切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11401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 zuw6YY8kQ  
    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! i fUgj8i_  
     
    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 z?7pn}-  
    function z = zernfun(n,m,r,theta,nflag) ]`%cTdpLj  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 9kcAMk1K  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N &W1c#]q@r  
    %   and angular frequency M, evaluated at positions (R,THETA) on the !^w+<p  
    %   unit circle.  N is a vector of positive integers (including 0), and @<_4Nb  
    %   M is a vector with the same number of elements as N.  Each element 3/iGSG`  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) q*>`HTPcU  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 9!tRM-  
    %   and THETA is a vector of angles.  R and THETA must have the same v q|W&  
    %   length.  The output Z is a matrix with one column for every (N,M) d bw`E"g  
    %   pair, and one row for every (R,THETA) pair. m6s32??m  
    %  C+_ NG  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike L49`=p<  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), w1[F]|  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral rQU;?[y  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ^j@,N&W:lG  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized > #SQDVFf  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *+5AN306  
    % bx1'  
    %   The Zernike functions are an orthogonal basis on the unit circle. VgOj#Z?K  
    %   They are used in disciplines such as astronomy, optics, and OX!9T.j  
    %   optometry to describe functions on a circular domain. 9k1n-po  
    % Lf3:' n  
    %   The following table lists the first 15 Zernike functions. Gt'%:9r  
    % z"|^Y|`m  
    %       n    m    Zernike function           Normalization C;_10Rb2ut  
    %       -------------------------------------------------- Eg>MG87  
    %       0    0    1                                 1 6tVB}UKs  
    %       1    1    r * cos(theta)                    2 m3,i{  
    %       1   -1    r * sin(theta)                    2 |+%K89W  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) |iJ37QIM  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ~b*f2UVs  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) +h"RXwlBM  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) F]x o*  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) V#zDYrp  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ygh*oVHO  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) D{~I  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) WI ' ;e4  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {2A/@$?  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 7i`8 c =.  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -W2 !_  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) r\Zz=~![<  
    %       -------------------------------------------------- >J+hu;I5  
    % pno]B ld'z  
    %   Example 1: 5P [b/.n  
    % >zVj+  
    %       % Display the Zernike function Z(n=5,m=1) 4 QD.'+ L  
    %       x = -1:0.01:1; WOiw 0  
    %       [X,Y] = meshgrid(x,x); 7YrX3Hx 8  
    %       [theta,r] = cart2pol(X,Y); D3N\$D  
    %       idx = r<=1; gq!| 0  
    %       z = nan(size(X)); /aP4'U8ov  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); crG+BFi  
    %       figure #}3$n/  
    %       pcolor(x,x,z), shading interp zQ&`|kS  
    %       axis square, colorbar a~jM^b;VN  
    %       title('Zernike function Z_5^1(r,\theta)') q,A;d^g  
    % $5Jo %K%  
    %   Example 2: X H,1\J-S  
    % jNBvy1  
    %       % Display the first 10 Zernike functions Mt"j< ]EW  
    %       x = -1:0.01:1; G$ ( B26  
    %       [X,Y] = meshgrid(x,x); Q E1DTU  
    %       [theta,r] = cart2pol(X,Y); F6`$5%$M;?  
    %       idx = r<=1; 4*&_h g)h  
    %       z = nan(size(X)); }j;*7x8(  
    %       n = [0  1  1  2  2  2  3  3  3  3]; $n.oY5=\  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; RX3P %xZ  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; gZ8n[zxf6  
    %       y = zernfun(n,m,r(idx),theta(idx)); )OpB\k  
    %       figure('Units','normalized') $9)|cO  
    %       for k = 1:10 W{B)c?G]  
    %           z(idx) = y(:,k); S2T~7-  
    %           subplot(4,7,Nplot(k)) *EY^t=  
    %           pcolor(x,x,z), shading interp )2~Iqzc4  
    %           set(gca,'XTick',[],'YTick',[]) R9%Um6  
    %           axis square lu2"?y[2  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) QpbyC_:;$4  
    %       end XR3=Y0YDf  
    % lky{<jZ%  
    %   See also ZERNPOL, ZERNFUN2. KsZd.Rf=@  
    h2<Y*j  
    %   Paul Fricker 11/13/2006 wC{?@ h  
    (r78AZ  
    w_4/::K*  
    % Check and prepare the inputs: ]#x!mZ!  
    % ----------------------------- ?Zu2=<DU  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ='"DUQH|*  
        error('zernfun:NMvectors','N and M must be vectors.') #[=%+*Q  
    end (CDwl,  
    VjsQy>5m  
    if length(n)~=length(m) >,;, 6|S  
        error('zernfun:NMlength','N and M must be the same length.') Uskz~~}G  
    end T-S6`^_L  
    !0p_s;uu,W  
    n = n(:); G>Uam TM  
    m = m(:); *PEk+e  
    if any(mod(n-m,2)) &b~if}vcb  
        error('zernfun:NMmultiplesof2', ... z86[_l:  
              'All N and M must differ by multiples of 2 (including 0).') 6'E3Q=}d  
    end d'y\~M9(  
    fdk]i/*)  
    if any(m>n) NWx.l8G  
        error('zernfun:MlessthanN', ... Ut|G.%1Vd%  
              'Each M must be less than or equal to its corresponding N.') *?dw`j_b >  
    end oVDqX=G  
    0H;,~ WY  
    if any( r>1 | r<0 ) >Z-f</v03  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') J)yNp,V  
    end EkWipF(  
    (4ueO~jb $  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) \l$gcFXb  
        error('zernfun:RTHvector','R and THETA must be vectors.') 5ctH=t0  
    end [3&Y* W  
    LNm{}VJ%  
    r = r(:); 6h8fzqRzc  
    theta = theta(:); 3JnBKh\n  
    length_r = length(r); BM6 J  
    if length_r~=length(theta) .~>Uh3S  
        error('zernfun:RTHlength', ... e(t,~(  
              'The number of R- and THETA-values must be equal.') b d!|/Lk  
    end  B6| g2Tt  
    z^xrB$8 u  
    % Check normalization: f/!^QL{  
    % -------------------- X0IXj%\N  
    if nargin==5 && ischar(nflag) srX" vF  
        isnorm = strcmpi(nflag,'norm'); ~k}>CNTr  
        if ~isnorm TttD}`\.  
            error('zernfun:normalization','Unrecognized normalization flag.') yDk|ad|  
        end /<J&ZoeJB  
    else iPnu *29  
        isnorm = false; 7*D*nY4+  
    end DH@})TN*O  
    ?=C?3R  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% eJh4hp;x  
    % Compute the Zernike Polynomials {OK+d#=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% JYQ.Y!X1O  
    ^7cZ9/3  
    % Determine the required powers of r: S w<V/t  
    % ----------------------------------- !%pY)69gv  
    m_abs = abs(m); kB`t_`7f  
    rpowers = []; ?hW?w$C  
    for j = 1:length(n) [;IW'cXNq  
        rpowers = [rpowers m_abs(j):2:n(j)]; ~[E@P1  
    end Woo2hg-ti  
    rpowers = unique(rpowers); H\Bh Af  
    xUp[)B6?:  
    % Pre-compute the values of r raised to the required powers, GoVB1)  
    % and compile them in a matrix: )T4%}$(  
    % ----------------------------- oN0p$/La  
    if rpowers(1)==0 Ib$*w)4:  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); |(mr&7O  
        rpowern = cat(2,rpowern{:}); y(I_ 6+B^  
        rpowern = [ones(length_r,1) rpowern]; S}0W<H P  
    else rkp 1tv  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ulcm  
        rpowern = cat(2,rpowern{:}); !>%U8A  
    end st8=1}:&\  
    q(N2 #di  
    % Compute the values of the polynomials: cy0 %tsB|  
    % -------------------------------------- Sk*-B@!S  
    y = zeros(length_r,length(n)); QH_0U`3  
    for j = 1:length(n) R`7v3{  
        s = 0:(n(j)-m_abs(j))/2; )+' De  
        pows = n(j):-2:m_abs(j); OK=lp4X  
        for k = length(s):-1:1 vY+{zGF  
            p = (1-2*mod(s(k),2))* ... 0zSRk]i.f  
                       prod(2:(n(j)-s(k)))/              ... .I6:iB  
                       prod(2:s(k))/                     ... lu]Z2xSv  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... )p,uZ`~v  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ]e*Zx;6oi  
            idx = (pows(k)==rpowers); H|E{n/g  
            y(:,j) = y(:,j) + p*rpowern(:,idx);  |7ga9  
        end /zB;1%m-  
         pHW Qk z(  
        if isnorm Q}a, f75  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); aD2+9?m  
        end .d\<}\zZ7J  
    end zjyj,jP  
    % END: Compute the Zernike Polynomials r*-e~  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [G(}`u8w"  
    w"-'  
    % Compute the Zernike functions: Qv3g 4iJ  
    % ------------------------------ NPN*k].  
    idx_pos = m>0; @,b:s+]rp  
    idx_neg = m<0; -jBk  
    F vJJpPS  
    z = y; uO4kCK<7C  
    if any(idx_pos) n?@3+wG  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); )gO=5_^u*o  
    end Z*/*P4\  
    if any(idx_neg) 9u3~s <  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); r_sZw@lqJ  
    end c1v,5c6d j  
    qr5ME/)z  
    % 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) -:O~J#D  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. =c)O8  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 1' #%U A  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive DYvi1X6  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Cm4$&?  
    %   and THETA is a vector of angles.  R and THETA must have the same ?K<m.+4b*y  
    %   length.  The output Z is a matrix with one column for every P-value, 8}Pd- .se  
    %   and one row for every (R,THETA) pair. |+?ABPk"  
    % /]/3)@wT  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike !fFmQ\|)4S  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) +6vm4(3?  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) @IaK:  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 {.W$<y (j7  
    %   for all p. IN? A`A  
    % vXUrS+~x  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Sb&sW?M  
    %   Zernike functions (order N<=7).  In some disciplines it is !}sYPz]7!  
    %   traditional to label the first 36 functions using a single mode O/gBBTB  
    %   number P instead of separate numbers for the order N and azimuthal |?c v5l7E  
    %   frequency M. Dh*>361y-  
    % H}}]Gh.T  
    %   Example: AJRfl%3  
    % J~Gq#C^e  
    %       % Display the first 16 Zernike functions E' p5  
    %       x = -1:0.01:1; O,Xf.O1c  
    %       [X,Y] = meshgrid(x,x); M5C}*c9  
    %       [theta,r] = cart2pol(X,Y); 05 ".;(  
    %       idx = r<=1; =2VM(GtK>  
    %       p = 0:15; s'LY)_n  
    %       z = nan(size(X)); ~%Y*2i f  
    %       y = zernfun2(p,r(idx),theta(idx)); cw 3JSz9  
    %       figure('Units','normalized') 6gS<h \h0  
    %       for k = 1:length(p) H'uRgBjWJ  
    %           z(idx) = y(:,k); )Cuc ]>SC  
    %           subplot(4,4,k) A[lkGQtS4  
    %           pcolor(x,x,z), shading interp e_6@oh2s-  
    %           set(gca,'XTick',[],'YTick',[]) &~ g||rq  
    %           axis square YaTJKgi"0  
    %           title(['Z_{' num2str(p(k)) '}']) $'}:nwq6x  
    %       end 5S'89 r3m  
    % ]Mv.Rul?~  
    %   See also ZERNPOL, ZERNFUN. 4|fI9.  
    N|eus3\E  
    %   Paul Fricker 11/13/2006 d+KLtvB%M  
    S#{e@ C  
    umXa   
    % Check and prepare the inputs: _20nOg`o  
    % ----------------------------- /H+j6*}r  
    if min(size(p))~=1 Mn(:qQo^&`  
        error('zernfun2:Pvector','Input P must be vector.') PTe8,cD>  
    end ya]CxnKR3  
    7O{c>@\  
    if any(p)>35 n!mtMPH$  
        error('zernfun2:P36', ... >Pv#)qtm  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... bet?5Dk  
               '(P = 0 to 35).']) 4oLrCQZ\  
    end C=}YKsi|R|  
    3O<:eS~  
    % Get the order and frequency corresonding to the function number: z65|NO6JW.  
    % ---------------------------------------------------------------- vMJv.O>HW  
    p = p(:); 1btQ[a6j  
    n = ceil((-3+sqrt(9+8*p))/2); _X{i hf  
    m = 2*p - n.*(n+2); //VgPl  
    =LHE_ AA  
    % Pass the inputs to the function ZERNFUN: 8>G3KZ3  
    % ---------------------------------------- ev>gh0  
    switch nargin 5nIm7vlQm  
        case 3 0_Tr>hz  
            z = zernfun(n,m,r,theta); vX})6O  
        case 4 [QA@XBy6  
            z = zernfun(n,m,r,theta,nflag); xZt]s3?  
        otherwise )sG`sET]`f  
            error('zernfun2:nargin','Incorrect number of inputs.') hKb-l`KO  
    end ']1j M n  
    m sS5"Qr  
    % 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) U/&!F  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 8D[P*?O  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 1Qui.],c  
    %   order N and frequency M, evaluated at R.  N is a vector of QbqEe/*$_  
    %   positive integers (including 0), and M is a vector with the @#Jc!p7)  
    %   same number of elements as N.  Each element k of M must be a GQ ZEMy7  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) QkUq%}_0  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is y/rmxQtP  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix :9q=o|T6D  
    %   with one column for every (N,M) pair, and one row for every Q PgM<ns  
    %   element in R. ;>Y,b4B;  
    % j K?GB  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- to9 u%d8  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is *A.E?9pL\  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to D Cx3_  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1  Ko9"mHNB  
    %   for all [n,m]. 6-U|e|e  
    % oe,L&2Jz@  
    %   The radial Zernike polynomials are the radial portion of the l'HrU 1_7Y  
    %   Zernike functions, which are an orthogonal basis on the unit w+ gA3Dg  
    %   circle.  The series representation of the radial Zernike #>C.61Fx  
    %   polynomials is 2/O/h  
    %  M]:4X_  
    %          (n-m)/2 @x?7J@:  
    %            __ v.H00}[.  
    %    m      \       s                                          n-2s 6"C$]kF?  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r v??}d   
    %    n      s=0 XZhuV<  
    % vBAds  
    %   The following table shows the first 12 polynomials. rr>*_67-:  
    % ,2S w6u  
    %       n    m    Zernike polynomial    Normalization wND0KiwH  
    %       --------------------------------------------- W-zD1q~0?  
    %       0    0    1                        sqrt(2) Ol3$!x9  
    %       1    1    r                           2 5CH9m[S  
    %       2    0    2*r^2 - 1                sqrt(6) (5rH 72g(  
    %       2    2    r^2                      sqrt(6) E$]7w4,n  
    %       3    1    3*r^3 - 2*r              sqrt(8) 4.2qt  
    %       3    3    r^3                      sqrt(8) &NiDv   
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) lR_ 4iyqb  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) [xq"[*Evv  
    %       4    4    r^4                      sqrt(10) 0<75G6wd  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) DJP)V8]!B  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) U,S&"`a  
    %       5    5    r^5                      sqrt(12) <X~ X#9V  
    %       --------------------------------------------- YqwDvJWX  
    % &*iar+vr  
    %   Example: H^sPC{6+pf  
    % k ]C+/  
    %       % Display three example Zernike radial polynomials T(%U$ea-S  
    %       r = 0:0.01:1; [ah%>&u  
    %       n = [3 2 5]; WL(u'%5  
    %       m = [1 2 1]; jrT5Rw_}q  
    %       z = zernpol(n,m,r); }8J77[>/  
    %       figure Rdao  
    %       plot(r,z) g-j`Ex%  
    %       grid on &> 43l+  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') *kE2d{h^=C  
    % \ a18Hp|%  
    %   See also ZERNFUN, ZERNFUN2. "MZj}}l  
    $$`E@\5P  
    % A note on the algorithm. @bU(z$eB  
    % ------------------------ v`#T)5gl-  
    % The radial Zernike polynomials are computed using the series l&cYN2T b  
    % representation shown in the Help section above. For many special e#]=-^  
    % functions, direct evaluation using the series representation can 3lYM(DT  
    % produce poor numerical results (floating point errors), because @q2If{Tk  
    % the summation often involves computing small differences between 'PTWC.C?9  
    % large successive terms in the series. (In such cases, the functions 2ijw g~_@  
    % are often evaluated using alternative methods such as recurrence (wLzkV/6  
    % relations: see the Legendre functions, for example). For the Zernike (r,tU(  
    % polynomials, however, this problem does not arise, because the c-8Pc ]+g  
    % polynomials are evaluated over the finite domain r = (0,1), and TMZg GUn  
    % because the coefficients for a given polynomial are generally all sXxF5&AF0  
    % of similar magnitude. >J>4g;Y  
    % O{hGh{y  
    % ZERNPOL has been written using a vectorized implementation: multiple =;Gy"F1 dp  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 'V=w?G 5  
    % values can be passed as inputs) for a vector of points R.  To achieve s8iJl+Jm  
    % this vectorization most efficiently, the algorithm in ZERNPOL ^50#R< Ny  
    % involves pre-determining all the powers p of R that are required to d[;=X.fZ2  
    % compute the outputs, and then compiling the {R^p} into a single Fn0Rq9/@  
    % matrix.  This avoids any redundant computation of the R^p, and >Z ZX]#=I  
    % minimizes the sizes of certain intermediate variables. 6'd=% V  
    % -ZqN~5>j)  
    %   Paul Fricker 11/13/2006 J@'}lG  
    sxcpWSGA^  
    bJ8G5QU  
    % Check and prepare the inputs: }wzU<(Rx  
    % ----------------------------- TIlBT{A<  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) H@Dj$U  
        error('zernpol:NMvectors','N and M must be vectors.') c`I`@Bed  
    end Qk?;nF  
    aV`4M VWOz  
    if length(n)~=length(m) ._JM3o}F  
        error('zernpol:NMlength','N and M must be the same length.') A`2l;MW  
    end F<y$Q0Z}  
    :=x-b3U  
    n = n(:); JJlwzH  
    m = m(:); Ftu~nh}  
    length_n = length(n); KZ^W@*`D  
    _ Js & _d  
    if any(mod(n-m,2)) }ybveZxv5A  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') xe5|pBT  
    end 8d.5D&  
    qXO@FW]  
    if any(m<0) $8#zPJR&  
        error('zernpol:Mpositive','All M must be positive.') 7@"J&><w!  
    end gd3~R+Kd  
    S;[g0j  
    if any(m>n) F/;uN5{o  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') {2?o:  
    end _:F0>=$  
    afY~Y?PJ<  
    if any( r>1 | r<0 ) g2p/#\D\J  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') `[x`#irD  
    end f%ude@E3  
    8+m;zvDSU  
    if ~any(size(r)==1) <}x_F)E[t  
        error('zernpol:Rvector','R must be a vector.') GQ~wx1jj1  
    end a& b75.-  
    I_xvg >i  
    r = r(:); .T0w2Dv/  
    length_r = length(r); lN"%~n?  
    dbF?#s~u  
    if nargin==4 P}B{FIpNG  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); mEsOYIu{  
        if ~isnorm Ia-`x/r*m  
            error('zernpol:normalization','Unrecognized normalization flag.') ZaYux-0]kF  
        end ?.66B9Lld  
    else :^".cs?g  
        isnorm = false; P+00wbx0  
    end ab4(?-'-  
    J WaI[n}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% WO{V,<;  
    % Compute the Zernike Polynomials E _d^&{j  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% w yi n  
    <RzGxhT  
    % Determine the required powers of r: D0Cs g39  
    % ----------------------------------- 3B|?{U~  
    rpowers = []; Eshc"U  
    for j = 1:length(n) ir^%9amh  
        rpowers = [rpowers m(j):2:n(j)]; fW^\G2Fk  
    end  ;ew j  
    rpowers = unique(rpowers); Cz@[l=-T7  
    zFVNb  
    % Pre-compute the values of r raised to the required powers, Q _Yl:c  
    % and compile them in a matrix: y ;W|)  
    % ----------------------------- ?CSc5b`eo  
    if rpowers(1)==0 #4uuT?!  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); R7YL I1ov  
        rpowern = cat(2,rpowern{:}); dd4g?):  
        rpowern = [ones(length_r,1) rpowern]; &K Ti[  
    else ~RdJP'YF-  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); !Cse,6/Z  
        rpowern = cat(2,rpowern{:}); := OdjfhY  
    end I75>$"$<  
    .ET@J`"M  
    % Compute the values of the polynomials: LRNgpjE}  
    % -------------------------------------- n^Co  
    z = zeros(length_r,length_n); <)01]lKH  
    for j = 1:length_n {s^vAD<~x3  
        s = 0:(n(j)-m(j))/2; ]z5`!e)L  
        pows = n(j):-2:m(j); sp%EA=: E  
        for k = length(s):-1:1 1&\ A#  
            p = (1-2*mod(s(k),2))* ... C>\0 "}iD  
                       prod(2:(n(j)-s(k)))/          ... \ZSZ(p#1  
                       prod(2:s(k))/                 ... $ {29[hO  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Z!\xVCG"q  
                       prod(2:((n(j)+m(j))/2-s(k))); 09FHE/L  
            idx = (pows(k)==rpowers); 1u4)  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 'F3cvpc`  
        end Z x&gr|)}  
         A UCk]  
        if isnorm [,;h1m ~iX  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); F$[1KjS  
        end tuZA q;X  
    end M6yzqAh  
    %"#%/>U4  
    % 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
    在线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)  d&!;uzOx  
    WFl, u!"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。 XI4le=^EM  
    ~OuKewr\  
    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)