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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 n%h^o   
    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! 1}hIW":3Sr  
     
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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 BL@:!t  
    function z = zernfun(n,m,r,theta,nflag) F~ Lx|)0M  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~>9_(L  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N t 6v/sZ{F  
    %   and angular frequency M, evaluated at positions (R,THETA) on the KfF!{g f  
    %   unit circle.  N is a vector of positive integers (including 0), and U%0Ty|$Y   
    %   M is a vector with the same number of elements as N.  Each element )M2F4[vcb  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) z;@*r}H  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, y qtKy  
    %   and THETA is a vector of angles.  R and THETA must have the same -i-?.:  
    %   length.  The output Z is a matrix with one column for every (N,M) V I% 6.6D  
    %   pair, and one row for every (R,THETA) pair. Y^<bl2"y8  
    % !3T&4t  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike mf'V)  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), h gJ[LU|>  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral f6$b s+oP  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, <w3!!+oK"  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized \"hJCP?,  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ;c$J=h]  
    % {v3P9s(  
    %   The Zernike functions are an orthogonal basis on the unit circle. e%W$*f  
    %   They are used in disciplines such as astronomy, optics, and QeF3qXI  
    %   optometry to describe functions on a circular domain. Cu6%h>@K$  
    % 4&l10fR5  
    %   The following table lists the first 15 Zernike functions. U*.0XNKp{  
    % X$/2[o#g  
    %       n    m    Zernike function           Normalization Haqm^Ky$  
    %       -------------------------------------------------- m,fAeln  
    %       0    0    1                                 1 Jmx Ko+-  
    %       1    1    r * cos(theta)                    2 s+>:,U<A  
    %       1   -1    r * sin(theta)                    2 V59(Z  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) -W>'^1cR  
    %       2    0    (2*r^2 - 1)                    sqrt(3) _V`DWR *  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) (5\N B0  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Z0l+1iMx  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ?6'rBH/w  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) [=~pe|8:  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) $ImrOf^qt  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) qe5feky  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) V^;jJ']  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) :6%Z]tt  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6-O_\Cq8  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ?IpLf\n-  
    %       -------------------------------------------------- DK}"b}Fvq  
    % 43=,yz2Ef  
    %   Example 1: o=`C<}  
    % 2#k5+?-c61  
    %       % Display the Zernike function Z(n=5,m=1) F:aILx  
    %       x = -1:0.01:1; Q|@4bzi)  
    %       [X,Y] = meshgrid(x,x); z?35=%~w   
    %       [theta,r] = cart2pol(X,Y); 6uR^%W8]  
    %       idx = r<=1; +@r*}  
    %       z = nan(size(X)); -lv)tHs<  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 5 (A5Y-B  
    %       figure JfPD}w  
    %       pcolor(x,x,z), shading interp P9 Z}H(?C  
    %       axis square, colorbar 0V?F'<qy  
    %       title('Zernike function Z_5^1(r,\theta)') 6^DR0sO  
    % iTaWup  
    %   Example 2: *z7dl5xJ  
    % jmeRrnC}  
    %       % Display the first 10 Zernike functions RD.V'`n"  
    %       x = -1:0.01:1; c/ uNM  
    %       [X,Y] = meshgrid(x,x); 2PG [7u^  
    %       [theta,r] = cart2pol(X,Y); 4f<$4d^md  
    %       idx = r<=1; - |gmQG  
    %       z = nan(size(X)); rXHv`k y  
    %       n = [0  1  1  2  2  2  3  3  3  3]; B/n[m@O  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; $kQ~d8 O  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; )rixMl &[  
    %       y = zernfun(n,m,r(idx),theta(idx)); .aflsUD  
    %       figure('Units','normalized') CJhL)0Cs  
    %       for k = 1:10 0Zg%+)iy@  
    %           z(idx) = y(:,k); 9H%X2#:fH  
    %           subplot(4,7,Nplot(k)) a`0=AQ  
    %           pcolor(x,x,z), shading interp :Lz\yARpk  
    %           set(gca,'XTick',[],'YTick',[]) )(@Hd  
    %           axis square &zo|Lfe  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) gmm.{%1_I;  
    %       end y a_<^O 9  
    % GQ-Rtn4v  
    %   See also ZERNPOL, ZERNFUN2. )YqXRm  
    ,#8e_3Z$  
    %   Paul Fricker 11/13/2006 c ;'[W60  
     Sr?#S  
    `HBf&Z  
    % Check and prepare the inputs: xL" |)A =  
    % ----------------------------- +[tP_%/r'^  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) dc rSz4E|>  
        error('zernfun:NMvectors','N and M must be vectors.') KSrx[q  
    end |ely|U. Tf  
    =J~ x  
    if length(n)~=length(m)  ^k\e8F/  
        error('zernfun:NMlength','N and M must be the same length.') jkvgoxY  
    end 3@]SKfoo1  
    LWt&3  
    n = n(:); &ZQJ>#~j^  
    m = m(:); } GiHjzsR  
    if any(mod(n-m,2)) u#@Q:tnN_  
        error('zernfun:NMmultiplesof2', ... Tq~=TSD  
              'All N and M must differ by multiples of 2 (including 0).') zi3\63D3eO  
    end Ct%x&m:  
    NrJKbk^4u/  
    if any(m>n) @|tL8?  
        error('zernfun:MlessthanN', ... ~JH:EB:  
              'Each M must be less than or equal to its corresponding N.') |u;v27  
    end ?pza G{  
    Y(d$  
    if any( r>1 | r<0 ) pt}X>ph{  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') f1(+ bE%  
    end jNC4_q&  
    0MdDXG-7  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 'Un " rts  
        error('zernfun:RTHvector','R and THETA must be vectors.') eET}r 24  
    end GbaEgA'fa  
    @#-q^}3  
    r = r(:); Vkc#7W(  
    theta = theta(:); L:'J Bhg  
    length_r = length(r); l c '=mA  
    if length_r~=length(theta) 2Roc|)-47  
        error('zernfun:RTHlength', ... 9\DQ>V TQ  
              'The number of R- and THETA-values must be equal.') TU 1I} ,  
    end 'uxX5k/D@t  
    W!&vul5  
    % Check normalization: O7$hYk  
    % -------------------- E\4 +_L_j  
    if nargin==5 && ischar(nflag) 6}oXP_0U  
        isnorm = strcmpi(nflag,'norm'); yT,.z 0  
        if ~isnorm d8x\  
            error('zernfun:normalization','Unrecognized normalization flag.') PxS8 n?y  
        end ;y2/-tL?  
    else v*[.a#1^  
        isnorm = false; JC3m.)/  
    end se>MQM5 )  
    A, LuD.8  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :B:"NyPA  
    % Compute the Zernike Polynomials 8UVmv=T  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% E0?iXSJ  
    % V ;?  
    % Determine the required powers of r: 2j[&=R/.  
    % ----------------------------------- UTH_^HAN#G  
    m_abs = abs(m); k4 [|'Dk?  
    rpowers = []; ]h5Yg/sms  
    for j = 1:length(n) }-sdov<<  
        rpowers = [rpowers m_abs(j):2:n(j)]; C-H@8p?T  
    end  W0]gLw9*  
    rpowers = unique(rpowers); ?C A,  
    EL9]QI  
    % Pre-compute the values of r raised to the required powers, #: [<iSk  
    % and compile them in a matrix: W!>.$4Q9  
    % ----------------------------- oT>(V]*5  
    if rpowers(1)==0 L');!/:  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); |YY_^C`"-  
        rpowern = cat(2,rpowern{:}); fu]s/'8B  
        rpowern = [ones(length_r,1) rpowern]; $ . 9V&  
    else 3^6 d]f  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); t8+X%-r  
        rpowern = cat(2,rpowern{:}); )(384@'"u  
    end Rw:*'1  
    ~S7 D>D3S  
    % Compute the values of the polynomials: ^i} L-QR  
    % -------------------------------------- tyqT  
    y = zeros(length_r,length(n)); / :n#`o=;  
    for j = 1:length(n) dKxyA"@  
        s = 0:(n(j)-m_abs(j))/2; 9uA>N  
        pows = n(j):-2:m_abs(j); J:zU,IIJ  
        for k = length(s):-1:1 {*I``T_+  
            p = (1-2*mod(s(k),2))* ...  q;][5  
                       prod(2:(n(j)-s(k)))/              ... 7M<'/s  
                       prod(2:s(k))/                     ... ZU%[guf  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... -K3^BZ HI  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); *=I}Qh(1  
            idx = (pows(k)==rpowers); |='z{WS  
            y(:,j) = y(:,j) + p*rpowern(:,idx); c5D)   
        end @8pp EFw  
         W)f/0QX}W  
        if isnorm r 0iK  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); S9~ +c  
        end Bx4w)9+3  
    end Z*= $8 e@  
    % END: Compute the Zernike Polynomials }%_ b$  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~3WF,mW  
    P<a)25be/  
    % Compute the Zernike functions: O#S;q5L@  
    % ------------------------------ /! "|_W|n  
    idx_pos = m>0; qfMo7e@6*  
    idx_neg = m<0; B=^)Ub5'  
    HV{wI1  
    z = y; h1B16)  
    if any(idx_pos) AN/;)wc  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); c_'OPJ  
    end 2;DuHO1  
    if any(idx_neg) sE Q=dcK  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); #\ X#w<\?  
    end y+c|vdW%  
    4O)1uF;  
    % 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) Af'" 6BS  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. hog=ut  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated WSp  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive ;U.hxh;+  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ;h*K}U  
    %   and THETA is a vector of angles.  R and THETA must have the same FrL]^59a  
    %   length.  The output Z is a matrix with one column for every P-value, LE9(fe) fe  
    %   and one row for every (R,THETA) pair. B"TAjB& *  
    % 7!EBH(,z  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike #t: S.A@  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) &:dH,  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 3L_\`Ia9  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 kt["m.  
    %   for all p. =}DR) 9  
    % LWz&YF#T-  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ,!Z *5  
    %   Zernike functions (order N<=7).  In some disciplines it is 'E/^8md>  
    %   traditional to label the first 36 functions using a single mode w[S pw<Z  
    %   number P instead of separate numbers for the order N and azimuthal qB0E_y)a  
    %   frequency M. ?fUlgQ }N  
    % WJkZ!O$"j  
    %   Example: 19Mu61  
    % 'B 43_  
    %       % Display the first 16 Zernike functions `_`QxM  
    %       x = -1:0.01:1; :U3kW8;UMP  
    %       [X,Y] = meshgrid(x,x); vd 0ljA  
    %       [theta,r] = cart2pol(X,Y); .<x&IJ /  
    %       idx = r<=1;  D#il*  
    %       p = 0:15; GT}F9F~  
    %       z = nan(size(X)); XWS%zLaK  
    %       y = zernfun2(p,r(idx),theta(idx)); ,9}h  
    %       figure('Units','normalized') 723bkJw V  
    %       for k = 1:length(p) #\.,?A}9  
    %           z(idx) = y(:,k); JORGj0v  
    %           subplot(4,4,k) Jq&uF*!  
    %           pcolor(x,x,z), shading interp .TND  a&  
    %           set(gca,'XTick',[],'YTick',[]) zr+zhpp  
    %           axis square u09:Z{tL;@  
    %           title(['Z_{' num2str(p(k)) '}']) >L;eO'D  
    %       end ]b7zJUz  
    % E*V`":efS  
    %   See also ZERNPOL, ZERNFUN. K1r#8Q!t  
    @eD):Y  
    %   Paul Fricker 11/13/2006 ~sl{|E  
    b\vKJ2  
    h|VeG3H  
    % Check and prepare the inputs: F)&@P-9+  
    % ----------------------------- (@<lRA ^  
    if min(size(p))~=1 5!DBmAB  
        error('zernfun2:Pvector','Input P must be vector.') P9^-6;'Y  
    end p^%YBY#,H  
     -xSA  
    if any(p)>35 7 %Oa;]|  
        error('zernfun2:P36', ... .S(TxksCz  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... m?pstuUK(  
               '(P = 0 to 35).']) ,SynnE68  
    end *1p|5!4c  
    KIui(n#/  
    % Get the order and frequency corresonding to the function number: Co (.:z~  
    % ---------------------------------------------------------------- /y _O 4  
    p = p(:); 5p<ItU$pnL  
    n = ceil((-3+sqrt(9+8*p))/2); e+$p9k~  
    m = 2*p - n.*(n+2); k2c}3 MeP  
    k7?N ?7w  
    % Pass the inputs to the function ZERNFUN: S M0~fAtE  
    % ---------------------------------------- J pFfzb  
    switch nargin #!aN{nK0  
        case 3 gVq;m>\|F  
            z = zernfun(n,m,r,theta); UDL!43K  
        case 4 x(hE3S#+  
            z = zernfun(n,m,r,theta,nflag); e,F1Xi #d  
        otherwise Q1O}ly}JS  
            error('zernfun2:nargin','Incorrect number of inputs.') ,k{#S?:b  
    end @.b+av4J  
    iF-6Y0~8  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) 1T y<\bZ=  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. }4Tc  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of bSfpbo4(  
    %   order N and frequency M, evaluated at R.  N is a vector of `tHvD=`m.  
    %   positive integers (including 0), and M is a vector with the _A+s)]}  
    %   same number of elements as N.  Each element k of M must be a !lf|7  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) %MrWeYd1  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is OUeyklw  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix MaRi+3F  
    %   with one column for every (N,M) pair, and one row for every 73V|6tmgY  
    %   element in R. /4a._@1h[y  
    % rqdE6y+^  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- dRj|g  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is R A KFU  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to :p]'32FA!  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 M,/mE~  
    %   for all [n,m]. u=/{cOJI6  
    % (yF:6$:#  
    %   The radial Zernike polynomials are the radial portion of the *pAV2V(!23  
    %   Zernike functions, which are an orthogonal basis on the unit #0ETY\}ZD  
    %   circle.  The series representation of the radial Zernike ] 8Q4BW  
    %   polynomials is \hBG<nH{0  
    % nQ q=7Gu  
    %          (n-m)/2 `S6x<J&T\/  
    %            __ RRRCS]y7$t  
    %    m      \       s                                          n-2s 3gv|9T  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r <\NY<QIwFw  
    %    n      s=0 ?Cl%{2omO  
    % &d"G/6  
    %   The following table shows the first 12 polynomials. .q9 $\wM/  
    % ( M7pT  
    %       n    m    Zernike polynomial    Normalization -i)ZQCE  
    %       --------------------------------------------- D+>4AqG  
    %       0    0    1                        sqrt(2) y^%n'h{  
    %       1    1    r                           2 R{ a"Y$  
    %       2    0    2*r^2 - 1                sqrt(6) 8-BflejX  
    %       2    2    r^2                      sqrt(6) W_kHj}dj,p  
    %       3    1    3*r^3 - 2*r              sqrt(8) p1&b!*o-&  
    %       3    3    r^3                      sqrt(8) BReJ!|{m}  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) kKAP"'v  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) (vb SM}P  
    %       4    4    r^4                      sqrt(10) f>W -  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) _(h&7P9  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) K{[%7AM  
    %       5    5    r^5                      sqrt(12) 'R c,Mq'  
    %       --------------------------------------------- c17_2 @N  
    % ~NQ72wph{  
    %   Example: NMa} <  
    % TMig-y*[  
    %       % Display three example Zernike radial polynomials 73xAG1D$r  
    %       r = 0:0.01:1; 0URji~?|x  
    %       n = [3 2 5]; |962G1.  
    %       m = [1 2 1]; 5<UVD:~z  
    %       z = zernpol(n,m,r); huJ&]"C  
    %       figure .u4 W /  
    %       plot(r,z) f ` R/ i  
    %       grid on KTP8?Q"n0  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') Jh ]i]7r  
    % G5C I<KRK#  
    %   See also ZERNFUN, ZERNFUN2. 13@|w1/Z  
    m06ALD_  
    % A note on the algorithm. C}_ ojcR  
    % ------------------------ .w0s%T,8}^  
    % The radial Zernike polynomials are computed using the series Q aS\(_  
    % representation shown in the Help section above. For many special MO n  
    % functions, direct evaluation using the series representation can a>GyO&+Dkg  
    % produce poor numerical results (floating point errors), because zxC#0@qX07  
    % the summation often involves computing small differences between k~jP'aD  
    % large successive terms in the series. (In such cases, the functions 9D7+[`r(-  
    % are often evaluated using alternative methods such as recurrence  \4v]7SV  
    % relations: see the Legendre functions, for example). For the Zernike 8xccp4  
    % polynomials, however, this problem does not arise, because the fp+gyTnd3  
    % polynomials are evaluated over the finite domain r = (0,1), and _i20|v   
    % because the coefficients for a given polynomial are generally all b)=[1g/=L  
    % of similar magnitude. k} |   
    % RLLTw ?]$  
    % ZERNPOL has been written using a vectorized implementation: multiple T`5bZu^c  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Bh;7C@dq  
    % values can be passed as inputs) for a vector of points R.  To achieve OoA|8!CFa  
    % this vectorization most efficiently, the algorithm in ZERNPOL hFH*B~*:#  
    % involves pre-determining all the powers p of R that are required to X22[tqg;&  
    % compute the outputs, and then compiling the {R^p} into a single yF"1#{*y  
    % matrix.  This avoids any redundant computation of the R^p, and %?p1d!  
    % minimizes the sizes of certain intermediate variables. yuat" Pg  
    % i*#-I3  
    %   Paul Fricker 11/13/2006 Z@]e{zO  
    $shoasSuI  
    xHz[t6;4;  
    % Check and prepare the inputs: X pBj%e:  
    % ----------------------------- 4}4Pyjh  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) %F-ZN^R  
        error('zernpol:NMvectors','N and M must be vectors.') xwJH(_-  
    end 2vAQ  
    F W/W%^  
    if length(n)~=length(m) :'~ Y  
        error('zernpol:NMlength','N and M must be the same length.') ( 5tvfz%  
    end *#tJM.Z  
    Y#u}tE d  
    n = n(:); gx\&_) w N  
    m = m(:); W9D86]3Y  
    length_n = length(n); r=X}%~_8X  
    HN&Z2v   
    if any(mod(n-m,2)) 2JJ"O|Ibz  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') }M3fmAP}  
    end [^U#Qj)hL  
    763v  
    if any(m<0) gG-BVl"59  
        error('zernpol:Mpositive','All M must be positive.') Z; A`oKd  
    end YN!>}  
    -Xxqm%([71  
    if any(m>n) Axe8n1*y  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') \H=&`?  
    end bpKZ3}U  
    nij!1z|M  
    if any( r>1 | r<0 ) `eIenA  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') &:, dJ  
    end ?sMP~RHQ  
    rz@=pR :  
    if ~any(size(r)==1) b+f'[;  
        error('zernpol:Rvector','R must be a vector.') lJE93rXU  
    end LAd\Tvms  
    ZE2$I^DY-  
    r = r(:); 20Z8HwQi  
    length_r = length(r); a^=-Mp  
    AO=h 23ZI  
    if nargin==4 BI $   
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); TW[_Ko86  
        if ~isnorm /XhIx\40 l  
            error('zernpol:normalization','Unrecognized normalization flag.') )tl.s)"N  
        end ,:Lb7bFv>  
    else (Nx;0"5IX  
        isnorm = false; OU/MiyP2  
    end qeL5D*  
    *X, /7C   
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `.x Fiyc  
    % Compute the Zernike Polynomials m<0&~rg   
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fv Jd8kV  
    ,B0_MDA +  
    % Determine the required powers of r: OujCb^Rm  
    % ----------------------------------- {6~l$  
    rpowers = []; D,g1<:<  
    for j = 1:length(n) R0L&*Bjm  
        rpowers = [rpowers m(j):2:n(j)]; DBT&DS  
    end pGK;1gVj  
    rpowers = unique(rpowers); ~R &;v3  
    kn>$lTHQ  
    % Pre-compute the values of r raised to the required powers, 86\S?=J-b  
    % and compile them in a matrix: {WPobP"  
    % ----------------------------- T+<A`k: -  
    if rpowers(1)==0 Fm # w2o  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); tWoh''@#  
        rpowern = cat(2,rpowern{:}); |'{zri|A"  
        rpowern = [ones(length_r,1) rpowern]; TH$N5w%  
    else $.GOZqMs  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ;\\@q"n%<  
        rpowern = cat(2,rpowern{:}); vsr[ur[eP  
    end sH?/E6  
    HfvTxaK  
    % Compute the values of the polynomials: S}ECW,K  
    % -------------------------------------- !V,{_(LT  
    z = zeros(length_r,length_n); YBP:q2H  
    for j = 1:length_n stk9Ah  
        s = 0:(n(j)-m(j))/2; ?zsB6B?;  
        pows = n(j):-2:m(j); =?s 3iP  
        for k = length(s):-1:1 8kIR y   
            p = (1-2*mod(s(k),2))* ... 'qF#<1&  
                       prod(2:(n(j)-s(k)))/          ... ty*@7g0k  
                       prod(2:s(k))/                 ... X0M1(BJgGo  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... n Ml%'[u  
                       prod(2:((n(j)+m(j))/2-s(k))); 0#YX=vjX7  
            idx = (pows(k)==rpowers); jD`d#R  
            z(:,j) = z(:,j) + p*rpowern(:,idx); O=dJi9;`#_  
        end {nvLPUL  
         f4guz  
        if isnorm sPb=82~z  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); =pk)3<GwF  
        end %bD}m!  
    end @bnG:np  
    {!K-E9_,S  
    % 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)  {[(pWd%J  
    X-B8MoG|  
    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。 %So] 3;'  
    dZGbC9  
    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)