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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 h1QrFPQnu  
    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! J]$]zD  
     
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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 5vqh09-FB  
    function z = zernfun(n,m,r,theta,nflag) ">"B  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. !6DH6<HC  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N b0lZb'  
    %   and angular frequency M, evaluated at positions (R,THETA) on the jij-pDQnv  
    %   unit circle.  N is a vector of positive integers (including 0), and Vh5Z'4N  
    %   M is a vector with the same number of elements as N.  Each element 2sNV09id  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) "*0h=x$  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, uUI@!)@2  
    %   and THETA is a vector of angles.  R and THETA must have the same x"n)y1y  
    %   length.  The output Z is a matrix with one column for every (N,M) /&g~*AL  
    %   pair, and one row for every (R,THETA) pair. 0N4+6k|  
    % @}iY(-V  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike jp P'{mc  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), b;Uqyc  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral qr_:zXsob_  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, EiWsVic[  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ''~#tK f  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ca!DZ%y  
    % n>:|K0u"  
    %   The Zernike functions are an orthogonal basis on the unit circle. a) 5;Od  
    %   They are used in disciplines such as astronomy, optics, and QPT%CW61M  
    %   optometry to describe functions on a circular domain. 8:)itYE  
    % 0X[uXf  
    %   The following table lists the first 15 Zernike functions. xO2CgqEb  
    % x^P~+(g  
    %       n    m    Zernike function           Normalization <c$K3  
    %       -------------------------------------------------- \?rBtD(  
    %       0    0    1                                 1 ^Y- S"Ks  
    %       1    1    r * cos(theta)                    2 ?PST.+l  
    %       1   -1    r * sin(theta)                    2 l!YjDm{E  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) S67>yqha  
    %       2    0    (2*r^2 - 1)                    sqrt(3) v'H\KR-;  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ^=V b'g3P~  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) $@ Fvl-lK  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) z]O,Vqpl?  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) No G`J$D  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) |>L|7>J{<d  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) G tSvb6UNn  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) hj|P*yKV  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Ec;{N  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +1Ua`3dWN_  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) -cW 'g  
    %       -------------------------------------------------- Vv3{jn6%  
    % XDcA&cM}p  
    %   Example 1:  (:ObxJ*  
    % ?ta(`+"  
    %       % Display the Zernike function Z(n=5,m=1) wEJ) h1=)^  
    %       x = -1:0.01:1; BmGY#D,  
    %       [X,Y] = meshgrid(x,x); 8O0E;6b  
    %       [theta,r] = cart2pol(X,Y); .S =^)  
    %       idx = r<=1; #Kd^t =k  
    %       z = nan(size(X)); ^jxV  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Zr U9oy&!C  
    %       figure p{BBqKv  
    %       pcolor(x,x,z), shading interp %qj8*1  
    %       axis square, colorbar g8^YDrH  
    %       title('Zernike function Z_5^1(r,\theta)') ^~Dmb2h  
    % }HC6m{vH(  
    %   Example 2: Gcz@z1a=n  
    % }E%#g#  
    %       % Display the first 10 Zernike functions bQFMg41*w7  
    %       x = -1:0.01:1; 3Sb'){.MT+  
    %       [X,Y] = meshgrid(x,x); FJl_2  
    %       [theta,r] = cart2pol(X,Y); }g\1JSJ%H  
    %       idx = r<=1; X[{tD#  
    %       z = nan(size(X)); /:],bNb  
    %       n = [0  1  1  2  2  2  3  3  3  3]; G^Q8B^Lg  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; X|iWnz+^  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ~p0M|  
    %       y = zernfun(n,m,r(idx),theta(idx)); !uwZ%Ux z  
    %       figure('Units','normalized') ;5(ptXX1W  
    %       for k = 1:10 '**dD2 n  
    %           z(idx) = y(:,k); >|S&@<  
    %           subplot(4,7,Nplot(k)) cB ,l=/?  
    %           pcolor(x,x,z), shading interp [)E.T,fjMQ  
    %           set(gca,'XTick',[],'YTick',[]) 9< $n'g  
    %           axis square B<p -.tv  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 1ae,s{|  
    %       end y$7vJl.uS/  
    % 5!pof\/a  
    %   See also ZERNPOL, ZERNFUN2. <*4BT}r,^2  
    ;I^+u0ga  
    %   Paul Fricker 11/13/2006 5RysN=czA  
    dvl'Sq<  
    9h$08l  
    % Check and prepare the inputs: yK3b^  
    % ----------------------------- /P>t3E2c  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) " A}S92  
        error('zernfun:NMvectors','N and M must be vectors.') 'q_^28rK  
    end qij<XNZU"&  
    th?w&;L  
    if length(n)~=length(m) 5UgxuuP4  
        error('zernfun:NMlength','N and M must be the same length.') ev}ugRxt|k  
    end xR`W9Z5  
    bkvm-$/  
    n = n(:); +"i|)yUYy}  
    m = m(:); N#Y|MfLc  
    if any(mod(n-m,2)) WX9ABh&5  
        error('zernfun:NMmultiplesof2', ... sBLf(Q,  
              'All N and M must differ by multiples of 2 (including 0).') >Yf)]e-  
    end Z@G[\"  
    d}Y\; '2,  
    if any(m>n) _,?<r&>v6  
        error('zernfun:MlessthanN', ... Q2L>P<87T  
              'Each M must be less than or equal to its corresponding N.') H`:2J8   
    end ,@#))2<RK  
    Duu)8ru  
    if any( r>1 | r<0 ) Q^H8gsv  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ~g|Z6-?4Jj  
    end 5S EyAhB  
    pmIOV~K  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) R|&Rq(ow"  
        error('zernfun:RTHvector','R and THETA must be vectors.') rEF0A&5  
    end fy6<KEea  
    @|jLw($Ly  
    r = r(:); .EF(<JC?  
    theta = theta(:); =G9 9U/  
    length_r = length(r); T.}wcQf&*  
    if length_r~=length(theta) /qd5{%:  
        error('zernfun:RTHlength', ... bl8EzO  
              'The number of R- and THETA-values must be equal.') !*tV[0 i2  
    end ,DZoE~  
    8nj^x?bn  
    % Check normalization: U $2"ZyFii  
    % -------------------- s.#%hPX{  
    if nargin==5 && ischar(nflag) XB.xIApmy  
        isnorm = strcmpi(nflag,'norm'); 1LK`    
        if ~isnorm 0'3f^Ajf  
            error('zernfun:normalization','Unrecognized normalization flag.') Ki,SFww8r  
        end cR *5iqA  
    else vR)f'+_Nz  
        isnorm = false; 3b d(.he2u  
    end RnaxRnXVR  
    F+m%PVW:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j TyR+#Wn  
    % Compute the Zernike Polynomials ev'` K=n8  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /TndB7l"3  
    ]VkM)< +  
    % Determine the required powers of r: 6${=N}3Kw  
    % ----------------------------------- 'e(]woe  
    m_abs = abs(m); X"k:+  
    rpowers = []; Sf>#Zqj/  
    for j = 1:length(n) cs]h+yE  
        rpowers = [rpowers m_abs(j):2:n(j)]; hb.^ &  
    end #B!HPlrv  
    rpowers = unique(rpowers); (2J\o  
    =.48^$LWx  
    % Pre-compute the values of r raised to the required powers, x_+-TC4IXn  
    % and compile them in a matrix: vH?rln  
    % ----------------------------- $SOFq+-T  
    if rpowers(1)==0 F<+!28&h  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ]J(BaX4  
        rpowern = cat(2,rpowern{:}); E^`-:L(_  
        rpowern = [ones(length_r,1) rpowern]; 4F`&W*x  
    else $A;%p6PO)  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); */6lyODf  
        rpowern = cat(2,rpowern{:});  CK"OHjR  
    end gJZH??b  
    dHsI<:T#  
    % Compute the values of the polynomials: [2P6XoI#  
    % -------------------------------------- Mp7X+o/  
    y = zeros(length_r,length(n)); r6Qsh CA"  
    for j = 1:length(n) gWu<5Y=C  
        s = 0:(n(j)-m_abs(j))/2; KPrH1 [VU  
        pows = n(j):-2:m_abs(j); WbWEgd%8.  
        for k = length(s):-1:1 WqJrDj~  
            p = (1-2*mod(s(k),2))* ... Z_h-5VU-  
                       prod(2:(n(j)-s(k)))/              ... (UB?UJc  
                       prod(2:s(k))/                     ... jf^BEz5  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... p>l:^ -N;f  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Q3I^(Ll"L  
            idx = (pows(k)==rpowers); t }YT+S  
            y(:,j) = y(:,j) + p*rpowern(:,idx); N`HiNb [  
        end /os,s[w  
         /U 3Uuk:  
        if isnorm ,(A $WT@e  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); y}U}AUt  
        end |JLXgwML  
    end q|g>;_  
    % END: Compute the Zernike Polynomials x^7 9s_h5  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {[)n<.n[g  
    ^~;"$=Wf  
    % Compute the Zernike functions: ;O7Vl5R  
    % ------------------------------ eBWgAf.k  
    idx_pos = m>0; ]Zz.n5c  
    idx_neg = m<0; ,rS?^"h9  
    :2.<JUDM  
    z = y; E 1`TQA  
    if any(idx_pos) b+CJRB1  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); v&qL r+_7  
    end  :Y Ki  
    if any(idx_neg) S J2l6  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ?2%;VKN4  
    end  tE#;$Ss  
    kMx)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) jU-aa+  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. q B IekQT  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated /i~x.i3  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive |\W53,n9  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, +e, c'.  
    %   and THETA is a vector of angles.  R and THETA must have the same & 'CUc/,  
    %   length.  The output Z is a matrix with one column for every P-value, N% ?R(  
    %   and one row for every (R,THETA) pair. >=4('  
    % I^nDO\m <  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike :(\JY?+w   
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) @QMy!y_K~m  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) R nwFxFIQ  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 }<YU4EW  
    %   for all p. +0?1"2  
    % ez5J+  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ?qb35  
    %   Zernike functions (order N<=7).  In some disciplines it is fTso[r:F.  
    %   traditional to label the first 36 functions using a single mode SpImd IpD  
    %   number P instead of separate numbers for the order N and azimuthal >(-A"jf  
    %   frequency M. `{|w*)mD  
    % 0'HQ=pP  
    %   Example: !,dp/5 V  
    % mITNx^p4f  
    %       % Display the first 16 Zernike functions ,kf.'N  
    %       x = -1:0.01:1; zE<Iv\Q  
    %       [X,Y] = meshgrid(x,x); 14"+ctq  
    %       [theta,r] = cart2pol(X,Y); $}Ab R:z  
    %       idx = r<=1; 1BEs> Sm  
    %       p = 0:15; v2I? 5?j  
    %       z = nan(size(X)); xKl1DIN[  
    %       y = zernfun2(p,r(idx),theta(idx)); $}.+}'7$  
    %       figure('Units','normalized') KIcIYCBz  
    %       for k = 1:length(p) BN `2UVH  
    %           z(idx) = y(:,k); ;*$e8y2  
    %           subplot(4,4,k) KIi:5Y  
    %           pcolor(x,x,z), shading interp L$ i:~6  
    %           set(gca,'XTick',[],'YTick',[]) c6lCF &  
    %           axis square WQ}wQ:]  
    %           title(['Z_{' num2str(p(k)) '}']) $4^SWT.  
    %       end @gfW*PNjlP  
    % d!UxFY@  
    %   See also ZERNPOL, ZERNFUN. qDG2rFu&[  
    lm 96:S  
    %   Paul Fricker 11/13/2006 c -PZG|<C[  
    P6O\\,B1A  
    ,l6,k<   
    % Check and prepare the inputs: 14" 57Jt8  
    % ----------------------------- P%)r4+at  
    if min(size(p))~=1 t,Ka] /I  
        error('zernfun2:Pvector','Input P must be vector.') <gFa@at  
    end |1 "&[ .  
    @ ?M\[qeF@  
    if any(p)>35 2QRO$NieV  
        error('zernfun2:P36', ... |?#JCG  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... J_ h.7V  
               '(P = 0 to 35).']) M2-`p  
    end 4qz+cB_  
    ROmmak(y8  
    % Get the order and frequency corresonding to the function number: :09NZ !!  
    % ---------------------------------------------------------------- l'/`2Y1  
    p = p(:); vUVFW'-  
    n = ceil((-3+sqrt(9+8*p))/2); _FYA? d}  
    m = 2*p - n.*(n+2); `!/[9Y#Hp  
    ~1%*w*  
    % Pass the inputs to the function ZERNFUN: ]c~yMA+]FZ  
    % ---------------------------------------- L FkDb}  
    switch nargin K^U ="  
        case 3 D>[Sib/@  
            z = zernfun(n,m,r,theta); K<wFr-z  
        case 4 fj;ZGbg-O  
            z = zernfun(n,m,r,theta,nflag); **L&I5Hhm  
        otherwise |67Jw2  
            error('zernfun2:nargin','Incorrect number of inputs.') NX,m6u  
    end Q{|%kU"  
    W06#|8,{v  
    % 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) 3e&+[j  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. X|T|iB,vT  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 5[Vr {^)  
    %   order N and frequency M, evaluated at R.  N is a vector of dGYR  'x  
    %   positive integers (including 0), and M is a vector with the M5ZH6X@5  
    %   same number of elements as N.  Each element k of M must be a 5[jcw`  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) T0tX%_6`  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 7.hBc;%2u  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 2L~Vr4eHG  
    %   with one column for every (N,M) pair, and one row for every 7]vmtlL  
    %   element in R. x 4+WZYv3  
    % -/pz3n  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- fA6IW(_bi  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is s#s">hMrI  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to j~k,d.17M  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 *~)6 sm  
    %   for all [n,m]. b!]O]dk#  
    % (<eLj Q  
    %   The radial Zernike polynomials are the radial portion of the yMz#e0k  
    %   Zernike functions, which are an orthogonal basis on the unit N.JR($N$  
    %   circle.  The series representation of the radial Zernike {Nl?  
    %   polynomials is k sv]  
    % Iw`tb N L[  
    %          (n-m)/2 6kH6"  
    %            __ 9fEe={ B+  
    %    m      \       s                                          n-2s ;#85 _/  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r V1]GOmXz  
    %    n      s=0 [f_^B U&  
    % FMVAXOO  
    %   The following table shows the first 12 polynomials. >[;W ~*  
    % Ic2?1<IZA  
    %       n    m    Zernike polynomial    Normalization ,a{85HLr]  
    %       --------------------------------------------- ,p d -hu  
    %       0    0    1                        sqrt(2) &s+l/;3  
    %       1    1    r                           2 ']1n?K=A  
    %       2    0    2*r^2 - 1                sqrt(6) BYq80Vk%@  
    %       2    2    r^2                      sqrt(6) UH!(`Z\C  
    %       3    1    3*r^3 - 2*r              sqrt(8) TmG);B}  
    %       3    3    r^3                      sqrt(8) y|6n:<o  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ddfGR/1X  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) &>zH.6%$  
    %       4    4    r^4                      sqrt(10) NfvPE]S  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ,W}:vdC  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) \jiE :Qt  
    %       5    5    r^5                      sqrt(12) Y"mFUW4  
    %       --------------------------------------------- efXnF*Z  
    % L]kBY2c  
    %   Example: k`:zQd^T  
    % m.K cTM%j  
    %       % Display three example Zernike radial polynomials |(.\J`_e  
    %       r = 0:0.01:1; wb$uq/|  
    %       n = [3 2 5]; CeYhn\m5K0  
    %       m = [1 2 1]; |UB$^)Twb  
    %       z = zernpol(n,m,r); +K1M&(  
    %       figure :0r,.)  
    %       plot(r,z) ,.<mj !YE  
    %       grid on ~!$"J}d}<  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') CPOH qK`k  
    % 3+6Ed;P  
    %   See also ZERNFUN, ZERNFUN2. (Mk7"FC7  
    9hA`I tS  
    % A note on the algorithm. 1"H;Tr|  
    % ------------------------ 0nb%+],pX  
    % The radial Zernike polynomials are computed using the series nQiZ6[L  
    % representation shown in the Help section above. For many special Gm|-[iUTG]  
    % functions, direct evaluation using the series representation can B->AY.&j  
    % produce poor numerical results (floating point errors), because _9h$8(wjn  
    % the summation often involves computing small differences between 8FuxN2  
    % large successive terms in the series. (In such cases, the functions d6J/)nl  
    % are often evaluated using alternative methods such as recurrence aD: #AmbJ  
    % relations: see the Legendre functions, for example). For the Zernike Ixa0;nxj  
    % polynomials, however, this problem does not arise, because the ;QBS0x\f@  
    % polynomials are evaluated over the finite domain r = (0,1), and DG;7+2U  
    % because the coefficients for a given polynomial are generally all 8%9 C<+.R  
    % of similar magnitude. dVbFMQ&  
    % &&&-P\3  
    % ZERNPOL has been written using a vectorized implementation: multiple +x2JC' -H  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] UY(T>4H+h  
    % values can be passed as inputs) for a vector of points R.  To achieve \qG ?'Iy  
    % this vectorization most efficiently, the algorithm in ZERNPOL <A,V/']  
    % involves pre-determining all the powers p of R that are required to pkn^K+<n,  
    % compute the outputs, and then compiling the {R^p} into a single Cy;UyZ  
    % matrix.  This avoids any redundant computation of the R^p, and c]^P$F8U  
    % minimizes the sizes of certain intermediate variables. 9+@h2"|N4*  
    % T-] {gc  
    %   Paul Fricker 11/13/2006 WE) *~5  
    +hN>Q $E  
    cw<I L  
    % Check and prepare the inputs: 27SHj9I  
    % ----------------------------- Ev7.!  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ;u<Ah?w=Z  
        error('zernpol:NMvectors','N and M must be vectors.') ^QS`H@+Z  
    end 2z-Nw <bA  
    :{%[6lE^G  
    if length(n)~=length(m) ]E/0iM5  
        error('zernpol:NMlength','N and M must be the same length.') %,@pV%2  
    end  >. K  
    yT7$6x  
    n = n(:); Fl`U{03  
    m = m(:); kvwnqaX  
    length_n = length(n); p>tkRA?lk  
    Dj\nsc@e3  
    if any(mod(n-m,2)) m4/qxm"Dx:  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') pAy4%|(  
    end `p()ko  
    u9"=t  
    if any(m<0) \bARp z?a  
        error('zernpol:Mpositive','All M must be positive.') A6]:BuP;c  
    end &ksuk9M  
    >PA*L(Dh%  
    if any(m>n) ,U\ s89  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') a}y b~:TC  
    end `>sOOA  
    =dGp&9K,fw  
    if any( r>1 | r<0 ) K%J?'-  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') *)E${\1'<  
    end 5Y *4a%"  
    .y s_'F-]0  
    if ~any(size(r)==1) 1$0Kvvg[  
        error('zernpol:Rvector','R must be a vector.') Rt#QW*h\|i  
    end  LSC[S:  
    " aG6u^%  
    r = r(:); }B-$}  
    length_r = length(r); "-&K!Vfs  
    u}%OC43  
    if nargin==4 w % Hj'  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); V}s/knd  
        if ~isnorm "u6pl);G  
            error('zernpol:normalization','Unrecognized normalization flag.') H,% bKl#  
        end a&%aads  
    else ia5%  
        isnorm = false; E+/Nicn=  
    end "KIY+7@S}  
    bLg!LZ|S0s  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p7|I>8ur.  
    % Compute the Zernike Polynomials > Z+*tq  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 64u(X^i  
    o ]*yI[\  
    % Determine the required powers of r: 63 oe0T&  
    % ----------------------------------- 5=?P 6I_$G  
    rpowers = []; B=cA$620  
    for j = 1:length(n) TQ(q [:>  
        rpowers = [rpowers m(j):2:n(j)]; "B3jq^  
    end ;U'\"N9  
    rpowers = unique(rpowers); liS'  
    5sJ>+Rg  
    % Pre-compute the values of r raised to the required powers, d_`MS@2  
    % and compile them in a matrix: )Eozo4~  
    % ----------------------------- q9vND[BQ  
    if rpowers(1)==0 q1VKoKb6\:  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ZD&F ,2v  
        rpowern = cat(2,rpowern{:}); "p O  
        rpowern = [ones(length_r,1) rpowern]; *F( qg%1+  
    else p(RF   
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); g4^-B  
        rpowern = cat(2,rpowern{:}); ^&<M""Z  
    end g@'XmT="_  
    pb97S^K[  
    % Compute the values of the polynomials: &| (K#|^@  
    % -------------------------------------- Nw. )O  
    z = zeros(length_r,length_n); :<aGZ\R5  
    for j = 1:length_n uj3`M9  
        s = 0:(n(j)-m(j))/2; *P' X[z  
        pows = n(j):-2:m(j); _#K|g#p5  
        for k = length(s):-1:1 X`g<"Ka  
            p = (1-2*mod(s(k),2))* ... EGD{nE  
                       prod(2:(n(j)-s(k)))/          ... `/WxEu3  
                       prod(2:s(k))/                 ... yP]>eLTSd  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ->I.D?p  
                       prod(2:((n(j)+m(j))/2-s(k))); Cj>HMB}  
            idx = (pows(k)==rpowers); b]u=I za  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ~3s ?.[}d  
        end q_[y|ETJ]  
         x_7$g<n  
        if isnorm ;}Jv4Z  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); %X_A#9  
        end 7u"Q1n(h/  
    end GSRf/::I}4  
    O]XRalkEM  
    % 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)  aEUEy:.  
    (<`> B  
    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。 SP  =8v0  
    }uD*\.  
    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)