    % ------------------------------ 2^ZPO4|  
    idx_pos = m>0; Aq]'.J =4  
    idx_neg = m<0; GXK?7S0H  
    3M*[a~  
    z = y; 4KSN;G  
    if any(idx_pos) <_q/ +x]8  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); BF [?* b  
    end vm^# aoDB  
    if any(idx_neg) h GXD u;{  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); |M>k &p,B-  
    end knzED~ v@(  
    OYp8r  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) #p`7gFl  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. d$~b`  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 8J1.(Mwb?  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive -y*+G&  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ,ToEK Id  
    %   and THETA is a vector of angles.  R and THETA must have the same =I}V PxhE7  
    %   length.  The output Z is a matrix with one column for every P-value, 8N_rJ)f  
    %   and one row for every (R,THETA) pair. .Awq(  
    % Q^w]Nj(e_  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike }0>/G?2Yp  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) /n7,B}  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) *~^^A9C8  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 K+OU~SED%F  
    %   for all p. CWYJ<27v{  
    % /k"P4\P`+Q  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ;_&L^)~P$  
    %   Zernike functions (order N<=7).  In some disciplines it is Y,\mrW}K   
    %   traditional to label the first 36 functions using a single mode x,_Ucc.  
    %   number P instead of separate numbers for the order N and azimuthal 1&"1pH  
    %   frequency M. l'\b(3JF  
    % NmF2E+'  
    %   Example: Sxu v}y\  
    % ;rL$z;}8  
    %       % Display the first 16 Zernike functions )ixE  
    %       x = -1:0.01:1; F[Sat;Sll  
    %       [X,Y] = meshgrid(x,x); rWqA)j*!  
    %       [theta,r] = cart2pol(X,Y); oU?X"B9  
    %       idx = r<=1; }TvAjLIS6  
    %       p = 0:15;  E/;YhFb[  
    %       z = nan(size(X)); !:{_<C"D  
    %       y = zernfun2(p,r(idx),theta(idx)); ]#.#]}=  
    %       figure('Units','normalized') ;gV8f{X{Z  
    %       for k = 1:length(p) O8o18m8UH  
    %           z(idx) = y(:,k); e5KF~0`  
    %           subplot(4,4,k) cfS]C_6d  
    %           pcolor(x,x,z), shading interp .r'.5RI A  
    %           set(gca,'XTick',[],'YTick',[]) \s'6)_  
    %           axis square >WX'oP(<  
    %           title(['Z_{' num2str(p(k)) '}']) ^]gl#&"D  
    %       end oX ,M;;Yq  
    % q% 9oGYjvQ  
    %   See also ZERNPOL, ZERNFUN. gHshG;z*  
    +-d>Sl (  
    %   Paul Fricker 11/13/2006 miSC'!  
    Njjeg9f  
    kzXW<V9  
    % Check and prepare the inputs: }3lF;k(2g  
    % ----------------------------- r+u\jZ  
    if min(size(p))~=1 7RJW  
        error('zernfun2:Pvector','Input P must be vector.') f}eVfAf  
    end ApB0)N  
    p<34}iZ  
    if any(p)>35 74#@F{w  
        error('zernfun2:P36', ... k<H&4Z)d9  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... B,T.bgp\  
               '(P = 0 to 35).']) $*R9LPpk+  
    end @oNrR$7  
    oZtz"B  
    % Get the order and frequency corresonding to the function number: Cj9Tj'0@I+  
    % ---------------------------------------------------------------- BW 7[JD  
    p = p(:); rfoCYsX'  
    n = ceil((-3+sqrt(9+8*p))/2); "RA$Twhj  
    m = 2*p - n.*(n+2); ^(:Rbsl  
    i,T{SV  
    % Pass the inputs to the function ZERNFUN: Rw`s O:eZ  
    % ---------------------------------------- H l@rS  
    switch nargin M(f'qFY=K  
        case 3 _P:P5H8  
            z = zernfun(n,m,r,theta); 9qA_5x%"%u  
        case 4 [:qX3"B  
            z = zernfun(n,m,r,theta,nflag); {r)M@@[  
        otherwise W+X zU"l  
            error('zernfun2:nargin','Incorrect number of inputs.') ^*OA%wg3=h  
    end add-]2`  
    }I]q$3 .  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) {ex]_V>  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. plq\D.C  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of '4rgIs3=x"  
    %   order N and frequency M, evaluated at R.  N is a vector of o%a$m9I  
    %   positive integers (including 0), and M is a vector with the sBwgl9  
    %   same number of elements as N.  Each element k of M must be a nj  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) D[mYrWHpn  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is m[f\I^ \%8  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix |Th{*IJ <,  
    %   with one column for every (N,M) pair, and one row for every Lnzhs;7L  
    %   element in R.  a4yU[KK  
    % i]v!o$7  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ;1nd~0o  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is LFC k6 R  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to tj3p71%  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 <#UvLll  
    %   for all [n,m]. 1tDN$rM5  
    % K5$ y  
    %   The radial Zernike polynomials are the radial portion of the z,tax`O  
    %   Zernike functions, which are an orthogonal basis on the unit XV&3h>5  
    %   circle.  The series representation of the radial Zernike |8B[yr.b  
    %   polynomials is ^RyrUb  
    % 1X5\VY>S`h  
    %          (n-m)/2 *K;~V  
    %            __ @."R9s  
    %    m      \       s                                          n-2s R+\5hI@ >i  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r A{QS+fa/  
    %    n      s=0 R~i<*  
    % \0$?r4A  
    %   The following table shows the first 12 polynomials. 9SPu 4i  
    % {f)p|)  
    %       n    m    Zernike polynomial    Normalization X:A\{^ ~  
    %       --------------------------------------------- Dz?F,g_  
    %       0    0    1                        sqrt(2) ktCh*R[`  
    %       1    1    r                           2 7j Q`i;L}Y  
    %       2    0    2*r^2 - 1                sqrt(6) Z8x(_ft5  
    %       2    2    r^2                      sqrt(6) . XmD[=  
    %       3    1    3*r^3 - 2*r              sqrt(8) 6=   
    %       3    3    r^3                      sqrt(8) o|+tRl  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10)  7;XdTx  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) D|xSO~M5  
    %       4    4    r^4                      sqrt(10) ~LVa#  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) (n< xoV[e  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) w*+rBp,f  
    %       5    5    r^5                      sqrt(12) {XVSHUtw  
    %       --------------------------------------------- Ul=`]@]]  
    % 1#'wR3[+  
    %   Example: g%Z;rDfi  
    % &"BKue~q@p  
    %       % Display three example Zernike radial polynomials G*QQpSp  
    %       r = 0:0.01:1; Na=q(OKN  
    %       n = [3 2 5]; qRUz;M4  
    %       m = [1 2 1]; %63<Iz"  
    %       z = zernpol(n,m,r); 9u9#&xx  
    %       figure CB~&!MdMr  
    %       plot(r,z) qAm%h\  
    %       grid on PtHT>  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') hi37p1t   
    % +,smjg:O  
    %   See also ZERNFUN, ZERNFUN2. MV/JZ;55  
    !} 1p:@  
    % A note on the algorithm. M! s&<Bi  
    % ------------------------ fROhn}<**[  
    % The radial Zernike polynomials are computed using the series `);`E_'U k  
    % representation shown in the Help section above. For many special  I{E10;  
    % functions, direct evaluation using the series representation can Z#F2<*+Pe  
    % produce poor numerical results (floating point errors), because cv^^NgQ  
    % the summation often involves computing small differences between ]`MRH[{  
    % large successive terms in the series. (In such cases, the functions }, ]W/  
    % are often evaluated using alternative methods such as recurrence B *:6U+I  
    % relations: see the Legendre functions, for example). For the Zernike !u^(<.xJ   
    % polynomials, however, this problem does not arise, because the rO-Tr  
    % polynomials are evaluated over the finite domain r = (0,1), and Sh"} c2  
    % because the coefficients for a given polynomial are generally all X 6>Pq  
    % of similar magnitude. cD{[rI E3  
    % 9 Hm!B )Y  
    % ZERNPOL has been written using a vectorized implementation: multiple Tkd4nRo~  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] _uRgKoiy  
    % values can be passed as inputs) for a vector of points R.  To achieve O9opX\9  
    % this vectorization most efficiently, the algorithm in ZERNPOL XpM#0hm  
    % involves pre-determining all the powers p of R that are required to jtZ@`io  
    % compute the outputs, and then compiling the {R^p} into a single /_LUys/0  
    % matrix.  This avoids any redundant computation of the R^p, and W2n%D& PE  
    % minimizes the sizes of certain intermediate variables. ~<|xS  
    % BqR8%F  
    %   Paul Fricker 11/13/2006 b2Ct^`|M5  
    c=ZX7U  
    %DiZ&}^Ck  
    % Check and prepare the inputs: Jx 'p\*  
    % ----------------------------- -8-Aqh8|  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) L%<1cE))  
        error('zernpol:NMvectors','N and M must be vectors.') e'u 9 SpJ  
    end Nf3L  
    I@[.W!w  
    if length(n)~=length(m) H*l8,*M}  
        error('zernpol:NMlength','N and M must be the same length.') gllXJM^ -  
    end JK,k@RE y]  
    WhvO-WF  
    n = n(:); BM:je(*p  
    m = m(:); B&tl6?7h  
    length_n = length(n); lT*Hj.  
    +lE 9*Gs_$  
    if any(mod(n-m,2)) b-ZvEDCR  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') }4+S_b  
    end R,tR{| 8  
    x3)qK6,\  
    if any(m<0) #A?U_32z/2  
        error('zernpol:Mpositive','All M must be positive.') +`\C_i-  
    end @ j' I  
    YT#" HYO  
    if any(m>n) `57ffQR9  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') H]YPMG<  
    end M,6m*  
    3pm;?6i6  
    if any( r>1 | r<0 ) BjJ+~R  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ca-|G'q  
    end !30BZM^  
    xez~Yw2  
    if ~any(size(r)==1) {V8 v  
        error('zernpol:Rvector','R must be a vector.') kIM* K%L}  
    end =eYrz@,  
    Xa+ u>1"2"  
    r = r(:); .|cQ0:B[  
    length_r = length(r); ?-J\~AXL  
    Haiuf)a  
    if nargin==4 '@rGX+"  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); y1f&+y9e  
        if ~isnorm OZ0q6"  
            error('zernpol:normalization','Unrecognized normalization flag.') wn5CaP(]8  
        end {R]4N]l>  
    else 2,'m]`;GNr  
        isnorm = false; N }Z"$4  
    end }7k+tJ<   
    I%;xMt Y1o  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e<>(c7bF  
    % Compute the Zernike Polynomials d:Wh0y}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f0}+8JW5h  
    <[kdF")  
    % Determine the required powers of r: Fb VtyQz  
    % ----------------------------------- aw {?UvL&  
    rpowers = []; z1_\P) M  
    for j = 1:length(n) $ !ka8) ~  
        rpowers = [rpowers m(j):2:n(j)]; vfm |?\  
    end KE6[u*\  
    rpowers = unique(rpowers); C%0|o/Wi  
    A)&OR]0[  
    % Pre-compute the values of r raised to the required powers, Tw);`&Ulo  
    % and compile them in a matrix: 9psD"=/"  
    % ----------------------------- D8qZh1w%A|  
    if rpowers(1)==0 2t*@P"e!  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); T)qD}hl  
        rpowern = cat(2,rpowern{:}); G)M9to  
        rpowern = [ones(length_r,1) rpowern]; H5{d;L1[  
    else ZWxq<& Cg  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); +m]$P,yMt  
        rpowern = cat(2,rpowern{:}); +t})tDPXw  
    end :6./yj(  
    O3PE w4yA  
    % Compute the values of the polynomials: la[xbv   
    % -------------------------------------- $-EbJ  
    z = zeros(length_r,length_n); &AG,]#  
    for j = 1:length_n sTU`@}}  
        s = 0:(n(j)-m(j))/2; *O+G}_}  
        pows = n(j):-2:m(j); ;*[nZV>  
        for k = length(s):-1:1 ]]J2#mN:n  
            p = (1-2*mod(s(k),2))* ... 6$lj$8\  
                       prod(2:(n(j)-s(k)))/          ... bT2b)nf  
                       prod(2:s(k))/                 ... XL1v&'HLV  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 49E<`f0  
                       prod(2:((n(j)+m(j))/2-s(k))); U5[xW  
            idx = (pows(k)==rpowers); ^ duNEu0*  
            z(:,j) = z(:,j) + p*rpowern(:,idx); #%rXDGDS  
        end ! jm>  
         }1f@>'o  
        if isnorm 4#{i  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ML^c-xY(  
        end 7aj|-gZ  
    end 9# IKb:9k  
    Y.$ '<1  
    % 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)  +76{S_CZ  
    g=w,*68vuy  
    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;1/.`NPB  
    gD&/ k  
    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)