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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 9$,x^Qx  
    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! O7@CAr  
     
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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 *]2LN$  
    function z = zernfun(n,m,r,theta,nflag) v-6" *EP  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. TJ(PTB;  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Hj ]$  
    %   and angular frequency M, evaluated at positions (R,THETA) on the l,uYp"F,ps  
    %   unit circle.  N is a vector of positive integers (including 0), and ~82[pY  
    %   M is a vector with the same number of elements as N.  Each element ]2G5ng' @  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) \~xI#S@  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 8Ml&lfn_8  
    %   and THETA is a vector of angles.  R and THETA must have the same y e!Bfz>  
    %   length.  The output Z is a matrix with one column for every (N,M) <4jQbY;  
    %   pair, and one row for every (R,THETA) pair. zb9^ii$g  
    % RAR0LKGX  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike fs4pAB#F  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), %VYQz)yW  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 5zJkPki  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, HE&,?vioy  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized T=cSTS!P;q  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ln.kEhQ3B  
    % GF~^-5  
    %   The Zernike functions are an orthogonal basis on the unit circle. xO'I*)  
    %   They are used in disciplines such as astronomy, optics, and (^GVy=  
    %   optometry to describe functions on a circular domain. lJ]r %YlF  
    % '|^LNAx  
    %   The following table lists the first 15 Zernike functions. N_<sCRd]9  
    % /^96|  
    %       n    m    Zernike function           Normalization -Hzn7L  
    %       -------------------------------------------------- FzmCS@yA  
    %       0    0    1                                 1 >(z{1'f{  
    %       1    1    r * cos(theta)                    2 8 o8FL~&]  
    %       1   -1    r * sin(theta)                    2 o;Ijv\Em  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) RAKQ+Y"nl  
    %       2    0    (2*r^2 - 1)                    sqrt(3) A/N*Nc  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) XuJwZN!(  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) %sC,;^wla'  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) sBuJK'  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) mOwgk7s[ J  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 1_:1cF{w  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ]i*q*]x2u  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) rh2pVDS  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) g$VcT\X  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)  .Pq8C  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) hM E|=\  
    %       -------------------------------------------------- ub6\m=Y7  
    % =f@O~nGm  
    %   Example 1: ?97MW a   
    % dgssX9g37  
    %       % Display the Zernike function Z(n=5,m=1) !mBsDn(J  
    %       x = -1:0.01:1; 0kgK~\^,.O  
    %       [X,Y] = meshgrid(x,x); LoHWkNZ5:  
    %       [theta,r] = cart2pol(X,Y); |Ix6D  
    %       idx = r<=1; Bir }X  
    %       z = nan(size(X)); Y^LFJB|b4  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); G_5sF|(mq  
    %       figure d_J?i]AP|'  
    %       pcolor(x,x,z), shading interp cNC\w%  
    %       axis square, colorbar [2w3c4K  
    %       title('Zernike function Z_5^1(r,\theta)') pALB[;9g  
    % L:YsAv  
    %   Example 2: QOuy(GY  
    % GQqw(2Ub}  
    %       % Display the first 10 Zernike functions 1E$Z]5C9  
    %       x = -1:0.01:1; S "oUE_>  
    %       [X,Y] = meshgrid(x,x); 2`5(XpYe  
    %       [theta,r] = cart2pol(X,Y); $Br^c< y  
    %       idx = r<=1; s cR-|GuZ  
    %       z = nan(size(X)); &o"Hb=k<  
    %       n = [0  1  1  2  2  2  3  3  3  3]; .u7d  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ?3SlvKI}H`  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; +azPpGZ=  
    %       y = zernfun(n,m,r(idx),theta(idx)); +^YV>;  
    %       figure('Units','normalized') UQ|0Aqwq  
    %       for k = 1:10 _zh}%#6L  
    %           z(idx) = y(:,k); =@pm-rI|-  
    %           subplot(4,7,Nplot(k)) e::5|6x  
    %           pcolor(x,x,z), shading interp Y@eHp-[  
    %           set(gca,'XTick',[],'YTick',[]) i?=3RdP/R1  
    %           axis square };oRx)  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) {J})f>x<xM  
    %       end O#O~A |  
    % w2]1ftY  
    %   See also ZERNPOL, ZERNFUN2. ^'EEry  
    uNd;; X  
    %   Paul Fricker 11/13/2006 0IDHoNaT<  
    8YkP57Y%[Z  
    gEKJrAA  
    % Check and prepare the inputs: WY 2b  
    % ----------------------------- 5B'-&.Aj+  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ) 2S0OY.  
        error('zernfun:NMvectors','N and M must be vectors.') Xz]}cRQ[  
    end DDAqgx  
    fS#/-wugOB  
    if length(n)~=length(m) ^Jnp\o>  
        error('zernfun:NMlength','N and M must be the same length.') .6O>P2m]a_  
    end m.K"IXD  
    Rp`}"x9  
    n = n(:); P]Gsc  
    m = m(:); 9k7|B>LT  
    if any(mod(n-m,2)) 7h&xfrSrD  
        error('zernfun:NMmultiplesof2', ... 0?p_|X'_  
              'All N and M must differ by multiples of 2 (including 0).') ,6t0w|@-k  
    end Fg#*rzA  
    }$qy_Esl  
    if any(m>n) u x:,io  
        error('zernfun:MlessthanN', ... gFDP:I/`  
              'Each M must be less than or equal to its corresponding N.') |lJXI:G G  
    end (3]7[h7  
    1&jX~'  
    if any( r>1 | r<0 ) R63"j\0  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') GQ8I |E  
    end K?I@'B'  
    t:=Ui/!q  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Nq*\{rb  
        error('zernfun:RTHvector','R and THETA must be vectors.') a*SJHBB  
    end *[.\ S3K`  
    $%1[<}<  
    r = r(:); 2Y wV}  
    theta = theta(:); SF.,sCk  
    length_r = length(r); {ReAl_Cm  
    if length_r~=length(theta) ORtl~V'  
        error('zernfun:RTHlength', ... TP^.]I O-  
              'The number of R- and THETA-values must be equal.') 8kMMQES  
    end +!_^MBkk  
    /o|@]SAe.  
    % Check normalization: 7FMHz.ZRE  
    % -------------------- 9MHb<~F  
    if nargin==5 && ischar(nflag) PFPfLxna  
        isnorm = strcmpi(nflag,'norm'); H[>_LYZ8  
        if ~isnorm x[(2}Qd  
            error('zernfun:normalization','Unrecognized normalization flag.') -q+Fj;El  
        end mD)Nh  
    else J=\Y4- "  
        isnorm = false; *f4KmiQ~ %  
    end :=i0$k<E/  
    8|d[45*q  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vxqMo9T  
    % Compute the Zernike Polynomials <M$hj6.tn  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q!AS}rV  
    -Q$$2QW!  
    % Determine the required powers of r: QGshc  
    % ----------------------------------- wV-cpJ,}  
    m_abs = abs(m); pg>P]a{  
    rpowers = []; CiMy_`H  
    for j = 1:length(n) iOJgZuP  
        rpowers = [rpowers m_abs(j):2:n(j)]; Tl=vgs1  
    end B]Zsn`n  
    rpowers = unique(rpowers); {X"X.`p  
    ax 3:rl  
    % Pre-compute the values of r raised to the required powers, '6xn!dK  
    % and compile them in a matrix: QPFpGS{d  
    % ----------------------------- 0 \h2&  
    if rpowers(1)==0 (O<lVz@8  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); }XXE hOO  
        rpowern = cat(2,rpowern{:}); 9s7B1Pf  
        rpowern = [ones(length_r,1) rpowern]; XkK16aLE  
    else J@Orrz2q#  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); [{zekF~)@  
        rpowern = cat(2,rpowern{:}); qlgh$9  
    end <v2R6cj5  
    ED$gnFa3I  
    % Compute the values of the polynomials: `nizGg~1  
    % -------------------------------------- SU#|&_wtr!  
    y = zeros(length_r,length(n)); BryMq !  
    for j = 1:length(n) }Ns_RS$  
        s = 0:(n(j)-m_abs(j))/2; ~(&xBtg:}  
        pows = n(j):-2:m_abs(j); f a\cLC  
        for k = length(s):-1:1 /NkZ;<uxJ  
            p = (1-2*mod(s(k),2))* ... ]3I_H+hU  
                       prod(2:(n(j)-s(k)))/              ... T4f:0r;^f*  
                       prod(2:s(k))/                     ... [2FXs52  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ~cZ1=,P  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); []Fy[G.)H  
            idx = (pows(k)==rpowers); snK9']WXo  
            y(:,j) = y(:,j) + p*rpowern(:,idx); I+<;D sp  
        end ##n\9ipD  
         Qy$QOtrv  
        if isnorm Z7f~|}  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); t)m4"p7  
        end ?_^9e  
    end J`V6zGgW  
    % END: Compute the Zernike Polynomials V2y[IeSQ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DMf9wB  
    ]':C~-RV{  
    % Compute the Zernike functions: jxoEOEA  
    % ------------------------------ ze ua`jQ  
    idx_pos = m>0; -Kc-eU-&q  
    idx_neg = m<0; x[?_F  
    eU12*(  
    z = y; /J6CSk  
    if any(idx_pos) EP8LJzd"  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 1rKR=To  
    end  I&v B\A  
    if any(idx_neg) m2}&5vD8-  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); *PI3L/*  
    end D H.ljGb  
    [Ytia#Vv  
    % 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) 6RF01z|~_  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. }E=kfMu  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated J\%:jg( m  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive z6!X+`&  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, OYzJE@r^  
    %   and THETA is a vector of angles.  R and THETA must have the same A1@-;/H3  
    %   length.  The output Z is a matrix with one column for every P-value, z=xHk|+'  
    %   and one row for every (R,THETA) pair. @Yg7F>s  
    % \x+DEy'4;5  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike z~BB|-kp1  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ]@f6O *&=  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) m<yA] ';s  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 s=$7lYX  
    %   for all p. p"JSYF 9]  
    % eS"gHldz  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 OBZ|W**N"  
    %   Zernike functions (order N<=7).  In some disciplines it is =UV?Pi*M>  
    %   traditional to label the first 36 functions using a single mode ,'9tR&S$_  
    %   number P instead of separate numbers for the order N and azimuthal VgdkCdWRm_  
    %   frequency M. .$yw;go3  
    % 06`__$@h  
    %   Example: Z:*U/_G  
    % {)[i\=,`{  
    %       % Display the first 16 Zernike functions -3V~YhG  
    %       x = -1:0.01:1; =.%ZF]Oe+#  
    %       [X,Y] = meshgrid(x,x); cC[n~OV  
    %       [theta,r] = cart2pol(X,Y); *HC8kD a%$  
    %       idx = r<=1; {7wvC)WW  
    %       p = 0:15; V ;6M[ic}  
    %       z = nan(size(X)); bDkE*4SRX  
    %       y = zernfun2(p,r(idx),theta(idx)); ZChY:I$<  
    %       figure('Units','normalized') R*y[/Aw  
    %       for k = 1:length(p) 1^;h:,e6  
    %           z(idx) = y(:,k); d{he  
    %           subplot(4,4,k) :}-u`K*  
    %           pcolor(x,x,z), shading interp 0 mQ3P.9  
    %           set(gca,'XTick',[],'YTick',[]) w?*KO?K  
    %           axis square yjO7/< 2  
    %           title(['Z_{' num2str(p(k)) '}']) sHuz10  
    %       end KFhn}C3 i  
    % D7. P  
    %   See also ZERNPOL, ZERNFUN. K8 Y/XEK  
    @Otc$hj  
    %   Paul Fricker 11/13/2006 +,[3a%c)H  
    q+z\Y?  
    ]~zJ7I  
    % Check and prepare the inputs: pd1m/:  
    % ----------------------------- )eEvyU  
    if min(size(p))~=1 {|gJC>f@  
        error('zernfun2:Pvector','Input P must be vector.') U{_s1  
    end >2:Sv1T  
    ^zJ. W  
    if any(p)>35 `|w#K28t"  
        error('zernfun2:P36', ... "{k )nr+7U  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 8_m9CQ6 i  
               '(P = 0 to 35).']) t/1NTa  
    end PPPwDsJ  
    wUeOD.;#F  
    % Get the order and frequency corresonding to the function number: 9/M!S[N9  
    % ---------------------------------------------------------------- >>cd3)b  
    p = p(:); DRoxw24  
    n = ceil((-3+sqrt(9+8*p))/2); <`3(i\-X  
    m = 2*p - n.*(n+2); qlJOb}$ I  
    [J`G`s!  
    % Pass the inputs to the function ZERNFUN: Zsogx}i-  
    % ---------------------------------------- B|=maz:_  
    switch nargin WS/+Yl  
        case 3 \Sby(l  
            z = zernfun(n,m,r,theta); 55LF  
        case 4 ss{=::#  
            z = zernfun(n,m,r,theta,nflag); D:E_h  
        otherwise Q $0%~`t  
            error('zernfun2:nargin','Incorrect number of inputs.') ]M(f^   
    end sri#L+I  
    -C}59G8  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) Ju7C?)x  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. h&M RQno  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Qz(T[H5%W  
    %   order N and frequency M, evaluated at R.  N is a vector of (OcNC/9  
    %   positive integers (including 0), and M is a vector with the !TL}~D:J  
    %   same number of elements as N.  Each element k of M must be a xO-U]%oq  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) )UZ0gfx  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is /)?P>!#;\  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix r&3o~!  
    %   with one column for every (N,M) pair, and one row for every Fg\| e%  
    %   element in R. ^s~n[  
    % E9B*K2l^{  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- `ab\i`g9  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ([CnYv  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ?U/Wio$@  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 O;e8ft '|  
    %   for all [n,m]. P"uHtHK  
    % Owpg]p yVD  
    %   The radial Zernike polynomials are the radial portion of the LL[#b2CKa  
    %   Zernike functions, which are an orthogonal basis on the unit .hlQ?\  
    %   circle.  The series representation of the radial Zernike n~ >h4=h  
    %   polynomials is #G  +  
    % Ipz 1+ #s'  
    %          (n-m)/2 \*%i#]wO@  
    %            __ W+f&%En  
    %    m      \       s                                          n-2s +#uNQ`1v  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r  LbX6p  
    %    n      s=0 a5}44/%  
    % '<&EPUO  
    %   The following table shows the first 12 polynomials. "-Q Rkif  
    % b;J0'o^G|  
    %       n    m    Zernike polynomial    Normalization @>Ghfh>~D  
    %       --------------------------------------------- myWmU0z/  
    %       0    0    1                        sqrt(2) QPe9s[Y  
    %       1    1    r                           2 Q_kT}6#(J=  
    %       2    0    2*r^2 - 1                sqrt(6) Vo 6y8@\  
    %       2    2    r^2                      sqrt(6) -RH4y 2  
    %       3    1    3*r^3 - 2*r              sqrt(8) Cj !i)-  
    %       3    3    r^3                      sqrt(8) =,d* {m~A  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) h*#2bS~nl-  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) !0OD(XT  
    %       4    4    r^4                      sqrt(10) ~1=.?Ho  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) :q>oD-b$}  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) `  -[Bo  
    %       5    5    r^5                      sqrt(12) S#h'\/S  
    %       --------------------------------------------- 5hJYy`h~  
    % B)"#/@!bHH  
    %   Example: RO%tuU,-  
    % up &NCX  
    %       % Display three example Zernike radial polynomials -4vHK!l  
    %       r = 0:0.01:1;  ^%5~ ;  
    %       n = [3 2 5]; 6MQs \J6.  
    %       m = [1 2 1]; ii_|)udz  
    %       z = zernpol(n,m,r); O2q=gYX>\  
    %       figure MvZ+n  
    %       plot(r,z) 4+5OR&kxZ  
    %       grid on N[,VSO&  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') UH 47e  
    % AB2mt:^  
    %   See also ZERNFUN, ZERNFUN2. Q7 uAf3  
    &e-#|p#v  
    % A note on the algorithm. `J>E9p<  
    % ------------------------ O&#S4]Y   
    % The radial Zernike polynomials are computed using the series ~m?74^ i  
    % representation shown in the Help section above. For many special jr, &=C(  
    % functions, direct evaluation using the series representation can {d 1N&  
    % produce poor numerical results (floating point errors), because 3?.1~"-J  
    % the summation often involves computing small differences between s; B j7]  
    % large successive terms in the series. (In such cases, the functions M<{5pH(K  
    % are often evaluated using alternative methods such as recurrence YY#s=  
    % relations: see the Legendre functions, for example). For the Zernike S2rEy2\}:  
    % polynomials, however, this problem does not arise, because the ?iPZsV  
    % polynomials are evaluated over the finite domain r = (0,1), and }uF[Ra  
    % because the coefficients for a given polynomial are generally all sf |oNOz  
    % of similar magnitude. ( zn_8s  
    % I&TTr7  
    % ZERNPOL has been written using a vectorized implementation: multiple Wl& >6./{  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] (s}Rj)V[^  
    % values can be passed as inputs) for a vector of points R.  To achieve 2^)D .&  
    % this vectorization most efficiently, the algorithm in ZERNPOL t] r,9df'  
    % involves pre-determining all the powers p of R that are required to cBz!U 8(  
    % compute the outputs, and then compiling the {R^p} into a single g08*}0-k  
    % matrix.  This avoids any redundant computation of the R^p, and pqyWv;  
    % minimizes the sizes of certain intermediate variables. z5XYpi_;[  
    % zwM"`z  
    %   Paul Fricker 11/13/2006 r{t. c?/  
    ,|T*|2Gm  
    xwTijSj  
    % Check and prepare the inputs: S}oG.r 9  
    % ----------------------------- = n+q_.A  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) "gXxRHTX  
        error('zernpol:NMvectors','N and M must be vectors.') rNxrQ  
    end + ,@ FxZl  
    &`9j)3^J.  
    if length(n)~=length(m) t8+?U^j  
        error('zernpol:NMlength','N and M must be the same length.') Nk96"P$P  
    end xS= _yO9-  
    O&`U5w  
    n = n(:); Uc_jQ4e_  
    m = m(:); [J a)<!]<  
    length_n = length(n); /xl4ohL$a  
    E]?)FH<oP  
    if any(mod(n-m,2)) 3jvx2  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ]i-P-9PA4  
    end VZ\O9lD  
    B$cOssl  
    if any(m<0) F(#haJ$>  
        error('zernpol:Mpositive','All M must be positive.') _Zh2eXWdjM  
    end iI3v[S  
    >vUB%OLyP  
    if any(m>n) h/,R{A2mO  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') &Fw[YGJayz  
    end CPVzX%=  
    sW }<zGYd  
    if any( r>1 | r<0 ) hHcJN  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') Z}|TW~J=  
    end 8]S,u:E:N  
    x>}B#  
    if ~any(size(r)==1) YTQ5sFuGM  
        error('zernpol:Rvector','R must be a vector.') ,Z^Ca15z  
    end O`cdQu  
    k/%#>  
    r = r(:); he"L*p*H  
    length_r = length(r); `YPe^!` $  
    GxxDY]!  
    if nargin==4 +wipfL~&S  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); m;dm|4L^  
        if ~isnorm Vzpt(_><  
            error('zernpol:normalization','Unrecognized normalization flag.') <"<Mbbp  
        end UcgG  
    else 5?Bc Y ;  
        isnorm = false; )D;*DUtMVm  
    end VM7 !0  
    )CJES!! W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yH^f\u0  
    % Compute the Zernike Polynomials Q8p=!K  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% cgyp5\*>+  
    5L F/5`  
    % Determine the required powers of r: hR-K@fS%l'  
    % ----------------------------------- @<2d8ed  
    rpowers = []; D}-o+6TI?  
    for j = 1:length(n) xq2V0Jp1u  
        rpowers = [rpowers m(j):2:n(j)]; W;4Lkk$  
    end 3QW_k5o  
    rpowers = unique(rpowers); ylu2R0] (  
    5y]io Jc9-  
    % Pre-compute the values of r raised to the required powers, [u`6^TycP  
    % and compile them in a matrix: Y(_KizBY  
    % ----------------------------- Wbe0ZnM]  
    if rpowers(1)==0 9RH"d[%yc}  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); $xT1 1 ^  
        rpowern = cat(2,rpowern{:}); L7]]ZAH!1  
        rpowern = [ones(length_r,1) rpowern]; $/+so;KD  
    else ,of]J|  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); @D~B{Hg  
        rpowern = cat(2,rpowern{:}); t0q_>T-kt  
    end [F+,YV%t  
    \@K~L4>  
    % Compute the values of the polynomials: Di>rO038  
    % -------------------------------------- fxd0e;NAAh  
    z = zeros(length_r,length_n); 6g"C#&{@  
    for j = 1:length_n ?R|th Z  
        s = 0:(n(j)-m(j))/2; PnA?+u2m  
        pows = n(j):-2:m(j); S/.^7R7{f  
        for k = length(s):-1:1 KVN"XqE4  
            p = (1-2*mod(s(k),2))* ... h./P\eDc  
                       prod(2:(n(j)-s(k)))/          ... eZH~je{1  
                       prod(2:s(k))/                 ... w~|1Wd<v  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 'Dath>Y=  
                       prod(2:((n(j)+m(j))/2-s(k))); z}+i=cAN  
            idx = (pows(k)==rpowers); L2fZ{bgy  
            z(:,j) = z(:,j) + p*rpowern(:,idx); %?9Ok  
        end *)'Vvu<  
         3-C\2  
        if isnorm {:bN/zV#  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); )#C mQXgG  
        end E)W@{?.o#  
    end T  p<s1'"  
    e4\dpvL  
    % 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)  x+8_4>,>Y7  
    Ysq'2  
    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。 sV"UI  
    -VxTx^)>  
    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)