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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 xpb,Nzwt^  
    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! 9Un3La8PX  
     
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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 KXQ &u{[<  
    function z = zernfun(n,m,r,theta,nflag) %]2hxTV  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. =41g9UQ  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N iE~][_%U  
    %   and angular frequency M, evaluated at positions (R,THETA) on the /3VSO"kcZ  
    %   unit circle.  N is a vector of positive integers (including 0), and w[5uX>  
    %   M is a vector with the same number of elements as N.  Each element I:ag}L8`  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) zXop@"(e  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, (SEE(G35  
    %   and THETA is a vector of angles.  R and THETA must have the same aw\\oN*  
    %   length.  The output Z is a matrix with one column for every (N,M) >;$C@  
    %   pair, and one row for every (R,THETA) pair. k"kGQk4  
    % x?aNK$A~X  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike G`_LD+  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), t+ ,'  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral GV+K] KDI  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, e|t@"MxvC  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 1kd\Fq^z$  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ]d4`PXI  
    % y*BS %xTF  
    %   The Zernike functions are an orthogonal basis on the unit circle. [eb?Fd~WB]  
    %   They are used in disciplines such as astronomy, optics, and y&-1SP<  
    %   optometry to describe functions on a circular domain. W7F1o[  
    % 95wi~^^  
    %   The following table lists the first 15 Zernike functions. o*[n[\cR  
    % [{i"Au]  
    %       n    m    Zernike function           Normalization ?F^$4:  
    %       --------------------------------------------------  ^n5rUwS>  
    %       0    0    1                                 1 n0ZrgTVJ  
    %       1    1    r * cos(theta)                    2 z f rEM  
    %       1   -1    r * sin(theta)                    2 9_h  V1:  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) _+OnH!G0  
    %       2    0    (2*r^2 - 1)                    sqrt(3) -KuC31s_W  
    %       2    2    r^2 * sin(2*theta)             sqrt(6)  4 Wb^$i!  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) j5rB+  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) kE8\\}B7  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Z~?1xJ&  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) sRMz[n 5k  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ($h`Y;4  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) yobcAV`  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) KWq&<X5  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) DR%16y<h  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) V1P]mUs{1  
    %       -------------------------------------------------- 'P:u/Sq?m  
    % sU|\? pJ  
    %   Example 1: =ObI  
    % a_GnN\kX^Z  
    %       % Display the Zernike function Z(n=5,m=1) Z8Jrt3l{2  
    %       x = -1:0.01:1; ef Moi'v  
    %       [X,Y] = meshgrid(x,x); f"{|c@%  
    %       [theta,r] = cart2pol(X,Y); JNJ96wnX1  
    %       idx = r<=1; K1gZ>FEY|N  
    %       z = nan(size(X)); (}#8$ )  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 4kNiS^h  
    %       figure $ouw *|<  
    %       pcolor(x,x,z), shading interp G2 E4  
    %       axis square, colorbar --> ~<o  
    %       title('Zernike function Z_5^1(r,\theta)') 9GV1@'<Y]  
    % mBrH`!  
    %   Example 2: tF/)DZ.to  
    % yMd<<:Ap  
    %       % Display the first 10 Zernike functions e}PJN6"5  
    %       x = -1:0.01:1; ]UMt  
    %       [X,Y] = meshgrid(x,x); 6XFLWN-)  
    %       [theta,r] = cart2pol(X,Y); Fu z'!  
    %       idx = r<=1; (/^s?`1{N?  
    %       z = nan(size(X)); `hVi!Q]*P  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ]RvFn~E!s  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; mr 6~8 I  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ~OE1Sd:2  
    %       y = zernfun(n,m,r(idx),theta(idx)); '&;s32']}  
    %       figure('Units','normalized') wDvG5  
    %       for k = 1:10  UZV\]Y  
    %           z(idx) = y(:,k); NKSK+ll2  
    %           subplot(4,7,Nplot(k)) F%]Z yO9  
    %           pcolor(x,x,z), shading interp }x9D;%)/  
    %           set(gca,'XTick',[],'YTick',[]) OpNxd]"T  
    %           axis square zUIh^hbFf  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Z)7|m  
    %       end !bq3c(d  
    % s2X<b `  
    %   See also ZERNPOL, ZERNFUN2. DH>>u  
    w<P$)~6  
    %   Paul Fricker 11/13/2006 J-k/#A4o  
    rP7[{'%r  
    Od,P,t9  
    % Check and prepare the inputs: ?=dp]E{  
    % ----------------------------- O6[ 4=4L  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) -Gyj]v5y`c  
        error('zernfun:NMvectors','N and M must be vectors.') V#P`FX  
    end %*A|hK+G:W  
    }t FRl  
    if length(n)~=length(m) Qf .ASC   
        error('zernfun:NMlength','N and M must be the same length.') hHsN(v  
    end C]bre^q  
    y!kU0  
    n = n(:); m+a\NXWR?N  
    m = m(:); (Ev=kO  
    if any(mod(n-m,2)) J6C/`)+w  
        error('zernfun:NMmultiplesof2', ... |b+ZKRW  
              'All N and M must differ by multiples of 2 (including 0).') MV?#g-5  
    end 0^m`jD  
    .*k$abb  
    if any(m>n) yP^C)  
        error('zernfun:MlessthanN', ... {@7xOOAw  
              'Each M must be less than or equal to its corresponding N.') \+T U{vr  
    end {2v,J]v_[  
    N fBH  
    if any( r>1 | r<0 ) Sp]u5\  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Mjj5~by:  
    end QUO'{;,  
    iU/v; T(  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) &*YFK/]  
        error('zernfun:RTHvector','R and THETA must be vectors.')  v[+ ]  
    end ]=Dzr<*v  
    sd,KB+)  
    r = r(:); #WlTE&  
    theta = theta(:); Klj -dz  
    length_r = length(r); Py~1xf/  
    if length_r~=length(theta) Jmml2?V-c  
        error('zernfun:RTHlength', ... ~#];&WE  
              'The number of R- and THETA-values must be equal.') crbph.0  
    end H 2JKQm_  
    4Nl3"@<$  
    % Check normalization: W' Y?X]xr  
    % -------------------- Kk\TW1w3  
    if nargin==5 && ischar(nflag) &`%J1[dy  
        isnorm = strcmpi(nflag,'norm'); dI?x&#(vw  
        if ~isnorm \n<9R8g5  
            error('zernfun:normalization','Unrecognized normalization flag.') F^Y%Q(Dd7w  
        end pdySip<  
    else R^?9 V=Y<T  
        isnorm = false; Ju@8_ ?8=  
    end gjL+8Rk  
    Ow50M;E  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^iqy|zNtn  
    % Compute the Zernike Polynomials % 4Gt^:J"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _O3X;U7rc  
    EpCF/i?9:  
    % Determine the required powers of r: %:!ILN  
    % ----------------------------------- {%+UQ!]d8  
    m_abs = abs(m); H#/Hs#  
    rpowers = []; W QqOXF  
    for j = 1:length(n) qORL 7?{  
        rpowers = [rpowers m_abs(j):2:n(j)]; WYm<_1  
    end \OW.?1d  
    rpowers = unique(rpowers); H{4_,2h =m  
    ;Xl {m`E+  
    % Pre-compute the values of r raised to the required powers, }Y!v"DO#Q*  
    % and compile them in a matrix: }B ?_>0  
    % ----------------------------- TXS{=  
    if rpowers(1)==0 tNG[|Bi#  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); jRN>^Ur;g  
        rpowern = cat(2,rpowern{:}); .G-L/*&%  
        rpowern = [ones(length_r,1) rpowern]; 7DPxz'7):  
    else q|sT4} =  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); JLak>MS  
        rpowern = cat(2,rpowern{:}); Ke^9R-jP  
    end Vtv~jJ{m  
    Ei4Iv#Oi`  
    % Compute the values of the polynomials: %z6_,|%  
    % -------------------------------------- <8ih >s(C  
    y = zeros(length_r,length(n)); ENy$sS6[D  
    for j = 1:length(n) vc C"  
        s = 0:(n(j)-m_abs(j))/2; DEW;0ic  
        pows = n(j):-2:m_abs(j); :(YFIW`59  
        for k = length(s):-1:1 &fW'_,-  
            p = (1-2*mod(s(k),2))* ... wv  
                       prod(2:(n(j)-s(k)))/              ... {8"Uxj_6V  
                       prod(2:s(k))/                     ... "$.B@[iY@  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... CI1K:K AM  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); pv,z$3Q  
            idx = (pows(k)==rpowers); =wMq!mBd  
            y(:,j) = y(:,j) + p*rpowern(:,idx); +y^'\KN  
        end =9;b|Y"aQ  
         uN=f( -"  
        if isnorm d*dPi^JjC  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); #y f  
        end Tm2+/qO,  
    end uT>"(wnJ|  
    % END: Compute the Zernike Polynomials (QS 0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a1shP};pK  
    pf&U$oR4  
    % Compute the Zernike functions: i_:#][nWX  
    % ------------------------------ 3X#Cep20a  
    idx_pos = m>0; 8Oa+,?<0x  
    idx_neg = m<0; Sq x'nXgO  
    deEc;IAo  
    z = y; \A6 }=  
    if any(idx_pos) !p Q*m`Xo  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ,0eXg  
    end kDG?/j90D  
    if any(idx_neg) :<v@xOzxx  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); R[l~E![!j  
    end x *a_43`  
    .O SQ8W }  
    % 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) \ 0Ba?  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. }@~+%_;  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated *;l[|  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive UgD)O:xaU  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, zYM0?O8pJ~  
    %   and THETA is a vector of angles.  R and THETA must have the same j8%Y[:~D  
    %   length.  The output Z is a matrix with one column for every P-value, 5lyHg{iqD  
    %   and one row for every (R,THETA) pair. wRZS+^hx  
    % /]of @  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike u $B24Cy.  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) `/9I` <y  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) a`zHx3Yg  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 eIOMW9Ivt  
    %   for all p. LP/SblE  
    % Sbeq%Iwm.  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 4y!GFhMh  
    %   Zernike functions (order N<=7).  In some disciplines it is ?J-D6;  
    %   traditional to label the first 36 functions using a single mode 1~E;@eK'  
    %   number P instead of separate numbers for the order N and azimuthal >Bu _NoM  
    %   frequency M. Lt i2KY}/%  
    % $~\Tl:!#?  
    %   Example: m}f{o  
    % T-]UAN"O  
    %       % Display the first 16 Zernike functions ge1U1o  
    %       x = -1:0.01:1; 6R*eJICN  
    %       [X,Y] = meshgrid(x,x); +:W?:\  
    %       [theta,r] = cart2pol(X,Y); p.H`lbVY  
    %       idx = r<=1; iBmvy 7S?  
    %       p = 0:15; b}*q*Bq  
    %       z = nan(size(X)); <sX_hIA^Fx  
    %       y = zernfun2(p,r(idx),theta(idx)); 1tTY )Evf  
    %       figure('Units','normalized') Asy2jw\V  
    %       for k = 1:length(p) q\<NW%KtX  
    %           z(idx) = y(:,k); x3F94+<n{  
    %           subplot(4,4,k) QU(Lv(/O  
    %           pcolor(x,x,z), shading interp HBs 6:[q  
    %           set(gca,'XTick',[],'YTick',[]) B1]FB|0's  
    %           axis square 1^ iLs  
    %           title(['Z_{' num2str(p(k)) '}']) /O`R9+;  
    %       end )<Mo.  
    % r?dkE=B  
    %   See also ZERNPOL, ZERNFUN. {:KPEN  
    foB&H;A4oC  
    %   Paul Fricker 11/13/2006 0.c9 6&  
    D |fo:Xp,  
    @Y<fj^]k  
    % Check and prepare the inputs: Q.\vN-(  
    % ----------------------------- v- p8~u1N  
    if min(size(p))~=1 :a:m>S<~  
        error('zernfun2:Pvector','Input P must be vector.') R]RLy#j  
    end bJkFCI/  
    :XTxrYt28  
    if any(p)>35 \Ym!5,^o  
        error('zernfun2:P36', ... vl?fCO  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 2/Ye<.#  
               '(P = 0 to 35).']) 8#9OSupp  
    end E^m)&.+'M  
    o{ccO29H/  
    % Get the order and frequency corresonding to the function number: %e_){28 n  
    % ---------------------------------------------------------------- EW3--33s  
    p = p(:); ca,c+5  
    n = ceil((-3+sqrt(9+8*p))/2); &vIj(e9Y  
    m = 2*p - n.*(n+2);  v&7x ~!O  
    xK4E+^ b  
    % Pass the inputs to the function ZERNFUN: 5HTY ~&C  
    % ---------------------------------------- f# hmMa  
    switch nargin V343 IT\  
        case 3 >S S^qjh/  
            z = zernfun(n,m,r,theta); l E* .9T  
        case 4 yVJ)JhV  
            z = zernfun(n,m,r,theta,nflag); - s2Yhf  
        otherwise ;=@?( n  
            error('zernfun2:nargin','Incorrect number of inputs.') RB;2  
    end T+FlN-iy)  
    l1%*LyD  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) Hl8\*#;C&>  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. u!b0 <E  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of n:Dr< q .  
    %   order N and frequency M, evaluated at R.  N is a vector of tMo=q7ig  
    %   positive integers (including 0), and M is a vector with the a`Q-5* \;z  
    %   same number of elements as N.  Each element k of M must be a 6c}nP[6|  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) '[bw7T  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is H{_6e6`e.  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix y@Td]6|f  
    %   with one column for every (N,M) pair, and one row for every [kPl7[OL  
    %   element in R. w2K>k/v{-  
    % '%a:L^a?  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- P_gai7Xg  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 1W9uWkk_d  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to I#W J";kqB  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 :;Npk9P(N  
    %   for all [n,m]. (&/~q:a>   
    % v,US4C|^3i  
    %   The radial Zernike polynomials are the radial portion of the ;3Q3!+%j  
    %   Zernike functions, which are an orthogonal basis on the unit *4l6+#W  
    %   circle.  The series representation of the radial Zernike T\Jm=+]c!  
    %   polynomials is SWNT}{x]  
    % ^n\g,  
    %          (n-m)/2 <V#]3$(S  
    %            __ NF7+Gp6?q  
    %    m      \       s                                          n-2s (]5gYi  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ,@Elw>^  
    %    n      s=0 gNh4c{Al9  
    % F_V/&OV  
    %   The following table shows the first 12 polynomials. 0rc'SEl  
    % "h&[6-0'  
    %       n    m    Zernike polynomial    Normalization ^YEMR C  
    %       --------------------------------------------- Ul:M=8nE%  
    %       0    0    1                        sqrt(2) YO;@Tj2)x  
    %       1    1    r                           2 i"{znKz vD  
    %       2    0    2*r^2 - 1                sqrt(6) q]y{ 4"=5  
    %       2    2    r^2                      sqrt(6) >a: 6umY  
    %       3    1    3*r^3 - 2*r              sqrt(8) hP jL  
    %       3    3    r^3                      sqrt(8) AQ,%5MeqJ  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) t5S!j2E  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Sqla+L*  
    %       4    4    r^4                      sqrt(10) =8 DS~J{  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) U#4>GO;A  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 59(} D'lw>  
    %       5    5    r^5                      sqrt(12) @$?*UI6y  
    %       --------------------------------------------- >W'"xK|:  
    %  L$[1+*  
    %   Example: ~8[`(/hj  
    % oCB#i~|>a  
    %       % Display three example Zernike radial polynomials .3xf!E*  
    %       r = 0:0.01:1; [ _&z+  
    %       n = [3 2 5];  ;.~D!  
    %       m = [1 2 1]; W1O Y}2kj  
    %       z = zernpol(n,m,r); +qyx3c+  
    %       figure ^]$rh.7&  
    %       plot(r,z) Y,X0x-  
    %       grid on (KT+7j0^  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') P)UpUMt;k  
    % 'Y>@t6E4  
    %   See also ZERNFUN, ZERNFUN2. qkq^oHI  
    /qXP\ a  
    % A note on the algorithm. z-`4DlJUS  
    % ------------------------ !Ee&e~"  
    % The radial Zernike polynomials are computed using the series 4HyD=6V#  
    % representation shown in the Help section above. For many special zB+zw\ncN  
    % functions, direct evaluation using the series representation can 0y/P  
    % produce poor numerical results (floating point errors), because Q7?[@2HN  
    % the summation often involves computing small differences between aOW$H:b  
    % large successive terms in the series. (In such cases, the functions qWI8 >my11  
    % are often evaluated using alternative methods such as recurrence r5uX?^mJ0  
    % relations: see the Legendre functions, for example). For the Zernike +< c(;Ucl?  
    % polynomials, however, this problem does not arise, because the IHe?/oUL"b  
    % polynomials are evaluated over the finite domain r = (0,1), and iMp_1EXe  
    % because the coefficients for a given polynomial are generally all <*djtO  
    % of similar magnitude. *0%G`Q  
    % tnw6[U!rh=  
    % ZERNPOL has been written using a vectorized implementation: multiple S!7|vb*ko  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] r7  *'s  
    % values can be passed as inputs) for a vector of points R.  To achieve `AhTER  
    % this vectorization most efficiently, the algorithm in ZERNPOL WD#7Q&T(;  
    % involves pre-determining all the powers p of R that are required to \xOv9(  
    % compute the outputs, and then compiling the {R^p} into a single HQrx9CXE  
    % matrix.  This avoids any redundant computation of the R^p, and < #7j~<  
    % minimizes the sizes of certain intermediate variables. i7xBi:Si  
    % qLm g18  
    %   Paul Fricker 11/13/2006 [L>AU; :  
    BE," lX  
    r1|;V~ a$~  
    % Check and prepare the inputs: 6?3\P>`3Y  
    % ----------------------------- c]/&xRd  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) UjS,<>fm  
        error('zernpol:NMvectors','N and M must be vectors.') 7gT^ZL  
    end O@,9a~Ghd  
    /??nO Vvt  
    if length(n)~=length(m) (rjv3=9\3  
        error('zernpol:NMlength','N and M must be the same length.') uJL[m(G  
    end etH]-S  
    "A&HNkRz  
    n = n(:); ]llvG \  
    m = m(:); Bl v @u?  
    length_n = length(n); %ZJ;>a#  
    gJuA*^  
    if any(mod(n-m,2)) CWM_J9f  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ]08 ~"p  
    end |kkg1M#  
    0-zIohSJdQ  
    if any(m<0) P:v|JER   
        error('zernpol:Mpositive','All M must be positive.') dL>8|  
    end TllIs&MCe  
    " IC0v9  
    if any(m>n) C2W&*W*  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') R]Pv=fn  
    end mdukl!_x  
    w:o,mzuXK  
    if any( r>1 | r<0 ) 2<Q3-|/i  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') i 9wk)  
    end _tpqo>  
    @wOX</_g  
    if ~any(size(r)==1) h$q=NTV  
        error('zernpol:Rvector','R must be a vector.') +(J{~A~  
    end i?CXDuL  
    ~> |o3&G{  
    r = r(:); wdTjJf r  
    length_r = length(r); c*`= o( S  
    Kv-4VWh  
    if nargin==4 o"@GYc["  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); j_HwR9^fd,  
        if ~isnorm \Dr?}D  
            error('zernpol:normalization','Unrecognized normalization flag.') Kq2,J&Ca3  
        end o<8=@ ^T  
    else MLn\ b0  
        isnorm = false; $&[}+??  
    end fs wQ*  
    T_b$8GYfCY  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AH#klYK  
    % Compute the Zernike Polynomials ) v^;"q"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% lq\/E`fc`  
    OMVK\_oXo  
    % Determine the required powers of r: <hSrx7o  
    % ----------------------------------- QIZbAnn_  
    rpowers = []; W: vw.  
    for j = 1:length(n) bClMM  
        rpowers = [rpowers m(j):2:n(j)]; t^-yK;`?q:  
    end HCTjFW>C  
    rpowers = unique(rpowers); 86N"EuH$  
    COk;z.Kn  
    % Pre-compute the values of r raised to the required powers, |] 8Hh>  
    % and compile them in a matrix: ]S#m o  
    % ----------------------------- 4` zfrT^  
    if rpowers(1)==0 UuvI?D  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); V[T`I a\  
        rpowern = cat(2,rpowern{:}); l2LUcI$ x  
        rpowern = [ones(length_r,1) rpowern]; (r+#}z}  
    else pv #uLo  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); |T@\ -8Ok  
        rpowern = cat(2,rpowern{:}); 98AX=%8  
    end &?mJL0fy  
    vkQkU,q  
    % Compute the values of the polynomials: ;.4A,7w#  
    % -------------------------------------- b 5X~^L  
    z = zeros(length_r,length_n); '8b/TL  
    for j = 1:length_n pk0C x  
        s = 0:(n(j)-m(j))/2; 1hn4YcHb  
        pows = n(j):-2:m(j); "=97:H{!  
        for k = length(s):-1:1 "]M]pR/j  
            p = (1-2*mod(s(k),2))* ... g1s\6%g  
                       prod(2:(n(j)-s(k)))/          ... %t:pG}A>:C  
                       prod(2:s(k))/                 ... xVn"xk  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ToWtltCD  
                       prod(2:((n(j)+m(j))/2-s(k))); 8Vb.%f &I  
            idx = (pows(k)==rpowers); bijE]:<AE7  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 8NRc+@f|m  
        end 39(]UO6^;  
         /X_g[*]?  
        if isnorm bEJz>oyW"  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 05cyWg9a  
        end J<4 egk4  
    end QXcSDJ  
    #gL$~.1  
    % 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)  "f/Su(6{0  
    Qi 3di  
    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。 u&q RK>wLa  
    f^P:eBgpx  
    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)