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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 \B D'"  
    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! <@:LONe<  
     
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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 _P*<T6\J>  
    function z = zernfun(n,m,r,theta,nflag) uM<6][^`  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. QcDWVM'v  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N aPMqJ#fIr  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ZNvnVW<  
    %   unit circle.  N is a vector of positive integers (including 0), and 0cm+:  
    %   M is a vector with the same number of elements as N.  Each element px1{=~V/  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ;/8oP ;X2  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, r&t)%R@q  
    %   and THETA is a vector of angles.  R and THETA must have the same Q}MS $[y  
    %   length.  The output Z is a matrix with one column for every (N,M) j7)Xm,wI8  
    %   pair, and one row for every (R,THETA) pair. S@a#,,\[  
    % v8xNtUxN  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike N{<=s]I%x  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), &[hq !v  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral R~],5_|  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, N3Jfp3_b@  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized <([1(SY2e  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. FaNH+LPe  
    % Y(4#b`k3  
    %   The Zernike functions are an orthogonal basis on the unit circle. :+SpZ>  
    %   They are used in disciplines such as astronomy, optics, and >}*i Qq  
    %   optometry to describe functions on a circular domain. {{?[b^  
    % |? !Ew# w  
    %   The following table lists the first 15 Zernike functions. FN&.PdRT  
    % {yy ^DlHb  
    %       n    m    Zernike function           Normalization IZ;%lV7t  
    %       -------------------------------------------------- EQkv&k5X  
    %       0    0    1                                 1 . ` OdnLGy  
    %       1    1    r * cos(theta)                    2 Zq--m/  
    %       1   -1    r * sin(theta)                    2 MU^7(s="  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 9LkP*$2"M<  
    %       2    0    (2*r^2 - 1)                    sqrt(3) UpgY}pf}  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) wyk4v}  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) c%aY6dQG&%  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) DdTTWp/  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) hN6j5.x%  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) {@u;F2?  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) xFpMn}CD  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) n:GK0wu.s  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 9IKFrCO9,  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )jK"\'cK  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) {ZH9W  
    %       -------------------------------------------------- &P%3'c}G  
    % L[d 7@  
    %   Example 1: W^W^5-'"D,  
    % `/'Hq9$F<"  
    %       % Display the Zernike function Z(n=5,m=1) zA&lJD $0  
    %       x = -1:0.01:1; 1.0S>+^JE  
    %       [X,Y] = meshgrid(x,x); {|%N  
    %       [theta,r] = cart2pol(X,Y); ?L$ Dk5-W  
    %       idx = r<=1; Vc3tKuMsiX  
    %       z = nan(size(X)); *f:^6h  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 2B7X~t>8a  
    %       figure Z@=1-l  
    %       pcolor(x,x,z), shading interp }!\ZJoa  
    %       axis square, colorbar cjU*  
    %       title('Zernike function Z_5^1(r,\theta)') =Uta5$\a)  
    % tt`j!!  
    %   Example 2: yAoJ?<4^W  
    % @8TD^ub  
    %       % Display the first 10 Zernike functions 8 kw`=wSH>  
    %       x = -1:0.01:1;  M SU|T  
    %       [X,Y] = meshgrid(x,x); k~u$&a  
    %       [theta,r] = cart2pol(X,Y); #J]u3*T n|  
    %       idx = r<=1; 0hXI1@8]`  
    %       z = nan(size(X)); e%&2tf4  
    %       n = [0  1  1  2  2  2  3  3  3  3]; cs7T AX  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; A('=P}I^  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; nsqs*$  
    %       y = zernfun(n,m,r(idx),theta(idx)); _PrK6M@"L  
    %       figure('Units','normalized') &AmTXW  
    %       for k = 1:10 Ql> DS~a  
    %           z(idx) = y(:,k); sn&y;Vc[$  
    %           subplot(4,7,Nplot(k)) "#2z 'J  
    %           pcolor(x,x,z), shading interp zg&<HJO  
    %           set(gca,'XTick',[],'YTick',[]) o+SD(KVn-  
    %           axis square ja}_u}:  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) A1:<-TF6^p  
    %       end D0tmNV@  
    % ;BqYhi  
    %   See also ZERNPOL, ZERNFUN2. OS6 l*S('  
    V<AT"vU[  
    %   Paul Fricker 11/13/2006 ua*k{0[  
    [Z|R-{"  
    gvO}u2.:  
    % Check and prepare the inputs: U[=VW0  
    % ----------------------------- (Bd8@}\u_  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) bE.,)GY  
        error('zernfun:NMvectors','N and M must be vectors.') *,~d!Fc  
    end v' 7,(.E  
    m UpLD+-j  
    if length(n)~=length(m) q-gN0"z^6$  
        error('zernfun:NMlength','N and M must be the same length.') \5 IB/ *  
    end $*^Ms>Pa_  
    |ul25/B B  
    n = n(:); d0~F|j\#  
    m = m(:); .v%H%z~Rl#  
    if any(mod(n-m,2)) 0'`>20Y  
        error('zernfun:NMmultiplesof2', ... Cfu]umZLn  
              'All N and M must differ by multiples of 2 (including 0).') >S3iP?V7  
    end `uy)][j-  
    6wx;grt'Z  
    if any(m>n) *z;4. OX  
        error('zernfun:MlessthanN', ... -`gqA%#+  
              'Each M must be less than or equal to its corresponding N.') D ::),,  
    end Juj"cjob  
    `;4P?!WG  
    if any( r>1 | r<0 ) abkl)X>k  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') e.jrX;;$!&  
    end Mib(J+Il  
    Lt0JUUa0  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) #N_C| v/  
        error('zernfun:RTHvector','R and THETA must be vectors.') 2`I" QU  
    end "S.5_@?  
    &U ]L@ ]x  
    r = r(:); x?Doe`/6?  
    theta = theta(:); f/RzE  
    length_r = length(r); 72R|zR  
    if length_r~=length(theta) hIu;\dfwk  
        error('zernfun:RTHlength', ... A;n3""  
              'The number of R- and THETA-values must be equal.') 7N,E%$QL  
    end I}Uj"m`>  
    ;<d("Yz:@Z  
    % Check normalization: ?47q0C  
    % -------------------- ra=U,  
    if nargin==5 && ischar(nflag) Cqy84!Z<  
        isnorm = strcmpi(nflag,'norm'); % 1ZJi}~  
        if ~isnorm U|. kAI*  
            error('zernfun:normalization','Unrecognized normalization flag.') 1@sy:{ d`  
        end Y3+DTR0|'  
    else +<7~yZ[Z8  
        isnorm = false; u8L%R[#o  
    end ?U.+SQ  
    hAtf)  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9HrT>{@  
    % Compute the Zernike Polynomials FIhq>L.q4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% HpY-7QTPJ~  
    S[(Tpk2_  
    % Determine the required powers of r: U;u@\E@2  
    % ----------------------------------- UZ7Zzc#g  
    m_abs = abs(m); Jt5\  
    rpowers = []; @dei} !e  
    for j = 1:length(n) 5H#f;L\k  
        rpowers = [rpowers m_abs(j):2:n(j)]; ;"46H'>!  
    end }A,9`  
    rpowers = unique(rpowers); N,fEta6  
    !qk+>6~A,  
    % Pre-compute the values of r raised to the required powers, jsB%RvX  
    % and compile them in a matrix: w%F~4|F  
    % ----------------------------- a)w *  
    if rpowers(1)==0 *P2_l Q=  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); I^S gWC  
        rpowern = cat(2,rpowern{:}); tb36c<U-  
        rpowern = [ones(length_r,1) rpowern]; @=JOAo  
    else j=b?WNK  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ScOiOz:Ha  
        rpowern = cat(2,rpowern{:}); -P#PyZEH&I  
    end z6 T3vw  
    vpnQs#8O  
    % Compute the values of the polynomials: hZ@frbuowk  
    % -------------------------------------- L~9Q7 6w  
    y = zeros(length_r,length(n)); ;PM(q<@\  
    for j = 1:length(n) W;%$7&+0  
        s = 0:(n(j)-m_abs(j))/2; ,5}%_  
        pows = n(j):-2:m_abs(j); ZNWo:N8;  
        for k = length(s):-1:1 j#4 Iu&YJ  
            p = (1-2*mod(s(k),2))* ... ZcJa:  
                       prod(2:(n(j)-s(k)))/              ... b>g&Pf#N!  
                       prod(2:s(k))/                     ... |Z6M?n  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... LFvO[&  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 8i$quHd&x  
            idx = (pows(k)==rpowers); *iLlBE  
            y(:,j) = y(:,j) + p*rpowern(:,idx); VPOzt7:  
        end u}_,4J  
         /`6Y-8e2  
        if isnorm 2S%[YR>>  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); >Sc)?[H  
        end b0X<)1O  
    end rdj_3Utv  
    % END: Compute the Zernike Polynomials WXq=FZ-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }-`N^  
    `Xs3^FJt  
    % Compute the Zernike functions: .M(')$\U  
    % ------------------------------ gR5 EK$  
    idx_pos = m>0; ZVu_E.4.  
    idx_neg = m<0; 4)Jtc2z7Z\  
    au=A+  
    z = y; wPr9N}rf  
    if any(idx_pos) #BPJRNXd  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); T'i^yd }*v  
    end 8Dy5g  
    if any(idx_neg) '%Fg+cZN\  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); \NZ(Xk  
    end # <?igtUO  
    OdKfU^  
    % 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) KIRCye  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. eW<|I  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated V><,.p8  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive a^vTBJXo  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, %[cZ,F=  
    %   and THETA is a vector of angles.  R and THETA must have the same UWXl c  
    %   length.  The output Z is a matrix with one column for every P-value, T0;8koj^_  
    %   and one row for every (R,THETA) pair. PEr &|H2  
    % ohJo1}{  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike @> |3d  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) :~^_*:  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) d6+$[4w  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 +,Ud 3iS  
    %   for all p. W(jOD,QMB  
    % fzdWM:g  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ""f'L,`{.  
    %   Zernike functions (order N<=7).  In some disciplines it is :{LNr!I?I  
    %   traditional to label the first 36 functions using a single mode F$sDmk#  
    %   number P instead of separate numbers for the order N and azimuthal _|Uv7>}J^  
    %   frequency M. nYnB WDnV  
    % F~$ay@g  
    %   Example: ( 5 BZZ  
    % $L4h'(s  
    %       % Display the first 16 Zernike functions j.ZXLe~  
    %       x = -1:0.01:1; PX- PVW  
    %       [X,Y] = meshgrid(x,x); Pi hpo  
    %       [theta,r] = cart2pol(X,Y); Fhrj$  
    %       idx = r<=1; zCdcwTe  
    %       p = 0:15; oLS/  
    %       z = nan(size(X)); $KVCEe!X  
    %       y = zernfun2(p,r(idx),theta(idx)); KG=57=[  
    %       figure('Units','normalized') b5S4C2Ynq  
    %       for k = 1:length(p) dw>1Ut{"3  
    %           z(idx) = y(:,k); oCxy(q'y  
    %           subplot(4,4,k) yBRYEqS+  
    %           pcolor(x,x,z), shading interp Q_)$Ha{>H,  
    %           set(gca,'XTick',[],'YTick',[]) Qt\^h/zjG  
    %           axis square O)!S[5YI  
    %           title(['Z_{' num2str(p(k)) '}']) _+9o'<#u(  
    %       end ny"z<N&}/  
    % QuIZpP=  
    %   See also ZERNPOL, ZERNFUN. $jOp:R&I^3  
    &Nw[J5-"k  
    %   Paul Fricker 11/13/2006 O+?vQ$z  
    J(:y-U  
    U1W8f|u  
    % Check and prepare the inputs: Bm5\*Xd1(  
    % ----------------------------- ^GS\(egt  
    if min(size(p))~=1  u]OYu  
        error('zernfun2:Pvector','Input P must be vector.') %i.Prckrb  
    end B|"-Ed  
    UP7?9\  
    if any(p)>35 f~R+Q/Gtz`  
        error('zernfun2:P36', ...  20]p<  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... NM"5.   
               '(P = 0 to 35).']) nT%ko7~-  
    end Kk).KgR  
    "r~/E|Da<  
    % Get the order and frequency corresonding to the function number: ^ X-6j[".  
    % ---------------------------------------------------------------- &~c`p[  
    p = p(:); NX5$x/uz  
    n = ceil((-3+sqrt(9+8*p))/2); 81H9d6hqcD  
    m = 2*p - n.*(n+2); 2;K2|G7  
    @*roW{?!  
    % Pass the inputs to the function ZERNFUN: L_tjclk0J  
    % ---------------------------------------- DKF` xuJP  
    switch nargin M0T z('~s  
        case 3 {rwT4]4  
            z = zernfun(n,m,r,theta); Qff.QI,  
        case 4 &xpvHKJl  
            z = zernfun(n,m,r,theta,nflag); (((|vI3 <  
        otherwise M.HMn N#  
            error('zernfun2:nargin','Incorrect number of inputs.') =Qf{  
    end |\>Ifv%{  
    4Y{;%;-i  
    % 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) Y^nm{;G+  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. F B:nkUR`  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of U^eos;:s8  
    %   order N and frequency M, evaluated at R.  N is a vector of |+KwyHE`9  
    %   positive integers (including 0), and M is a vector with the '\GU(j  
    %   same number of elements as N.  Each element k of M must be a $fB j}\o  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) UZs'H"K  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is pSI8"GwQ  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix K&,";9c  
    %   with one column for every (N,M) pair, and one row for every *<[zG7+&[  
    %   element in R. J"Fp),  
    % Qm=iCZ|E^!  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly-  fZ&' _  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ExMd$`gW  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to BOh^oQh  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 (tX)r4VU  
    %   for all [n,m]. ]G D` f  
    % )Ay9 0Wt  
    %   The radial Zernike polynomials are the radial portion of the 1J72*`4OK  
    %   Zernike functions, which are an orthogonal basis on the unit I~6 o<HO  
    %   circle.  The series representation of the radial Zernike !{{gL=_@  
    %   polynomials is 6`vW4]zu  
    % pp@B]We  
    %          (n-m)/2 yn"4qC#Z  
    %            __ AW E ab  
    %    m      \       s                                          n-2s $7ix(WL<%  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Pjn{3/*wi  
    %    n      s=0 nt+OaXe5D  
    % i(OeE"YA  
    %   The following table shows the first 12 polynomials. oam;hmw  
    % qGX#(,E9;  
    %       n    m    Zernike polynomial    Normalization ZzjCS2U  
    %       --------------------------------------------- 4 R(m$!E!  
    %       0    0    1                        sqrt(2) |2%|=   
    %       1    1    r                           2 q3#+G:nh  
    %       2    0    2*r^2 - 1                sqrt(6) &r~s3S{pQ  
    %       2    2    r^2                      sqrt(6) RKE"}|i +S  
    %       3    1    3*r^3 - 2*r              sqrt(8) x(xi%?G  
    %       3    3    r^3                      sqrt(8) dGi HO  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 3!osQ1  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ~%C F3?e6  
    %       4    4    r^4                      sqrt(10) _Gf-s51s  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) p:K%-^  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) y4LUC;[n  
    %       5    5    r^5                      sqrt(12) k& +gkJm  
    %       --------------------------------------------- T1*.3_wtP  
    % |q 8N$m  
    %   Example: ]@ke_' "  
    % l)dE7$H  
    %       % Display three example Zernike radial polynomials \D(3~y>  
    %       r = 0:0.01:1; UAT\ .  
    %       n = [3 2 5]; Je6wio- 4  
    %       m = [1 2 1]; L<0_e^8  
    %       z = zernpol(n,m,r); ,a,2I  
    %       figure > *VvV/UU  
    %       plot(r,z) y`j=(|DV  
    %       grid on zSQy  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') DyCzRkH  
    % gwQMy$  
    %   See also ZERNFUN, ZERNFUN2. <n06(9BF  
    fZ5 UFq_~s  
    % A note on the algorithm. Su"Z3gm5Kw  
    % ------------------------ c9fz x  
    % The radial Zernike polynomials are computed using the series bT&{8a  
    % representation shown in the Help section above. For many special Q}OloA(+  
    % functions, direct evaluation using the series representation can #IZh}*$  
    % produce poor numerical results (floating point errors), because BZHoRd{EH  
    % the summation often involves computing small differences between \U]K!K=  
    % large successive terms in the series. (In such cases, the functions @$n $f  
    % are often evaluated using alternative methods such as recurrence kx?Yin8K  
    % relations: see the Legendre functions, for example). For the Zernike kj[box N  
    % polynomials, however, this problem does not arise, because the 0bM_EC  
    % polynomials are evaluated over the finite domain r = (0,1), and b<~-s sL7a  
    % because the coefficients for a given polynomial are generally all ^@O 7d1&y  
    % of similar magnitude. D|*w6p("z  
    % g:a[N%[C  
    % ZERNPOL has been written using a vectorized implementation: multiple 'JJKnE zQ  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] SFd_k9  
    % values can be passed as inputs) for a vector of points R.  To achieve f"P866@oWn  
    % this vectorization most efficiently, the algorithm in ZERNPOL aG^E^^Y  
    % involves pre-determining all the powers p of R that are required to 1' U  
    % compute the outputs, and then compiling the {R^p} into a single ;Vat\,45pg  
    % matrix.  This avoids any redundant computation of the R^p, and vm_]X{80;  
    % minimizes the sizes of certain intermediate variables. 1xd6p  
    % a-Y6ghs  
    %   Paul Fricker 11/13/2006 U364'O8_  
    xZ P SUEG  
    J ( d[05x0  
    % Check and prepare the inputs: }7 +%k/  
    % ----------------------------- r8:"\%"f>  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 1Ub=RyB  
        error('zernpol:NMvectors','N and M must be vectors.') Q|h$D~  
    end #jg-q|nd  
    KFHZ3HZ:>  
    if length(n)~=length(m) {+hABusq  
        error('zernpol:NMlength','N and M must be the same length.') <I34@;R c  
    end ub-e!{  
    452kE@=49  
    n = n(:); QrK%DN  
    m = m(:); sV,Yz3E<u$  
    length_n = length(n); $Y_S`#c@i  
    q%])dZ!lE  
    if any(mod(n-m,2)) \DS*G7.A+&  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Od~uYOL/B  
    end V<S6 a  
    4~h 0/H"  
    if any(m<0) (LmU\Pe%  
        error('zernpol:Mpositive','All M must be positive.') $.4A?,d  
    end n4S`k%CI  
    zgEN2d  
    if any(m>n) C0N}B1-MU  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') tt?`,G.(]  
    end )~=8Ssu  
    2K};-}eW  
    if any( r>1 | r<0 ) &lSNI5l  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') L7buY(F(  
    end ?8H{AuLB  
    `i cs2po  
    if ~any(size(r)==1) M9R'ONYAa  
        error('zernpol:Rvector','R must be a vector.') (JlPe)Q5  
    end ?SFQx \/  
    {24>&<p  
    r = r(:); %6uZb sa  
    length_r = length(r); z~vcwiYAP  
    "[?DS  
    if nargin==4 -`XS2  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ]dNNw`1\V  
        if ~isnorm u'#/vT#l  
            error('zernpol:normalization','Unrecognized normalization flag.') 0}NDi|o  
        end yPtE5"(o  
    else TYGI f4z  
        isnorm = false; /}~=)QHH  
    end -fFtHw:kHh  
    '[liZCg  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a)pc+w#  
    % Compute the Zernike Polynomials 07:V[@'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -V6caVlg  
    w6!97x  
    % Determine the required powers of r: E8r6P:5d`  
    % ----------------------------------- y@~ VE5N  
    rpowers = []; |8CxMs  
    for j = 1:length(n) OJa(Gds  
        rpowers = [rpowers m(j):2:n(j)]; <A Hzs  
    end *4%pXm;  
    rpowers = unique(rpowers); 8b'@_s!_  
    ,M{G X  
    % Pre-compute the values of r raised to the required powers, 4Z%1eOR9V  
    % and compile them in a matrix: bI:W4y>I=  
    % ----------------------------- tcXXo&ZS  
    if rpowers(1)==0 o!+%|V8Y  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); p2 1|  
        rpowern = cat(2,rpowern{:}); ugTsI~aE  
        rpowern = [ones(length_r,1) rpowern]; 2YZ>nqy  
    else QyVAs;  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); GB Yy^wjU  
        rpowern = cat(2,rpowern{:}); N!~]D[D  
    end &j1-Ouy  
    d"Zu10  
    % Compute the values of the polynomials: I m I$~q'  
    % -------------------------------------- ?HPAX  
    z = zeros(length_r,length_n); pt.V^a  
    for j = 1:length_n 2T+-[}*  
        s = 0:(n(j)-m(j))/2; \O "`o4  
        pows = n(j):-2:m(j); *`"+J_   
        for k = length(s):-1:1 :po6%}hn  
            p = (1-2*mod(s(k),2))* ... \_,p@r]Q  
                       prod(2:(n(j)-s(k)))/          ... -V{"Lzrfug  
                       prod(2:s(k))/                 ... >Vt2@Ee  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... tIDN~[1  
                       prod(2:((n(j)+m(j))/2-s(k))); 7\%JJw6h  
            idx = (pows(k)==rpowers); Cs>`f, o  
            z(:,j) = z(:,j) + p*rpowern(:,idx); R&Nl!QTJj  
        end [5L?#Y  
         g=nb-A{#  
        if isnorm y*^UGJC:  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); *tj(,:!  
        end n?cC]k;P~  
    end ZAfuW^r  
    GgY8\>u  
    % 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)  /^Lo@672  
    iw EHEi%  
    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。 pvd9wKz  
    q/YO5>s15  
    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)