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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: d`L+tcc  
    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! ztgSd8GGE  
     
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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 "4<RMYQ  
    function z = zernfun(n,m,r,theta,nflag) Yev] Lp  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 4&r[`gL  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ONH!ms(kb  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 6y)TXp  
    %   unit circle.  N is a vector of positive integers (including 0), and @i'RIL}  
    %   M is a vector with the same number of elements as N.  Each element 9E'fM  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ^HgQ"dD <  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, "9'~6b  
    %   and THETA is a vector of angles.  R and THETA must have the same $5yH(Z[[  
    %   length.  The output Z is a matrix with one column for every (N,M) )a AKO`  
    %   pair, and one row for every (R,THETA) pair. q_bE?j{  
    % %PRG;kR  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike wzVx16Rvc  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 2&MIt(\-  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 2BZYC5jy  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ROlef;/A  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ~b}a|K  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. S96H`kedZo  
    % ~P8 6=Vw  
    %   The Zernike functions are an orthogonal basis on the unit circle. f4UnLig  
    %   They are used in disciplines such as astronomy, optics, and m?'H 7cFR  
    %   optometry to describe functions on a circular domain. a\;1%2a  
    % cyrVz4_a  
    %   The following table lists the first 15 Zernike functions. +Z> Y//  
    % $mdmuUIy-3  
    %       n    m    Zernike function           Normalization GKT2x '(e  
    %       -------------------------------------------------- YXz*B5R  
    %       0    0    1                                 1 <Hd8Jd4f  
    %       1    1    r * cos(theta)                    2 vT/e&8w  
    %       1   -1    r * sin(theta)                    2 BIGln`;,f  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) B7 PkCS&X  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Hdvtgss!  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) \9"   
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) b-? wJSf|  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) q^!_jMN5  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ],`xd_=]=  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ]oT8H?%*Y  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) eTg8I/ )%B  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) R[_Q}W'HG  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) p7{2/m j  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6/-]  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) !8g419Yg  
    %       -------------------------------------------------- uqz HS>GM  
    % E_q/*}]pE  
    %   Example 1: ))T@U?r  
    % r ^=rs!f@  
    %       % Display the Zernike function Z(n=5,m=1) pbCj ^  
    %       x = -1:0.01:1; bG0 |+k3O  
    %       [X,Y] = meshgrid(x,x); 1G}f83yR  
    %       [theta,r] = cart2pol(X,Y); V}3'0  
    %       idx = r<=1; z*V 8l*  
    %       z = nan(size(X)); 5!QT }Um  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 6a*?m{  
    %       figure *wk?{ U  
    %       pcolor(x,x,z), shading interp 3Pu8IXW  
    %       axis square, colorbar A<h^.{  
    %       title('Zernike function Z_5^1(r,\theta)') z`"*60b  
    % v)p'0F#6A  
    %   Example 2: (k^% j  
    % V` T l$EF  
    %       % Display the first 10 Zernike functions 8Y]% S9.  
    %       x = -1:0.01:1; mjQZ"h0  
    %       [X,Y] = meshgrid(x,x); a(J@]X>'  
    %       [theta,r] = cart2pol(X,Y); vjL +fH<0:  
    %       idx = r<=1; <x8I<K  
    %       z = nan(size(X)); Ucx"\/"  
    %       n = [0  1  1  2  2  2  3  3  3  3]; hW!2C6  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; (*M*muk  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; eMV{rFmT  
    %       y = zernfun(n,m,r(idx),theta(idx)); ZaBmH|k  
    %       figure('Units','normalized') yhkKakg,)  
    %       for k = 1:10 HA J[Y3d<  
    %           z(idx) = y(:,k); <{+U- ^rzR  
    %           subplot(4,7,Nplot(k)) zfD@/kU  
    %           pcolor(x,x,z), shading interp GlHP`&;UH  
    %           set(gca,'XTick',[],'YTick',[]) =F2`X#x_j  
    %           axis square 4Q!|fn0Sv  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) pO/vD~C>  
    %       end HUAbq }  
    % y| 0!sNg  
    %   See also ZERNPOL, ZERNFUN2. QuP)j1"X  
    i[?VF\Y(  
    %   Paul Fricker 11/13/2006 W ]$/qyc&J  
    V'tqsKQ!  
    Q:j~ kutS|  
    % Check and prepare the inputs: K.=5p/^a  
    % ----------------------------- !{4'=+  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) cij8'( "+!  
        error('zernfun:NMvectors','N and M must be vectors.') [dR#!"6t  
    end fP[S.7F+No  
    2z.~K&+x  
    if length(n)~=length(m) a)4%sX*I  
        error('zernfun:NMlength','N and M must be the same length.') &"?99E>  
    end 1S(n3(KRk$  
    ')}itS8  
    n = n(:); ^D]J68)#a  
    m = m(:); !g`I*ZE+e  
    if any(mod(n-m,2)) RH:vd|q+  
        error('zernfun:NMmultiplesof2', ... v k= |TE  
              'All N and M must differ by multiples of 2 (including 0).') ?K;l 5$?%  
    end 6yBd9=3K  
    Z)IF3{*  
    if any(m>n) Fg 8lX9L  
        error('zernfun:MlessthanN', ... ,a?oGi  
              'Each M must be less than or equal to its corresponding N.') FrUqfTi+W  
    end sYo&@~T  
    `a:3S@n(}  
    if any( r>1 | r<0 ) #s81 k@#X  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') V%)Tu{L  
    end $}r.fji,c  
     }JWkV1  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) o<iU;15  
        error('zernfun:RTHvector','R and THETA must be vectors.') *sZH3:  
    end ZjMnGRP  
    M| j=J{r  
    r = r(:); t1U+7nM  
    theta = theta(:); :s&dn%5N"  
    length_r = length(r); ;2\6U;  
    if length_r~=length(theta) %k32:qe  
        error('zernfun:RTHlength', ... 'e' p`*  
              'The number of R- and THETA-values must be equal.') ,~&HL7 v  
    end a;^lOU|L{  
    $i6z)]rjg  
    % Check normalization: $.kJBRgV*  
    % -------------------- 8PH4v\tJEK  
    if nargin==5 && ischar(nflag) uDXV@;6<  
        isnorm = strcmpi(nflag,'norm'); '2i !RT-  
        if ~isnorm L'S,=NYXY  
            error('zernfun:normalization','Unrecognized normalization flag.') b=xn(HE8|  
        end 9( q(;|;Hp  
    else 23i2yT  
        isnorm = false; NU?<bIQ  
    end T]Ai{@i  
    &mmaoWR  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :>;F4gGVG  
    % Compute the Zernike Polynomials ![_0GFbT  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;5.&TQT  
    /z4c>)fV  
    % Determine the required powers of r: M*ZN]9{^.  
    % ----------------------------------- Q2- lHn^L:  
    m_abs = abs(m); .kg 3>*  
    rpowers = []; dH;2OWM  
    for j = 1:length(n) w6cl3J&  
        rpowers = [rpowers m_abs(j):2:n(j)]; oC49c~`8  
    end OVDuF&0  
    rpowers = unique(rpowers); 6(d6Uwc`  
    IF0!@f  
    % Pre-compute the values of r raised to the required powers, QwWd"Of  
    % and compile them in a matrix: ed#fDMXGQ%  
    % ----------------------------- Vez8 ~r3  
    if rpowers(1)==0 fxPg"R!1i  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); #8"oqqYi  
        rpowern = cat(2,rpowern{:}); -Q@f),  
        rpowern = [ones(length_r,1) rpowern]; I ]ZZN6"  
    else 15Mtlb  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); }8Y! -qX  
        rpowern = cat(2,rpowern{:}); } #H,oy;Dz  
    end )/>BgXwH  
    O%\cRn8m  
    % Compute the values of the polynomials: ftxL-7y%  
    % -------------------------------------- h>\C2Q  
    y = zeros(length_r,length(n)); (b f IS  
    for j = 1:length(n) e6j1Fa9  
        s = 0:(n(j)-m_abs(j))/2; vnvpb! @Q  
        pows = n(j):-2:m_abs(j); J@I>m N1\  
        for k = length(s):-1:1 n >y,{"J{  
            p = (1-2*mod(s(k),2))* ... 1$vGQ  
                       prod(2:(n(j)-s(k)))/              ... Ia#!T"]@W6  
                       prod(2:s(k))/                     ... C(G.yd  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 49QsT5b)  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); k9rws  
            idx = (pows(k)==rpowers); S"h;u=5it  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ct3i^,i  
        end /\/^= j  
         R<&Euph  
        if isnorm cWkg.ri-x  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ;b0Q%TDh  
        end M co:eE  
    end 2 L4[~>  
    % END: Compute the Zernike Polynomials _:m70%i  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% rc8HZ  
    Qjj }k)  
    % Compute the Zernike functions: L K #A  
    % ------------------------------  +x 3x  
    idx_pos = m>0; 3$ ! QP N  
    idx_neg = m<0; `Ow]@flLI  
    k2D*`\ D  
    z = y; I_ZJnu<  
    if any(idx_pos) PuP"( M  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); D", L.  
    end %1A8m-u]M  
    if any(idx_neg) ,zoHmV1Wd+  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); %]LoR$|Y  
    end e`4mrBtz|  
    #'&-S@/nQs  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) M#^q <K %  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Gk5'|s  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated GcRH$,<XG  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive [ 6VM4l"  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, <o()14  
    %   and THETA is a vector of angles.  R and THETA must have the same Mt4]\pMUb  
    %   length.  The output Z is a matrix with one column for every P-value, 0t!ZMH  
    %   and one row for every (R,THETA) pair. 7{U[cG+a#  
    % &pI\VIx ?  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike .K~V DUu  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) :j+E]|d(~6  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) S<]k0bC  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 x}a?B  
    %   for all p. aN"YEL>w  
    % [Uk cG9  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 eJqx,W5MK]  
    %   Zernike functions (order N<=7).  In some disciplines it is ;VCV%=W<  
    %   traditional to label the first 36 functions using a single mode o Rk'I  
    %   number P instead of separate numbers for the order N and azimuthal xqk(id\&  
    %   frequency M. bzpi7LKN  
    % ^e "4@O"  
    %   Example: rs4:jS$)  
    % f|~'(~Sr  
    %       % Display the first 16 Zernike functions WClprSl8  
    %       x = -1:0.01:1; }k AE  
    %       [X,Y] = meshgrid(x,x); K'8o'S_bF  
    %       [theta,r] = cart2pol(X,Y); ShRMzU  
    %       idx = r<=1; 7202N?a {  
    %       p = 0:15; oBai9 [+  
    %       z = nan(size(X)); si%V63^lN  
    %       y = zernfun2(p,r(idx),theta(idx)); "nJMS6HJ[  
    %       figure('Units','normalized') aEQrBs  
    %       for k = 1:length(p) $m>( kd1  
    %           z(idx) = y(:,k); g0~m[[  
    %           subplot(4,4,k) +:#g6(P]  
    %           pcolor(x,x,z), shading interp r_ 9"^Er  
    %           set(gca,'XTick',[],'YTick',[]) ;,/G*`81B  
    %           axis square .}R'(gN\6  
    %           title(['Z_{' num2str(p(k)) '}']) /q]fG  
    %       end !N:w?zsp  
    % VKXB)-'L  
    %   See also ZERNPOL, ZERNFUN. "}uu-5]3  
    RionKiN  
    %   Paul Fricker 11/13/2006 '6WZi|(a  
    qyE*?73W  
    ciH TnC  
    % Check and prepare the inputs: T/L\|_:'  
    % ----------------------------- ]~m=b` o  
    if min(size(p))~=1 l9 rN!Q|  
        error('zernfun2:Pvector','Input P must be vector.') ,({% t  
    end {yBd{x<>/  
    5',b~Pp  
    if any(p)>35 2JHF*zvO-  
        error('zernfun2:P36', ... 78mJ3/?rC  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... tN)Vpb\J  
               '(P = 0 to 35).']) )/v`k>E  
    end 4=ha$3h$  
    { 6*UtG  
    % Get the order and frequency corresonding to the function number: B8-Y)u1G  
    % ---------------------------------------------------------------- %+$!ctn  
    p = p(:); No/D"S#  
    n = ceil((-3+sqrt(9+8*p))/2); i\L7z)u  
    m = 2*p - n.*(n+2); ~;uW) [  
    KU0;}GSNX}  
    % Pass the inputs to the function ZERNFUN: M<)Vtn  
    % ---------------------------------------- ;xE1#ZT  
    switch nargin Eid~4a  
        case 3 Uv3Fe%>  
            z = zernfun(n,m,r,theta); k:E+]5  
        case 4 EO.}{1m=hx  
            z = zernfun(n,m,r,theta,nflag); cy_zEJjbD  
        otherwise M#]URS2h<O  
            error('zernfun2:nargin','Incorrect number of inputs.') ) ]]PhGX~  
    end 1\Vp[^#Vx  
    T+<OlXpL  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) h/F,D_O>ZO  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 2(sq*!tX  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of L%O( I  
    %   order N and frequency M, evaluated at R.  N is a vector of d#G H4+C  
    %   positive integers (including 0), and M is a vector with the #AUz.WHD  
    %   same number of elements as N.  Each element k of M must be a (|<.7K N  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) VN9C@ ;'$  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is `JL&x|q o  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Fs EPM"&?h  
    %   with one column for every (N,M) pair, and one row for every kUG3_ *1 .  
    %   element in R. K2R[u#Q  
    % bfjtNF*^  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- SLG3u;Ab  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is & ]/Z~Vt  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to <+c6CM$#}V  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1  $VCWc#  
    %   for all [n,m]. SQB[d3f  
    % jFBnP,WQ  
    %   The radial Zernike polynomials are the radial portion of the c);(+b  
    %   Zernike functions, which are an orthogonal basis on the unit UE9r1g`z  
    %   circle.  The series representation of the radial Zernike `o~9a N  
    %   polynomials is }Myi0I<  
    % <v)Ai;l,  
    %          (n-m)/2 ,4wZ/r> d  
    %            __ S?5z  
    %    m      \       s                                          n-2s wb }W;C@  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r v5 yOh5  
    %    n      s=0 FFNv'\)  
    % b)A$lP%`  
    %   The following table shows the first 12 polynomials. v?7.)2XcX  
    % /L1qdkG  
    %       n    m    Zernike polynomial    Normalization v?\bvg\E  
    %       --------------------------------------------- TY=BP!s  
    %       0    0    1                        sqrt(2) K/y#hP  
    %       1    1    r                           2 l*]L=rC  
    %       2    0    2*r^2 - 1                sqrt(6) uL!{xuN  
    %       2    2    r^2                      sqrt(6) TWC^M{e  
    %       3    1    3*r^3 - 2*r              sqrt(8) PT|W{RlNl  
    %       3    3    r^3                      sqrt(8) zX!zG<<K  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) SLc6 ]?  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) bQ=R,  
    %       4    4    r^4                      sqrt(10) R4"g? e  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Zjt3U;Y  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) %cD7}o:u  
    %       5    5    r^5                      sqrt(12) $H/3t?6h`  
    %       --------------------------------------------- L!-@dz  
    % `?Wak =]g  
    %   Example: 5w`v 3o  
    % k[N46=u  
    %       % Display three example Zernike radial polynomials & h9ji[  
    %       r = 0:0.01:1; 3_IuK 6K2  
    %       n = [3 2 5]; X;T(?,,  
    %       m = [1 2 1]; ]7ROCJ;  
    %       z = zernpol(n,m,r); ,k/*f+t  
    %       figure @7%nMTZ@&v  
    %       plot(r,z) S@zsPzw  
    %       grid on L&lNpMT  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') pHY~_^B4&  
    % 8p7Uvn+m*  
    %   See also ZERNFUN, ZERNFUN2. r}9qK%C G.  
    g^1M]1.f  
    % A note on the algorithm. QdG_zK>|e  
    % ------------------------ {y'c*NS  
    % The radial Zernike polynomials are computed using the series H.j(hc'  
    % representation shown in the Help section above. For many special 8(D>ws$  
    % functions, direct evaluation using the series representation can 5>q|c`&}E  
    % produce poor numerical results (floating point errors), because T<DQi  
    % the summation often involves computing small differences between ]E88zWDY`  
    % large successive terms in the series. (In such cases, the functions N>7INK  
    % are often evaluated using alternative methods such as recurrence z:|4S@9  
    % relations: see the Legendre functions, for example). For the Zernike ld4QhZia  
    % polynomials, however, this problem does not arise, because the )`^t,x<S  
    % polynomials are evaluated over the finite domain r = (0,1), and +BM(0M+  
    % because the coefficients for a given polynomial are generally all 7f'9Dm`  
    % of similar magnitude. veAGUE %3  
    % g% #" 5Kr  
    % ZERNPOL has been written using a vectorized implementation: multiple #5N#^#r"  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] gc7S_D~;  
    % values can be passed as inputs) for a vector of points R.  To achieve q/?#+d  
    % this vectorization most efficiently, the algorithm in ZERNPOL I2q C,Nkk  
    % involves pre-determining all the powers p of R that are required to S-npJh 6  
    % compute the outputs, and then compiling the {R^p} into a single AIF?+i%H}  
    % matrix.  This avoids any redundant computation of the R^p, and X26gl 'U  
    % minimizes the sizes of certain intermediate variables.  Jt.dR6,  
    % Xp{+){Iu  
    %   Paul Fricker 11/13/2006 ~a4htj  
    ^LE`Y>&m  
    _LFZ0  
    % Check and prepare the inputs: Z+=WgEu1  
    % ----------------------------- QSW62]=vV  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) kz(%8qi8&  
        error('zernpol:NMvectors','N and M must be vectors.') x1]^].#Eo  
    end xXE/pIXw  
    V{O,O,*  
    if length(n)~=length(m) =GTltFqI1  
        error('zernpol:NMlength','N and M must be the same length.') }>=k!l{  
    end KRf$VbuL  
    KTD# a1W  
    n = n(:); `$jc=ZLm  
    m = m(:); 0"(5\T  
    length_n = length(n); '`#2'MXG  
    ``)ys^V  
    if any(mod(n-m,2)) S " R]i  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') FylL7n  
    end 6&S;Nrg9  
    CDQ}C=4  
    if any(m<0) 3XYCtp8  
        error('zernpol:Mpositive','All M must be positive.') ,OsFv}v7  
    end p [4/Nq,c  
    6e(|t2^  
    if any(m>n) Zj99]4?9  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') Dj'aWyW'  
    end Fd":\7p  
    3,+)3,N  
    if any( r>1 | r<0 ) 47ra`*  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') s9>f5u?dK  
    end , \ |S BS  
    4d#w}  
    if ~any(size(r)==1) j-8v$ 0'  
        error('zernpol:Rvector','R must be a vector.') o7)<pfif  
    end ?F1NZA[%t  
    l5F>v!NA  
    r = r(:); l)< '1dqe  
    length_r = length(r); ~CgKU8  
    s!esk%h{K  
    if nargin==4 mD{<Lp=  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); &^3KF0\Q  
        if ~isnorm 0OO$(R*  
            error('zernpol:normalization','Unrecognized normalization flag.') R]O!F)_/'  
        end v!n\A}^:  
    else aG.j0`)%  
        isnorm = false; >Ig%|4Hw  
    end S*-n%D0q5  
    NWWag}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M3c!SXx\  
    % Compute the Zernike Polynomials M>W-lp^3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~SgW+sDF u  
    Bf33%I~  
    % Determine the required powers of r: }`#OA]NZ  
    % ----------------------------------- /nEt%YYh;x  
    rpowers = []; Y [Jt+p]  
    for j = 1:length(n) iU3)4(R  
        rpowers = [rpowers m(j):2:n(j)]; B7r={P!0  
    end M:x?I_JG8  
    rpowers = unique(rpowers); os6p1"_\f  
    $MD|YW5  
    % Pre-compute the values of r raised to the required powers, y+xw`gR:  
    % and compile them in a matrix: pwo5Ij,~q  
    % ----------------------------- td!WgL,m  
    if rpowers(1)==0 )D\cm7WX^[  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); (O{5L(  
        rpowern = cat(2,rpowern{:}); .2?tx OKh  
        rpowern = [ones(length_r,1) rpowern]; /*3[9,  
    else :K&>  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); _/;k ;$gDp  
        rpowern = cat(2,rpowern{:}); ^.bYLF  
    end KWtLrZ(j  
    3GVE/GtU  
    % Compute the values of the polynomials: 6&/H XqP  
    % -------------------------------------- sVP[7&vr~  
    z = zeros(length_r,length_n); !z@QoD  
    for j = 1:length_n 'Gwa[ |6i  
        s = 0:(n(j)-m(j))/2; ,Q|[Yr  
        pows = n(j):-2:m(j); = 7U^pT  
        for k = length(s):-1:1 5Pmmt&#/Z  
            p = (1-2*mod(s(k),2))* ... 7oY}=281  
                       prod(2:(n(j)-s(k)))/          ... 4T#B7wVoM  
                       prod(2:s(k))/                 ... g7*cwu  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... V9xZH5T8^  
                       prod(2:((n(j)+m(j))/2-s(k))); &8!* u3  
            idx = (pows(k)==rpowers); jU')8m[  
            z(:,j) = z(:,j) + p*rpowern(:,idx); S"?py=7  
        end :%GxU;<E{  
         {l&6= z  
        if isnorm -:9E+b  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); RFF&-M]  
        end =XWew*  
    end \ZdV|23  
    ^bpxhf x  
    % 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)  c~ l$_A  
    fC.-* r  
    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。 x^8xz5:O  
    &E.0!BuqV  
    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)