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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 $CP_oEb  
    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! Udl8?EVSz  
     
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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 Ka|, qkb  
    function z = zernfun(n,m,r,theta,nflag) P{ HYZg  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. y)]L>o~  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ^j>w<ljzz  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 3sF^6<E  
    %   unit circle.  N is a vector of positive integers (including 0), and t~(|2nTO5  
    %   M is a vector with the same number of elements as N.  Each element Exu>%  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 6<>T{2b:(p  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Yp(F}<f?  
    %   and THETA is a vector of angles.  R and THETA must have the same .QVZ!  
    %   length.  The output Z is a matrix with one column for every (N,M)  SE;Yb'  
    %   pair, and one row for every (R,THETA) pair. lS!uL9t.  
    % RwyRPc _  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike h-+GS%  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), z [9f  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral f&ri=VJY\T  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 75?z" i  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized $7 FT0?kG  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. i?0+f }5<p  
    % ;.EW7`)Z  
    %   The Zernike functions are an orthogonal basis on the unit circle. 2n|]&D3V"'  
    %   They are used in disciplines such as astronomy, optics, and |jT^[q(z  
    %   optometry to describe functions on a circular domain. \[yg f6#[  
    % XjINRC8^4  
    %   The following table lists the first 15 Zernike functions. B;=-h(E}vJ  
    % kD.KZV  
    %       n    m    Zernike function           Normalization 9Impp5`/B  
    %       -------------------------------------------------- U\~9YX8  
    %       0    0    1                                 1 H)VzPe#{  
    %       1    1    r * cos(theta)                    2 'wm :Xa  
    %       1   -1    r * sin(theta)                    2 &upM,Jsr*  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) L$rMfe S  
    %       2    0    (2*r^2 - 1)                    sqrt(3) >xB[k-C4  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) .`@)c/<0  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) :+*q,lX8  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) i$ CN{c*  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 6G0Y,B7&  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) YRRsbm{  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) TpIx!R9  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) pB0p?D)n  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) $vjl-1x&  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {2,vxGi  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) YggeKN  
    %       -------------------------------------------------- _`-trE.  
    % ":!7R<t  
    %   Example 1: 'ugc=-0pd  
    % MFzJ 8^.1R  
    %       % Display the Zernike function Z(n=5,m=1) [QZ g=."  
    %       x = -1:0.01:1; su\iUi  
    %       [X,Y] = meshgrid(x,x); R;l;;dC=  
    %       [theta,r] = cart2pol(X,Y); K~6,xZlDWM  
    %       idx = r<=1; bbe$6xwi  
    %       z = nan(size(X)); 1r?hRJ:'  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); vmsrypm  
    %       figure 734f &2  
    %       pcolor(x,x,z), shading interp 2>+(OL4l  
    %       axis square, colorbar 1=U NA :t<  
    %       title('Zernike function Z_5^1(r,\theta)') s:ZYiZ-  
    % Q}6!t$Vk  
    %   Example 2: qSA]61U&  
    % /9@[gv A  
    %       % Display the first 10 Zernike functions ms%RNxU4:  
    %       x = -1:0.01:1; qEJ#ce]G  
    %       [X,Y] = meshgrid(x,x); EJ@&vuDd$  
    %       [theta,r] = cart2pol(X,Y); 0Fc^c[  
    %       idx = r<=1; }huFv*<@'  
    %       z = nan(size(X)); CR8szMa  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ATzFs]~K;  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; V]Z!x.x"=y  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; RzOcz=A}  
    %       y = zernfun(n,m,r(idx),theta(idx)); dtx3;d<NsJ  
    %       figure('Units','normalized') kJ[r.)HU  
    %       for k = 1:10 {16]8-pe  
    %           z(idx) = y(:,k); ? dh  
    %           subplot(4,7,Nplot(k)) AC&)FY  
    %           pcolor(x,x,z), shading interp ;1AX u/  
    %           set(gca,'XTick',[],'YTick',[]) -\[H>)z]RB  
    %           axis square  $+  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) r\T'_wo  
    %       end f>hA+  
    % Ek6z[G` O  
    %   See also ZERNPOL, ZERNFUN2. hZ`<ID  
    /N9ct4 {^  
    %   Paul Fricker 11/13/2006 m"/ o4  
    Aw$+Ew[8 2  
    Lvd es.0|  
    % Check and prepare the inputs: q5xF~SQGw2  
    % ----------------------------- w<&R|= 93  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Lm3~< vP1e  
        error('zernfun:NMvectors','N and M must be vectors.') .L@gq/x)  
    end z3Zo64V~7  
    g^: & Dh  
    if length(n)~=length(m) :i9=Wj  
        error('zernfun:NMlength','N and M must be the same length.') 0PD=/fh[  
    end A9_} RJ9  
    l0w<NZ F  
    n = n(:); IhjZ{oV/@  
    m = m(:); hN^,'O  
    if any(mod(n-m,2)) z_8lf_N  
        error('zernfun:NMmultiplesof2', ... PC!g?6J  
              'All N and M must differ by multiples of 2 (including 0).') lG5KZ[/Or  
    end b.j$Gna>Q  
    R8-=N+hX  
    if any(m>n) 6,cJ3~!48  
        error('zernfun:MlessthanN', ... Ef$a&*)PH  
              'Each M must be less than or equal to its corresponding N.') g{^~g  
    end GIZw/L7Yb  
    $1 t IC_  
    if any( r>1 | r<0 ) E?- ~*T  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') =Hbf()cN)  
    end v>0I=ut  
    C2{*m{ D  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) i__f%j`!W  
        error('zernfun:RTHvector','R and THETA must be vectors.') MfZamu5+F  
    end (YM2Cv{4  
    hVIv->  
    r = r(:); A<_{7F9  
    theta = theta(:); MY}/h@  
    length_r = length(r); )G),iy  
    if length_r~=length(theta) "`NAg  
        error('zernfun:RTHlength', ... ua E,F^p  
              'The number of R- and THETA-values must be equal.') (q@%eor&}  
    end )FN\jo!!.  
    6WX?Xc]$3  
    % Check normalization: -AN5LE9-  
    % -------------------- d$^ @$E2f  
    if nargin==5 && ischar(nflag) a<J< Oc!  
        isnorm = strcmpi(nflag,'norm'); IIN,Da;hD  
        if ~isnorm <JIqkGeAi  
            error('zernfun:normalization','Unrecognized normalization flag.') ,rV;T";r  
        end L*OG2liJ  
    else `S+n,,l  
        isnorm = false; 1 -$+@Xl  
    end Eh^gR`I  
    : { iK 5  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5"y)<VLJX  
    % Compute the Zernike Polynomials T+q5~~\d  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zs6rd83#  
    B@v (ZY  
    % Determine the required powers of r: orOq5?3  
    % ----------------------------------- aLl=L_  
    m_abs = abs(m); k t'[  
    rpowers = []; d_!}9  
    for j = 1:length(n) !jf!\Uu[U  
        rpowers = [rpowers m_abs(j):2:n(j)]; {#~A `crO  
    end V-3;7  
    rpowers = unique(rpowers); AZf69z  
    YYL3a=;`a  
    % Pre-compute the values of r raised to the required powers, c/^l2CJ0  
    % and compile them in a matrix: +koW3>  
    % ----------------------------- yuC|_nL  
    if rpowers(1)==0 M3Qi]jO98  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); l$[,V:N  
        rpowern = cat(2,rpowern{:}); Sk:x.oOZ  
        rpowern = [ones(length_r,1) rpowern]; 0"Euf41  
    else "[-W(=  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); I5)$M{#a  
        rpowern = cat(2,rpowern{:}); e,Z[Nox  
    end 5 `@yX[G  
    /;vHAtt;f  
    % Compute the values of the polynomials: nch#DE8 2  
    % -------------------------------------- el\xMe^SY  
    y = zeros(length_r,length(n)); )3R5cq  
    for j = 1:length(n) YeVo=hYH@  
        s = 0:(n(j)-m_abs(j))/2; 2'@D0L  
        pows = n(j):-2:m_abs(j); );h  
        for k = length(s):-1:1 u@P1`E1Q  
            p = (1-2*mod(s(k),2))* ... aK_k'4YTm  
                       prod(2:(n(j)-s(k)))/              ... :;c`qO4  
                       prod(2:s(k))/                     ... 7kITssVHI  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... tt CC] Q  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); O\gVB!x  
            idx = (pows(k)==rpowers); qA[cF$CIl)  
            y(:,j) = y(:,j) + p*rpowern(:,idx); )c?nh3D  
        end 8)2M%R\THn  
         z`eMb  
        if isnorm 24 .'+3  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); f3 imkZ(  
        end R](cko=  
    end *K& $9fah  
    % END: Compute the Zernike Polynomials Bz|/TV?X(  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]omBq<ox'Y  
    6$kh5$[  
    % Compute the Zernike functions: |j{]6Nu  
    % ------------------------------ fQwLx  
    idx_pos = m>0; $Yp.BE<}  
    idx_neg = m<0; lIZ&' z  
    k2.k}?w!JO  
    z = y; |WpJen*?Y  
    if any(idx_pos) Sx (E'?]  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); :6Tv4ZUvcG  
    end So75h*e  
    if any(idx_neg) 4?+jvVq  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); KfYT  
    end jW4>WDN:  
    qq_ZkU@xg  
    % EOF zernfun
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) RTDplv; ]  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. P2 qC[1hYH  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated kY6_n4  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Zz]/4 4t  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ITEf Q@#jU  
    %   and THETA is a vector of angles.  R and THETA must have the same +EqL|  
    %   length.  The output Z is a matrix with one column for every P-value, \ rg;xZa5  
    %   and one row for every (R,THETA) pair. B/^o$i  
    % :zvAlt'q=  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike d0f(Uk  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) o*"Q{Xh#Qd  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) M _lLP8W}  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 !4<A|$mQ  
    %   for all p. cM4{ e^  
    % E1`_[=8a9  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 2$VSH&  
    %   Zernike functions (order N<=7).  In some disciplines it is e**'[3Y  
    %   traditional to label the first 36 functions using a single mode #?eMEws  
    %   number P instead of separate numbers for the order N and azimuthal >6@,L+-6r  
    %   frequency M. dTlEEgR  
    % Kb-m  
    %   Example: _34%St!lg  
    % GU9p'E  
    %       % Display the first 16 Zernike functions Pj_DI)^  
    %       x = -1:0.01:1; oIMS >&  
    %       [X,Y] = meshgrid(x,x); -w8?Ur1x:  
    %       [theta,r] = cart2pol(X,Y); tA'5ufj*:  
    %       idx = r<=1; -^;,m=4{3  
    %       p = 0:15; ]scr@e  
    %       z = nan(size(X)); a<>cbP  
    %       y = zernfun2(p,r(idx),theta(idx)); wlslG^^(!  
    %       figure('Units','normalized') I3izLi  
    %       for k = 1:length(p) %K7;ePu  
    %           z(idx) = y(:,k); aGws?<1$  
    %           subplot(4,4,k) ='C;^ Bk  
    %           pcolor(x,x,z), shading interp D0MW~Y6{  
    %           set(gca,'XTick',[],'YTick',[]) =<zlg~i  
    %           axis square ,t9CP  
    %           title(['Z_{' num2str(p(k)) '}']) $2blF)uYE  
    %       end yS[HYq  
    % vq-;wdq?2  
    %   See also ZERNPOL, ZERNFUN. qK~]au:C  
    o]&P0 b  
    %   Paul Fricker 11/13/2006 C7}iwklcsa  
    hst Ge>f[6  
    ~ahu{A4Bw  
    % Check and prepare the inputs: '"ze Im~  
    % ----------------------------- SJi;_bVf  
    if min(size(p))~=1 ] \!,yiVeU  
        error('zernfun2:Pvector','Input P must be vector.') e 0Z2B2  
    end ]"YXa~b  
    +&J1D8  
    if any(p)>35 TV0Y{x*~iH  
        error('zernfun2:P36', ... wyAh%'V  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 8493O x4 O  
               '(P = 0 to 35).']) 0AoWw-H6V  
    end ljz=u;O)  
    f.E{s*z>  
    % Get the order and frequency corresonding to the function number: !1]jk(Z  
    % ---------------------------------------------------------------- *A")A.R  
    p = p(:); \yLFV9P}EL  
    n = ceil((-3+sqrt(9+8*p))/2); -lq`EB +  
    m = 2*p - n.*(n+2); Tn(uH17  
    PpNG`_O  
    % Pass the inputs to the function ZERNFUN: 1|>bG#|  
    % ---------------------------------------- +JXn   
    switch nargin /rK/ l  
        case 3 MU:v& sk  
            z = zernfun(n,m,r,theta); !|9k&o  
        case 4 f'`y-]"V5)  
            z = zernfun(n,m,r,theta,nflag); -rHqU|  
        otherwise qw)Ou]L=  
            error('zernfun2:nargin','Incorrect number of inputs.') CWB<I  
    end "+ k}#<P4\  
    m")p]B&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) kK0zb{  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. "avG#rsH  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 9J<vkxG9`  
    %   order N and frequency M, evaluated at R.  N is a vector of ' 8Q }pp`  
    %   positive integers (including 0), and M is a vector with the 5a2;@ }%V  
    %   same number of elements as N.  Each element k of M must be a ygK,t*T20  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) xf|C{XV@H  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is %/!f^PIwX  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix "{~^EQq,  
    %   with one column for every (N,M) pair, and one row for every bhfKhXh8  
    %   element in R. 8k.#4}fP  
    % 4CS$%Cu\?w  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- w7\ \m9  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ] {0OPU  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to +vV?[e  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ,.;{J|4P  
    %   for all [n,m]. 9c5DEq  
    % Tq6\oIBkV  
    %   The radial Zernike polynomials are the radial portion of the 0a,B&o1  
    %   Zernike functions, which are an orthogonal basis on the unit ws U@hqS  
    %   circle.  The series representation of the radial Zernike 'c >^Aai  
    %   polynomials is U0N6\+  
    % q!Z{qt*`um  
    %          (n-m)/2 Mi}k>5VT  
    %            __ zW[HGI6w  
    %    m      \       s                                          n-2s Sg\+al7  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r y~VLa  
    %    n      s=0 a,n#E!zT?w  
    % TpHzf3.I  
    %   The following table shows the first 12 polynomials. @Q!Tvw/  
    % $2Bll5!]  
    %       n    m    Zernike polynomial    Normalization A+fXt`YNM  
    %       --------------------------------------------- r*FAUb`bG  
    %       0    0    1                        sqrt(2) "wxyY^"  
    %       1    1    r                           2 _!?a9  
    %       2    0    2*r^2 - 1                sqrt(6) T]\'D&P~D  
    %       2    2    r^2                      sqrt(6) xF 3Z>  
    %       3    1    3*r^3 - 2*r              sqrt(8) dMI G2log  
    %       3    3    r^3                      sqrt(8) Odw9]`,T  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) EK\xc'6M  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) }5Km \OI  
    %       4    4    r^4                      sqrt(10) :1v.Jk  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ke2M&TV  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) w 5t|C>  
    %       5    5    r^5                      sqrt(12) jm'^>p,9G  
    %       --------------------------------------------- {GGP8  
    % 0])[\O`j  
    %   Example: Pa?C-Xn^  
    % FU)=+m  
    %       % Display three example Zernike radial polynomials ih : XC  
    %       r = 0:0.01:1; fW=eB'Sl  
    %       n = [3 2 5]; f$--y|=  
    %       m = [1 2 1]; oS<*\!&D  
    %       z = zernpol(n,m,r); D&DbxTi  
    %       figure "}S6a?]V  
    %       plot(r,z) &&zsUAkS  
    %       grid on j&q%@%Gm  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') \]3[Xw-$  
    % E+$D$a  
    %   See also ZERNFUN, ZERNFUN2. ~CHVU3  
    0u +_D8G  
    % A note on the algorithm. m@",Zr `f=  
    % ------------------------ {9cjitl  
    % The radial Zernike polynomials are computed using the series w=5<mw  
    % representation shown in the Help section above. For many special l4U  
    % functions, direct evaluation using the series representation can 5!Ovd O}g  
    % produce poor numerical results (floating point errors), because )`mBvS.}  
    % the summation often involves computing small differences between Tz&h[+6`  
    % large successive terms in the series. (In such cases, the functions '*<I<? z;  
    % are often evaluated using alternative methods such as recurrence }-T,cA_H|  
    % relations: see the Legendre functions, for example). For the Zernike l]~IZTC  
    % polynomials, however, this problem does not arise, because the zu 7Fq]zD  
    % polynomials are evaluated over the finite domain r = (0,1), and AP ]`'C  
    % because the coefficients for a given polynomial are generally all 1I40N[PE)  
    % of similar magnitude. U&#`5u6'j  
    % bas1(/|S  
    % ZERNPOL has been written using a vectorized implementation: multiple 9|m:2["|?  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] v^Rw9*w{  
    % values can be passed as inputs) for a vector of points R.  To achieve +fQJ#?N2n  
    % this vectorization most efficiently, the algorithm in ZERNPOL wEQZ9?\  
    % involves pre-determining all the powers p of R that are required to UtR wZ(09  
    % compute the outputs, and then compiling the {R^p} into a single eYevj[c;  
    % matrix.  This avoids any redundant computation of the R^p, and %8xKBL]J  
    % minimizes the sizes of certain intermediate variables. KxWm63"  
    % 1Vs>G  
    %   Paul Fricker 11/13/2006 v4XEp   
    }hcY5E-n  
    oqzWL~  
    % Check and prepare the inputs: ,Kt51vGi  
    % ----------------------------- [$#G|>x  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Lb{.}  
        error('zernpol:NMvectors','N and M must be vectors.') }i^$ li@  
    end 1\g r ;b  
    oc#hAjB.  
    if length(n)~=length(m) 5=8t<v1Bn  
        error('zernpol:NMlength','N and M must be the same length.') 2rb@Md]dx  
    end %D~Mij  
    %AmyT  
    n = n(:); lbC,*U^  
    m = m(:); !'B='].  
    length_n = length(n); R@U4Ae{+  
    | /n  
    if any(mod(n-m,2)) g{f7 } gTG  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') QlS_{XV  
    end DWN9_*{  
    9TwKd0AT$&  
    if any(m<0) #WS>Z3AY  
        error('zernpol:Mpositive','All M must be positive.') EK&0Cn3z  
    end *,~L_)vWO  
    c0;rvw7  
    if any(m>n) "2p\/VfA  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') L\V`ou  
    end u'T-}95 V  
    }$ Kd-cj+  
    if any( r>1 | r<0 ) E'NS$,h  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') \[]?9Z=n  
    end /rky  
    U+C ^"[B  
    if ~any(size(r)==1) ) $0>L5d:  
        error('zernpol:Rvector','R must be a vector.') {|B[[W\TN  
    end l]gW_wUQd  
    Xz9[0;Q  
    r = r(:); &9"Y:),  
    length_r = length(r); :Gew8G  
    Z] x6np  
    if nargin==4 8H`L8: CM  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ~O]{m,)n  
        if ~isnorm ?'tRu !~  
            error('zernpol:normalization','Unrecognized normalization flag.') kt=& mq/B  
        end 0Ue~dVrM(?  
    else z 4;@"B  
        isnorm = false; p}5413z5Z=  
    end L\t_zf_0  
    |o'r?"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j(k}NWPH  
    % Compute the Zernike Polynomials ) .KMZ]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p#_ 5w  
    Zo }^"u  
    % Determine the required powers of r: X *:,|  
    % ----------------------------------- _O ;4>  
    rpowers = []; pMAP/..+2  
    for j = 1:length(n) sZEa8  
        rpowers = [rpowers m(j):2:n(j)];  nF<xJs  
    end pM}~/  
    rpowers = unique(rpowers); >#Xz~xI/I  
    F^`+.G\  
    % Pre-compute the values of r raised to the required powers, ael] {'h]  
    % and compile them in a matrix: L ./c#b!{  
    % ----------------------------- `xx3JQv[  
    if rpowers(1)==0 /+8VW;4|I  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); cbs ;  
        rpowern = cat(2,rpowern{:}); '@Yp@ _  
        rpowern = [ones(length_r,1) rpowern]; pLys%1hg  
    else WtaOf_  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); -GM"gkz  
        rpowern = cat(2,rpowern{:}); f=u +G  
    end O .-n&U9  
    hJD3G |E  
    % Compute the values of the polynomials: &Yc'X+'4  
    % -------------------------------------- 5jUy[w @  
    z = zeros(length_r,length_n); scYqU7$%T  
    for j = 1:length_n =8%*Rrj^  
        s = 0:(n(j)-m(j))/2; sriDta?Cz  
        pows = n(j):-2:m(j); j>uu3ADd2  
        for k = length(s):-1:1 xplV6q`  
            p = (1-2*mod(s(k),2))* ... 9qgs*]J  
                       prod(2:(n(j)-s(k)))/          ... s@{~8cHgU  
                       prod(2:s(k))/                 ... _[-W*,xJ)  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... xSNGf@1b  
                       prod(2:((n(j)+m(j))/2-s(k))); K=nDC.  
            idx = (pows(k)==rpowers); 2eA.04F  
            z(:,j) = z(:,j) + p*rpowern(:,idx); pnyu&@e  
        end ewHs ]V+U  
         r| )45@  
        if isnorm "v( pluN|  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); kFC*,  
        end XgM&0lVT  
    end BG= J8  
    oif|X7H;  
    % 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)  4s9@4  
    {R(CGrI  
    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。 b3R( O|  
    vr4r,[B6y  
    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)