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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 6Z2|j~  
    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! (0.JoeA`y  
     
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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 RL/7>YQ  
    function z = zernfun(n,m,r,theta,nflag) FeQo,a  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. jsr)  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N m qUDve(  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Fm6]mz%~u#  
    %   unit circle.  N is a vector of positive integers (including 0), and 0Js5 ' 9}H  
    %   M is a vector with the same number of elements as N.  Each element gTB|IcOs  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Tyb'p9  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, QtW e,+WWV  
    %   and THETA is a vector of angles.  R and THETA must have the same 99,=dzm  
    %   length.  The output Z is a matrix with one column for every (N,M) '&K' 0qG  
    %   pair, and one row for every (R,THETA) pair. ,!g/1m  
    % 9f5~hBlo  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .*>C[^  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), u|u)8;'9(  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ~| ZAS]  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, H1KXAy`&  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Gv }  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. :eB+t`M  
    % O&~ @ior  
    %   The Zernike functions are an orthogonal basis on the unit circle. nU\.`.39 +  
    %   They are used in disciplines such as astronomy, optics, and B9cWxe4R#  
    %   optometry to describe functions on a circular domain. *ezft&{)`  
    % T?=]&9Y'  
    %   The following table lists the first 15 Zernike functions. <mTo54g  
    % >c5   
    %       n    m    Zernike function           Normalization b].U/=Hs  
    %       -------------------------------------------------- [eTEK W]  
    %       0    0    1                                 1 xy$aFPH!-  
    %       1    1    r * cos(theta)                    2 e7&RZ+s#wZ  
    %       1   -1    r * sin(theta)                    2 +>[zn  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) *`/4KMrq  
    %       2    0    (2*r^2 - 1)                    sqrt(3) -ik((qx_  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) NE) w$>0M  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) h<PS<  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Nt?=0X|M  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) :6EX-Xyj  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ]6|?H6'/`v  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) (dO0`wfM  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) REi"Aj=  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) YZnFU( j  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f.oY:3h:  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 2_?VR~mA#  
    %       -------------------------------------------------- hjk]?MC  
    % e},:QL0X  
    %   Example 1: m c@Z+t'  
    % Y(EF )::  
    %       % Display the Zernike function Z(n=5,m=1) =T?Xph{  
    %       x = -1:0.01:1; 5b I4' ;  
    %       [X,Y] = meshgrid(x,x); EBQ_c@  
    %       [theta,r] = cart2pol(X,Y); ! /|B4Yv  
    %       idx = r<=1; v{*2F  
    %       z = nan(size(X)); }v_|N"@  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); dpt P(H  
    %       figure r( wtuD23q  
    %       pcolor(x,x,z), shading interp n%~r^ C_  
    %       axis square, colorbar )fS6H<*  
    %       title('Zernike function Z_5^1(r,\theta)') P#8lO%;  
    % Y(K`3? A  
    %   Example 2: Py+ B 2G|  
    % WUQa2$.  
    %       % Display the first 10 Zernike functions <&)zT#"  
    %       x = -1:0.01:1; @j%@Z  
    %       [X,Y] = meshgrid(x,x); f6Y-ss;'  
    %       [theta,r] = cart2pol(X,Y); dI=&gz  
    %       idx = r<=1; j-FMWEp  
    %       z = nan(size(X)); ~HtD]|7  
    %       n = [0  1  1  2  2  2  3  3  3  3]; o4z|XhLr  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 1UB.2}/:  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Zx6h%l,%  
    %       y = zernfun(n,m,r(idx),theta(idx)); "EWq{l_I5$  
    %       figure('Units','normalized') 9j5Z!Vsy  
    %       for k = 1:10 jC?l :m?  
    %           z(idx) = y(:,k); BuC\Bd^0  
    %           subplot(4,7,Nplot(k)) N"~P$B1 X  
    %           pcolor(x,x,z), shading interp ^d(gC%+!u  
    %           set(gca,'XTick',[],'YTick',[]) Bw[IW[(~!  
    %           axis square XZ8]se"C  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) I_`NjJ;61  
    %       end jgkY^l  
    % X"HVK+  
    %   See also ZERNPOL, ZERNFUN2. { W5 _KX  
    |&bucG=  
    %   Paul Fricker 11/13/2006 4)L};B=  
    ;vpq0t`  
    "uyr@u0b  
    % Check and prepare the inputs: V;~\+@  
    % ----------------------------- I;, n|o  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ;MlPP)*k  
        error('zernfun:NMvectors','N and M must be vectors.') G2|G}#E  
    end #D >:'ezm  
    p2+K-/}ApP  
    if length(n)~=length(m) Ggv*EsN/cC  
        error('zernfun:NMlength','N and M must be the same length.') #AO}JP  
    end $"0`2C  
    wg:\$_Og  
    n = n(:); uOd1:\%*  
    m = m(:); Zl]@;*u  
    if any(mod(n-m,2)) F2)KAIl  
        error('zernfun:NMmultiplesof2', ... eOZ~p  
              'All N and M must differ by multiples of 2 (including 0).') tWTC'Gx-J  
    end jOK !k  
    ;2sP3!*  
    if any(m>n) tejpY  
        error('zernfun:MlessthanN', ... t [G7&ovj  
              'Each M must be less than or equal to its corresponding N.') RYl\Q,#  
    end jz\>VYi(7  
    f&$$*a  
    if any( r>1 | r<0 ) k6\&[BQs  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 7|!Zx-}  
    end w2r* $Q  
    3 rLc\rK  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) h 3  J&  
        error('zernfun:RTHvector','R and THETA must be vectors.') ]2[\E~^KU  
    end XuU>.T$]c  
    Z 2$S'}F  
    r = r(:); IiX2O(*ZE  
    theta = theta(:); ~BnmAv$m[  
    length_r = length(r); h]VC<BD6S  
    if length_r~=length(theta) IZd~Am3f  
        error('zernfun:RTHlength', ... %UV"@I+  
              'The number of R- and THETA-values must be equal.') r -uu`=,  
    end VArMFP)cz  
    =65XT^  
    % Check normalization: 7Q&S [])  
    % -------------------- #!r>3W&  
    if nargin==5 && ischar(nflag) VZ9`Kbu  
        isnorm = strcmpi(nflag,'norm'); =~21.p  
        if ~isnorm X7MA>j3m  
            error('zernfun:normalization','Unrecognized normalization flag.') x Y}.mP  
        end Ffd;aZ4n  
    else FJW,G20L  
        isnorm = false; )E6E}  
    end KHeeB`V>J  
    1ZvXRJ)%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B? XK;*])  
    % Compute the Zernike Polynomials tC7 4=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zIU6bMMT3u  
    Go[anf  
    % Determine the required powers of r: I.%EYAai  
    % ----------------------------------- m\|EM'@k  
    m_abs = abs(m); ~cfvL*~5  
    rpowers = []; xi)M8\K  
    for j = 1:length(n) 5mm&l+N)  
        rpowers = [rpowers m_abs(j):2:n(j)]; }0OQm?xh  
    end X%`:waR  
    rpowers = unique(rpowers); QS-X_  
    @U =~ c9  
    % Pre-compute the values of r raised to the required powers, $vn x)#r3  
    % and compile them in a matrix: Z)}2bJwA  
    % ----------------------------- G P ' -  
    if rpowers(1)==0 D\DwBZ>  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); |HwEwL+  
        rpowern = cat(2,rpowern{:}); V7 hO}  
        rpowern = [ones(length_r,1) rpowern]; veS) j?4  
    else !0v3Lu ~j  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 6O*lZNN  
        rpowern = cat(2,rpowern{:}); NK%Ok  
    end C!Fi &~  
    >U]KPL[%  
    % Compute the values of the polynomials: ^Qxv5HS2  
    % -------------------------------------- r( zn1;zl  
    y = zeros(length_r,length(n)); Y&$puiH-j  
    for j = 1:length(n) /9?yw!  
        s = 0:(n(j)-m_abs(j))/2; (!9+QXb'  
        pows = n(j):-2:m_abs(j); _k(&<1i  
        for k = length(s):-1:1 SPtx_+ Q)S  
            p = (1-2*mod(s(k),2))* ... I(Vg  
                       prod(2:(n(j)-s(k)))/              ... pLMaXX~4_  
                       prod(2:s(k))/                     ... zbvV:9N  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... d$n<^ ~Z  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); -<(RYMk*)  
            idx = (pows(k)==rpowers); !y$+RA7\  
            y(:,j) = y(:,j) + p*rpowern(:,idx); hsYv=Tw3C  
        end U/h@Q\~U  
         Z,8t!Y  
        if isnorm #jPn7  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); BUyKiMW49  
        end J.c yb  
    end +HG*T[%/  
    % END: Compute the Zernike Polynomials }|Bs|$q  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F|8;Swb5  
    n`T4P$pt  
    % Compute the Zernike functions: ?^`fPH=  
    % ------------------------------ -_Kw3x  
    idx_pos = m>0; S[N9/2  
    idx_neg = m<0; BW"24JhF"  
    (?"z!dgc  
    z = y; y43ha  
    if any(idx_pos) J_9[ x mM  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); vD(:?M  
    end 8U!$()^?  
    if any(idx_neg) Ms-)S7tMz  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); \[ 4y  
    end |n~,{=  
    6r`Xi&  
    % 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) R]0`-_T  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ^1_CS*  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated $KlaZ>D h  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Fqh./@o  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, e&!8UYP  
    %   and THETA is a vector of angles.  R and THETA must have the same )UyJ.!Fly  
    %   length.  The output Z is a matrix with one column for every P-value, dqO]2d  
    %   and one row for every (R,THETA) pair. zV(aw~CbZ  
    % Ty7)j]b"zl  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike l+X\>,  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) s^Xs*T@~h  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Z$zX%w  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 r`< x@,  
    %   for all p. 0f_A"K  
    % xC}'"``s  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 U} w@,6  
    %   Zernike functions (order N<=7).  In some disciplines it is wc&D[M]-/  
    %   traditional to label the first 36 functions using a single mode {SD%{  
    %   number P instead of separate numbers for the order N and azimuthal ,LDL%<7t  
    %   frequency M. W_,7hvE?"H  
    % ~ H/ZiBL@  
    %   Example: JVr8O`>T  
    % c c/nzB  
    %       % Display the first 16 Zernike functions M}q;\}  
    %       x = -1:0.01:1; L!,@_   
    %       [X,Y] = meshgrid(x,x); b~@+6 ?  
    %       [theta,r] = cart2pol(X,Y); OXn-!J90P  
    %       idx = r<=1; hTmJ ~m'J  
    %       p = 0:15; yB 'C9wEH  
    %       z = nan(size(X)); ;' H\s  
    %       y = zernfun2(p,r(idx),theta(idx)); 9vSKIq  
    %       figure('Units','normalized') }Z< Sca7  
    %       for k = 1:length(p) }w-M .  
    %           z(idx) = y(:,k); G5RdytK  
    %           subplot(4,4,k) .?LRt  
    %           pcolor(x,x,z), shading interp ?e,:x ]\L  
    %           set(gca,'XTick',[],'YTick',[]) 1kR. .p<"  
    %           axis square AWssDbh/[  
    %           title(['Z_{' num2str(p(k)) '}']) %s^1de  
    %       end ;zV<63tW  
    % 3i'01z  
    %   See also ZERNPOL, ZERNFUN. 'f.k'2T  
    PsD)]V9%:  
    %   Paul Fricker 11/13/2006 H4j1yD(d  
    *'\HG  
    ZX8@/8sv  
    % Check and prepare the inputs: 5HE5$S  
    % ----------------------------- 69apTx  
    if min(size(p))~=1 r adP%W-U  
        error('zernfun2:Pvector','Input P must be vector.') ~t ZB1+%)  
    end "fUNrhCx  
    6a_U[-a9;  
    if any(p)>35 MUGoW;}v )  
        error('zernfun2:P36', ... }[h]z7e2S  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... g1.u1}  
               '(P = 0 to 35).']) lnLy"f"zV  
    end 99CK [G  
    FK`:eP{  
    % Get the order and frequency corresonding to the function number: >Gk<a  
    % ---------------------------------------------------------------- lyyf&?2  
    p = p(:); NL;sn"  
    n = ceil((-3+sqrt(9+8*p))/2); P#`M8k  
    m = 2*p - n.*(n+2); OE Xa}K#  
    A1`6+8}o;b  
    % Pass the inputs to the function ZERNFUN: <5P*uZ  
    % ---------------------------------------- &K(y%ieIJ  
    switch nargin dUl"w`3  
        case 3 )Q>Ao.  
            z = zernfun(n,m,r,theta); B& R?{y*  
        case 4 wu`+KUx  
            z = zernfun(n,m,r,theta,nflag); >]C/ Q6  
        otherwise $5&~gHc,  
            error('zernfun2:nargin','Incorrect number of inputs.') I,HtW),  
    end V\opC6*L_e  
    !H{>c@i  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) yNn=r;FZQ  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. v+`'%E  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of S<*IoZ?T  
    %   order N and frequency M, evaluated at R.  N is a vector of yjH'<  
    %   positive integers (including 0), and M is a vector with the r]D U  
    %   same number of elements as N.  Each element k of M must be a ZH8w^}  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) #s15AyKz5  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Xw<;)m  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ksuePMIK  
    %   with one column for every (N,M) pair, and one row for every N-knhA  
    %   element in R. _~ei1 G.R  
    % |G$-5 7fk  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- jw {B8<@s  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 5|N`:h'9M  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ITTEUw~+o  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 "`cPV){]  
    %   for all [n,m]. 3o/f, }_  
    % d|7LCW+HW  
    %   The radial Zernike polynomials are the radial portion of the :yJ([  
    %   Zernike functions, which are an orthogonal basis on the unit XM*5I 4V  
    %   circle.  The series representation of the radial Zernike =>tkc/aa  
    %   polynomials is "VSx?74q  
    % %6 GM[1__  
    %          (n-m)/2 0)~c)B:5  
    %            __ 3oH/34jj  
    %    m      \       s                                          n-2s W} H~ka  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r N[Ei%I  
    %    n      s=0 n*na6rV\k  
    % QT^b-~^  
    %   The following table shows the first 12 polynomials. B@i%B+qCLv  
    % nGYi mRYO  
    %       n    m    Zernike polynomial    Normalization l"nS +z  
    %       --------------------------------------------- ~LV]cX2J(  
    %       0    0    1                        sqrt(2) yt5<J-m  
    %       1    1    r                           2 Ddg!1SF  
    %       2    0    2*r^2 - 1                sqrt(6) Wkjp:`(-$r  
    %       2    2    r^2                      sqrt(6) FdzdoMY  
    %       3    1    3*r^3 - 2*r              sqrt(8)  JJ}DYv  
    %       3    3    r^3                      sqrt(8) H)gc"aRe;Y  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ZAN~TG<n  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) %X %zK1  
    %       4    4    r^4                      sqrt(10) n5.sx|bI?  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) {cIk-nG -_  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) h4|}BGO  
    %       5    5    r^5                      sqrt(12) ./Ek+p*96H  
    %       --------------------------------------------- R#;xBBt8  
    % FjtS  
    %   Example: :H m'o}  
    % ?2Z`xL9QT  
    %       % Display three example Zernike radial polynomials 4OgH+<G  
    %       r = 0:0.01:1; 6?KUS}nRS  
    %       n = [3 2 5]; F!)[H["_  
    %       m = [1 2 1]; wS#Uw_[  
    %       z = zernpol(n,m,r); K$/"I0YyI  
    %       figure 83/m^^F{]  
    %       plot(r,z) TaHcvjhR  
    %       grid on p!^K.P1 '  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 5  >0\=  
    % z+6PVQ  
    %   See also ZERNFUN, ZERNFUN2.  .nrbd#i-  
    NiW9/(;xB  
    % A note on the algorithm. iO?^y(phC  
    % ------------------------ ,&S0/j  
    % The radial Zernike polynomials are computed using the series S qb>a j  
    % representation shown in the Help section above. For many special n9={D  
    % functions, direct evaluation using the series representation can KhB775  
    % produce poor numerical results (floating point errors), because Q. O4R_H  
    % the summation often involves computing small differences between ov,s]g83  
    % large successive terms in the series. (In such cases, the functions UhS:tT]7  
    % are often evaluated using alternative methods such as recurrence K&NH?  
    % relations: see the Legendre functions, for example). For the Zernike 0LL0\ly]  
    % polynomials, however, this problem does not arise, because the : q%1Vi  
    % polynomials are evaluated over the finite domain r = (0,1), and 0q-lyVZ^X  
    % because the coefficients for a given polynomial are generally all }k%6X@  
    % of similar magnitude. ^ IuhHP  
    % 8&"Jlz |  
    % ZERNPOL has been written using a vectorized implementation: multiple = wDXlAQ  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] r/ g{j  
    % values can be passed as inputs) for a vector of points R.  To achieve u$[8Zmgzz  
    % this vectorization most efficiently, the algorithm in ZERNPOL 'hBnV xd&  
    % involves pre-determining all the powers p of R that are required to SF-"3M  
    % compute the outputs, and then compiling the {R^p} into a single 2!B|w8ar  
    % matrix.  This avoids any redundant computation of the R^p, and &k}B66  
    % minimizes the sizes of certain intermediate variables. Ul]7IUzsu  
    % fv8x7l7  
    %   Paul Fricker 11/13/2006 V^[&4  
    wW\@^5  
    b:Zh|-  
    % Check and prepare the inputs: ]3I a>i  
    % ----------------------------- qQ3Q4R\  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) \l /}` w  
        error('zernpol:NMvectors','N and M must be vectors.') 2h51zG#qd  
    end -A w]b} #v  
    rmkBp_i{|  
    if length(n)~=length(m) ~<VxtcEBz  
        error('zernpol:NMlength','N and M must be the same length.') ]j/= x2p  
    end _h}(j Ed!  
    T&pCLvkz  
    n = n(:); ]9w)0iH  
    m = m(:); _p0Yhju?  
    length_n = length(n); \z!lw  
    TA*}p=?6?!  
    if any(mod(n-m,2)) b=MW;]F  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ^\O*e)#*  
    end > VIFQ\  
    (b#M4ho*f  
    if any(m<0) _yN5sLLyb  
        error('zernpol:Mpositive','All M must be positive.') W1"NKg~4  
    end .p e3L7g  
    a}NB6E)-  
    if any(m>n) n8;L_43U  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') qfJ2iE|o2.  
    end f]%S FQ+  
    8el6z2  
    if any( r>1 | r<0 ) ~\NQkaBkY  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') R)Mkt8v  
    end ' abEY  
    \os"w "  
    if ~any(size(r)==1) r7R'beiH  
        error('zernpol:Rvector','R must be a vector.') 4_QfM}Fyp  
    end /fT"WaTEK  
    SQK82 /  
    r = r(:); #*CMf.OCh  
    length_r = length(r); Wgte.K> /  
    Pa d)|  
    if nargin==4 "QXnE^  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); mTX:?>  
        if ~isnorm J  Y8Rk=  
            error('zernpol:normalization','Unrecognized normalization flag.') ]h`*w  
        end \[[xyd  
    else klQmo30i  
        isnorm = false; =bD.5,F)  
    end (N&?Z]|yr  
    +?"F=.SZ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M}11 tUl  
    % Compute the Zernike Polynomials *> nOL  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% bv]SR_Tiq  
    TX$dxHSPK  
    % Determine the required powers of r: --l UEo~  
    % ----------------------------------- LhAW|];  
    rpowers = []; y ]@JkF(  
    for j = 1:length(n) *Xk5H,:  
        rpowers = [rpowers m(j):2:n(j)]; DQW)^j h  
    end [UzacXt  
    rpowers = unique(rpowers); hE=xS:6  
    T:{&e WH  
    % Pre-compute the values of r raised to the required powers, HJg&fkHn1  
    % and compile them in a matrix: P/ 6$TgQ  
    % ----------------------------- "0PsCr}!  
    if rpowers(1)==0 NYHK>u/5c  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); -|}?+W  
        rpowern = cat(2,rpowern{:}); UJqh~s  
        rpowern = [ones(length_r,1) rpowern]; &UnhYG{A  
    else v+{{j|x=  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 1K/ :  
        rpowern = cat(2,rpowern{:}); F%p DF\  
    end %Jh( 5  
    M.y!J  
    % Compute the values of the polynomials: "TaLvworb4  
    % -------------------------------------- l+2NA4s  
    z = zeros(length_r,length_n); Z|*#)<| ~  
    for j = 1:length_n %+Nng<_U\T  
        s = 0:(n(j)-m(j))/2; ,|yscp8  
        pows = n(j):-2:m(j); [8Y7Q5Had  
        for k = length(s):-1:1 |LC"1 k  
            p = (1-2*mod(s(k),2))* ... y{3+Un  
                       prod(2:(n(j)-s(k)))/          ... Fl($0}ER  
                       prod(2:s(k))/                 ... ldp9+7n~  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... a_S`$(7k  
                       prod(2:((n(j)+m(j))/2-s(k))); zOSUYn  
            idx = (pows(k)==rpowers); ?q4`&";{3  
            z(:,j) = z(:,j) + p*rpowern(:,idx); o>(<:^x9  
        end 1o\2\B=k{  
         fh)eL<I  
        if isnorm :35h0;8+  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); <?IDCOt ?  
        end iP9]b&  
    end :^`j:B  
    ^GM3nx$  
    % 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)  )!zg=}V  
    T 9}dgf  
    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。 '|tmmoY6a:  
    i-95>ff  
    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)