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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 XX|wle1Kg  
    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! C8bv%9  
     
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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 d?_LNSDo  
    function z = zernfun(n,m,r,theta,nflag) LwL\CE_6+  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~PAbtY9}U  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N {pof=G  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 6ISDY>p  
    %   unit circle.  N is a vector of positive integers (including 0), and b/ dyH  
    %   M is a vector with the same number of elements as N.  Each element ^vH3 -A;*  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ,H+LE$=  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, (!9ybH;T  
    %   and THETA is a vector of angles.  R and THETA must have the same OlI{VszR  
    %   length.  The output Z is a matrix with one column for every (N,M) %B{NH~  
    %   pair, and one row for every (R,THETA) pair. |L"!^Y#=D  
    % h]z>H~.<*  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike J)xc mK  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), gQ=g,X4  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral '5n67Hl 1  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 6}E C)j;Fw  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 9BM 8  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. `!$I6KxT  
    % %: .{?FB_  
    %   The Zernike functions are an orthogonal basis on the unit circle. s*0PJ\E2  
    %   They are used in disciplines such as astronomy, optics, and Cw_XLMY%V1  
    %   optometry to describe functions on a circular domain. CN"hx-f  
    % z nc'  
    %   The following table lists the first 15 Zernike functions. w 9mi2=  
    % -n`igC  
    %       n    m    Zernike function           Normalization 1TvR-.e  
    %       -------------------------------------------------- SdTJ?P+m  
    %       0    0    1                                 1 /\_wDi+#  
    %       1    1    r * cos(theta)                    2 Cp@' k;(  
    %       1   -1    r * sin(theta)                    2 'l}T_7g  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) xX ktMlI  
    %       2    0    (2*r^2 - 1)                    sqrt(3) bqt*d)$  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) $"/xi `  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) "7k 82dw  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 4,|A\dXE  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) r6Hdp  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Pkbx /\  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 8,,$C7"EP  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 8C{mV^cn~  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) De(\ <H#  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) z$>_c "D  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) x{X(Y]*1S  
    %       -------------------------------------------------- <6s?M1J  
    % a3<.F&c+c  
    %   Example 1: 9p#Laei].  
    % wf<=r W'  
    %       % Display the Zernike function Z(n=5,m=1) AIvIQ$6}  
    %       x = -1:0.01:1; K;u<-?En  
    %       [X,Y] = meshgrid(x,x); {5=Iu\e  
    %       [theta,r] = cart2pol(X,Y); bJo)rM :m  
    %       idx = r<=1; \V#2K><  
    %       z = nan(size(X)); Qw{LD+r(  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); .#,!&Lt  
    %       figure |-HV@c]  
    %       pcolor(x,x,z), shading interp oT4A|M  
    %       axis square, colorbar [`~E)B1Y  
    %       title('Zernike function Z_5^1(r,\theta)') !c+Nf2I7S  
    % p. eq N  
    %   Example 2: H?~|Uj 6  
    % v: Av 2y  
    %       % Display the first 10 Zernike functions #-_';Er\  
    %       x = -1:0.01:1; )5}=^aqd  
    %       [X,Y] = meshgrid(x,x); Gyak?.@R  
    %       [theta,r] = cart2pol(X,Y); cu4&*{  
    %       idx = r<=1; ] {r*Z6bs  
    %       z = nan(size(X)); }hralef #N  
    %       n = [0  1  1  2  2  2  3  3  3  3]; *Op;].>E  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; iINd*eXb^  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; (6R^/*-o  
    %       y = zernfun(n,m,r(idx),theta(idx)); RnN]m!"5  
    %       figure('Units','normalized') 3iHUG^sLW  
    %       for k = 1:10 y\DR,$Py  
    %           z(idx) = y(:,k); +0016UgS#  
    %           subplot(4,7,Nplot(k)) bqHR~4 #IR  
    %           pcolor(x,x,z), shading interp BULf@8~(  
    %           set(gca,'XTick',[],'YTick',[]) (5s$vcK  
    %           axis square 0^41dfdE  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) +rw?k/  
    %       end qn VxP&  
    % %T hY6y(  
    %   See also ZERNPOL, ZERNFUN2. >~-8RM  
    *{qW7x.6h  
    %   Paul Fricker 11/13/2006 o5 UM)g  
    hjVct r  
    jP?YV  
    % Check and prepare the inputs: Wj"\nT4  
    % ----------------------------- ^t&S?_DSZ  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) CbmT aEaP  
        error('zernfun:NMvectors','N and M must be vectors.') ~C1lbn b  
    end *C81DQ  
    Y40`~  
    if length(n)~=length(m) =.=4P~T&  
        error('zernfun:NMlength','N and M must be the same length.') "@1e0`n Q  
    end 39p&M"Yo  
    #-xsAKi  
    n = n(:); DQ '=$z  
    m = m(:); t$NK{Mw5_  
    if any(mod(n-m,2)) &b[ .bf  
        error('zernfun:NMmultiplesof2', ... &vf9Gp+MK  
              'All N and M must differ by multiples of 2 (including 0).') DJxe3<  
    end g.wp }fz  
    Y}<w)b1e|  
    if any(m>n) `nAR/Ye  
        error('zernfun:MlessthanN', ... .+|HJ(  
              'Each M must be less than or equal to its corresponding N.') _l`d+ \#  
    end >K }j}M%  
    ^I=W<  
    if any( r>1 | r<0 ) D=hy[sDBw  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') d5zv8?|X+  
    end G:$Ta6=  
    Tm!pAD  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Sz_bjhyT}  
        error('zernfun:RTHvector','R and THETA must be vectors.') ({XB,Rm  
    end [D !-~]5  
    [$PW {d8|  
    r = r(:); ~#z8Q{!O  
    theta = theta(:); 7jss3^.wA  
    length_r = length(r); en6Kdqe  
    if length_r~=length(theta) eI?|Ps{S  
        error('zernfun:RTHlength', ... {+`'ZU6C  
              'The number of R- and THETA-values must be equal.') ;DQ{6(  
    end #&fi[|%X$  
    -~ w5 yd  
    % Check normalization: eIZ7uSl  
    % -------------------- cK( )_RB#  
    if nargin==5 && ischar(nflag) |;~kHc$W  
        isnorm = strcmpi(nflag,'norm'); v5 |XyN"  
        if ~isnorm tM&O<6Y  
            error('zernfun:normalization','Unrecognized normalization flag.') /WvF}y  
        end 'o D31\@I  
    else K90wX1&  
        isnorm = false; L="ipM:Z  
    end 0:NCIsIm<  
    :Ma=P\J W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vpt*?eR  
    % Compute the Zernike Polynomials OvL@@SX |  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $KSdNFtM)A  
    R,+Pcn$ws  
    % Determine the required powers of r: uu5AW=j  
    % ----------------------------------- 5Q)hl.<{o7  
    m_abs = abs(m); (R'GrN>  
    rpowers = []; 1 u[a713O  
    for j = 1:length(n) JQi+y;  
        rpowers = [rpowers m_abs(j):2:n(j)]; ??\1eo2gB  
    end ;Jh=7wx  
    rpowers = unique(rpowers); *$%ch=  
    xIOYwVC  
    % Pre-compute the values of r raised to the required powers, `S`,H  
    % and compile them in a matrix: Ijg //=  
    % ----------------------------- , %8keGhl  
    if rpowers(1)==0 E#?Bn5-uBs  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); =1kE2u  
        rpowern = cat(2,rpowern{:}); N>zpx U {  
        rpowern = [ones(length_r,1) rpowern]; 2p^Jqp`$  
    else @2yoy&IO  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )JNUfauyT  
        rpowern = cat(2,rpowern{:}); H0!LiazA>  
    end ":qhO0  
    xE$>;30b_  
    % Compute the values of the polynomials: DGc5Lol~  
    % -------------------------------------- MNuBZnO  
    y = zeros(length_r,length(n)); V(lxkEu/Fj  
    for j = 1:length(n) 0mt lM(  
        s = 0:(n(j)-m_abs(j))/2; n]%T>\gw  
        pows = n(j):-2:m_abs(j); x=S8UKUx  
        for k = length(s):-1:1 +'-i(]@!'  
            p = (1-2*mod(s(k),2))* ... TnuaP'xZ  
                       prod(2:(n(j)-s(k)))/              ...  1{fu  
                       prod(2:s(k))/                     ... g-C)y 06  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Oax6_kmOj  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); QIK;kjr*A3  
            idx = (pows(k)==rpowers); #F|q->2`o  
            y(:,j) = y(:,j) + p*rpowern(:,idx); iBqxz:PHN(  
        end bjL8Wpk  
         eNHSfq  
        if isnorm &c AFKYt  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); bZ5cKQ\6  
        end T{CCZ"Fv  
    end KUV(vAY,  
    % END: Compute the Zernike Polynomials M~?2g.o'D  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b41f7t=  
    )yl;i  
    % Compute the Zernike functions: =q\Ghqj1  
    % ------------------------------ 9}*Pb6  
    idx_pos = m>0; \kR:GZ`{UV  
    idx_neg = m<0; >s%&t[r6  
    L*(!P4S%}  
    z = y; za,JCI  
    if any(idx_pos) I)(@'^)  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); JK%UaEut=  
    end *3!#W|#=]N  
    if any(idx_neg) }J^+66{  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); -f-@[;D  
    end 6)]zt  
    O0Pb"ou_h.  
    % 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) P+Q}bTb8  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. '}, 8x?  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated !+)5?o  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive @Rw]boC  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, /J5)_> R:  
    %   and THETA is a vector of angles.  R and THETA must have the same .At^b4#(  
    %   length.  The output Z is a matrix with one column for every P-value, -Q MO*PY  
    %   and one row for every (R,THETA) pair. DedY(JOvB  
    % ^Z>Nbzr{  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike <-(n48  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) CQns:.`$`  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ukDaX  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 PLueH/gC.  
    %   for all p. MC~<jJ,  
    % :>*0./hG  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 O!k C  
    %   Zernike functions (order N<=7).  In some disciplines it is 3Hi[Y[O`%P  
    %   traditional to label the first 36 functions using a single mode le150;7  
    %   number P instead of separate numbers for the order N and azimuthal jmJeu@(  
    %   frequency M. ] L6LB \  
    % *%n(t+'q  
    %   Example: V'8Rz#Gc5  
    % 5_+pgJL  
    %       % Display the first 16 Zernike functions s(8e)0Tl  
    %       x = -1:0.01:1; VT2f\d[Q  
    %       [X,Y] = meshgrid(x,x); )ZMR4U$+v  
    %       [theta,r] = cart2pol(X,Y); .H}#,pQ}l  
    %       idx = r<=1; .YlhK=d4  
    %       p = 0:15; giH WC%/  
    %       z = nan(size(X)); *q*$%H  
    %       y = zernfun2(p,r(idx),theta(idx)); \qkb8H  
    %       figure('Units','normalized') V|vXxWm/  
    %       for k = 1:length(p) ]-{A"tJ  
    %           z(idx) = y(:,k); D}OhmOu 3  
    %           subplot(4,4,k) >9Z7l63+}  
    %           pcolor(x,x,z), shading interp 2v`Q;%7O  
    %           set(gca,'XTick',[],'YTick',[]) K)#6&\0tT  
    %           axis square BV)) #D9  
    %           title(['Z_{' num2str(p(k)) '}']) xs^wRE_  
    %       end :NynNu'  
    % 7_~_$I~g*  
    %   See also ZERNPOL, ZERNFUN. z#GrwE,r   
    sf Zb$T J  
    %   Paul Fricker 11/13/2006 34I;DUdcE  
    N gagzsJ=  
    rCd*'Qg  
    % Check and prepare the inputs: F13vc~$Ky  
    % ----------------------------- [e7nW9\l  
    if min(size(p))~=1 Lt_A&  
        error('zernfun2:Pvector','Input P must be vector.') fbW<c`LH  
    end $ qTv2)W1{  
    w ,-4A o2x  
    if any(p)>35 NL-V",gI-~  
        error('zernfun2:P36', ... A-l[f\  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ,!QtViA7  
               '(P = 0 to 35).']) /pL'G`  
    end FVWHiwRU,  
    3oM&#a  
    % Get the order and frequency corresonding to the function number: SedVp cb+  
    % ---------------------------------------------------------------- V)c.AX5  
    p = p(:); Qov*xRO6  
    n = ceil((-3+sqrt(9+8*p))/2); %+oV-o\ #A  
    m = 2*p - n.*(n+2); XB<Q A>dLh  
    ;~Gez;AhK  
    % Pass the inputs to the function ZERNFUN: $msf~M*  
    % ---------------------------------------- scPvuHzl  
    switch nargin vlo!D9zsV3  
        case 3 BFQ`Ab+  
            z = zernfun(n,m,r,theta); XblZlWP#  
        case 4 %&!B2z}  
            z = zernfun(n,m,r,theta,nflag); Vo%DoZg  
        otherwise .>NPgd I  
            error('zernfun2:nargin','Incorrect number of inputs.') km29]V=}  
    end 0Om<+]).R  
    z{nd4qOsD  
    % 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) O )INM  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. $*C'{&2  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of IJBIO>Z/  
    %   order N and frequency M, evaluated at R.  N is a vector of ?I7%ueFY  
    %   positive integers (including 0), and M is a vector with the .50ql[En  
    %   same number of elements as N.  Each element k of M must be a pDt45   
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) vP^V3  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is AS a)xf9  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix @O8X )  
    %   with one column for every (N,M) pair, and one row for every AQ)J|i  
    %   element in R. }^azj>p5  
    % ddEV@2F  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- }Io5&ww:U  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is [E0.4FLT!  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Dyh|F\T  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 $ spk.j  
    %   for all [n,m]. Hx NoV.q  
    % 0A F}wz>  
    %   The radial Zernike polynomials are the radial portion of the *#j_nNM4  
    %   Zernike functions, which are an orthogonal basis on the unit ddw^oU  
    %   circle.  The series representation of the radial Zernike g5t`YcL  
    %   polynomials is ;ibOd~  
    % ?L6pB]l8b  
    %          (n-m)/2 HJ;!'@  
    %            __ [#;CBs5o  
    %    m      \       s                                          n-2s S&NWZ:E3[  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r `It3X.^}  
    %    n      s=0 VJgYXPE `  
    % 40%<E  
    %   The following table shows the first 12 polynomials. `{xKU8j^  
    % n W:Bo#  
    %       n    m    Zernike polynomial    Normalization H4uHCkj  
    %       --------------------------------------------- y0,>_MS  
    %       0    0    1                        sqrt(2) !_>o2  
    %       1    1    r                           2  e,T^8_>  
    %       2    0    2*r^2 - 1                sqrt(6) xo#K_"E  
    %       2    2    r^2                      sqrt(6) mj&$+zM>  
    %       3    1    3*r^3 - 2*r              sqrt(8) c1 Hp  
    %       3    3    r^3                      sqrt(8) Rln% Y  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) WntolYd  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) V:2{LR<R8  
    %       4    4    r^4                      sqrt(10) K$5mDScoJ  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) i)7B :uA  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ]r>m{"~E  
    %       5    5    r^5                      sqrt(12) fzzk#jU  
    %       --------------------------------------------- b ; U  
    % YFeL#)5y  
    %   Example: 9)D9'/{L#  
    % NB3ar&.$S  
    %       % Display three example Zernike radial polynomials !&R|P|7qN}  
    %       r = 0:0.01:1; Dmr3r[  
    %       n = [3 2 5]; YA~`R~9d  
    %       m = [1 2 1]; VCa`|S?2  
    %       z = zernpol(n,m,r); Z*YS7 ~  
    %       figure 8BX9JoDi  
    %       plot(r,z) N V`=T?1[5  
    %       grid on N[;R8S P  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 4f'!,Q ;  
    % KU;J2Kt  
    %   See also ZERNFUN, ZERNFUN2. b 4A1M  
    [vOk=  
    % A note on the algorithm. =" pNE#  
    % ------------------------ R6\|:mI,$  
    % The radial Zernike polynomials are computed using the series A5RM&y  
    % representation shown in the Help section above. For many special 6yd?xeD  
    % functions, direct evaluation using the series representation can p:3 V-$4X  
    % produce poor numerical results (floating point errors), because synueg  
    % the summation often involves computing small differences between x| r#  
    % large successive terms in the series. (In such cases, the functions dUkZ_<5''  
    % are often evaluated using alternative methods such as recurrence ~=?^v[T1  
    % relations: see the Legendre functions, for example). For the Zernike 0](V@F"~  
    % polynomials, however, this problem does not arise, because the r:H.VAD  
    % polynomials are evaluated over the finite domain r = (0,1), and NGmXF_kqN  
    % because the coefficients for a given polynomial are generally all d)L,kzN  
    % of similar magnitude. hI,+J>  
    % 7E;`1lh7  
    % ZERNPOL has been written using a vectorized implementation: multiple |l:,EA_v|  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] VlS`m,:{  
    % values can be passed as inputs) for a vector of points R.  To achieve 'SKq<X%R;  
    % this vectorization most efficiently, the algorithm in ZERNPOL yZ,S$tSR  
    % involves pre-determining all the powers p of R that are required to E=t^I/f)E  
    % compute the outputs, and then compiling the {R^p} into a single L | #"Yn  
    % matrix.  This avoids any redundant computation of the R^p, and Tfw5i,{  
    % minimizes the sizes of certain intermediate variables. 76b2 3|  
    % w exa\o  
    %   Paul Fricker 11/13/2006 U3t) yr h  
    Pa"[&{:  
    K[i&!Z&  
    % Check and prepare the inputs: BQ(sjJ$v6F  
    % ----------------------------- ';I(#J6  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Vs(D(d,  
        error('zernpol:NMvectors','N and M must be vectors.') /l;_ xs  
    end R_b)2FU1y  
    :a_MT  
    if length(n)~=length(m) vWjHHw  
        error('zernpol:NMlength','N and M must be the same length.') @^nE^;  
    end ;R^=($X  
    /~P4<1  
    n = n(:); E+~1GKd  
    m = m(:); fnK H<  
    length_n = length(n); j){0>O.V  
    9eEA80i7  
    if any(mod(n-m,2)) +5H1n(6)  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Z._%T$8aJv  
    end T 2Gscey  
    a#m T@l\  
    if any(m<0) ,$"T/yYer  
        error('zernpol:Mpositive','All M must be positive.') Y.E]U!i*  
    end *ch7z|wo.  
    nk2H^RM^  
    if any(m>n) DlQ*'PX7  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') VEBvS>i*  
    end rDC=rG  
    Gg6<4T1  
    if any( r>1 | r<0 ) ltOsl-OpR  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') LNM#\fb  
    end 3<}r+,j  
    a ~F\ 2`Q  
    if ~any(size(r)==1) {r:5\  
        error('zernpol:Rvector','R must be a vector.') F,@uYMQs  
    end ?F9c6$|  
    N`+@_.iBX  
    r = r(:); 7$"n.cr :  
    length_r = length(r); #fq&yjl#A  
    Sb?HRoe_  
    if nargin==4 z W*Z  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ef:YYt{|q  
        if ~isnorm L+.-aB2!d  
            error('zernpol:normalization','Unrecognized normalization flag.') !~te&ccPE  
        end {r_x\VC=p  
    else ||'A9  
        isnorm = false; 53l!$#o  
    end j "e]Ui  
    2 xt$w%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }nMp.7b  
    % Compute the Zernike Polynomials DVw 04ay%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7a4h7/  
    D()tP  
    % Determine the required powers of r: g.COKA  
    % ----------------------------------- Ev,b5KelD  
    rpowers = []; tWA<OOl  
    for j = 1:length(n) no7Q%O9  
        rpowers = [rpowers m(j):2:n(j)]; C@rIyBj1g  
    end \)2~o N  
    rpowers = unique(rpowers); sYd)r%%AU  
    @c;:D`\p1C  
    % Pre-compute the values of r raised to the required powers, B=|m._OL]n  
    % and compile them in a matrix: oe{,-<yck  
    % ----------------------------- H+ 7Fw'u  
    if rpowers(1)==0 h8:5[;e  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); :-e[$6}S  
        rpowern = cat(2,rpowern{:}); 73kI%nNB  
        rpowern = [ones(length_r,1) rpowern]; x k&# fW^r  
    else 8GT4U5c ;  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); A(ZtA[G  
        rpowern = cat(2,rpowern{:}); M6z$*? <  
    end SAokW,  
    7loIjT7  
    % Compute the values of the polynomials: [*d<LAnuWP  
    % -------------------------------------- m&k l_f7  
    z = zeros(length_r,length_n); }PxP J$o  
    for j = 1:length_n KdLj1T  
        s = 0:(n(j)-m(j))/2; \04 (V'`U  
        pows = n(j):-2:m(j); ^2"3h$DJfS  
        for k = length(s):-1:1 ,W5!=\Gg(  
            p = (1-2*mod(s(k),2))* ... 'b Kc;\  
                       prod(2:(n(j)-s(k)))/          ... ,`ju(ac!  
                       prod(2:s(k))/                 ... i`7:^v;  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Aw=GvCo<  
                       prod(2:((n(j)+m(j))/2-s(k))); 6U%F mE@  
            idx = (pows(k)==rpowers); lh*!f$2 ~  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Sv[$.^mb  
        end ]TSzT"_r~~  
         |/~ISB  
        if isnorm xs$.EY:k  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); QR h %S{  
        end $B?IE#7S4  
    end m " c6^)U  
    @_Es|(4  
    % 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)  lr&O@ 5"oy  
    =@ "'aCU/  
    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。 h,)UB1  
    1[H1l;  
    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)