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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 \_m\U.*  
    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! pFpQ\xc9$  
     
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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 A&NC0K}G!  
    function z = zernfun(n,m,r,theta,nflag) ifJv~asp   
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 8k+q7  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ,$MWk(S  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Xm"w,J&  
    %   unit circle.  N is a vector of positive integers (including 0), and 'Yaf\Hp  
    %   M is a vector with the same number of elements as N.  Each element _/QKWk&j  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ~>}dse  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, I,],?DQX2)  
    %   and THETA is a vector of angles.  R and THETA must have the same Gx(KN57D  
    %   length.  The output Z is a matrix with one column for every (N,M) 7 SjF9x  
    %   pair, and one row for every (R,THETA) pair. OBKC$e6I  
    % t7C!}'g&'  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike |g7nh[  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 'mBLf&fB  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral M(.uu`B  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 5p]urfN-f  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized X <ba|(  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 3oppV_^JdT  
    % uZqu xu.  
    %   The Zernike functions are an orthogonal basis on the unit circle. O" X!S_R  
    %   They are used in disciplines such as astronomy, optics, and G:h;C].  
    %   optometry to describe functions on a circular domain. \jF" nl  
    % r 48;_4d)D  
    %   The following table lists the first 15 Zernike functions. 2HvTM8  
    % WL)_8!  
    %       n    m    Zernike function           Normalization J[& 7,}  
    %       -------------------------------------------------- {|Mxvp*Hg  
    %       0    0    1                                 1 k$$S!qi#  
    %       1    1    r * cos(theta)                    2 E5Snl#Gl\0  
    %       1   -1    r * sin(theta)                    2 =#POMK".6  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ~ X]"P4 u  
    %       2    0    (2*r^2 - 1)                    sqrt(3) D*d 3w  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) i h`y0(<  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 1eE]4Z4Q  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Y-neD?VN  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) evndw>  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) X_0{*!v8  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) m &3HFf  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Sq?6R}q%  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 6?<`wGs(  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Q?Bj q>  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ) <}VP&:X  
    %       -------------------------------------------------- .=b +O~  
    % XqE55Jclp  
    %   Example 1: QRg"/62WCD  
    % Y>dg10=  
    %       % Display the Zernike function Z(n=5,m=1) %CsTB0Y7n,  
    %       x = -1:0.01:1; N) V7yo?  
    %       [X,Y] = meshgrid(x,x); 2t]! {L  
    %       [theta,r] = cart2pol(X,Y); 9|G=KN)P:  
    %       idx = r<=1; 8,H#t@+MT  
    %       z = nan(size(X)); RBv=  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 9sO{1rF  
    %       figure 0-t4+T  
    %       pcolor(x,x,z), shading interp R+ #.bQg  
    %       axis square, colorbar )K\k6HC.  
    %       title('Zernike function Z_5^1(r,\theta)') QX.F1T 2e?  
    % Be14$7r  
    %   Example 2: x%:> Ol  
    % VvM U)  
    %       % Display the first 10 Zernike functions <4!&iU+;  
    %       x = -1:0.01:1; vU \w3  
    %       [X,Y] = meshgrid(x,x); !Lg}q!*%>V  
    %       [theta,r] = cart2pol(X,Y); n_xQSVI0F  
    %       idx = r<=1; [r/Seg"  
    %       z = nan(size(X)); JI[rIL \Ey  
    %       n = [0  1  1  2  2  2  3  3  3  3]; fbx;-He!  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; d'g{K]=tF  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; @=<TA0;LL  
    %       y = zernfun(n,m,r(idx),theta(idx)); $CQwBsYb=  
    %       figure('Units','normalized') `X.=uG+m  
    %       for k = 1:10 d=+Lv<  
    %           z(idx) = y(:,k); rY_C3;B  
    %           subplot(4,7,Nplot(k)) a,0o{* (u$  
    %           pcolor(x,x,z), shading interp 7"CH\*%  
    %           set(gca,'XTick',[],'YTick',[]) EH!EyNNb  
    %           axis square yS.fe[  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) }&C!^v o  
    %       end 82@;.%  
    % |z<wPJ,;2  
    %   See also ZERNPOL, ZERNFUN2. ^)0{42!]  
    2G:{FY  
    %   Paul Fricker 11/13/2006 ! ,(bXa\^  
    x_H7=\pX]  
    n`I jG  
    % Check and prepare the inputs: OTFu4"]M  
    % ----------------------------- 8Jy1=R*S  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 3xCA\*  
        error('zernfun:NMvectors','N and M must be vectors.') ^J5V!i$  
    end [2j (\vC!  
    WCfe!P?g  
    if length(n)~=length(m) ,w58n%)H  
        error('zernfun:NMlength','N and M must be the same length.') szsZFyW )+  
    end / jL{JF>I  
    . =foXN  
    n = n(:); HI?~t| [y  
    m = m(:); %Pvb>U(Xs  
    if any(mod(n-m,2)) U+}9X^  
        error('zernfun:NMmultiplesof2', ... 1.d9{LO[-  
              'All N and M must differ by multiples of 2 (including 0).') X9`C2fyVd  
    end :~A1Ud4c  
    2.&V  
    if any(m>n) \3Ald.EqtM  
        error('zernfun:MlessthanN', ... #]\G*>{  
              'Each M must be less than or equal to its corresponding N.') uxJiec`&  
    end [,A'  
    b%~3+c  
    if any( r>1 | r<0 ) ^5@"|m1  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 9 0if:mYA  
    end m&z %kVsg]  
    Zz*mf+  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 9kg>)ty@  
        error('zernfun:RTHvector','R and THETA must be vectors.') ,c %gwzU  
    end 0v)mgrl=,  
    fD}]Mi:V  
    r = r(:); ;@-5lCvC(+  
    theta = theta(:); C%7)sLWjJS  
    length_r = length(r); +n~rM'^4/  
    if length_r~=length(theta) ps;o[gB@5  
        error('zernfun:RTHlength', ... A kQFb2|ir  
              'The number of R- and THETA-values must be equal.') -Aym+N9  
    end J1ro\"  
    V^5k> `A  
    % Check normalization: <.B > LU  
    % -------------------- M,U=zNPnk  
    if nargin==5 && ischar(nflag) cZ2, u,4  
        isnorm = strcmpi(nflag,'norm'); "=TTsxyM6P  
        if ~isnorm #w?%&,Kp  
            error('zernfun:normalization','Unrecognized normalization flag.') A(sx5Ynp  
        end 5Fm? ,^  
    else nk,Mo5iqV  
        isnorm = false; n[S*gX0  
    end ..{^"`FQ  
    .0;k|&eBD  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Us~wv"L=UX  
    % Compute the Zernike Polynomials tfIBsw.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6]A\8Ty  
    |B WK"G  
    % Determine the required powers of r: ' g!_Flk  
    % ----------------------------------- f! Nc+  
    m_abs = abs(m); <1%XN  
    rpowers = []; _Ws k3AP  
    for j = 1:length(n)  X_S]8Aa  
        rpowers = [rpowers m_abs(j):2:n(j)]; \ bmboNe  
    end q$*_C kT  
    rpowers = unique(rpowers); w}<I\*\`!  
    UdgI<a~`k6  
    % Pre-compute the values of r raised to the required powers, }nERQq&A  
    % and compile them in a matrix: ]`U?<9~Ob  
    % ----------------------------- 1,D ^,  
    if rpowers(1)==0 u"$HWB~@z  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); >a~FSZf  
        rpowern = cat(2,rpowern{:}); hUvH t+d  
        rpowern = [ones(length_r,1) rpowern]; wm[d5A4  
    else =U|SK"oO  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3/<^R}w\  
        rpowern = cat(2,rpowern{:}); ?bi^h/ f  
    end l zkn B  
    A+ *(Pds  
    % Compute the values of the polynomials: bv"({:x  
    % -------------------------------------- .tZ$a_O  
    y = zeros(length_r,length(n)); /P}tgcs  
    for j = 1:length(n) l),13"?C(  
        s = 0:(n(j)-m_abs(j))/2; hpKc_|un  
        pows = n(j):-2:m_abs(j); ~OfKn1D  
        for k = length(s):-1:1 / UBAQ8TR  
            p = (1-2*mod(s(k),2))* ... SvJ8Kl OV  
                       prod(2:(n(j)-s(k)))/              ... j`hbQp\`  
                       prod(2:s(k))/                     ... dL"i\5#%A  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... K`2DhJC  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); JH,bSb  
            idx = (pows(k)==rpowers); r/:'}os;  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Efd[ZJxS6  
        end 4tKf  
         FJ. :*K[  
        if isnorm 3{E}^ve  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); pDN,(Ip  
        end 1#RA+d(  
    end 6% axbB  
    % END: Compute the Zernike Polynomials ?E+XD'~  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'm ((G4  
    }Ec"&  
    % Compute the Zernike functions: Qp Vm  
    % ------------------------------ DzOJ{dF  
    idx_pos = m>0; 7nIMIkT:  
    idx_neg = m<0; q@> m~R  
    |,f6c Om f  
    z = y; >qZRIDE5$  
    if any(idx_pos) j KK48S  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); @35]IxD  
    end y5 +&P  
    if any(idx_neg) 2AE|N_v8W  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); [a6lE"yr  
    end Fm{y.URo  
    Lj\<qF~n  
    % 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) ea/6$f9^  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 3e:y?hpeL  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated KcE=m\h  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive <9vkiEo  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, eSf:[^  
    %   and THETA is a vector of angles.  R and THETA must have the same "b;?2_w:E  
    %   length.  The output Z is a matrix with one column for every P-value, `WL*Jb  
    %   and one row for every (R,THETA) pair. ,kI1"@Tu  
    % x;/3_"$9>\  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike B7C6Mau  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) XO>Y*7rO  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) yuq E  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ,.B8hr@H6-  
    %   for all p. HC$cK+,ZU}  
    % ,$>Z= ~x*  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Z )I4U  
    %   Zernike functions (order N<=7).  In some disciplines it is ^ TS\x/P  
    %   traditional to label the first 36 functions using a single mode fC[gu$f][  
    %   number P instead of separate numbers for the order N and azimuthal }W J`q`g  
    %   frequency M. 7 #`:m|$  
    % XafyI*pOX  
    %   Example: 7;V5hul  
    % uq_SF.a'v  
    %       % Display the first 16 Zernike functions /:)4tIV  
    %       x = -1:0.01:1; 'Z[R*Ikzq  
    %       [X,Y] = meshgrid(x,x); / e,lD)  
    %       [theta,r] = cart2pol(X,Y); ZBWe,Xvq  
    %       idx = r<=1; O)?0G$0  
    %       p = 0:15; =v}.sJ V?  
    %       z = nan(size(X)); 1['A1 ,  
    %       y = zernfun2(p,r(idx),theta(idx));  qn .  
    %       figure('Units','normalized') EOiKwhrV  
    %       for k = 1:length(p) P:o<kRj1  
    %           z(idx) = y(:,k);  u[u=:Y+  
    %           subplot(4,4,k) Phczf  
    %           pcolor(x,x,z), shading interp B^Q#@[T   
    %           set(gca,'XTick',[],'YTick',[]) e# DAa  
    %           axis square = zSrre  
    %           title(['Z_{' num2str(p(k)) '}']) <f%9w]  
    %       end 6r`g+Js/  
    % ~*qGH  
    %   See also ZERNPOL, ZERNFUN. HD>{UU?  
    c}lgWu~  
    %   Paul Fricker 11/13/2006 <5 +?&i  
    w@4+&v>O  
    5VN4A<))  
    % Check and prepare the inputs: oT'XcMn  
    % ----------------------------- Z'~5L_.]Ai  
    if min(size(p))~=1 XN Y(@  
        error('zernfun2:Pvector','Input P must be vector.') ME(!xI//JZ  
    end TFhj]r^ {  
    H0S7k`.  
    if any(p)>35 BdTj0{S1u  
        error('zernfun2:P36', ... Co M8  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... q:'(1y~  
               '(P = 0 to 35).']) JmK+#o  
    end hf '3yEm  
    mMR[(  
    % Get the order and frequency corresonding to the function number: ;Mc}If*  
    % ---------------------------------------------------------------- 0-FbV,:;  
    p = p(:); *VpQ("  
    n = ceil((-3+sqrt(9+8*p))/2); tPUQ"S  
    m = 2*p - n.*(n+2); LTF%b AQ,  
    !(]|!F[m  
    % Pass the inputs to the function ZERNFUN: -%A6eRShk  
    % ---------------------------------------- ,/KHKLY7  
    switch nargin z<ek?0?yS  
        case 3 9:Y\D.M  
            z = zernfun(n,m,r,theta); FR&RIFy  
        case 4 `4o;Lz~  
            z = zernfun(n,m,r,theta,nflag); Vo\d&}Q  
        otherwise * PZ=$>r  
            error('zernfun2:nargin','Incorrect number of inputs.') ZE9*i}r  
    end 4DNZ y2`  
    k$hWR;U  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) ScI9.{  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. !vsUL-  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 1q*3V8  
    %   order N and frequency M, evaluated at R.  N is a vector of k~?@~xm,R  
    %   positive integers (including 0), and M is a vector with the >Nov9<p  
    %   same number of elements as N.  Each element k of M must be a 'HC4Q{b`  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) xGA%/dy,;  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 2@ad! h  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix i^n&K:6  
    %   with one column for every (N,M) pair, and one row for every |h3 YL!  
    %   element in R. g><sZqj8tt  
    % 6PTD%Rf\  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ]!f=b\-Av  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is #):FXB$a  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 67#;.}4a  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 rsP1?Hxq  
    %   for all [n,m]. $`uL^ hlj]  
    % ~h1'_0t   
    %   The radial Zernike polynomials are the radial portion of the |ey6Czm  
    %   Zernike functions, which are an orthogonal basis on the unit vX{]_  
    %   circle.  The series representation of the radial Zernike <EE)d@%>v  
    %   polynomials is 4Fnr8 r8W  
    % 5r.{vQ  
    %          (n-m)/2 kweypIB  
    %            __ 9@!`,Co  
    %    m      \       s                                          n-2s ub-ZrC'  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r KQEnC`Nz  
    %    n      s=0 <)rol  
    % N~a?0x  
    %   The following table shows the first 12 polynomials. N[AX29  
    % 8&3G|m1-2  
    %       n    m    Zernike polynomial    Normalization n\d-^ml  
    %       --------------------------------------------- 2cww7z/B  
    %       0    0    1                        sqrt(2) TEY%OI zU+  
    %       1    1    r                           2 [Y5B$7|s<  
    %       2    0    2*r^2 - 1                sqrt(6) 9XS'5AXN  
    %       2    2    r^2                      sqrt(6) s:Memvf  
    %       3    1    3*r^3 - 2*r              sqrt(8) 2?HLEiI1  
    %       3    3    r^3                      sqrt(8) or0f%wAF  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) {| Tl3  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) R7vO,kZ6Q  
    %       4    4    r^4                      sqrt(10) B[8  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Jz3u r)|  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) A9[l5E  
    %       5    5    r^5                      sqrt(12) >R :Bkf-  
    %       --------------------------------------------- ]m YY1%H8M  
    % <zrGPwk  
    %   Example: wVp  
    % G !wFG-Y}  
    %       % Display three example Zernike radial polynomials 6VIi nuOW  
    %       r = 0:0.01:1; 40mgB4I  
    %       n = [3 2 5]; @'dtlY5;  
    %       m = [1 2 1]; 6tj +  
    %       z = zernpol(n,m,r); yw2sK7  
    %       figure IRD?.K]*  
    %       plot(r,z) bz,C%HFA  
    %       grid on `O*+%/(  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') U#-89.x  
    % $o5<#g"/T  
    %   See also ZERNFUN, ZERNFUN2. { "=d7i  
    Eufw1vDa  
    % A note on the algorithm. 1^$ vmULj  
    % ------------------------ E{|j  
    % The radial Zernike polynomials are computed using the series um ,Zt  
    % representation shown in the Help section above. For many special r65/O5F  
    % functions, direct evaluation using the series representation can cjp H hoW  
    % produce poor numerical results (floating point errors), because n YWS'i@  
    % the summation often involves computing small differences between 6f(K'v  
    % large successive terms in the series. (In such cases, the functions fn]f$n*`  
    % are often evaluated using alternative methods such as recurrence O6 bB CF;  
    % relations: see the Legendre functions, for example). For the Zernike "1yXOy^2  
    % polynomials, however, this problem does not arise, because the !3E33  
    % polynomials are evaluated over the finite domain r = (0,1), and xXQDHc -Ba  
    % because the coefficients for a given polynomial are generally all a}EO7tcg,  
    % of similar magnitude. }HRM6fR1S  
    % <S <@V?h  
    % ZERNPOL has been written using a vectorized implementation: multiple ]+Ik/+Nz  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] [HLXWu3  
    % values can be passed as inputs) for a vector of points R.  To achieve @WEDXB  
    % this vectorization most efficiently, the algorithm in ZERNPOL 5Ay\s:hb[u  
    % involves pre-determining all the powers p of R that are required to ET.c8K1f  
    % compute the outputs, and then compiling the {R^p} into a single OLg=kF[[  
    % matrix.  This avoids any redundant computation of the R^p, and #+>8gq^5  
    % minimizes the sizes of certain intermediate variables. AT+7!UGL  
    % / p}^ Tpu  
    %   Paul Fricker 11/13/2006 rI23e[  
    `2.[8%6  
    W^v3pH-y#  
    % Check and prepare the inputs: L/t'|<m  
    % ----------------------------- E>NRC\^@  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ^`?2g[AA  
        error('zernpol:NMvectors','N and M must be vectors.') w7+3?'L  
    end [Wf%iwB  
    ER-X1fD  
    if length(n)~=length(m) L"e8S%UqX  
        error('zernpol:NMlength','N and M must be the same length.') AXFQd@#  
    end 'So,*>]63  
    G |033(j  
    n = n(:); `\Z7It?aDs  
    m = m(:); V $Y=JK@  
    length_n = length(n); .ww~'5b0  
    #2{H!jr  
    if any(mod(n-m,2))  <m7m  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') tX)l_ ?jVH  
    end Okxuhzn>"  
    X"lPXoCN  
    if any(m<0) U|yXJ.Z3  
        error('zernpol:Mpositive','All M must be positive.') ~?E.U,R  
    end 9 M>.9~  
    _[IOPHa"  
    if any(m>n) s aY;[bz}  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') v{tw;Z#  
    end g4z*6L,u  
    N=%4V  
    if any( r>1 | r<0 ) ePLpGT  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') # O<,  
    end U4s)3jDw  
    |0^~S  
    if ~any(size(r)==1) lFJDdf2:$C  
        error('zernpol:Rvector','R must be a vector.') .! 'SG6 q  
    end EnW}>XN  
    :yFUlO:  
    r = r(:); |f67aN  
    length_r = length(r); VkW N1A  
    r8%"#<]/  
    if nargin==4 I) ]"`2w2w  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); :}0>IPW-V  
        if ~isnorm 5TynAiSD_>  
            error('zernpol:normalization','Unrecognized normalization flag.') :[\M|iAo  
        end Bl$Hg,in-  
    else .s-V:k5  
        isnorm = false; j;TXZ`|(  
    end "WF@T  
    fmgXh)=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?q{HS&k  
    % Compute the Zernike Polynomials  4>R)2g  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  2-$O$&s.  
    *TyLB&<t  
    % Determine the required powers of r: &d sXK~9M>  
    % ----------------------------------- A  r,fmq  
    rpowers = []; Ie"eqO!  
    for j = 1:length(n) (pv6V2i  
        rpowers = [rpowers m(j):2:n(j)]; BS*Y3$  
    end %^KNY ;E  
    rpowers = unique(rpowers); Ah :d2*SR4  
    4"^v]&I  
    % Pre-compute the values of r raised to the required powers, Yx[B*] 2  
    % and compile them in a matrix: 5do49H_  
    % ----------------------------- ZVIlVuZ}  
    if rpowers(1)==0 eHE?#r16Z  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); &RB{0Qhx  
        rpowern = cat(2,rpowern{:}); <rI8O;\H  
        rpowern = [ones(length_r,1) rpowern]; g>*P}r~;^b  
    else +?9. &<?  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); \DMZ M  
        rpowern = cat(2,rpowern{:}); .o(S60iH!(  
    end qw<~v?{|C  
    &zV; p  
    % Compute the values of the polynomials: ,z5B"o{Et  
    % -------------------------------------- wN]]t~K)Q  
    z = zeros(length_r,length_n); wNm1H[{  
    for j = 1:length_n b}HwvS:  
        s = 0:(n(j)-m(j))/2; b|Sjh;  
        pows = n(j):-2:m(j); y^:N^Gt  
        for k = length(s):-1:1 jJqq:.XqB8  
            p = (1-2*mod(s(k),2))* ... M$Or|HTG  
                       prod(2:(n(j)-s(k)))/          ... tRYi q  
                       prod(2:s(k))/                 ... hqc)Ydg_%  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... }cO}H2m  
                       prod(2:((n(j)+m(j))/2-s(k))); ]k)h<)nY  
            idx = (pows(k)==rpowers); A}W}H;8x  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 2fFGS.l  
        end  ovsI2  
         $s<bKju  
        if isnorm 7El:$H  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); q?$<{Z"  
        end ?#gYu %7DN  
    end tB#-}Gf  
    >Pwu>  
    % 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)   g1B[RSWv  
    @WfX{485  
    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。 w;4FN'  
    J-)9>~[E<  
    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)