切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11181阅读
    • 9回复

    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

    上一主题 下一主题
    离线niuhelen
     
    发帖
    19
    光币
    28
    光券
    0
    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 +SzU  
    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! J[&@PUy  
     
    分享到
    离线phility
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    19
    光币
    28
    光券
    0
    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 Eh`7X=Z7E  
    function z = zernfun(n,m,r,theta,nflag) m,28u3@r  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 1#g2A0U,  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N X56q-|  
    %   and angular frequency M, evaluated at positions (R,THETA) on the T.F!+  
    %   unit circle.  N is a vector of positive integers (including 0), and 5<k"K^0QS  
    %   M is a vector with the same number of elements as N.  Each element yf)%%&  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) yF:1( 4  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, T~?Ff|qFC  
    %   and THETA is a vector of angles.  R and THETA must have the same S>+|OCl";  
    %   length.  The output Z is a matrix with one column for every (N,M) OKZV{Gja  
    %   pair, and one row for every (R,THETA) pair. TprTWod2]t  
    % tIi&;tw]  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike eeg)N1\  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), R-wp9^  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral mUC)gA/  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, H'5)UX@LP  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized NX.6px17  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. f)rq%N &  
    % Ib!RD/  
    %   The Zernike functions are an orthogonal basis on the unit circle. B IEO,W|  
    %   They are used in disciplines such as astronomy, optics, and 4B;=kL_f  
    %   optometry to describe functions on a circular domain. s+Pq&<nV-  
    % F;EwQjTF  
    %   The following table lists the first 15 Zernike functions. CkC^'V)  
    % atH*5X6d  
    %       n    m    Zernike function           Normalization Q}JOU  
    %       -------------------------------------------------- XW H5d-  
    %       0    0    1                                 1 _ye |Y  
    %       1    1    r * cos(theta)                    2 /62!cp/F/D  
    %       1   -1    r * sin(theta)                    2 w "F 9l  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 5I;&mW`1,`  
    %       2    0    (2*r^2 - 1)                    sqrt(3) j;Gtu  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 539>WyG5  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ]mq|w  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) g-k|>-h  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) @;4zrzQi7  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) `hm-.@f,9  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) z9Mfd#5?>P  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) s^TZXCyF o  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) \K{ z  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 3*bU6$|5FP  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) >uB?rGcM  
    %       -------------------------------------------------- ~/U 1xk%  
    % P;no?  
    %   Example 1: ;1=1:S8  
    % XJB)rP  
    %       % Display the Zernike function Z(n=5,m=1) dQX6(J j  
    %       x = -1:0.01:1; 0> E r=,e  
    %       [X,Y] = meshgrid(x,x); O\tb R=  
    %       [theta,r] = cart2pol(X,Y); ~P qM]^  
    %       idx = r<=1; M0"_^?  
    %       z = nan(size(X)); nW:C/{n2tG  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); =%O6:YM   
    %       figure MJ)RvNF  
    %       pcolor(x,x,z), shading interp 8W7J3{d  
    %       axis square, colorbar DfD&)tsMQ  
    %       title('Zernike function Z_5^1(r,\theta)') B-Hrex]  
    % hfB%`x#akQ  
    %   Example 2: ty!`T+3  
    % (,2S XV  
    %       % Display the first 10 Zernike functions LOYk9m  
    %       x = -1:0.01:1; BOX2O.Pm  
    %       [X,Y] = meshgrid(x,x); |-ALklXr  
    %       [theta,r] = cart2pol(X,Y); e%M;?0j  
    %       idx = r<=1; d1T!+I  
    %       z = nan(size(X)); ,qwuLBW  
    %       n = [0  1  1  2  2  2  3  3  3  3]; R\f+SvE  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; cVpp-Z|s8  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; j;r-NCBnz  
    %       y = zernfun(n,m,r(idx),theta(idx)); +`0k Fbx  
    %       figure('Units','normalized') G_JA-@i%  
    %       for k = 1:10 q?:dCFw$x5  
    %           z(idx) = y(:,k); RB\uK 1+  
    %           subplot(4,7,Nplot(k)) Jpq~  
    %           pcolor(x,x,z), shading interp (9 d&  
    %           set(gca,'XTick',[],'YTick',[]) r5/0u(\LB  
    %           axis square 29b9`NXt  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) f~[7t:WD*  
    %       end gJ{)-\  
    % 6MW{,N  
    %   See also ZERNPOL, ZERNFUN2. ajT*/L!0_  
    kTB 0b*V  
    %   Paul Fricker 11/13/2006 B6 ;|f'e!  
    n@i HFBb  
    r6qj7}\  
    % Check and prepare the inputs: X?',n 1  
    % ----------------------------- ?V=ZIGj  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) o|:b;\)b  
        error('zernfun:NMvectors','N and M must be vectors.') |df Pki{  
    end n>XdU%&  
    =WATyY:s  
    if length(n)~=length(m) #!# l45p6  
        error('zernfun:NMlength','N and M must be the same length.') J8(lIk:e  
    end '<<t]kK[N  
    ]m<$}  
    n = n(:); aXYY:;  
    m = m(:); G` A4|+W"  
    if any(mod(n-m,2)) e !Y~Qy  
        error('zernfun:NMmultiplesof2', ... P@B]  
              'All N and M must differ by multiples of 2 (including 0).') tNI^@xdim1  
    end GxxW&y  
    LL!Dx%JZ  
    if any(m>n) m s \}  
        error('zernfun:MlessthanN', ... fr3d  
              'Each M must be less than or equal to its corresponding N.') WT=;:j  
    end <'*LRd$1  
    7$=In K  
    if any( r>1 | r<0 ) w@E3ZL^  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') eMsd37J  
    end aFYIM`?(  
    GVn!O1jio  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) IJ"q~r$  
        error('zernfun:RTHvector','R and THETA must be vectors.') ,"ZMRq  
    end a=2%4Wmz  
    Q &JUt(  
    r = r(:); T8g$uFo  
    theta = theta(:); z:*|a+cy  
    length_r = length(r); ~?BXti<!  
    if length_r~=length(theta) ZE}}W _  
        error('zernfun:RTHlength', ... lo+A%\1  
              'The number of R- and THETA-values must be equal.') 8Z~EwY*  
    end C'x&Py/#  
    ga+dt  
    % Check normalization: 3w'tH4C[Y  
    % -------------------- GTd,n=  
    if nargin==5 && ischar(nflag) 77Y/!~kd  
        isnorm = strcmpi(nflag,'norm'); f:} x7_Q  
        if ~isnorm ]=BB#  
            error('zernfun:normalization','Unrecognized normalization flag.') z} #JK? u  
        end 0H:X3y+  
    else ;=z:F<Y  
        isnorm = false; ~DwpoeYX  
    end 1qA;/-Zr<o  
    UK!(G  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9'B `]/L  
    % Compute the Zernike Polynomials h_'*XWd@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9.#<b |g  
    h376Be{P  
    % Determine the required powers of r: zb3t IRH  
    % ----------------------------------- 75lA%| *X  
    m_abs = abs(m); Bzf^ivT3L  
    rpowers = []; ^cWnF0)j.  
    for j = 1:length(n)  ob]w;"  
        rpowers = [rpowers m_abs(j):2:n(j)]; R|(a@sL  
    end \FaP|28h  
    rpowers = unique(rpowers); ih3n<gXF  
    ? r4>"[  
    % Pre-compute the values of r raised to the required powers, ^\m![T\bX  
    % and compile them in a matrix: !N^@4*  
    % ----------------------------- }SZd  
    if rpowers(1)==0 i%?*@uj  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); +}AI@+  
        rpowern = cat(2,rpowern{:}); Kg]J/|0\  
        rpowern = [ones(length_r,1) rpowern]; ~xTt204S  
    else h(DTa  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); <P<z N~i9j  
        rpowern = cat(2,rpowern{:}); x8|J-8A(  
    end y~V(aih}D  
    [}m[)L\  
    % Compute the values of the polynomials: pxi3PY?  
    % -------------------------------------- !4!~L k=  
    y = zeros(length_r,length(n)); 6y<EgYzdE  
    for j = 1:length(n) HzJz+ x:  
        s = 0:(n(j)-m_abs(j))/2; L~3Pm%{@A  
        pows = n(j):-2:m_abs(j); >$7B wO  
        for k = length(s):-1:1 7tp36TE  
            p = (1-2*mod(s(k),2))* ... <_+X 88  
                       prod(2:(n(j)-s(k)))/              ...  M6TD"-  
                       prod(2:s(k))/                     ... WIGi51yC.x  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... K 8O|?x]  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); E{(;@PzE  
            idx = (pows(k)==rpowers); eMzk3eOJ  
            y(:,j) = y(:,j) + p*rpowern(:,idx); *^`Vz?g<  
        end j>kqz>3  
         Zd+bx*rD  
        if isnorm t{>q|0  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); wd6owr  
        end  D%Z|  
    end dh\P4  
    % END: Compute the Zernike Polynomials ,zc(t<|-y  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |W^IlqTH  
    l,).p  
    % Compute the Zernike functions: cwL_tq  
    % ------------------------------ dRMx[7jVA  
    idx_pos = m>0; \)e'`29;  
    idx_neg = m<0; ,,r>,Xq 6  
    5r0YA IJ  
    z = y; KPki}'GO  
    if any(idx_pos) q(w(Sd)#L  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); *1"+%Z^  
    end Vvo 7C!$z  
    if any(idx_neg) Dv6}bx(  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); +C)~bb*  
    end qP ,EBE  
    VEH>]-0K  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) 9H~n _   
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 3' 'me  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated =pr7G+_u  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive s#MPX3itK  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, *^r}"in  
    %   and THETA is a vector of angles.  R and THETA must have the same }B^tL$k  
    %   length.  The output Z is a matrix with one column for every P-value, |BYRe1l6l  
    %   and one row for every (R,THETA) pair. #K&Gp-  
    % X-/]IH DN  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike AFn7uW!9Gw  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) mZBo~(}  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) @+DX.9  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 3$/IC@+  
    %   for all p. g{LP7 D;6  
    % MfkZ  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 A(XKyEx  
    %   Zernike functions (order N<=7).  In some disciplines it is r|Z{-*`  
    %   traditional to label the first 36 functions using a single mode {G-kNU  
    %   number P instead of separate numbers for the order N and azimuthal 4!$"ayGv;D  
    %   frequency M. <naz+QK'  
    % 8EY:t zw  
    %   Example: |a@L}m  
    % ,u m|1dh  
    %       % Display the first 16 Zernike functions Ca-j?bb!  
    %       x = -1:0.01:1; [Qr"cR^  
    %       [X,Y] = meshgrid(x,x); [hs ds\  
    %       [theta,r] = cart2pol(X,Y); @|!z9Y*  
    %       idx = r<=1; 4K74=r),i  
    %       p = 0:15; P2Y^d#jO  
    %       z = nan(size(X)); n@w%Zl  
    %       y = zernfun2(p,r(idx),theta(idx)); ?ubro0F:  
    %       figure('Units','normalized') cCX*D_kCB  
    %       for k = 1:length(p) rlD8D|ZG  
    %           z(idx) = y(:,k); a{e4it  
    %           subplot(4,4,k) =H~j,K  
    %           pcolor(x,x,z), shading interp 2rMpgV5  
    %           set(gca,'XTick',[],'YTick',[]) ,?3G;-  
    %           axis square T C"<g  
    %           title(['Z_{' num2str(p(k)) '}']) jdBLsy@  
    %       end Gh$^{  
    % .V*^|UXbHi  
    %   See also ZERNPOL, ZERNFUN. ?Ob3tUz2  
    g&.=2uP  
    %   Paul Fricker 11/13/2006 iQ{VY ^ 0  
    r*Xuj=  
    @pxcpXCy  
    % Check and prepare the inputs: gZ5 |UR<  
    % ----------------------------- hOeRd#AQK  
    if min(size(p))~=1 F!do~Z  
        error('zernfun2:Pvector','Input P must be vector.') "5 A! jq  
    end f!"w5qC^  
    +h$ 9\  
    if any(p)>35 m kexc~l  
        error('zernfun2:P36', ... @WB@]-+J T  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ,vDbp?)'U  
               '(P = 0 to 35).']) ##{taR8  
    end y)*RV;^  
    <uJ@:oWG7  
    % Get the order and frequency corresonding to the function number: ctUp=po  
    % ---------------------------------------------------------------- Y$zSQ_k;U  
    p = p(:); +n)9Tz5  
    n = ceil((-3+sqrt(9+8*p))/2); OKV8zO  
    m = 2*p - n.*(n+2); j39wA~ K  
    g+l CMW\  
    % Pass the inputs to the function ZERNFUN: ;nGa.= "L  
    % ---------------------------------------- v2?ZQeHr_(  
    switch nargin Lr<cMK<  
        case 3 /E>e"tvss  
            z = zernfun(n,m,r,theta); F5Va+z,jg  
        case 4 y)pk6d   
            z = zernfun(n,m,r,theta,nflag); ix$bRdl  
        otherwise Y0>y8U V  
            error('zernfun2:nargin','Incorrect number of inputs.') ;bG>ZqJCVA  
    end {8OCXus3m  
    ]?*wbxU0  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) 9my^ Y9B  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. OH88n69  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Qd-A.{[h  
    %   order N and frequency M, evaluated at R.  N is a vector of ~V-XEQA  
    %   positive integers (including 0), and M is a vector with the g ?k=^C  
    %   same number of elements as N.  Each element k of M must be a <m m[S  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) z}@7'_iJ  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is `g,..Ns-r  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix N$DkX)Z  
    %   with one column for every (N,M) pair, and one row for every #?E"x/$Y6  
    %   element in R. p[-O( 3Y  
    % :svq E+2  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- +:f"Y0  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is KP"+e:a%  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to +%&yJ4-  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 yr6V3],Tp  
    %   for all [n,m]. <[phnU^ 8  
    % @oNXZRg6  
    %   The radial Zernike polynomials are the radial portion of the ?(PKeq6  
    %   Zernike functions, which are an orthogonal basis on the unit IcEdG(  
    %   circle.  The series representation of the radial Zernike =I4lL]>  
    %   polynomials is d1*<Ll9K  
    % /mHqurB  
    %          (n-m)/2 GeqPRah  
    %            __ qLCR] _*  
    %    m      \       s                                          n-2s 7 &\yj9  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r y9;Yiv r)  
    %    n      s=0 u!s2 BC0}N  
    % [Zrr)8A  
    %   The following table shows the first 12 polynomials. ;`Z{7'^U  
    % %C0Dw\A*:  
    %       n    m    Zernike polynomial    Normalization ~[ F`"  
    %       --------------------------------------------- N;R^h? '  
    %       0    0    1                        sqrt(2) n|hNM?v  
    %       1    1    r                           2 b' y%n   
    %       2    0    2*r^2 - 1                sqrt(6) i1085ztN  
    %       2    2    r^2                      sqrt(6) 5N]"~w*  
    %       3    1    3*r^3 - 2*r              sqrt(8) HsWk*L `y  
    %       3    3    r^3                      sqrt(8) /efUjkP  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) i@q&5;%%  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) wq{hF<  
    %       4    4    r^4                      sqrt(10) *hrvYil2b  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) }qUX=s GG  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) {_}I!`opr$  
    %       5    5    r^5                      sqrt(12) r^ XVB`v  
    %       --------------------------------------------- gr{ DWCK  
    % |:o4w  
    %   Example: _GPe<H  
    % 3R/bz0 V>  
    %       % Display three example Zernike radial polynomials fJ\[*5eiS  
    %       r = 0:0.01:1; vI?, 47Hj+  
    %       n = [3 2 5]; @CoIaUVP  
    %       m = [1 2 1]; V+\Wb[zDJ  
    %       z = zernpol(n,m,r); TvM~y\s  
    %       figure WAqINLdX  
    %       plot(r,z) K:M8h{Ua  
    %       grid on +t.b` U`-  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') yauvXosX  
    % ]|@^1we  
    %   See also ZERNFUN, ZERNFUN2. /QQ*8o8  
    ^ 9sjj  
    % A note on the algorithm. jdN` mosJ  
    % ------------------------ TpaInXR  
    % The radial Zernike polynomials are computed using the series K"6vXv4QO  
    % representation shown in the Help section above. For many special ,6/V" kqIP  
    % functions, direct evaluation using the series representation can TC('H[ ]  
    % produce poor numerical results (floating point errors), because Sdo-nt  
    % the summation often involves computing small differences between V9vTsmo(  
    % large successive terms in the series. (In such cases, the functions $qiya[&G4  
    % are often evaluated using alternative methods such as recurrence B~mj 8l4  
    % relations: see the Legendre functions, for example). For the Zernike wzA$'+Mb  
    % polynomials, however, this problem does not arise, because the aXVFc5C\  
    % polynomials are evaluated over the finite domain r = (0,1), and 0K+ne0I  
    % because the coefficients for a given polynomial are generally all dr(*T  
    % of similar magnitude. kstIgcI  
    % #E[0ys1O  
    % ZERNPOL has been written using a vectorized implementation: multiple Xvv6~  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] F [M,]?   
    % values can be passed as inputs) for a vector of points R.  To achieve !i50QA|(G  
    % this vectorization most efficiently, the algorithm in ZERNPOL >?b!QU* a  
    % involves pre-determining all the powers p of R that are required to PCvWS.{  
    % compute the outputs, and then compiling the {R^p} into a single txpgO1  
    % matrix.  This avoids any redundant computation of the R^p, and 0sqFF[i  
    % minimizes the sizes of certain intermediate variables. }C:r 9? T  
    % W!X@  
    %   Paul Fricker 11/13/2006 9x8fhAy}4  
    ,}PgOJZ  
    XX@ZQcN  
    % Check and prepare the inputs: ' %qr.T %  
    % ----------------------------- uH]OEz\H'  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) eRYK3W  
        error('zernpol:NMvectors','N and M must be vectors.') )4OxY[2J  
    end ixFi{_  
    +0&/g&a\R  
    if length(n)~=length(m) ` A>@]d  
        error('zernpol:NMlength','N and M must be the same length.') 6<]lW  
    end x Ar\gu  
     g(052]  
    n = n(:); S!UaH>Rh  
    m = m(:); H)?z #x  
    length_n = length(n); Wri<h:1  
    )UR7i8]!0  
    if any(mod(n-m,2)) I0 -MRU~[K  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ?}0,o.  
    end KwS@D9bok  
    +R&gqja  
    if any(m<0) vt8By@]:  
        error('zernpol:Mpositive','All M must be positive.') l;Wj]  
    end 2 nCA<&  
    6t$8M[0-U  
    if any(m>n) rH-23S  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') \85i+q:LuA  
    end "[J^YKoF  
    N['  .BN  
    if any( r>1 | r<0 ) yAt ^;  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') [~HN<>L@C  
    end siI;"?  
    bw7@5=?;  
    if ~any(size(r)==1) $mILoy B,  
        error('zernpol:Rvector','R must be a vector.') QV!up^Zso  
    end v+XJ*N[W  
    5+'<R8{:,  
    r = r(:); RP"kC4~1  
    length_r = length(r); ueudRb  
    r&CiSMS*  
    if nargin==4 K-4PI+qQ\  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); t_^4`dW`  
        if ~isnorm Y7|EIAU5Y  
            error('zernpol:normalization','Unrecognized normalization flag.') +%'(!A?*`  
        end (zk"~Ud  
    else (>Em^(&  
        isnorm = false; d0D] Q  
    end rp$'L7lrX  
    kmW4:EA%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s<Ziegmw|g  
    % Compute the Zernike Polynomials Ac@VGT:9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c)J%`i$  
    qPNR`%}Q  
    % Determine the required powers of r: ?4,T}@P  
    % ----------------------------------- DQ3<$0  
    rpowers = []; TOt dUO  
    for j = 1:length(n) ;l+Leex  
        rpowers = [rpowers m(j):2:n(j)]; LVGe]lD  
    end 2G7Wi!J  
    rpowers = unique(rpowers); .A|udZ,  
    1M6D3d_  
    % Pre-compute the values of r raised to the required powers, <I?Zk80  
    % and compile them in a matrix: IxU/?Zm  
    % ----------------------------- )7F/O3Tq  
    if rpowers(1)==0 dV_G1'  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); `?]k{ l1R  
        rpowern = cat(2,rpowern{:}); ye&;(30Oq  
        rpowern = [ones(length_r,1) rpowern]; kVgTGC"L=  
    else CJY$G}rk  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); P:c w|Q  
        rpowern = cat(2,rpowern{:}); Ml_^ `vn  
    end ?s01@f#  
    uRvP hkqm  
    % Compute the values of the polynomials: k[xSbs'D  
    % -------------------------------------- K+eM   
    z = zeros(length_r,length_n); L *wYx|  
    for j = 1:length_n tQ)qCk07  
        s = 0:(n(j)-m(j))/2; ftb\0,-   
        pows = n(j):-2:m(j); t.<i:#rj>l  
        for k = length(s):-1:1 Z.,MVcd  
            p = (1-2*mod(s(k),2))* ... @d'j zs  
                       prod(2:(n(j)-s(k)))/          ... XFl 6M~ c  
                       prod(2:s(k))/                 ... N~Jda o  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ...  B,@i  
                       prod(2:((n(j)+m(j))/2-s(k))); ?uu*L6  
            idx = (pows(k)==rpowers); #qki  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ch]IzdD  
        end }j Xfb@`K  
         :#Wd~~d  
        if isnorm O.? JmE  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); G|Ti4_w  
        end z{ dEC %  
    end MgZ/(X E  
    L(-4w+  
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    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
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
    发帖
    856
    光币
    846
    光券
    0
    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
    发帖
    4352
    光币
    5478
    光券
    1
    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)  OZT.=^:A  
    VX/#1StC  
    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。 k8Xm n6X  
    p7Cs.2>M>S  
    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)