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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 v .*fJ   
    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! nCwA8AG  
     
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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 *w. ":\P]  
    function z = zernfun(n,m,r,theta,nflag) \"RCJadK  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. _#v"sGmN  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N K"t?  
    %   and angular frequency M, evaluated at positions (R,THETA) on the j&/+/s9N  
    %   unit circle.  N is a vector of positive integers (including 0), and )N~ p4kp  
    %   M is a vector with the same number of elements as N.  Each element :4)x  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) &QD)1b[U  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Eo ^m; p5  
    %   and THETA is a vector of angles.  R and THETA must have the same >WZbb d-  
    %   length.  The output Z is a matrix with one column for every (N,M) @=AQr4&  
    %   pair, and one row for every (R,THETA) pair. LKI\(%ba#  
    % n6,YA2yZO  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike @,= pG  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ]!!?gnPd5  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral [O^/"Qk  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Q5dqn"?  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized FXY>o>K%h  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. V;RgO}  
    % U!Zj%H1XQ0  
    %   The Zernike functions are an orthogonal basis on the unit circle. 3f^jy(  
    %   They are used in disciplines such as astronomy, optics, and U5-8It2OR  
    %   optometry to describe functions on a circular domain. |.RyF@N`T  
    % $X-PjQb1Bb  
    %   The following table lists the first 15 Zernike functions. \ ;]{`  
    % <)LR  
    %       n    m    Zernike function           Normalization tb oQn~&4  
    %       -------------------------------------------------- b'SP,}s5"  
    %       0    0    1                                 1 )lt1I\n*k  
    %       1    1    r * cos(theta)                    2 (||qFu9a  
    %       1   -1    r * sin(theta)                    2 QGOkB  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ^{IZpT3  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 6~ y'  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) \WnTpl>B  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) S]%,g%6i  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) SX'NFdY  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) C[%&;\3S@  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Va.TUz4  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) =$bF[3D  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #E=8kbD7  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) vf>d{F^rv  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |[5;dt_U/  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) oI`Mn3N  
    %       -------------------------------------------------- YWd2bRb  
    % F[O147&C  
    %   Example 1: mh[,E8'd  
    % 3}phg  
    %       % Display the Zernike function Z(n=5,m=1) z8S]FpM6  
    %       x = -1:0.01:1; `EMGrw_  
    %       [X,Y] = meshgrid(x,x); Jia@HrLR  
    %       [theta,r] = cart2pol(X,Y); )S4ga  
    %       idx = r<=1; r6Vw!^]8u8  
    %       z = nan(size(X)); b p?TO]LH  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); c-NUD$  
    %       figure dVMl;{  
    %       pcolor(x,x,z), shading interp jCtk3No  
    %       axis square, colorbar Bx}"X?%S  
    %       title('Zernike function Z_5^1(r,\theta)') +?3RC$jyw  
    % `%#_y67v  
    %   Example 2: OOIp)=4  
    % A_ &IK;-go  
    %       % Display the first 10 Zernike functions Uv.Xw}q  
    %       x = -1:0.01:1; &-^*D%9  
    %       [X,Y] = meshgrid(x,x); WhH60/`  
    %       [theta,r] = cart2pol(X,Y); x4g6Qze  
    %       idx = r<=1; OA9 P"*  
    %       z = nan(size(X)); BHgs,  
    %       n = [0  1  1  2  2  2  3  3  3  3]; =Oh$pZRymu  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; P%yL{  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Z|UVH  
    %       y = zernfun(n,m,r(idx),theta(idx)); #k>n5cR@0  
    %       figure('Units','normalized') ("}Hs[  
    %       for k = 1:10 : Gi8Jo  
    %           z(idx) = y(:,k); X1o R  
    %           subplot(4,7,Nplot(k)) H*0g*(  
    %           pcolor(x,x,z), shading interp HES$. a  
    %           set(gca,'XTick',[],'YTick',[]) Fq+Cr?-  
    %           axis square D1>*ml  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) &u[F)|  
    %       end >a2[P"   
    % Citumc)E  
    %   See also ZERNPOL, ZERNFUN2. G] tT=X[  
    \j)c?1*$  
    %   Paul Fricker 11/13/2006 g]44|9x(W  
    B&59c*K  
    .L#4#IO  
    % Check and prepare the inputs: d72 yu3  
    % ----------------------------- RDQ]_wsyKG  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) kn3GgdU  
        error('zernfun:NMvectors','N and M must be vectors.') ^qC.bv]&  
    end `'r]Oe  
    r:0RvWif  
    if length(n)~=length(m) /M]P&Zb |  
        error('zernfun:NMlength','N and M must be the same length.') lc fAb@}2  
    end n 78!]O  
    U$a)lcJd  
    n = n(:); p*cyW l  
    m = m(:); (qc <'$o  
    if any(mod(n-m,2)) PPpaH!(D  
        error('zernfun:NMmultiplesof2', ... ^56D)A=  
              'All N and M must differ by multiples of 2 (including 0).') Lnn^j#n  
    end G5 )"%G.  
    4Vf-D% h>a  
    if any(m>n) 30Q77,Nsny  
        error('zernfun:MlessthanN', ... IWN18aaL?  
              'Each M must be less than or equal to its corresponding N.') $E:z*~ ?  
    end loq2+(  
    KU+u.J  
    if any( r>1 | r<0 ) Y@ ;/Sf$Q  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') HH(2  
    end zKYN5|17  
    ,T  3M  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) )  d*([!!i  
        error('zernfun:RTHvector','R and THETA must be vectors.') n3/ Bs  
    end {}" <  
    TK> ~)hc}  
    r = r(:); O6-';H:I]L  
    theta = theta(:); +['1~5  
    length_r = length(r); E){ODyk  
    if length_r~=length(theta) 9*n?V;E  
        error('zernfun:RTHlength', ... [["eK9 }0  
              'The number of R- and THETA-values must be equal.') LG("<CU  
    end HPO:aGU   
    #f=41d%  
    % Check normalization: M M @&QaK  
    % -------------------- Lq@uwiq!  
    if nargin==5 && ischar(nflag) ` -f\6r|:)  
        isnorm = strcmpi(nflag,'norm'); wz:,gpH  
        if ~isnorm !14v Ovj4{  
            error('zernfun:normalization','Unrecognized normalization flag.') l0',B*og  
        end @2$Uk!  
    else a[!:`o1U  
        isnorm = false; J<cY'?D  
    end ?LvxEQ-g  
    -"N vu  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &)!N5Veb  
    % Compute the Zernike Polynomials 6k37RpgH  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KlwB oC/{K  
    ldaT: er9  
    % Determine the required powers of r: [NGq$5  
    % ----------------------------------- R\6dvd  
    m_abs = abs(m); C6tfFS3bq  
    rpowers = []; A4L.bBl  
    for j = 1:length(n)  ? EhIK  
        rpowers = [rpowers m_abs(j):2:n(j)]; 56Lt "Z F  
    end bSTTr<W  
    rpowers = unique(rpowers); ZU 7u>  
    U:aaa  
    % Pre-compute the values of r raised to the required powers, %~Wr/TOt+  
    % and compile them in a matrix: X4bZ4U*  
    % ----------------------------- 1:I _ ;O_  
    if rpowers(1)==0 '?mky,:HT  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); [F27i#'I]  
        rpowern = cat(2,rpowern{:}); >(Wt  
        rpowern = [ones(length_r,1) rpowern]; b|.<rV'BTt  
    else }?U #@ h  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); N;cSR\Ng  
        rpowern = cat(2,rpowern{:}); P$/Y9o  
    end m\ @Q}  
    soB_j  
    % Compute the values of the polynomials: [&p/7  
    % -------------------------------------- %W2 o`W$  
    y = zeros(length_r,length(n)); w (odgD  
    for j = 1:length(n) kL -f@CD  
        s = 0:(n(j)-m_abs(j))/2; HNX/#?3  
        pows = n(j):-2:m_abs(j); 8(-N;<Ef2  
        for k = length(s):-1:1 ;l@Ge`&u  
            p = (1-2*mod(s(k),2))* ... NQd0$q  
                       prod(2:(n(j)-s(k)))/              ... RE;)#t?K  
                       prod(2:s(k))/                     ... Gfle"_4m8  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... OK:YnSk"  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); (6)X Fp&  
            idx = (pows(k)==rpowers); q:,ck@-4  
            y(:,j) = y(:,j) + p*rpowern(:,idx); e= ",58  
        end -wnBdL  
         C^ ~[b o  
        if isnorm #4& <d.aw'  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); sFRQFX0XoY  
        end @Wzr rCpj  
    end A^7}:[s20  
    % END: Compute the Zernike Polynomials vPu {xy  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~=Fp0l)#  
    ].N%A07  
    % Compute the Zernike functions: #4^D'r>pJ  
    % ------------------------------ ^F+7@*u  
    idx_pos = m>0; 4m_CPe  
    idx_neg = m<0; @p9YHLxLjQ  
    YD;"_yH  
    z = y; -$f$z(h  
    if any(idx_pos) \r\wqz7  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); =#?=Lh  
    end k NUNh[  
    if any(idx_neg) -lI6!a^  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); =K6{AmG$  
    end ']>/$[!  
    .!g  
    % EOF zernfun
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) @t,Y< )U  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. A@ 4Oq  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated G\H|\i  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Jnq}SUev  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 1(m[L=H5>  
    %   and THETA is a vector of angles.  R and THETA must have the same kBJx`tjtp  
    %   length.  The output Z is a matrix with one column for every P-value, h Ap(1h#m  
    %   and one row for every (R,THETA) pair. j{H,{x  
    % b:6e2|xf?  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Hu7WU;w  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) &v&e- |r8;  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Q~$hx{foN  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 K}Rq<z W  
    %   for all p. ;cW9NS3:  
    % 5^GrG|~  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Gbc2\A\  
    %   Zernike functions (order N<=7).  In some disciplines it is ]*pro|  
    %   traditional to label the first 36 functions using a single mode , Y cF~  
    %   number P instead of separate numbers for the order N and azimuthal E'F87P^>  
    %   frequency M. ,Q>wcE6v  
    % ?H(']3X5@  
    %   Example: +>o} R?xj  
    % iKe68kx  
    %       % Display the first 16 Zernike functions %&S :W%qm?  
    %       x = -1:0.01:1; 0z=^_Fb  
    %       [X,Y] = meshgrid(x,x); "|K D$CY  
    %       [theta,r] = cart2pol(X,Y); ,~qjL|9  
    %       idx = r<=1; |j~{gfpSE  
    %       p = 0:15; =F90SyzTy  
    %       z = nan(size(X)); ?M@ff0  
    %       y = zernfun2(p,r(idx),theta(idx)); Nd61ns(N  
    %       figure('Units','normalized') y>_*}>2,O  
    %       for k = 1:length(p) * odwg$  
    %           z(idx) = y(:,k); j\@osjUu  
    %           subplot(4,4,k) jL9to6 Hmr  
    %           pcolor(x,x,z), shading interp 3q:>NB<  
    %           set(gca,'XTick',[],'YTick',[]) *WZ?C|6+  
    %           axis square ub=Bz1._  
    %           title(['Z_{' num2str(p(k)) '}']) QAKA3{-(  
    %       end Sv|jR r'  
    % *S{fyYyM  
    %   See also ZERNPOL, ZERNFUN. WeRX~  
    k5]`:k6  
    %   Paul Fricker 11/13/2006 _16IP  
    |;(0]  
    @DA.$zn&  
    % Check and prepare the inputs:  wA7^   
    % ----------------------------- .3< sv  
    if min(size(p))~=1 ok<!/"RX$  
        error('zernfun2:Pvector','Input P must be vector.') !O*uQB  
    end Vrx3%_NkQ  
    C9%2}E3Z$)  
    if any(p)>35 qQx5n  
        error('zernfun2:P36', ... Z2hIoCT  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... |sklY0?l(  
               '(P = 0 to 35).']) ? _Y2'O  
    end Ob>M]udn  
    Iji9N!Yx  
    % Get the order and frequency corresonding to the function number: 2C_/T8  
    % ---------------------------------------------------------------- 7\sRf/  
    p = p(:); Mg76v<mv<  
    n = ceil((-3+sqrt(9+8*p))/2); bO\E)%zp  
    m = 2*p - n.*(n+2); $3Srr*  
    -iJ @K  
    % Pass the inputs to the function ZERNFUN: %_%/ym  
    % ---------------------------------------- 76rRF   
    switch nargin Or*e$uMIY  
        case 3 2P4$^G[  
            z = zernfun(n,m,r,theta); h,%b>JFo  
        case 4 E{B=%ZNnm  
            z = zernfun(n,m,r,theta,nflag); =[T_`*s&  
        otherwise Xj("  
            error('zernfun2:nargin','Incorrect number of inputs.') b Q6<R4  
    end `' "125T  
    >@wyiBU  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) DEcsFC/SK  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 2AK]x`GY  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of NHjZ`=J s  
    %   order N and frequency M, evaluated at R.  N is a vector of FG[YH5  
    %   positive integers (including 0), and M is a vector with the Yf=Puy}q  
    %   same number of elements as N.  Each element k of M must be a Q4vl  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) Q7vTTn\  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is A:-r 2;xB  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix _K5R?"H0  
    %   with one column for every (N,M) pair, and one row for every :xz,PeXo7  
    %   element in R. V<%eWT)x7C  
    % i^zncDMA  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 4$^\s5K  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is FhkS"y  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 50l! f7  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ,|r%tNh<8$  
    %   for all [n,m]. -lNq.pp3-$  
    % bb ]r  
    %   The radial Zernike polynomials are the radial portion of the Sb;=YW 1<  
    %   Zernike functions, which are an orthogonal basis on the unit wxx3']:  
    %   circle.  The series representation of the radial Zernike 2a 3RRP  
    %   polynomials is VX<jg#(  
    % l9"T"9C{  
    %          (n-m)/2 Bl"BmUn  
    %            __ &rmXz6 F  
    %    m      \       s                                          n-2s |{a`,%mw  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r iE;D_m.>`O  
    %    n      s=0 g} /efE  
    % ?Dr K2;q  
    %   The following table shows the first 12 polynomials. !iO%?nW;  
    % 'q_^28rK  
    %       n    m    Zernike polynomial    Normalization qij<XNZU"&  
    %       --------------------------------------------- )*wM DM5q  
    %       0    0    1                        sqrt(2) 5UgxuuP4  
    %       1    1    r                           2 ev}ugRxt|k  
    %       2    0    2*r^2 - 1                sqrt(6) 7|/Ct;oO:  
    %       2    2    r^2                      sqrt(6) #S*`7MvM  
    %       3    1    3*r^3 - 2*r              sqrt(8) hN3*]s;/6z  
    %       3    3    r^3                      sqrt(8) :p@.aD5  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) VoTnm   
    %       4    2    4*r^4 - 3*r^2            sqrt(10) \69h>h  
    %       4    4    r^4                      sqrt(10) ;;#_[Zl  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) +6$|No  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ~Gz b^  
    %       5    5    r^5                      sqrt(12) BM,]Wjfdj  
    %       --------------------------------------------- aA|<W g  
    % p!OCF]r  
    %   Example: ]#fmih^  
    % &P@dx=6d  
    %       % Display three example Zernike radial polynomials (1pR=  
    %       r = 0:0.01:1; B,_/'DneQK  
    %       n = [3 2 5]; m);0sb  
    %       m = [1 2 1]; {|E'  
    %       z = zernpol(n,m,r); $q iY)RE  
    %       figure L xg,BZV  
    %       plot(r,z) ;tZ;C(;<  
    %       grid on PXRkK63  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') b5u8j  
    % <U]!1  
    %   See also ZERNFUN, ZERNFUN2. 6Bop8B  
    UBm L:Qv  
    % A note on the algorithm. 0,z3A>C  
    % ------------------------ j^V r!y  
    % The radial Zernike polynomials are computed using the series T{"[Ih3Mbl  
    % representation shown in the Help section above. For many special e` QniTkT  
    % functions, direct evaluation using the series representation can p" ;5J+?(  
    % produce poor numerical results (floating point errors), because <*/IV<  
    % the summation often involves computing small differences between pXy'Ss@y  
    % large successive terms in the series. (In such cases, the functions <Pm!#)-g9  
    % are often evaluated using alternative methods such as recurrence JoCZ{MhM  
    % relations: see the Legendre functions, for example). For the Zernike ,Hzz:ce  
    % polynomials, however, this problem does not arise, because the zJ=lNb?q  
    % polynomials are evaluated over the finite domain r = (0,1), and <y}9Twdy  
    % because the coefficients for a given polynomial are generally all J_|LG rt})  
    % of similar magnitude. ]VCVV!G_=n  
    % ev'` K=n8  
    % ZERNPOL has been written using a vectorized implementation: multiple A5\00O~  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] "K9/^S_  
    % values can be passed as inputs) for a vector of points R.  To achieve :Rftn6!  
    % this vectorization most efficiently, the algorithm in ZERNPOL cS2PrsUx  
    % involves pre-determining all the powers p of R that are required to nr{#Krkb  
    % compute the outputs, and then compiling the {R^p} into a single i!a. 6Gq  
    % matrix.  This avoids any redundant computation of the R^p, and )-s9CWJv  
    % minimizes the sizes of certain intermediate variables. Z0'&@P$  
    % mM$|cge"  
    %   Paul Fricker 11/13/2006 Lhz*o6)  
    rsaN<6#_^Q  
    #hZ`r5GvTj  
    % Check and prepare the inputs: 9zL(PkC%\  
    % ----------------------------- @BmI1  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) li37*  
        error('zernpol:NMvectors','N and M must be vectors.') #aua6V!"  
    end N8E  
    Im g$D*BM  
    if length(n)~=length(m) wU5.t -|`  
        error('zernpol:NMlength','N and M must be the same length.') [KMNMg  
    end Dx5X6t9=  
    tgVMgu  
    n = n(:); LsI8T uv  
    m = m(:); nf0]<x2  
    length_n = length(n); Q;xJ/4 Z"  
    }`~n$OVx  
    if any(mod(n-m,2)) Ht"?ajW{  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') E Xxv  
    end Tpd|+60g  
    t+ vz=`  
    if any(m<0) ! }>CEE  
        error('zernpol:Mpositive','All M must be positive.') 0sA+5*mdM  
    end S0' ACt`  
    rQD^O4j R  
    if any(m>n) {ew; /;  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') `x]`<kS;  
    end ^?8/9 o  
    jcbq#  
    if any( r>1 | r<0 ) aJ"m`5]=%  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 0NF=7 j  
    end TYKs2+S6  
    o* ~aB_  
    if ~any(size(r)==1) N XCvS0/h  
        error('zernpol:Rvector','R must be a vector.') bP Q=88*  
    end ]SmN}Iq1  
    +,1 Ea )  
    r = r(:); +^DDWVp  
    length_r = length(r); .Im=-#EN  
    4:9N]1JCb  
    if nargin==4 ntntB{t  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); )~0TGy|  
        if ~isnorm ri%j*Kn  
            error('zernpol:normalization','Unrecognized normalization flag.') VTa%  
        end IG Ax+3V  
    else +# 3e<+!F  
        isnorm = false; al"=ld(  
    end U,K=(I7OBX  
    \^1S:z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ek"U q RY  
    % Compute the Zernike Polynomials iax0V  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {2)).g  
    P~M[i9 V  
    % Determine the required powers of r: f_2(`T#  
    % ----------------------------------- `&9iC 4P  
    rpowers = []; tZG l^mA"g  
    for j = 1:length(n) y_' 6bpb  
        rpowers = [rpowers m(j):2:n(j)]; 2){O&8A  
    end N8iLI`  
    rpowers = unique(rpowers); ` {qt4zd0  
    ~F^tLi!5  
    % Pre-compute the values of r raised to the required powers, >xXC=z+g]  
    % and compile them in a matrix: \n`/?\r.z  
    % ----------------------------- !QpOrg  
    if rpowers(1)==0 r )HZaq  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 9a]{|M9  
        rpowern = cat(2,rpowern{:}); npd:aGx  
        rpowern = [ones(length_r,1) rpowern]; TuEM  
    else W7. +  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); \(RD5@=!4#  
        rpowern = cat(2,rpowern{:}); Bi2 c5[3  
    end ^ L]e]<h(  
    3RanAT.nu:  
    % Compute the values of the polynomials:  wX5q=I  
    % -------------------------------------- Z5 p [*LMO  
    z = zeros(length_r,length_n);  fDloL  
    for j = 1:length_n -p?&vQDo`  
        s = 0:(n(j)-m(j))/2; l/,la]!T  
        pows = n(j):-2:m(j); fwvwmZW  
        for k = length(s):-1:1 n.rn+nuwv  
            p = (1-2*mod(s(k),2))* ... z-qbe97  
                       prod(2:(n(j)-s(k)))/          ... pztfm'  
                       prod(2:s(k))/                 ... Y]7503J  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... zu.B>INe  
                       prod(2:((n(j)+m(j))/2-s(k))); e=nvm'[h  
            idx = (pows(k)==rpowers); yVp,)T9  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ?5jLN&A3 G  
        end ;?k<L\zaw  
         2e-`V5{)b  
        if isnorm /wax5FS'I,  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); DJ DQH\&  
        end ?% [~J  
    end jo^c>ur  
    LP=y$B  
    % 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)  rkjnw@x\  
    }gkLO TJ/,  
    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。 { LvD\4h"  
    ]3O&8,  
    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)