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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 rH & ^SNc  
    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! kELV]iWb  
     
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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 y/Paq^Hd  
    function z = zernfun(n,m,r,theta,nflag) -n+ =[M  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. SfEgmp-m  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 48W$ ,  
    %   and angular frequency M, evaluated at positions (R,THETA) on the X\V1c$13CK  
    %   unit circle.  N is a vector of positive integers (including 0), and )jm}h7,  
    %   M is a vector with the same number of elements as N.  Each element bw&8"k>D?  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) [,yoFm%"  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Gdb6 U{  
    %   and THETA is a vector of angles.  R and THETA must have the same lN -vFna  
    %   length.  The output Z is a matrix with one column for every (N,M) j/ow8Jmc*  
    %   pair, and one row for every (R,THETA) pair. y)CnH4{  
    % nj]l'~Y0  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .T#h5[S2x  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ko2?q  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral zU}Ru&T9  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, |@!4BA  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Lzm9Kh;  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Mj2`p#5wKh  
    % N7=lSBm  
    %   The Zernike functions are an orthogonal basis on the unit circle. tHgu#k0  
    %   They are used in disciplines such as astronomy, optics, and  _xjw:  
    %   optometry to describe functions on a circular domain. F-D9nI4{X  
    % : M=0o<  
    %   The following table lists the first 15 Zernike functions. wxS.!9K  
    % 0go{gUI  
    %       n    m    Zernike function           Normalization :2ILN.&  
    %       -------------------------------------------------- Utd`T+AF*  
    %       0    0    1                                 1 ~ HN  
    %       1    1    r * cos(theta)                    2 $F2 A  
    %       1   -1    r * sin(theta)                    2 NGIt~"e7R4  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ;&RBg+Pr  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Ymt.>8L  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) }M7{~ov#s  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 3)cH\gsg9  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) (JenTL`%u  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) @ LPs.e  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) m~c6b{F3Z-  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) "{>BP$Jz  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) a=@]Ov/  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5)  -]n\|U<  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )09>#!*  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) uW;[FTcqy$  
    %       -------------------------------------------------- %'+}-w  
    % N(c`h  
    %   Example 1: :O)\+s-  
    % EC;R^)  
    %       % Display the Zernike function Z(n=5,m=1) DL<b)# h#  
    %       x = -1:0.01:1; wB'GV1|jL  
    %       [X,Y] = meshgrid(x,x); U</Vcz  
    %       [theta,r] = cart2pol(X,Y); IX+!+XC"U  
    %       idx = r<=1; ERTjY%A  
    %       z = nan(size(X)); K4U_sCh#f  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); pz4lC=H%o  
    %       figure +6~ut^YiM.  
    %       pcolor(x,x,z), shading interp ~  p~  
    %       axis square, colorbar @3*S:;x  
    %       title('Zernike function Z_5^1(r,\theta)') {gT4Oq__  
    % Z; 6N7U  
    %   Example 2: "zE>+zRl  
    % ly9tI-E  
    %       % Display the first 10 Zernike functions `@3{}  
    %       x = -1:0.01:1; @V}!elV  
    %       [X,Y] = meshgrid(x,x); KwAc Ga}J  
    %       [theta,r] = cart2pol(X,Y); t4d^DZDh!  
    %       idx = r<=1; F% < ZEVm  
    %       z = nan(size(X)); .RW&=1D6  
    %       n = [0  1  1  2  2  2  3  3  3  3]; dp}s]`x+  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; DMdVE P"m  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; k^@dDLr"  
    %       y = zernfun(n,m,r(idx),theta(idx)); mE"(d*fe'  
    %       figure('Units','normalized') #=uV, dw  
    %       for k = 1:10 /$NR@56 \  
    %           z(idx) = y(:,k); D]=V6l=  
    %           subplot(4,7,Nplot(k)) 1`Z:/]hl  
    %           pcolor(x,x,z), shading interp do[w&`jw8  
    %           set(gca,'XTick',[],'YTick',[]) 7TW&=(  
    %           axis square W\EvMV"  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]WYddiF  
    %       end :8t;_f  
    % ZHM NG~!  
    %   See also ZERNPOL, ZERNFUN2. ]!>tP,<`'  
    B4/\=MXb  
    %   Paul Fricker 11/13/2006 \RS0mb  
    7I/a  
    hsAk7KC  
    % Check and prepare the inputs: :JXGgl<y  
    % ----------------------------- XwZR Kh\>=  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) lJ Jn@A  
        error('zernfun:NMvectors','N and M must be vectors.') U<|*V5   
    end \_)[FC@  
    Nt,:`o |  
    if length(n)~=length(m) \MDhm,H<  
        error('zernfun:NMlength','N and M must be the same length.')  CH$K_\  
    end "_0sW3rG  
    9\Md.>  
    n = n(:); >H5_,A}f  
    m = m(:);  G){A&F  
    if any(mod(n-m,2)) o&$Of  
        error('zernfun:NMmultiplesof2', ... 14`S9SL{V  
              'All N and M must differ by multiples of 2 (including 0).') \E1CQP-  
    end .6c Bx  
    p`Ok(C_  
    if any(m>n) 6!@p$ pm)a  
        error('zernfun:MlessthanN', ... ]+5Y\~I  
              'Each M must be less than or equal to its corresponding N.') G0u H6x?  
    end [(; .D  
    T"DG$R,Aj  
    if any( r>1 | r<0 ) |RH^|2:x9Q  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') *7{{z%5Pu  
    end N C3XJ 4  
    +h? Gps  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) " 1h~P,  
        error('zernfun:RTHvector','R and THETA must be vectors.') )}J}d)  
    end T"e"?JSRJ  
    RF [81/w]  
    r = r(:); _D{{C  
    theta = theta(:); J;k8 a2$_  
    length_r = length(r); [5PQrf~Mo  
    if length_r~=length(theta) a+B3`6  
        error('zernfun:RTHlength', ... Q sPZ dC  
              'The number of R- and THETA-values must be equal.') * $|9e  
    end swg*fhJFB  
    w&Z.rB?  
    % Check normalization: lvG+9e3+  
    % -------------------- h^f?rWD:nz  
    if nargin==5 && ischar(nflag) Ow {NI-^K  
        isnorm = strcmpi(nflag,'norm'); G%dzJpC(  
        if ~isnorm 0?''v>%  
            error('zernfun:normalization','Unrecognized normalization flag.') &23{(]eO  
        end +.a->SZ5"  
    else ?'si ^N  
        isnorm = false; be]Zx`)k  
    end l]L"Ex{  
    w x,gth*p  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  n[7=  
    % Compute the Zernike Polynomials (Bss%\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n],"!>=+  
    ${tBu#$-d  
    % Determine the required powers of r: dF^`6-K1  
    % ----------------------------------- *>T@3G.{Rm  
    m_abs = abs(m); VO<P9g$UD  
    rpowers = []; _o-01gu.  
    for j = 1:length(n) vv D515i  
        rpowers = [rpowers m_abs(j):2:n(j)]; A<-3u  
    end ^ x_+ &  
    rpowers = unique(rpowers); *X 2dS {  
    B7n1'?  
    % Pre-compute the values of r raised to the required powers, <%"CQT6g %  
    % and compile them in a matrix: qJK6S4O]  
    % ----------------------------- QaLVIsnfN  
    if rpowers(1)==0 !Xzy:  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); mpzm6I eu  
        rpowern = cat(2,rpowern{:}); {'o\#4 Wk  
        rpowern = [ones(length_r,1) rpowern]; Vah.tOU  
    else `<?((l%;R  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Zb2.o5#}  
        rpowern = cat(2,rpowern{:}); w_#5Na}>d  
    end O*1la/~m  
    9j1 tcT  
    % Compute the values of the polynomials: (o8?j^ -v  
    % -------------------------------------- b|U3\Fmc  
    y = zeros(length_r,length(n)); \P9HAz'6  
    for j = 1:length(n) Ns-3\~QSi  
        s = 0:(n(j)-m_abs(j))/2; k&o1z'<C  
        pows = n(j):-2:m_abs(j); 9]|G-cyt  
        for k = length(s):-1:1 2w:cdAv$  
            p = (1-2*mod(s(k),2))* ... ETaLE[T%1  
                       prod(2:(n(j)-s(k)))/              ... A w)P%r  
                       prod(2:s(k))/                     ... %loe8yt  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 1y.!x~Pi,  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); (C hL$!x  
            idx = (pows(k)==rpowers); =mh)b]].4\  
            y(:,j) = y(:,j) + p*rpowern(:,idx); !{4bC  
        end M9wj };vy  
         Mh5 =]O+  
        if isnorm #zKF/H|_R  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); oHx=Cg;  
        end ^4tz*i  
    end U=ie| 3  
    % END: Compute the Zernike Polynomials d+g+ {p>?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B1 [O9U:  
    /N`E4bKBR  
    % Compute the Zernike functions: 1 =9 Kwd  
    % ------------------------------ }'Yk#Q  
    idx_pos = m>0; QfsTUAfR  
    idx_neg = m<0; J/^|Y6  
    =#{i;CC%  
    z = y; 8(0q,7)y  
    if any(idx_pos) ] 73BJ  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Js!V,={iX  
    end GA@Zfcg  
    if any(idx_neg) ahm@ +/2  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); wQ/FJoB  
    end /(skIvE|  
    D[R<H((  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) Ga} &%  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. /=2  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 7ezf.[{R  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive @}@J$ g  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, :$>TeCm  
    %   and THETA is a vector of angles.  R and THETA must have the same 6dq*ncNin  
    %   length.  The output Z is a matrix with one column for every P-value, R2$;f?;:  
    %   and one row for every (R,THETA) pair. @q]{s+#Xf  
    % |lhVk\X  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike e(m#elX  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) G\:^9!nwY~  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) f*^)0Po  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 yp:_W@  
    %   for all p. ?Em*yc@WD  
    % *PJg~F%  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 4#BoS9d2I<  
    %   Zernike functions (order N<=7).  In some disciplines it is =+j>?Yi  
    %   traditional to label the first 36 functions using a single mode `* =Tf  
    %   number P instead of separate numbers for the order N and azimuthal YaDr.?  
    %   frequency M. (C EXPf  
    % Ge]2g0  
    %   Example: jTJ]: EN  
    % idr,s\$>  
    %       % Display the first 16 Zernike functions +\a`:QET  
    %       x = -1:0.01:1; xW;-=Q  
    %       [X,Y] = meshgrid(x,x); ?E?dg#yk  
    %       [theta,r] = cart2pol(X,Y); v8xNtUxN  
    %       idx = r<=1; N{<=s]I%x  
    %       p = 0:15; &[hq !v  
    %       z = nan(size(X)); R~],5_|  
    %       y = zernfun2(p,r(idx),theta(idx)); duKR;5:  
    %       figure('Units','normalized') t3)nG8> )  
    %       for k = 1:length(p) !`G7X  
    %           z(idx) = y(:,k); .V?i3  
    %           subplot(4,4,k) {^xp?zpV  
    %           pcolor(x,x,z), shading interp Ex ?)FL$4  
    %           set(gca,'XTick',[],'YTick',[]) / 1jb8w'  
    %           axis square &1DU]|RoT&  
    %           title(['Z_{' num2str(p(k)) '}']) K~_[[)14b  
    %       end 4u zyU_  
    % 0 pH qNlb  
    %   See also ZERNPOL, ZERNFUN. Dm1;mRS+  
    U'st\Dt  
    %   Paul Fricker 11/13/2006 z Lw=*  
    +FWkhmTv  
    /2&jId  
    % Check and prepare the inputs: %jK-}0Tu  
    % ----------------------------- P{eL;^I  
    if min(size(p))~=1 rZDlPp>BPZ  
        error('zernfun2:Pvector','Input P must be vector.') XQu~/{A=  
    end f.`noZN  
    lbv9 kk[  
    if any(p)>35 szC~?]<YY  
        error('zernfun2:P36', ... _-*Lj;^V  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... $e;_N4d^  
               '(P = 0 to 35).']) OgNt"Vg  
    end JU7EC~7|2c  
    O4kBNUI/  
    % Get the order and frequency corresonding to the function number: .Z}ySd:X  
    % ---------------------------------------------------------------- mJu;B3@  
    p = p(:); V@Z8t8  
    n = ceil((-3+sqrt(9+8*p))/2); `?Yh`P0  
    m = 2*p - n.*(n+2); h$p]#]uMb  
    xD;5z`A3  
    % Pass the inputs to the function ZERNFUN: 32=Gq5pOc  
    % ---------------------------------------- TE4{W4I  
    switch nargin Vc3tKuMsiX  
        case 3 *f:^6h  
            z = zernfun(n,m,r,theta); #5kQn>R  
        case 4 ]k%Yz@*S  
            z = zernfun(n,m,r,theta,nflag); _yyQ^M/  
        otherwise cjU*  
            error('zernfun2:nargin','Incorrect number of inputs.') =Uta5$\a)  
    end hbhh m  
    8? 4j-  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) K*Zf^g m  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. @kUCc1LT  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of &dZ-}. af  
    %   order N and frequency M, evaluated at R.  N is a vector of :04sB]H  
    %   positive integers (including 0), and M is a vector with the +qe!KPk2  
    %   same number of elements as N.  Each element k of M must be a ja}_u}:  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) zCT Wi  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 7_taqcj  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix p5c^dC{   
    %   with one column for every (N,M) pair, and one row for every <Brq7:n|  
    %   element in R. [y| "iSD  
    % AoL4#.r3H  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 1FUadSB5)  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is Mf%0Cx `  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Qwb=N  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 nnX,_5s  
    %   for all [n,m]. v2:A 4Pd:+  
    % Tm5]M$)  
    %   The radial Zernike polynomials are the radial portion of the @.T '>;izr  
    %   Zernike functions, which are an orthogonal basis on the unit FyX\S=  
    %   circle.  The series representation of the radial Zernike ;sCf2TD,_  
    %   polynomials is W~+ ] 7<  
    % N;7Xt9l  
    %          (n-m)/2 zlZ$t{[,  
    %            __ 3'6%P_S  
    %    m      \       s                                          n-2s Y  9]  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ) f9f_^;  
    %    n      s=0 tgH@|Kg  
    % |GtTz&  
    %   The following table shows the first 12 polynomials. ~isrE;N1|  
    % twU^ewO&  
    %       n    m    Zernike polynomial    Normalization r k;k:<c  
    %       --------------------------------------------- D ::),,  
    %       0    0    1                        sqrt(2) Juj"cjob  
    %       1    1    r                           2 Vu0jNKUV  
    %       2    0    2*r^2 - 1                sqrt(6) -;cZW.<  
    %       2    2    r^2                      sqrt(6) b;#3X)  
    %       3    1    3*r^3 - 2*r              sqrt(8) bsy\L|wd  
    %       3    3    r^3                      sqrt(8) [ps5;  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ]7n+|@3x  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ,j^ /~  
    %       4    4    r^4                      sqrt(10) @6 uB78U4O  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) vO!p8r F  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ZIQy}b'  
    %       5    5    r^5                      sqrt(12) L5! aLv#  
    %       --------------------------------------------- ;@GlJ '$;  
    % v{ Md4 p  
    %   Example: bmVksi2b  
    % #z_lBg. K  
    %       % Display three example Zernike radial polynomials B}8xA}<  
    %       r = 0:0.01:1; %719h>$  
    %       n = [3 2 5]; |u8IQR'B  
    %       m = [1 2 1]; @9g$+_"ZT  
    %       z = zernpol(n,m,r); J3gJSRT@P  
    %       figure Meo(|U  
    %       plot(r,z) ;75K:_  
    %       grid on Aq%TZ_m  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') rk:^^r>5Qi  
    % Z .VIb|  
    %   See also ZERNFUN, ZERNFUN2. }#5V t  
    >%5Ld`c:SD  
    % A note on the algorithm. G=Hvh=K(  
    % ------------------------ E|Mu1I]e  
    % The radial Zernike polynomials are computed using the series 3 ha^NjE  
    % representation shown in the Help section above. For many special S0\QZ/je  
    % functions, direct evaluation using the series representation can "s[wLclfG  
    % produce poor numerical results (floating point errors), because lJ;7sgQ#  
    % the summation often involves computing small differences between ,%7>%*nhk  
    % large successive terms in the series. (In such cases, the functions lYldq)qB{  
    % are often evaluated using alternative methods such as recurrence fTd=}zY  
    % relations: see the Legendre functions, for example). For the Zernike b{JcV  
    % polynomials, however, this problem does not arise, because the } M-^A{C\%  
    % polynomials are evaluated over the finite domain r = (0,1), and N|-M|1w96  
    % because the coefficients for a given polynomial are generally all ekC 1wN l  
    % of similar magnitude. X c~yr\%]  
    % H <41H;m  
    % ZERNPOL has been written using a vectorized implementation: multiple TG 9 a1q  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ,Z*&QR  
    % values can be passed as inputs) for a vector of points R.  To achieve Hc^q_{}"  
    % this vectorization most efficiently, the algorithm in ZERNPOL !m8MyZ}%  
    % involves pre-determining all the powers p of R that are required to OP0KK^#  
    % compute the outputs, and then compiling the {R^p} into a single 5r)ndW,aN  
    % matrix.  This avoids any redundant computation of the R^p, and I^S gWC  
    % minimizes the sizes of certain intermediate variables. tb36c<U-  
    % @=JOAo  
    %   Paul Fricker 11/13/2006 6BK-(>c(6  
    $P'Y  
    vOIK6-   
    % Check and prepare the inputs: &iR3]FNI  
    % ----------------------------- Ug(;\*yg  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) D! TFb E  
        error('zernpol:NMvectors','N and M must be vectors.') APY^A6^:j  
    end yu3: Hv}  
    7L3:d7=MIW  
    if length(n)~=length(m) mY6d+  
        error('zernpol:NMlength','N and M must be the same length.') ,5}%_  
    end ZNWo:N8;  
    ZYRZ$87jZ  
    n = n(:); ZcJa:  
    m = m(:); b>g&Pf#N!  
    length_n = length(n); |Z6M?n  
    LFvO[&  
    if any(mod(n-m,2)) 8i$quHd&x  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') *iLlBE  
    end VPOzt7:  
    u}_,4J  
    if any(m<0) /`6Y-8e2  
        error('zernpol:Mpositive','All M must be positive.') 2S%[YR>>  
    end V9KI?}q:W  
    :c75*h`  
    if any(m>n) mQL8ec_c  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') S7oPdzcU-  
    end _{Z!$q6,  
    Y=G9|7*lO  
    if any( r>1 | r<0 ) s"OP[YEke/  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') rK` x<  
    end v9*ugu[K9  
    HKB?G~  
    if ~any(size(r)==1) .,({&L  
        error('zernpol:Rvector','R must be a vector.') H){}28dX  
    end RBOb/.$  
    M~/Pk7CC  
    r = r(:); Z%&$_-yJ  
    length_r = length(r); ws/e~ T<c  
    xE>jlr?  
    if nargin==4 "Yp:{e  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); |ty?Ah,vb  
        if ~isnorm :zA/~/Wo  
            error('zernpol:normalization','Unrecognized normalization flag.') L i g7Ac,  
        end 5r2A^<)  
    else y  J|/^qs  
        isnorm = false; L<D<3g|4  
    end 1`sLbPW  
    90"&KDh  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }>93X0%r  
    % Compute the Zernike Polynomials Fal##6B  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wak:"B[  
    Ppton+?(  
    % Determine the required powers of r: !l6Ez_'  
    % ----------------------------------- k`we_$/Gw  
    rpowers = []; isBtJ7\Sc  
    for j = 1:length(n) 1  b&<De  
        rpowers = [rpowers m(j):2:n(j)]; 9ZJn 8ki  
    end )tvP|  
    rpowers = unique(rpowers); nsA}A~(E  
    $.+_f,tU  
    % Pre-compute the values of r raised to the required powers, omP\qOc  
    % and compile them in a matrix: :  I q  
    % ----------------------------- ~:JoKm`vU  
    if rpowers(1)==0 @> |3d  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); :~^_*:  
        rpowern = cat(2,rpowern{:}); d6+$[4w  
        rpowern = [ones(length_r,1) rpowern]; n 9>**&5L  
    else PtTL tiE~  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); fzdWM:g  
        rpowern = cat(2,rpowern{:}); 11^.oa+`  
    end MD):g @  
    \; voBU  
    % Compute the values of the polynomials: +^<s'  
    % -------------------------------------- Te6cw+6  
    z = zeros(length_r,length_n); B7QRG0  
    for j = 1:length_n =vs]Kmm  
        s = 0:(n(j)-m(j))/2; 6!Q,X Hs  
        pows = n(j):-2:m(j); 9oU1IT9   
        for k = length(s):-1:1 41v#|%\w  
            p = (1-2*mod(s(k),2))* ... <GWzdj?  
                       prod(2:(n(j)-s(k)))/          ... |1pD n7  
                       prod(2:s(k))/                 ... ^qY?x7mx1  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... S[;d\Z]~  
                       prod(2:((n(j)+m(j))/2-s(k))); =OeLF  
            idx = (pows(k)==rpowers); gs"w 0[$  
            z(:,j) = z(:,j) + p*rpowern(:,idx); p:NIRs  
        end OQ&'3hv{  
         "h5.^5E6  
        if isnorm e?7Oom  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); s'^sT=b  
        end 7_Op(C4,nC  
    end %a:>3! +  
    X \BxRgl},  
    % 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)  &8Z .m,s]  
    T|0+o+i  
    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。 @qYT/V*/  
    M%Ksyr9  
    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)