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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 b4 hIeBI\  
    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! 0k?Sq#7q  
     
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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 6+ 8mV8{-8  
    function z = zernfun(n,m,r,theta,nflag) BPWnck=%  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. hNO )~rt  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N [EGx  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ]xR4->eix  
    %   unit circle.  N is a vector of positive integers (including 0), and /Ri,>}n  
    %   M is a vector with the same number of elements as N.  Each element ?f@ 9nph  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) nx;$dxx_Ws  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, h]P/KVqR.  
    %   and THETA is a vector of angles.  R and THETA must have the same QUPf *3Oy  
    %   length.  The output Z is a matrix with one column for every (N,M) !~d'{sy6  
    %   pair, and one row for every (R,THETA) pair. E{gv,cUM  
    % 6z1\a  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike taCCw2s-8*  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), p1 4d ,}4W  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral sJ7sjrEp 1  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, WFj*nS^~l  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 6+Jry@  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. L *{QjH  
    % `r]TA]D R  
    %   The Zernike functions are an orthogonal basis on the unit circle. eKJ:?Lxv;  
    %   They are used in disciplines such as astronomy, optics, and fM{1Os  
    %   optometry to describe functions on a circular domain. iIB9j8  
    % 3"vRK5Bf  
    %   The following table lists the first 15 Zernike functions. ^5>du~d  
    % <Cr8V'c  
    %       n    m    Zernike function           Normalization F8 ?uQP8  
    %       -------------------------------------------------- gr\@sx?b  
    %       0    0    1                                 1 *N'hA5.z  
    %       1    1    r * cos(theta)                    2 <c\]Ct  
    %       1   -1    r * sin(theta)                    2 /4H[4m]I  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) :K;T Q  
    %       2    0    (2*r^2 - 1)                    sqrt(3) p6[#f96^u  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) (h|ch#  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 0T1ko,C!,e  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) X/wmKi  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) \2Xx%SX  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) I)rGOda{  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) \KN dZC?V2  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;'hi9L  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) shy  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) u x#. :C|  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) N)P((>S;  
    %       -------------------------------------------------- J& )#G@fRX  
    % w`0)x5 TGR  
    %   Example 1: & L3UlL  
    % ]xI?,('_m  
    %       % Display the Zernike function Z(n=5,m=1) bk0Y  
    %       x = -1:0.01:1; T|!D>l'  
    %       [X,Y] = meshgrid(x,x); mHHzCKE,  
    %       [theta,r] = cart2pol(X,Y); ?n&$m  
    %       idx = r<=1; L=,Y1nO:p  
    %       z = nan(size(X)); n3*UgNg%fK  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ) (+)Q'*  
    %       figure ;*.(.  
    %       pcolor(x,x,z), shading interp %P(;8sS  
    %       axis square, colorbar PlF!cr7:4  
    %       title('Zernike function Z_5^1(r,\theta)') {:3.27jQ  
    % q`cEA<~S  
    %   Example 2: ?LR"hZ>  
    % @Mzz2&(d U  
    %       % Display the first 10 Zernike functions Vj/fAHR`>'  
    %       x = -1:0.01:1; 90?,-6  
    %       [X,Y] = meshgrid(x,x); _ r~+p  
    %       [theta,r] = cart2pol(X,Y); % <^[j^j}o  
    %       idx = r<=1; z^gi[ mi  
    %       z = nan(size(X)); ~~ U<  
    %       n = [0  1  1  2  2  2  3  3  3  3]; L)1C'8 ).  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; U%h7h`=F?  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; z2.*#xTZn  
    %       y = zernfun(n,m,r(idx),theta(idx)); w[e0wh`.  
    %       figure('Units','normalized') \Oz,Qzr|  
    %       for k = 1:10 @T5YsX]qb7  
    %           z(idx) = y(:,k); \ibCR~W4  
    %           subplot(4,7,Nplot(k)) C?{D"f`[]  
    %           pcolor(x,x,z), shading interp cJSVT8  
    %           set(gca,'XTick',[],'YTick',[]) Gee~>:_Q{J  
    %           axis square "$]ls9-%n  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) T.J`S(oI  
    %       end 2rF?Q?$,B  
    % Sy4 mZ}:  
    %   See also ZERNPOL, ZERNFUN2. ^@M [t<  
    N?\bBt@  
    %   Paul Fricker 11/13/2006 (%6(5,   
    #"hJpyW 4V  
    -QN1oK@\mE  
    % Check and prepare the inputs: t3pZjdLJd  
    % ----------------------------- {ms,q_Zr  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ,Y$F7&  
        error('zernfun:NMvectors','N and M must be vectors.') ,tcP=f dk]  
    end 7WgIhQ~  
    JL?Cnk$!  
    if length(n)~=length(m) Tt{U"EFO  
        error('zernfun:NMlength','N and M must be the same length.') &fCP2]hj'  
    end -l\~p4U  
    uE"5cq'B/  
    n = n(:); Po'-z<}wS  
    m = m(:); :!(YEF#}  
    if any(mod(n-m,2)) N[0 xqQ  
        error('zernfun:NMmultiplesof2', ... S&5Q~}{,  
              'All N and M must differ by multiples of 2 (including 0).') L[CU  
    end SAd 97A:  
    @c6"RHG9  
    if any(m>n) P{"  WlJ  
        error('zernfun:MlessthanN', ... (m<R0  
              'Each M must be less than or equal to its corresponding N.') 7fap*  
    end /_ $~rW  
    6e-#XCR{  
    if any( r>1 | r<0 ) $7msL#E7  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') #L_@s d  
    end ? (fQ<i n  
    ;3 G~["DA  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) oP+kAV#]  
        error('zernfun:RTHvector','R and THETA must be vectors.') N8,EI^W8Z  
    end nu;} S!J  
    [B}1z  
    r = r(:); !S~,> ,yd  
    theta = theta(:); zY]Bu-S3  
    length_r = length(r); {z.[tvE8h  
    if length_r~=length(theta) 2=igS#h  
        error('zernfun:RTHlength', ... R#"U/8b>z  
              'The number of R- and THETA-values must be equal.') %y~`"l$-  
    end ]cx"  
    qgwv=5|  
    % Check normalization: zj~8>QnKk  
    % -------------------- I(z>)S'7r  
    if nargin==5 && ischar(nflag) xP8iz?6"V  
        isnorm = strcmpi(nflag,'norm'); N90\]dFmy  
        if ~isnorm B@ZqJw9J[  
            error('zernfun:normalization','Unrecognized normalization flag.') )$ ofl%+  
        end 2q`)GCES~  
    else bHhC56[M  
        isnorm = false; aeG#: Ln+{  
    end 2>!_B\%)H  
    >E//pr)_Km  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s,1pZT <E  
    % Compute the Zernike Polynomials "WF( 6z#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% skk-.9  
    n%83jep9  
    % Determine the required powers of r: @?<N +qdH>  
    % ----------------------------------- /W"Bf  
    m_abs = abs(m); hg[l{)Q  
    rpowers = []; tU+@1~ ~  
    for j = 1:length(n) D}zOuB,S  
        rpowers = [rpowers m_abs(j):2:n(j)]; GOv9 2$e  
    end }u(d'9u  
    rpowers = unique(rpowers); )z]q"s5 Y  
    anHBy SI3  
    % Pre-compute the values of r raised to the required powers, B'G*y2UnG  
    % and compile them in a matrix: 91-P)%?  
    % ----------------------------- 3v9gb,)y\  
    if rpowers(1)==0 5en [)3E  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); LP5eFl`|T  
        rpowern = cat(2,rpowern{:}); >uBV  
        rpowern = [ones(length_r,1) rpowern]; ?;0nJf  
    else t x:rj6 -z  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Rz<d%C;R  
        rpowern = cat(2,rpowern{:}); N&0uXrw  
    end jOoIF/So  
    ,omp F$%  
    % Compute the values of the polynomials: g5kYyE  
    % -------------------------------------- MZUF! B  
    y = zeros(length_r,length(n)); d8Q_6(Ar|  
    for j = 1:length(n) $\YLmG  
        s = 0:(n(j)-m_abs(j))/2; ;4-p upK~%  
        pows = n(j):-2:m_abs(j); AmT| %j&3  
        for k = length(s):-1:1 33#7U+~]@  
            p = (1-2*mod(s(k),2))* ... Ft%TnEp  
                       prod(2:(n(j)-s(k)))/              ... }S~ysQwT  
                       prod(2:s(k))/                     ... p|bc=`TD  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ()@.;R.Z  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); .LXh]I *  
            idx = (pows(k)==rpowers); b'Fx),  
            y(:,j) = y(:,j) + p*rpowern(:,idx); < "L){$  
        end [a>JG8[ ,t  
         <B ]i80.  
        if isnorm /%ODJ1M  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); } #\;np  
        end \U)2 Tg  
    end ~uhyROO,G"  
    % END: Compute the Zernike Polynomials M5cOz|j/*R  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zCBtD_@  
    \p>]G[g  
    % Compute the Zernike functions: an$ ]IN  
    % ------------------------------ WTV3p,;6a  
    idx_pos = m>0;  Vq .!(x  
    idx_neg = m<0; *!r\GGb  
    | Q1ub S  
    z = y; Wvut)T  
    if any(idx_pos) "W_jdE6v  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); .WL\:{G8;  
    end eB<V%,%N#  
    if any(idx_neg) o-Q]Dk1W  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)');  \pewbu5^  
    end rB.=f[aX[  
    !\&7oAs=I  
    % EOF zernfun
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) !ulLGmUn  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. T4HoSei  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated VJ6>3  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive j?f,~Y<k  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, s!j(nUd/  
    %   and THETA is a vector of angles.  R and THETA must have the same +QXYU8bYZ  
    %   length.  The output Z is a matrix with one column for every P-value, H4y1Hpa,  
    %   and one row for every (R,THETA) pair. F)E7(Un`8  
    % I*vj26qvg  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike <D;H} ef  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) CYFas:rPLT  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Kc9mI>uH  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 NqQ(X'W7  
    %   for all p. CQ8o9A/  
    % f1]AfH#  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 zNsL^;uT  
    %   Zernike functions (order N<=7).  In some disciplines it is DX%8. @  
    %   traditional to label the first 36 functions using a single mode Ghq'k:K,  
    %   number P instead of separate numbers for the order N and azimuthal +3o)L?:g  
    %   frequency M. St3(1mApl  
    % *(\;}JF-  
    %   Example: . ~A"Wyu\  
    % *nsnX/e(-  
    %       % Display the first 16 Zernike functions 2LxVt@_R!%  
    %       x = -1:0.01:1; ~kj(s>xP  
    %       [X,Y] = meshgrid(x,x); :`>+f.)  
    %       [theta,r] = cart2pol(X,Y); S"KTL*9D  
    %       idx = r<=1; -EkDG]my  
    %       p = 0:15; ?^yh5   
    %       z = nan(size(X)); jC/JiI  
    %       y = zernfun2(p,r(idx),theta(idx)); m|ERf2-  
    %       figure('Units','normalized') /H;kYx  
    %       for k = 1:length(p) @8<uAu%  
    %           z(idx) = y(:,k); e\ l,gQP  
    %           subplot(4,4,k) 4na4Jsq{  
    %           pcolor(x,x,z), shading interp IjB*myN.  
    %           set(gca,'XTick',[],'YTick',[]) i3kI2\bd/  
    %           axis square ~g4rGz  
    %           title(['Z_{' num2str(p(k)) '}']) Y^jnlS)h  
    %       end DO-K  
    % a5U2[Ko80  
    %   See also ZERNPOL, ZERNFUN. h-_0 A]  
    aD/,c1  
    %   Paul Fricker 11/13/2006 MY<!\4/  
    d T,m{[+  
    WlQ&Yau  
    % Check and prepare the inputs: _[OEE<(  
    % ----------------------------- 6dS1\Y  
    if min(size(p))~=1 ,~N+?k_  
        error('zernfun2:Pvector','Input P must be vector.') SKc T  
    end oIL+@}u7  
    $Z7|t  
    if any(p)>35 +} !F(c  
        error('zernfun2:P36', ... N>6yacTB  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... l3pW{p  
               '(P = 0 to 35).']) sO4}kxZ  
    end i;'X}KW  
    (+Kof  
    % Get the order and frequency corresonding to the function number: -TU{r_!Z(  
    % ---------------------------------------------------------------- H'h4@S  
    p = p(:); ]BQWA  
    n = ceil((-3+sqrt(9+8*p))/2); (Q]Y> '  
    m = 2*p - n.*(n+2); p:Ld)U*  
    seV;f^-hR  
    % Pass the inputs to the function ZERNFUN: RAuAIiQ  
    % ---------------------------------------- Z Lio8  
    switch nargin `E0.PV  
        case 3 D)~nAkVq  
            z = zernfun(n,m,r,theta); ) Q  
        case 4 > %cWTC  
            z = zernfun(n,m,r,theta,nflag); WWs[]zr  
        otherwise I'%H:53^0  
            error('zernfun2:nargin','Incorrect number of inputs.') >RqT7n8h  
    end 2hA66ar{$  
    fJ"~XTN}T  
    % 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) *rFbehfH  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. $No>-^ )  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of . N} }cJq  
    %   order N and frequency M, evaluated at R.  N is a vector of M^Ay,jK!  
    %   positive integers (including 0), and M is a vector with the ^]!1'xg  
    %   same number of elements as N.  Each element k of M must be a K2o0L5Lke  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) @P5@ &G  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 7(USp#"  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix {2*l :'  
    %   with one column for every (N,M) pair, and one row for every I 3,e)Z  
    %   element in R. )qP{X,Uf  
    % B';> Hk  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- \C2P{q/m  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is eh7r'DmAR  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ="[](X^ l  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 S9#N%{8P  
    %   for all [n,m]. KPe.AK,8  
    % BRzWZq%r3  
    %   The radial Zernike polynomials are the radial portion of the qg:I+"u  
    %   Zernike functions, which are an orthogonal basis on the unit M9jo<+  
    %   circle.  The series representation of the radial Zernike s=Q*|  
    %   polynomials is '2J6%Gg  
    % Vyq<T(5  
    %          (n-m)/2 *k]S{]Y  
    %            __ '{k Nbx51  
    %    m      \       s                                          n-2s XaS_3d  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 8*~:gZ7:  
    %    n      s=0 f4y;K>u7p  
    % z'D{:q  
    %   The following table shows the first 12 polynomials. Zy3&Zt  
    % x[~OVG0M*  
    %       n    m    Zernike polynomial    Normalization Fj('l  
    %       --------------------------------------------- o9d$ 4s@/  
    %       0    0    1                        sqrt(2) bYB}A :  
    %       1    1    r                           2 7b%Cl   
    %       2    0    2*r^2 - 1                sqrt(6) 4S EC4yO  
    %       2    2    r^2                      sqrt(6) EA E\Xv  
    %       3    1    3*r^3 - 2*r              sqrt(8) }w^ T9OC  
    %       3    3    r^3                      sqrt(8) j/mp.'P1k  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) +5|nCp6||j  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) D2 cIVx3:(  
    %       4    4    r^4                      sqrt(10) 2(J tD  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) LP6FSo~K  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Z?aR9OTP  
    %       5    5    r^5                      sqrt(12) 6|qvo+%  
    %       --------------------------------------------- $#W6z:  
    % VgTI2  
    %   Example: 'J0s%m|j  
    % XJ/ kB8  
    %       % Display three example Zernike radial polynomials "{"2h>o#D}  
    %       r = 0:0.01:1; >$52B9ie  
    %       n = [3 2 5]; u0hbM9U>  
    %       m = [1 2 1]; A1}+j-D7!y  
    %       z = zernpol(n,m,r); 4l UE(#kUM  
    %       figure KY&,(z   
    %       plot(r,z) Rj;e82%%N  
    %       grid on R:^?6f<Z}  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') QF "&~  
    % je3n'^m  
    %   See also ZERNFUN, ZERNFUN2. gH)B` @  
    3CPOZZ  
    % A note on the algorithm. nJH%pBc  
    % ------------------------ L;7mt 4H  
    % The radial Zernike polynomials are computed using the series cNc _ n<M  
    % representation shown in the Help section above. For many special A0u:Fm{E  
    % functions, direct evaluation using the series representation can Z=8CbS).  
    % produce poor numerical results (floating point errors), because 0)a?W,+O  
    % the summation often involves computing small differences between :FpBz~!a  
    % large successive terms in the series. (In such cases, the functions `b'J*4|oGo  
    % are often evaluated using alternative methods such as recurrence 7]zZh a4X  
    % relations: see the Legendre functions, for example). For the Zernike >F_Ne)}qTQ  
    % polynomials, however, this problem does not arise, because the DC7}Xly(  
    % polynomials are evaluated over the finite domain r = (0,1), and lD#1"$Coz  
    % because the coefficients for a given polynomial are generally all yP]W\W'  
    % of similar magnitude. ',7Z1O  
    % tSa%ZkS  
    % ZERNPOL has been written using a vectorized implementation: multiple t3JPxg]0k'  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] =:8=5tj  
    % values can be passed as inputs) for a vector of points R.  To achieve }AYSQ~:  
    % this vectorization most efficiently, the algorithm in ZERNPOL 6dp_R2zH~o  
    % involves pre-determining all the powers p of R that are required to CoXL;\  
    % compute the outputs, and then compiling the {R^p} into a single XQ;d ew+  
    % matrix.  This avoids any redundant computation of the R^p, and bl-s0Ax-  
    % minimizes the sizes of certain intermediate variables. wGX"R5  
    % e91d~  
    %   Paul Fricker 11/13/2006 9GaER+d|  
    gRI|rDC)B  
    P32'`!/:  
    % Check and prepare the inputs: 1V?)zp  
    % ----------------------------- --)[>6)I  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Y2&6xTh  
        error('zernpol:NMvectors','N and M must be vectors.') V@-GQP1  
    end &r!>2$B\  
    ?7a[| -  
    if length(n)~=length(m) W<7Bq_L[|  
        error('zernpol:NMlength','N and M must be the same length.') [_1G\z_iE  
    end dL)5~V8s  
    ;0q6 bp(<H  
    n = n(:); 5] %kWV>  
    m = m(:); 0k<%l6Bq  
    length_n = length(n); &H{>7q#r  
    kA`qExw%  
    if any(mod(n-m,2)) HX*U2<^  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') [' 1?'*  
    end f |5|n>*  
    x#j_}L!V;  
    if any(m<0) ,CF~UX% bU  
        error('zernpol:Mpositive','All M must be positive.') 5UyK1e))  
    end pl&GFf o  
    +?tNly`  
    if any(m>n) MWf%Lh;R  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') TA7w:<  
    end `<G+ N  
    UU`qI}Ys8F  
    if any( r>1 | r<0 ) 6fyW6xv[,  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') v8N1fuP}  
    end v}@ 6"\  
    &LYZQ?|  
    if ~any(size(r)==1) VE m[F/'  
        error('zernpol:Rvector','R must be a vector.') `#F>?g$2  
    end tWI hbt  
    0IuU4h5Fr  
    r = r(:); pUx@QyrI  
    length_r = length(r); 0@;E8^pa  
    c7_b^7h1  
    if nargin==4 uRg^:  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); <o}t-Bgg  
        if ~isnorm tnntHQ&b  
            error('zernpol:normalization','Unrecognized normalization flag.') }e)ltp|  
        end \W!<xE  
    else Uz_{jAhW]  
        isnorm = false; je\UfEo%  
    end %l,EA#89 s  
    ~8K~@e$./  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~Tolz H!  
    % Compute the Zernike Polynomials T^W8_rm *3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Emo]I[<&q  
    $[e*0!e  
    % Determine the required powers of r: J u7AxTf~  
    % ----------------------------------- e2v,#3Q\  
    rpowers = []; ZN^Q!v  
    for j = 1:length(n) '|.u*M,b  
        rpowers = [rpowers m(j):2:n(j)]; nS#;<p$\  
    end %' Fc%3  
    rpowers = unique(rpowers); NDi@x"];  
    URwFNOM2  
    % Pre-compute the values of r raised to the required powers, e^fjla5  
    % and compile them in a matrix: APy a&TG  
    % ----------------------------- NH/H+7,o  
    if rpowers(1)==0 ;2^=#7I?  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ;|c,  
        rpowern = cat(2,rpowern{:}); IiL?@pIq  
        rpowern = [ones(length_r,1) rpowern]; tW +I?  
    else 'tc$#f^:  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); -z@}:N-uR  
        rpowern = cat(2,rpowern{:}); %3cBh v[q4  
    end Z(FAQ\7  
    m-K6y7t  
    % Compute the values of the polynomials: TQ FD  
    % -------------------------------------- ^H>vJT  
    z = zeros(length_r,length_n); g.'4uqU  
    for j = 1:length_n 3e"G.0vJ  
        s = 0:(n(j)-m(j))/2; Ty5\zxC|  
        pows = n(j):-2:m(j); #t\Oq9}^  
        for k = length(s):-1:1 zuOIos  
            p = (1-2*mod(s(k),2))* ... rYT3oqpfT  
                       prod(2:(n(j)-s(k)))/          ... 'RhMzPmY>  
                       prod(2:s(k))/                 ... }x+{=%~N  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... h^4oy^9  
                       prod(2:((n(j)+m(j))/2-s(k))); OT zh=Z^r  
            idx = (pows(k)==rpowers); LY"/ Q  
            z(:,j) = z(:,j) + p*rpowern(:,idx); {.sF&(e   
        end *+iWB_  
         &*0V!+#6  
        if isnorm 'del|"h!M  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ='f>p+*c%  
        end rv^j&X+EH  
    end H7WKnn@  
    tcs Z! #  
    % 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)  HI*xk  
    {=VauF  
    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。 y".uu+hL`  
    /{#1w\  
    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)