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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 a dfR!&J  
    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! (46 {r}_O  
     
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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 V3N0Og3  
    function z = zernfun(n,m,r,theta,nflag) 4'pS*v  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. W`rNBfG>  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N yq[Cq=rBk  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 0'Z\O   
    %   unit circle.  N is a vector of positive integers (including 0), and H[Q_hY[>V  
    %   M is a vector with the same number of elements as N.  Each element EOKzzX7 S  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 5`[n8mU  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 5~ 'Ie<Y_  
    %   and THETA is a vector of angles.  R and THETA must have the same U]~^ZR  
    %   length.  The output Z is a matrix with one column for every (N,M) i8X`HbmN  
    %   pair, and one row for every (R,THETA) pair. ~ AQp|  
    % &NZfJs  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ;$j7H&UNQj  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), B6P|Z%E;D6  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral hqSJ(gs{  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, |aToUi.Q%  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Y$8JM  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. uYG^Pc^v  
    % f7de'^t9  
    %   The Zernike functions are an orthogonal basis on the unit circle. j5$BK[p.  
    %   They are used in disciplines such as astronomy, optics, and +V862R4,o  
    %   optometry to describe functions on a circular domain. ?dZt[vAMn  
    % T5Eseesp  
    %   The following table lists the first 15 Zernike functions. &:B<Q$g#  
    % 1n*W2:,z  
    %       n    m    Zernike function           Normalization pY8q=Kl  
    %       -------------------------------------------------- 6 &U+6gb  
    %       0    0    1                                 1 Mn:/1eY  
    %       1    1    r * cos(theta)                    2 -C7]qbT }  
    %       1   -1    r * sin(theta)                    2 "O>n@Q|  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) H&}ipaDO  
    %       2    0    (2*r^2 - 1)                    sqrt(3) p4u5mM  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) C_:k8?  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) $3+PbYY  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 7B9`<{!h  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 4b]a&_-}  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) { >{B`e`$  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) "$HbK @]!h  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) BZK`O/  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Q-TV*FD.  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) h( QYxI,|  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) }1 vT)  
    %       -------------------------------------------------- ewsKH\#  
    % nx":"LFI  
    %   Example 1: vm23U^VJ  
    % -]G(ms;}/Y  
    %       % Display the Zernike function Z(n=5,m=1) Z^KA  
    %       x = -1:0.01:1; a)-FG P^  
    %       [X,Y] = meshgrid(x,x); -5G)?J/*  
    %       [theta,r] = cart2pol(X,Y); w]j+9-._  
    %       idx = r<=1; ?`?T7w|3 y  
    %       z = nan(size(X)); {y kYW%3s  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); s0UFym 8  
    %       figure eKZ%2|+j!7  
    %       pcolor(x,x,z), shading interp .] 4W!])9  
    %       axis square, colorbar jZfx Jm  
    %       title('Zernike function Z_5^1(r,\theta)') iGXI6`F"  
    % m@Ev~~;  
    %   Example 2: 7J$b$P0}  
    % Y o0FUj  
    %       % Display the first 10 Zernike functions jLg@FDb~  
    %       x = -1:0.01:1; "7%:sty  
    %       [X,Y] = meshgrid(x,x); -PB[-CX  
    %       [theta,r] = cart2pol(X,Y); w&&2H8  
    %       idx = r<=1; 8Q`WB0E<|  
    %       z = nan(size(X)); sE(HZR1  
    %       n = [0  1  1  2  2  2  3  3  3  3]; %6j)=IOts  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; JEn3`B!*  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 6Q|k7*,B  
    %       y = zernfun(n,m,r(idx),theta(idx)); 3ucP(Ex@tg  
    %       figure('Units','normalized') #PLEPB  
    %       for k = 1:10 H!e 3~+)  
    %           z(idx) = y(:,k); R_P}~l  
    %           subplot(4,7,Nplot(k)) Tz&Y]#h_  
    %           pcolor(x,x,z), shading interp ^o?SM^  
    %           set(gca,'XTick',[],'YTick',[]) H( -Y  
    %           axis square <M?:  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) +WJ(QZEhD  
    %       end AS!6XT  
    % k4J8O3E  
    %   See also ZERNPOL, ZERNFUN2. USJ- e  
    pfuW  
    %   Paul Fricker 11/13/2006 gv15t'y9  
    -php6$|  
     84zTCX  
    % Check and prepare the inputs: td2/9|Q  
    % ----------------------------- <c[U#KrvJ  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~0"p*?^  
        error('zernfun:NMvectors','N and M must be vectors.') 8Chj w wB  
    end >>dm }X  
    #PvB/3  
    if length(n)~=length(m) Huw\&E  
        error('zernfun:NMlength','N and M must be the same length.') $V>98M>j  
    end 59uwB('|lH  
    [a[/_Sf{  
    n = n(:); ge3sU5iZ  
    m = m(:); .@ C{3$,VG  
    if any(mod(n-m,2)) Fh7'[>onw  
        error('zernfun:NMmultiplesof2', ... }0 hL~i  
              'All N and M must differ by multiples of 2 (including 0).') I&9S;I$  
    end Wx'Kp+9'  
    @*N )i?>  
    if any(m>n) @\_x'!R  
        error('zernfun:MlessthanN', ... _:n b&B  
              'Each M must be less than or equal to its corresponding N.')  1iT\df  
    end !33#. @[  
    hlZ@Dq%f  
    if any( r>1 | r<0 ) {Ee>n^1  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') {@}?k s5  
    end TZir>5  
    $5`!Z%>/  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) V+-$ jOh  
        error('zernfun:RTHvector','R and THETA must be vectors.') j Ib  
    end ~\nBjM2  
    v}G]X Z8  
    r = r(:); u.pxz8  
    theta = theta(:); 8 S`9dSc  
    length_r = length(r); 9ILIEm:  
    if length_r~=length(theta) 5pNY)>]t=  
        error('zernfun:RTHlength', ... @(``:)Z<b  
              'The number of R- and THETA-values must be equal.') ~H)4)r^  
    end M_ 0zC1  
    'J*<iA*W  
    % Check normalization: SQsSa1  
    % -------------------- WzW-pV]  
    if nargin==5 && ischar(nflag) O/%< }3Sq  
        isnorm = strcmpi(nflag,'norm'); ~cAZB9Fa  
        if ~isnorm !2CL1j0(  
            error('zernfun:normalization','Unrecognized normalization flag.') "o!{51!'  
        end :Br5a34q  
    else gsar[gZ  
        isnorm = false; %ugHhS!  
    end 5 v^yQ<70  
    7x]4`#u  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?71+ f{s  
    % Compute the Zernike Polynomials &WXY'A=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mAgF73,3  
    O40+M)e]  
    % Determine the required powers of r: wmNHT _  
    % ----------------------------------- 4Ph0:^i_  
    m_abs = abs(m); c;f!!3&  
    rpowers = []; pi(-A  
    for j = 1:length(n) 87!C@XlK_  
        rpowers = [rpowers m_abs(j):2:n(j)]; js^ ,(CS  
    end A % Q!^d  
    rpowers = unique(rpowers);  9DQ)cy  
    -!RtH |P  
    % Pre-compute the values of r raised to the required powers, J;t 7&Zpe  
    % and compile them in a matrix: ivO/;)=t  
    % ----------------------------- djQv[Vc {  
    if rpowers(1)==0 =*BIB5  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); pnyWcrBf  
        rpowern = cat(2,rpowern{:}); dBsX*}C  
        rpowern = [ones(length_r,1) rpowern]; JG`Q;K  
    else lA!"z~03*  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); RT/o$$  
        rpowern = cat(2,rpowern{:}); f8 /'%$N  
    end  I7+9~5p  
    snM Z0W  
    % Compute the values of the polynomials: )O+}T5c=  
    % -------------------------------------- t9gfU5?  
    y = zeros(length_r,length(n)); qIUfPA=/_  
    for j = 1:length(n) Z#d&|5Xj  
        s = 0:(n(j)-m_abs(j))/2; x}/,yaWZ  
        pows = n(j):-2:m_abs(j); tbo>%kn  
        for k = length(s):-1:1 \b V6@#,  
            p = (1-2*mod(s(k),2))* ... Bm$"WbOq*R  
                       prod(2:(n(j)-s(k)))/              ... KAA-G2%M  
                       prod(2:s(k))/                     ... 8VG!TpX/B  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... @tohNO>  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); <`X"}I3 ba  
            idx = (pows(k)==rpowers); B 3m_D"?  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Kemw^48ts  
        end W+wA_s2&D  
         ',3HlOJ:  
        if isnorm B0$:b !  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); l5%G'1w#,j  
        end VLsxdwHgb  
    end K`&oC8p  
    % END: Compute the Zernike Polynomials [u@Jc,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G2 ]H6G$M  
    A61^[Y,dX_  
    % Compute the Zernike functions:  UsGa  
    % ------------------------------ @}_WE,r  
    idx_pos = m>0; T#%/s?_>.  
    idx_neg = m<0; mOpTzg@  
    7qOa ;^T  
    z = y; rt3qdk5U  
    if any(idx_pos) .LVQx  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 3P~o"a>  
    end o56`  
    if any(idx_neg) n8=5-7UT  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); T lAR.cV  
    end Xdi:1wW@p  
    0`.^MC?  
    % 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) .)1_Ew  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. A,su;Q h  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated }nt* [:%  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive d@w~[b  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Vc^HVyAx@n  
    %   and THETA is a vector of angles.  R and THETA must have the same Yw _+`,W   
    %   length.  The output Z is a matrix with one column for every P-value, ]v[|B  
    %   and one row for every (R,THETA) pair. $'W}aER  
    % =_j vk.  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 5tQ1fJze  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) f>'Y(dJ'W  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) "~UUx"Y  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 gVeEdo`$<  
    %   for all p. Iff9'TE  
    % y(R? ,wa=]  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Va Z!.#(P  
    %   Zernike functions (order N<=7).  In some disciplines it is f}guv~K  
    %   traditional to label the first 36 functions using a single mode HM>lg`S  
    %   number P instead of separate numbers for the order N and azimuthal |6G m:jV  
    %   frequency M. Uax+dl   
    % |AZg*T3:W  
    %   Example: Cg*H.f%Mr  
    % 3+ >G#W~  
    %       % Display the first 16 Zernike functions sNun+xsf^  
    %       x = -1:0.01:1; +C7 ~b~ %  
    %       [X,Y] = meshgrid(x,x); A^Kbsc  
    %       [theta,r] = cart2pol(X,Y); kyR*D1N&)  
    %       idx = r<=1; ^Rm  
    %       p = 0:15; v}WR+)uFQ  
    %       z = nan(size(X)); B|o2K}%f  
    %       y = zernfun2(p,r(idx),theta(idx)); iy""(c  
    %       figure('Units','normalized') w[P4&?2:  
    %       for k = 1:length(p) 5SCKP<rb  
    %           z(idx) = y(:,k); $d?.2Kg  
    %           subplot(4,4,k) RN=` -*E1  
    %           pcolor(x,x,z), shading interp vb9OonE2  
    %           set(gca,'XTick',[],'YTick',[]) P%K4[c W~  
    %           axis square Spt ? >sm  
    %           title(['Z_{' num2str(p(k)) '}']) );-~j  
    %       end .`?@%{  
    % ETs>`#`6o  
    %   See also ZERNPOL, ZERNFUN. p~LrPWHSTP  
    boG_f@dv(  
    %   Paul Fricker 11/13/2006 F R|&^j6  
    6(P M'@i  
    `6+"Z=:  
    % Check and prepare the inputs: Ty88}V  
    % ----------------------------- A!^q J#  
    if min(size(p))~=1 .k,YlFvj  
        error('zernfun2:Pvector','Input P must be vector.') yDNOtC|  
    end yCCrK@{oo  
    FVh U^  
    if any(p)>35 2wF8 P)  
        error('zernfun2:P36', ... uw lr9nB  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ...  }-~l!  
               '(P = 0 to 35).']) dH( ('u[  
    end <FZ@Q[RP  
    LdJYE;k Ju  
    % Get the order and frequency corresonding to the function number: ws4cF N9P?  
    % ---------------------------------------------------------------- arf8xqR-U]  
    p = p(:); qWw\_S  
    n = ceil((-3+sqrt(9+8*p))/2); |JCU<_<  
    m = 2*p - n.*(n+2); \TU3rk&X  
    vm>b m  
    % Pass the inputs to the function ZERNFUN: O$4yAaD X  
    % ---------------------------------------- Mw9 \EhA  
    switch nargin 1'? 4m0W1  
        case 3 iYC9eEF  
            z = zernfun(n,m,r,theta); .bio7c6  
        case 4 Hc`A3SMR  
            z = zernfun(n,m,r,theta,nflag); ,0LU~AGe   
        otherwise B#9{-t3Vf  
            error('zernfun2:nargin','Incorrect number of inputs.') =hl}.p  
    end 4[N^>qt =  
    }f2r!7:x  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) 9?k_y ZV  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. @e{^`\l=<  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of i&n'N8D@  
    %   order N and frequency M, evaluated at R.  N is a vector of mx=BD'  
    %   positive integers (including 0), and M is a vector with the U,LTVYrO  
    %   same number of elements as N.  Each element k of M must be a PdvqDa8  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 7 x<i :x3  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is P~ &$l2  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix M8u<qj&<O  
    %   with one column for every (N,M) pair, and one row for every Tyck/ EO  
    %   element in R. GAAm0;  
    % Nv"EV;$  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ,UuH}E  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is r hfb ftw  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 3'X.}>o   
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 %'T #pz  
    %   for all [n,m]. K^R,Iu/M  
    % D|.ic!w'  
    %   The radial Zernike polynomials are the radial portion of the & GreN  
    %   Zernike functions, which are an orthogonal basis on the unit Pfs;0}h5  
    %   circle.  The series representation of the radial Zernike wiBVuj#  
    %   polynomials is \7*`}&  
    % T' ~!9Q  
    %          (n-m)/2 n..g~ $k  
    %            __ Y3=_ec3w  
    %    m      \       s                                          n-2s LlSZr)X  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r OD_W8!-  
    %    n      s=0 }C|dyyr  
    % uyY|v$FM  
    %   The following table shows the first 12 polynomials. fYW6b[lI  
    % -ne Kuj  
    %       n    m    Zernike polynomial    Normalization vEn4L0D  
    %       --------------------------------------------- &>Vfa  
    %       0    0    1                        sqrt(2) p l&Muv  
    %       1    1    r                           2 tzh1s i  
    %       2    0    2*r^2 - 1                sqrt(6) >i6yl5s  
    %       2    2    r^2                      sqrt(6) /Js7`r=Rx  
    %       3    1    3*r^3 - 2*r              sqrt(8) 9rA=pH%<>B  
    %       3    3    r^3                      sqrt(8) gatB QwJb9  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 4f;HQ-Iv  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) i _%Q`i  
    %       4    4    r^4                      sqrt(10) ',S'.U  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) G2FXrkU  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) YSe.t_K2C  
    %       5    5    r^5                      sqrt(12) ;"m ,:5%  
    %       --------------------------------------------- &sd}ulEg`  
    % ~T89_L  
    %   Example: P$-X)c$&  
    % n9xAPB }  
    %       % Display three example Zernike radial polynomials ,zTy?OQ  
    %       r = 0:0.01:1; Zg.&V  
    %       n = [3 2 5]; E^ SH\5B  
    %       m = [1 2 1]; Pp5^@A  
    %       z = zernpol(n,m,r); @W9x$  
    %       figure xagBORg+Bd  
    %       plot(r,z) a 7,C>%I  
    %       grid on z.I9wQ]X[  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') )d>!"JB-  
    % ,11H.E Z  
    %   See also ZERNFUN, ZERNFUN2. :]1 TGfS  
    ,xxR\}  
    % A note on the algorithm. :EA\)@^$R  
    % ------------------------ _zwUE  
    % The radial Zernike polynomials are computed using the series (/j); oSK  
    % representation shown in the Help section above. For many special +o51x'Ld*  
    % functions, direct evaluation using the series representation can Ht4;5?/y  
    % produce poor numerical results (floating point errors), because |x-S&-  
    % the summation often involves computing small differences between qhz]Wm P   
    % large successive terms in the series. (In such cases, the functions G"XVn~]  
    % are often evaluated using alternative methods such as recurrence  ({=gw9f  
    % relations: see the Legendre functions, for example). For the Zernike bAm(8nT7w  
    % polynomials, however, this problem does not arise, because the 9,r rQQD_  
    % polynomials are evaluated over the finite domain r = (0,1), and +@/"%9w  
    % because the coefficients for a given polynomial are generally all [ra_ 2R  
    % of similar magnitude. w:<W.7y?0  
    % k'5?M  
    % ZERNPOL has been written using a vectorized implementation: multiple v3jg~"!  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] \<)9?M :  
    % values can be passed as inputs) for a vector of points R.  To achieve PuZf/um  
    % this vectorization most efficiently, the algorithm in ZERNPOL ut I"\1hQ  
    % involves pre-determining all the powers p of R that are required to y7i*s^ys{  
    % compute the outputs, and then compiling the {R^p} into a single Os1>kwC  
    % matrix.  This avoids any redundant computation of the R^p, and ]h5Yg/sms  
    % minimizes the sizes of certain intermediate variables. vt9)pMs  
    % C-H@8p?T  
    %   Paul Fricker 11/13/2006  W0]gLw9*  
    ZXuv CI  
    mA=i)Ga  
    % Check and prepare the inputs: /x1![$oC0  
    % ----------------------------- OUNd@o  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2mGaD\?K  
        error('zernpol:NMvectors','N and M must be vectors.') &Ob!4+v/GP  
    end y`XU~B)J1  
    -xA2pYz"  
    if length(n)~=length(m) t?'!$6   
        error('zernpol:NMlength','N and M must be the same length.') % V8U (z  
    end SF<c0bR9  
    Xn%pNxUL  
    n = n(:); J:zU,IIJ  
    m = m(:); `I,,C,{C  
    length_n = length(n); j+,d^!  
    YS*9t Q{  
    if any(mod(n-m,2)) c:o]d)S  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Us0EG\Y  
    end -K3^BZ HI  
    "\o+v|;  
    if any(m<0) /q`xCS  
        error('zernpol:Mpositive','All M must be positive.') W)f/0QX}W  
    end wlqpn(XR  
    Dw<bn<e-  
    if any(m>n) I ,z3xU  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') NP/Gn6fr  
    end O#S;q5L@  
    h_+dT  
    if any( r>1 | r<0 ) \(^nSy&N  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') h1B16)  
    end <?riU\-]y  
    U$WxHYo  
    if ~any(size(r)==1) |6K+E6H  
        error('zernpol:Rvector','R must be a vector.') J:V6  
    end n O\"HLM  
    HoRg^Ai?\  
    r = r(:); q#,f 4P  
    length_r = length(r); $+Z)  
    3@M|m<_R$  
    if nargin==4 }}\vV}s  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); x:7b/ j-  
        if ~isnorm j/_ s"}m{  
            error('zernpol:normalization','Unrecognized normalization flag.') 2ZG5<"DQ"  
        end  ]%wVHC  
    else I+[>I=ewa  
        isnorm = false; @,cowar*  
    end `lO[x.[  
    +wN^c#~7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }%'?p<^M  
    % Compute the Zernike Polynomials X.f>'0i  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =/y]d<g  
    clL2k8VS  
    % Determine the required powers of r: \eT5flC  
    % ----------------------------------- Sjyoc<Uo  
    rpowers = []; t\{'F7  
    for j = 1:length(n) :U3kW8;UMP  
        rpowers = [rpowers m(j):2:n(j)]; HkUWehVm  
    end M<.d8?p )  
    rpowers = unique(rpowers); >*#1ZB_l  
    j.w@(<=x  
    % Pre-compute the values of r raised to the required powers, TDFkxB>  
    % and compile them in a matrix: v9w'!C)b  
    % ----------------------------- )Ch2E|C?=8  
    if rpowers(1)==0 ^S UPi  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 0=* 8  
        rpowern = cat(2,rpowern{:}); TZ(cu>  
        rpowern = [ones(length_r,1) rpowern]; Xm%D><CC8"  
    else ypKUkH/  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);  86 W9rR  
        rpowern = cat(2,rpowern{:}); (@<lRA ^  
    end #A1Z'y0  
    u frW\X  
    % Compute the values of the polynomials: wRcAX%n&  
    % -------------------------------------- Hfw q/Is  
    z = zeros(length_r,length_n); z3&]%Q&  
    for j = 1:length_n lef2X1w}!  
        s = 0:(n(j)-m(j))/2; L;VoJf  
        pows = n(j):-2:m(j); DoJ\ q+  
        for k = length(s):-1:1 +BVym~*^  
            p = (1-2*mod(s(k),2))* ... Xu< k3oD7  
                       prod(2:(n(j)-s(k)))/          ... 10<x.8fSP  
                       prod(2:s(k))/                 ... <49K>S9O  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Pz>s6 [ob  
                       prod(2:((n(j)+m(j))/2-s(k))); YQ+tDZY8`  
            idx = (pows(k)==rpowers); u<3HQ.:;  
            z(:,j) = z(:,j) + p*rpowern(:,idx); P3cRl']  
        end M9V q -U18  
         A )^`?m3  
        if isnorm o.w\l\  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); !/}O>v~o  
        end S4^vpY DeN  
    end 5mJJU  
    E~%jX }/  
    % 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)  XqW@rU  
    ~%SH3$  
    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。 c6X}2a'  
    763v  
    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)