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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 +F*h\4ry#  
    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! zUq(bD  
     
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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:2LqPC  
    function z = zernfun(n,m,r,theta,nflag) Uf MQ?(,  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. )5n:UD{f[#  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N (UCCEQq5  
    %   and angular frequency M, evaluated at positions (R,THETA) on the jFip-=T{4  
    %   unit circle.  N is a vector of positive integers (including 0), and /nv+*+Q?d  
    %   M is a vector with the same number of elements as N.  Each element eiXl"R^  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) c,O;B_}M]  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, bI`JG:^b  
    %   and THETA is a vector of angles.  R and THETA must have the same \&~YFjB  
    %   length.  The output Z is a matrix with one column for every (N,M) *Mb'y d/|  
    %   pair, and one row for every (R,THETA) pair. #eX<=H]  
    % R.DUfU"gp  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 6nR EuT'k  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), A3*(c3  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral X8ZO } X  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3rd8mh&l  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized M2c7 |  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. &=kb>*  
    % \ \Tz'>[\  
    %   The Zernike functions are an orthogonal basis on the unit circle. <EcxNj1  
    %   They are used in disciplines such as astronomy, optics, and e ;^}@X  
    %   optometry to describe functions on a circular domain. ,7k-LAA  
    % z"mpw mv5  
    %   The following table lists the first 15 Zernike functions. KV8<'g+2?  
    % \WbQS#Z9  
    %       n    m    Zernike function           Normalization s~bi#U;dF  
    %       -------------------------------------------------- 8Lgm50bs  
    %       0    0    1                                 1 w^("Pg`  
    %       1    1    r * cos(theta)                    2 0igB pHS  
    %       1   -1    r * sin(theta)                    2 ly35n`  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) r ;MFVj{  
    %       2    0    (2*r^2 - 1)                    sqrt(3) t72rCq QC  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) [~X&J#  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) $4~Z]-38#A  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ^\kH^   
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) $d!Vxm  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) >\3\&[#"  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) a s('ZD.9  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) uubIL +  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 8ZqLG a]  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) t1"#L_<e  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Zd%wX<hU"  
    %       -------------------------------------------------- biBMd(6  
    % &:IcwD&  
    %   Example 1: gujP{Z  
    % .Gvk5Wn  
    %       % Display the Zernike function Z(n=5,m=1) hqlQ-aytS  
    %       x = -1:0.01:1; i;s;:{cn  
    %       [X,Y] = meshgrid(x,x); Xx%<rsA>F  
    %       [theta,r] = cart2pol(X,Y); Jj\lF*B  
    %       idx = r<=1; &6 <a<S  
    %       z = nan(size(X)); [ p~,;%  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); H0sTL#/L\  
    %       figure QxGcRlpLK  
    %       pcolor(x,x,z), shading interp NdSuOkwwt  
    %       axis square, colorbar PgGUs4[  
    %       title('Zernike function Z_5^1(r,\theta)') a@ <-L  
    % ;gSRpTS:  
    %   Example 2: >C!^%e;m  
    % Hk@Gkx_  
    %       % Display the first 10 Zernike functions { V[}#Mf  
    %       x = -1:0.01:1; tq3Rc}  
    %       [X,Y] = meshgrid(x,x); *8m['$oyV  
    %       [theta,r] = cart2pol(X,Y); 'P" i9j  
    %       idx = r<=1;  o kA<  
    %       z = nan(size(X)); l"1D' Hk  
    %       n = [0  1  1  2  2  2  3  3  3  3]; CswKT 9  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; a!-J=\>9  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 1^E5VG1[  
    %       y = zernfun(n,m,r(idx),theta(idx)); Mqvo j7  
    %       figure('Units','normalized')  X(X[v]  
    %       for k = 1:10 6}e*!,2Xj  
    %           z(idx) = y(:,k); 8.8t$  
    %           subplot(4,7,Nplot(k)) *o4a<.hd2  
    %           pcolor(x,x,z), shading interp FVBAB>   
    %           set(gca,'XTick',[],'YTick',[]) x.wDA3ys  
    %           axis square Up'#OkTx  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])  k4dC  
    %       end S\< i`q  
    % dt,Z^z+" E  
    %   See also ZERNPOL, ZERNFUN2. ^]D1':  
    |\?u-O3  
    %   Paul Fricker 11/13/2006 $--+M D29Q  
    w4Df?)Z  
    ?6&8-zt1?  
    % Check and prepare the inputs: F;8Q`$n  
    % ----------------------------- -~lq <M  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) A)>#n)  
        error('zernfun:NMvectors','N and M must be vectors.') Es)|#0m\x@  
    end 1^X)vck  
    zU+q03l8Ur  
    if length(n)~=length(m) :op_J!;  
        error('zernfun:NMlength','N and M must be the same length.') 9jqsEd-SW  
    end /\h*v!:  
    Zx_ ^P:rL  
    n = n(:); 7 [1|(6$  
    m = m(:); Ec3tfcNhR  
    if any(mod(n-m,2)) 9 %4:eTcp  
        error('zernfun:NMmultiplesof2', ... z|D*ymz*EY  
              'All N and M must differ by multiples of 2 (including 0).') =urGs`\  
    end wN4#j}C  
    X_hDU~5{wC  
    if any(m>n) (BeJ,K7  
        error('zernfun:MlessthanN', ... -|KZOea  
              'Each M must be less than or equal to its corresponding N.') ,r;xH}tbi  
    end 'u;O2$  
    &k%>u[Bo  
    if any( r>1 | r<0 ) YnU)f@b#  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ,:A;4  
    end |oXd4  
    ][v]Nk  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) y"q>}5  
        error('zernfun:RTHvector','R and THETA must be vectors.') vBl:&99[/  
    end 60u_,@rV  
    a~$Y;C_#<  
    r = r(:); Lm2) 3;ei  
    theta = theta(:); 5HV+7zU5  
    length_r = length(r); ~<n.5q%Z  
    if length_r~=length(theta)  3}8o 9  
        error('zernfun:RTHlength', ... DI{*E  
              'The number of R- and THETA-values must be equal.') mA+:)?e5~  
    end ud$-A  
    3>@VPMi  
    % Check normalization: ^.!jD+=I  
    % -------------------- 71{jedT  
    if nargin==5 && ischar(nflag)  |50sGJE(  
        isnorm = strcmpi(nflag,'norm'); !EhKg)y=  
        if ~isnorm c Pf_B=  
            error('zernfun:normalization','Unrecognized normalization flag.') 4v hz`1  
        end Pa{  
    else src+z#  
        isnorm = false; Fds 11 /c7  
    end R/ZScOW[  
    % ERcFI]G  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +wmG5!%$|  
    % Compute the Zernike Polynomials ~E7IU<B  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% XH$r(@Z\7  
    $3g{9)}  
    % Determine the required powers of r: lFyDH{!  
    % ----------------------------------- S*V}1</L  
    m_abs = abs(m); |2u=3#Jp  
    rpowers = []; j,+]tHC-  
    for j = 1:length(n) tO3R&"{  
        rpowers = [rpowers m_abs(j):2:n(j)]; F`QViZ'n>#  
    end K%? g6j  
    rpowers = unique(rpowers); x1.S+:  
    p/HDG ^T:u  
    % Pre-compute the values of r raised to the required powers, !ka* rd  
    % and compile them in a matrix: rQVX^  
    % ----------------------------- 73D< wMgZF  
    if rpowers(1)==0 Lz'VQO1U=  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); '|zrzU=  
        rpowern = cat(2,rpowern{:}); 0<-E)\:[g  
        rpowern = [ones(length_r,1) rpowern]; bItcF$#!!!  
    else zl|z4j'Irc  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); GT&}Burl/n  
        rpowern = cat(2,rpowern{:}); c0gVW~I1  
    end #4msBax4  
    U\Wo&giP[  
    % Compute the values of the polynomials: #_wq#rF  
    % -------------------------------------- :eVZ5?F  
    y = zeros(length_r,length(n)); ){5Nod{}a  
    for j = 1:length(n) }oRBQP^&K  
        s = 0:(n(j)-m_abs(j))/2; ZNX38<3h  
        pows = n(j):-2:m_abs(j); h^yqrDyJ  
        for k = length(s):-1:1 pa[/6(  
            p = (1-2*mod(s(k),2))* ... qUkM No3  
                       prod(2:(n(j)-s(k)))/              ... N7+L@CC6T  
                       prod(2:s(k))/                     ... _5jT}I<k  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... _F;v3|`D@<  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Q!e560@  
            idx = (pows(k)==rpowers); ?BnU0R_r]  
            y(:,j) = y(:,j) + p*rpowern(:,idx); @Nek;xJ  
        end KhHFJo[8sf  
         "La;$7ds  
        if isnorm "]+g5G  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Xo34~V@(  
        end T }}2J/sj  
    end qz-QVY,  
    % END: Compute the Zernike Polynomials N T`S)P*?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~|V^IJZ22  
    j~Aq-8R=  
    % Compute the Zernike functions: h+FM?ct6}  
    % ------------------------------ f2i:I1 p("  
    idx_pos = m>0; sS>b}u+v#!  
    idx_neg = m<0; A9$x8x*Lt  
    tJ\ $%  
    z = y; /,Xl8<~#  
    if any(idx_pos) &]nx^C8V;  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); iVXt@[  
    end HC%Hbc~S_Q  
    if any(idx_neg) 7z b^Z]  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); xh;V4zK@`  
    end FZr/trP~  
    k6(7G@@}  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) vrnvv?HPrR  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ;CYoc4e  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated #De>EQ%  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive z5E%*]  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, /( Wq  
    %   and THETA is a vector of angles.  R and THETA must have the same T8XrmR&?PX  
    %   length.  The output Z is a matrix with one column for every P-value, Bm$|XS3cD  
    %   and one row for every (R,THETA) pair. `JySuP2~/  
    % {q-&!l|  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ".?4`@7F\  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) T ozx0??)  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) z*\_+u~u  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 b6D;98p  
    %   for all p. ;8sL  
    % B~p%pT S+  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 C8U3+ s  
    %   Zernike functions (order N<=7).  In some disciplines it is GWA"!~Hu  
    %   traditional to label the first 36 functions using a single mode ma.84~m  
    %   number P instead of separate numbers for the order N and azimuthal "4N&T#  
    %   frequency M. 6d-\+ t8  
    % xe@1H\7:  
    %   Example: ul~ux$a  
    % n5:uG'L\  
    %       % Display the first 16 Zernike functions w'S,{GW  
    %       x = -1:0.01:1; LOh2eZ"n  
    %       [X,Y] = meshgrid(x,x); .WSyL  
    %       [theta,r] = cart2pol(X,Y); '!HTE` Aj  
    %       idx = r<=1; I'/3_AX  
    %       p = 0:15; bJ ~H  
    %       z = nan(size(X)); WyN ;lId  
    %       y = zernfun2(p,r(idx),theta(idx)); `|dyT6V0I_  
    %       figure('Units','normalized') 3SVGx< ,2  
    %       for k = 1:length(p) M5dYcCDE  
    %           z(idx) = y(:,k); %Bs. XW,  
    %           subplot(4,4,k) pgU [di  
    %           pcolor(x,x,z), shading interp /0s1;?  
    %           set(gca,'XTick',[],'YTick',[]) spEdq}  
    %           axis square ci0A!wWD  
    %           title(['Z_{' num2str(p(k)) '}']) kK6O ZhLH  
    %       end G@]3EP  
    % hZ&KE78?  
    %   See also ZERNPOL, ZERNFUN. aJu&h2 G  
    d:=' Xs  
    %   Paul Fricker 11/13/2006 ){^J8]b7#  
    ++cS^ Lo  
    r&gvP|W%  
    % Check and prepare the inputs: @X==[gQ  
    % ----------------------------- NR4+&d  
    if min(size(p))~=1 w#A)B<Y/"  
        error('zernfun2:Pvector','Input P must be vector.') ~ao:9 ynY  
    end $y(;"hy  
    PX:#+bq1  
    if any(p)>35 djd/QAfSC  
        error('zernfun2:P36', ... 6u[fCGi%  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... w"hd_8cO  
               '(P = 0 to 35).']) ]Q^)9uE\D  
    end ;%e&6  
    M0V<Ay\%O  
    % Get the order and frequency corresonding to the function number: t{md&k4  
    % ---------------------------------------------------------------- f ,F X# _4  
    p = p(:); Vk2$b{VdF  
    n = ceil((-3+sqrt(9+8*p))/2); I2 [U#4n  
    m = 2*p - n.*(n+2); <c+.%ka  
    ?Ga8.0Z~KT  
    % Pass the inputs to the function ZERNFUN: X/5m}-6d]  
    % ---------------------------------------- C6!F6Stn]g  
    switch nargin bo0U  
        case 3 dlu*s(O"  
            z = zernfun(n,m,r,theta); O]N/(pe:d  
        case 4 fBSa8D3}`  
            z = zernfun(n,m,r,theta,nflag); d:kB Zrq  
        otherwise Bf{u:TCK  
            error('zernfun2:nargin','Incorrect number of inputs.') n`Q@<op  
    end j G8;p41  
     a_?sJ  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) wh$bDT Cj  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. l\<.*6r  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ;[<(4v$  
    %   order N and frequency M, evaluated at R.  N is a vector of j.E=WLKV*  
    %   positive integers (including 0), and M is a vector with the sJ6.3= c  
    %   same number of elements as N.  Each element k of M must be a A3]A5s6  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ^rWg:fb  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 4 m:h&^`N  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix (_FU3ZW!  
    %   with one column for every (N,M) pair, and one row for every &Ok1j0~~  
    %   element in R. -FZC|[is  
    % "\4W])30  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- F#Lo^ 8  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is yD+4YD  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ,54z9F`  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ] !/  
    %   for all [n,m]. L(y70T  
    % !XPjRdq  
    %   The radial Zernike polynomials are the radial portion of the zxR]+9Zh  
    %   Zernike functions, which are an orthogonal basis on the unit HP# SR';E  
    %   circle.  The series representation of the radial Zernike Af3|l  
    %   polynomials is @*z"Hi>4  
    % IO)B3,g  
    %          (n-m)/2 P6+ B!pY  
    %            __ *HoRYCL  
    %    m      \       s                                          n-2s ^Jp T8B}  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 4'QX1p  
    %    n      s=0 jTa\I&s,A  
    % hGtz[u#p  
    %   The following table shows the first 12 polynomials. ]]j^  
    % s6H.Q$3L  
    %       n    m    Zernike polynomial    Normalization wQ33Gc  
    %       --------------------------------------------- g=Z52y`N<  
    %       0    0    1                        sqrt(2) QjTSbHtH  
    %       1    1    r                           2 ibkB>n{(  
    %       2    0    2*r^2 - 1                sqrt(6) ifD WN*k6  
    %       2    2    r^2                      sqrt(6) "6B@V=d  
    %       3    1    3*r^3 - 2*r              sqrt(8) BH:A]#_{  
    %       3    3    r^3                      sqrt(8) |ADf~-AY  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) Oe5rRQ$O  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) o=rR^Z$G   
    %       4    4    r^4                      sqrt(10) ^{M$S0g|N  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) EGXvz)y  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) c p"K?)  
    %       5    5    r^5                      sqrt(12) Am, {Fj  
    %       --------------------------------------------- <6UXk[y  
    % sVHF\{<  
    %   Example: g,,wG k  
    % jQ{ @ol}n  
    %       % Display three example Zernike radial polynomials #^}H)>jWy  
    %       r = 0:0.01:1; Xg dBLb  
    %       n = [3 2 5]; q`xc h[H  
    %       m = [1 2 1]; C19N0=  
    %       z = zernpol(n,m,r); En\@d@j<u  
    %       figure Wga2).j6  
    %       plot(r,z) DNGyEC  
    %       grid on Y 9$jJ1V  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') T5:Q_o]  
    % 8pnD6Lp>  
    %   See also ZERNFUN, ZERNFUN2. od=hCQ1 >  
    yCQvo(V[F  
    % A note on the algorithm. OxHcoNrz  
    % ------------------------ C.?^] Y  
    % The radial Zernike polynomials are computed using the series ^lp=4C9  
    % representation shown in the Help section above. For many special ~4 fE`-O  
    % functions, direct evaluation using the series representation can H_&to3b(  
    % produce poor numerical results (floating point errors), because w)7y{ya$  
    % the summation often involves computing small differences between 7 yE\,  
    % large successive terms in the series. (In such cases, the functions 6kAAdy}ck  
    % are often evaluated using alternative methods such as recurrence \Oq2{S x\  
    % relations: see the Legendre functions, for example). For the Zernike Mt.Cj;h@^[  
    % polynomials, however, this problem does not arise, because the Y(UK:LZ'  
    % polynomials are evaluated over the finite domain r = (0,1), and ZID-~ 6  
    % because the coefficients for a given polynomial are generally all B_[efM<R$  
    % of similar magnitude. O#D{:H_dD>  
    % 2M x\D  
    % ZERNPOL has been written using a vectorized implementation: multiple cN@_5  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 3t-STk?  
    % values can be passed as inputs) for a vector of points R.  To achieve }H ~-oYMu  
    % this vectorization most efficiently, the algorithm in ZERNPOL d88A.Z3w  
    % involves pre-determining all the powers p of R that are required to (\ab%M   
    % compute the outputs, and then compiling the {R^p} into a single )0/9 L  
    % matrix.  This avoids any redundant computation of the R^p, and }u;K<<h:  
    % minimizes the sizes of certain intermediate variables. Jl_W6gY"Z  
    % bMK X9`*o  
    %   Paul Fricker 11/13/2006 f2e;N[D  
    d5^^h<'  
    Y%;J/4dd  
    % Check and prepare the inputs: qur2t8gnxq  
    % ----------------------------- [q|W*[B:@  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) v~ SM"ky#  
        error('zernpol:NMvectors','N and M must be vectors.') e@P(+.Ke  
    end +,,(8=5 g  
    lp(2"$nQ  
    if length(n)~=length(m) rt^~ I \V  
        error('zernpol:NMlength','N and M must be the same length.') _eGYwBm  
    end r R6}  
    O50_qu33ju  
    n = n(:); }||u {[  
    m = m(:); LK DfV  
    length_n = length(n); X):7#x@uy  
    >ZJ]yhbhK  
    if any(mod(n-m,2)) Hs)Cf)8u  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Nvd(?+c  
    end w=#'8ZuU  
    'LMj.#A<g  
    if any(m<0) @_kF&~  
        error('zernpol:Mpositive','All M must be positive.') lpXGsK H2  
    end [gT}<W  
    pVc+}Wzh  
    if any(m>n) j&X&&=   
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') fG1iq<~  
    end b- - tl@H  
    eA7 Iv{M  
    if any( r>1 | r<0 ) +ydd"`  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 3RaW\cWzg  
    end OMK,L:poC  
    'i%r  
    if ~any(size(r)==1) WkXgz6 P  
        error('zernpol:Rvector','R must be a vector.') x|m9?[ !_  
    end HQ@g6  
    joI)6c  
    r = r(:); >Lo\?X~  
    length_r = length(r); VgVDTWs7  
    ~vA{I%z5~  
    if nargin==4 "}v.>L<P  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 0Fb ];:a  
        if ~isnorm OTF/Pu$  
            error('zernpol:normalization','Unrecognized normalization flag.') '^TeV=  
        end /K|(O^nw  
    else p{5m5x  
        isnorm = false; sQ`G'<!  
    end @dv8 F "v  
    2{?]W/&fS  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f|,Kh1{e  
    % Compute the Zernike Polynomials @ZU$W9g  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7_\F$bp`  
    O2>c|=#  
    % Determine the required powers of r: u{DEOhtI4  
    % ----------------------------------- s $Vv  
    rpowers = []; +51heuu[o  
    for j = 1:length(n) cTGd<  
        rpowers = [rpowers m(j):2:n(j)]; 36{GZDGQ  
    end Wu 0:X*>}p  
    rpowers = unique(rpowers); ./ {79  
    U5kKT.M  
    % Pre-compute the values of r raised to the required powers, =dPokLXn  
    % and compile them in a matrix: 0{b} 1D  
    % ----------------------------- U,P>P+\@  
    if rpowers(1)==0 {V/>5pz4e  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); SO6)FiPy!n  
        rpowern = cat(2,rpowern{:}); 3aq'JVq   
        rpowern = [ones(length_r,1) rpowern]; Y5tyFi#w[  
    else f(pq`v^-n  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3A`Gx#  
        rpowern = cat(2,rpowern{:}); GCJ[xn(_  
    end B,\VLX  
    x.-d)]a!  
    % Compute the values of the polynomials: ?)1Y|W'Rv  
    % -------------------------------------- aG]>{(~cL  
    z = zeros(length_r,length_n); /-p!|T}w  
    for j = 1:length_n ZMQ=D!kT  
        s = 0:(n(j)-m(j))/2; . f_ A%  
        pows = n(j):-2:m(j); GS>[A b+  
        for k = length(s):-1:1 Jx5`0?  
            p = (1-2*mod(s(k),2))* ... 'O(=Pz  
                       prod(2:(n(j)-s(k)))/          ... yIL=jzm`7  
                       prod(2:s(k))/                 ... j"nOxs  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 83c2y;|8  
                       prod(2:((n(j)+m(j))/2-s(k))); W#bYz{s.  
            idx = (pows(k)==rpowers); ]"_c-=  
            z(:,j) = z(:,j) + p*rpowern(:,idx); O#U maNj/  
        end Qel)%|dOn  
         m'N AM%$}J  
        if isnorm -qj[ck(y  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); (j'\h/  
        end !uGfS' Vl  
    end W q<t+E[  
    w6s[|i)&  
    % 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)  =,*/Ph&  
    B[mZQ&Gz`a  
    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。 V+y|C[A F  
    W ~MNst?  
    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)