切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11497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 n<x NE %  
    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! 9bDxml1  
     
    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 )ow|n^D($M  
    function z = zernfun(n,m,r,theta,nflag) cD6$C31Y]  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~CQYF,[Th  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N H1,;Xrm  
    %   and angular frequency M, evaluated at positions (R,THETA) on the :VPZGzK4  
    %   unit circle.  N is a vector of positive integers (including 0), and o0>z6Ya<  
    %   M is a vector with the same number of elements as N.  Each element 3N) bJ  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 0ih=<@1K  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, [Hn4&PET  
    %   and THETA is a vector of angles.  R and THETA must have the same xQ `>\f  
    %   length.  The output Z is a matrix with one column for every (N,M) zkdyfl5  
    %   pair, and one row for every (R,THETA) pair. :bLLN  
    % xj/ +Z!,9  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike D]9I-|  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), uvK1gJrA)  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral !\a'GO[  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, b7It8  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized :QnN7&j|(w  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. TR8<=  
    % 1/Pou)D  
    %   The Zernike functions are an orthogonal basis on the unit circle. w. gI0`  
    %   They are used in disciplines such as astronomy, optics, and s@sr.'yU  
    %   optometry to describe functions on a circular domain. qV$\.T>x  
    % j*m7&wOE  
    %   The following table lists the first 15 Zernike functions. K.cMuh  
    % (u81p  
    %       n    m    Zernike function           Normalization We#u-#k_O  
    %       -------------------------------------------------- ,C88%k  
    %       0    0    1                                 1 .kSx>3  
    %       1    1    r * cos(theta)                    2 ZM/*cA!"  
    %       1   -1    r * sin(theta)                    2 ocCC63J  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) P1b5=/}:V  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 6mJa  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) y K~;LV  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) DFKU?#R  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) m}] bP  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) K@P5]}'#  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) $UMxO`F  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) }g:y!p k  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7hq*+e  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 0^4uZeW?  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ey$J.qw3  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) =PiDZS^"  
    %       -------------------------------------------------- ,v$gWA!l  
    % U+t|wK  
    %   Example 1: @Iz vObK  
    % e%w>QN`  
    %       % Display the Zernike function Z(n=5,m=1) k:nR'TI  
    %       x = -1:0.01:1; A.<HOx&#  
    %       [X,Y] = meshgrid(x,x); klduJ T >  
    %       [theta,r] = cart2pol(X,Y); W is_N3M  
    %       idx = r<=1; C`c;I7  
    %       z = nan(size(X)); $v?+X20  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); r3oAP[+n  
    %       figure -o<L%Y<n2  
    %       pcolor(x,x,z), shading interp 'f9 fw^  
    %       axis square, colorbar cg$@x\fJ  
    %       title('Zernike function Z_5^1(r,\theta)') F gM<2$h  
    % 6CBk,2DswI  
    %   Example 2: <JE-#i  
    % /238pg~Cw5  
    %       % Display the first 10 Zernike functions ]w.:K*_=  
    %       x = -1:0.01:1; hM")DmvB4  
    %       [X,Y] = meshgrid(x,x); 6'UtB!gr  
    %       [theta,r] = cart2pol(X,Y); h4x*C=?A  
    %       idx = r<=1; |'WaBy1  
    %       z = nan(size(X)); "Zd4e2>{M\  
    %       n = [0  1  1  2  2  2  3  3  3  3]; @O#4duM4Qz  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; pmd=3,D'u  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; JX,&im*BG  
    %       y = zernfun(n,m,r(idx),theta(idx)); >;}np F>  
    %       figure('Units','normalized') r- 0BLq]~{  
    %       for k = 1:10 Ml)~%ZbF  
    %           z(idx) = y(:,k); OI"vC1.5  
    %           subplot(4,7,Nplot(k)) <:)T7yVq  
    %           pcolor(x,x,z), shading interp a<l(zJptG  
    %           set(gca,'XTick',[],'YTick',[]) nYG$V)iCb  
    %           axis square ,Ju f  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _ETG.SYq  
    %       end A6Ttx{]  
    % =D.M}x qo  
    %   See also ZERNPOL, ZERNFUN2. ,@ A1eX}  
    _y&m4Vuu  
    %   Paul Fricker 11/13/2006 ab8uY.j  
    !={Z]J  
    6%Ap/zvCZ>  
    % Check and prepare the inputs: ZzPlIl}\  
    % ----------------------------- X96>N{C*>  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +HNY!fv9  
        error('zernfun:NMvectors','N and M must be vectors.') f^il|Obzl  
    end 4*W ??(=j  
    ,:UoE  
    if length(n)~=length(m) qy=4zOOD#  
        error('zernfun:NMlength','N and M must be the same length.') i8pM,Ppi~  
    end 1a4HThDXP  
    vt}+d StUm  
    n = n(:); I=X-e#HM?  
    m = m(:); /gh=+;{  
    if any(mod(n-m,2)) Qi`Lj5;\F  
        error('zernfun:NMmultiplesof2', ... yS0YWqv]6@  
              'All N and M must differ by multiples of 2 (including 0).') (yWU9q)5  
    end w!o[pvyR$  
    /gT$d2{  
    if any(m>n) _#xS1sD  
        error('zernfun:MlessthanN', ... [<P(S~J  
              'Each M must be less than or equal to its corresponding N.') S'qEBz  
    end h}fz`ti U  
    {#N](yUm  
    if any( r>1 | r<0 ) "1>I/CM  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') q y73  
    end x'_I{$C &  
    &1Dq3%$c  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) "']I.  
        error('zernfun:RTHvector','R and THETA must be vectors.') bI.LE/yk  
    end [YG\a5QK  
    $}&Y$w>S  
    r = r(:); p x1y#Q  
    theta = theta(:); IMLk{y%6  
    length_r = length(r); :4s{?IY)l  
    if length_r~=length(theta) Gqq%q!k&1  
        error('zernfun:RTHlength', ... XB,  2+  
              'The number of R- and THETA-values must be equal.') y?j#;n0  
    end J'9hzag  
    ~*RG|4#  
    % Check normalization: j*@^O`^v  
    % -------------------- $j*%}x~[  
    if nargin==5 && ischar(nflag) S 4uX utd  
        isnorm = strcmpi(nflag,'norm'); /tI8JXcUK  
        if ~isnorm qeLfO  
            error('zernfun:normalization','Unrecognized normalization flag.') x? 3U3\W  
        end _4F(WCco  
    else h<2o5c|  
        isnorm = false; <?'d \B  
    end ;Ak<O[  
    eS(\E0%QI  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [:<CgU9C  
    % Compute the Zernike Polynomials 7Y T%.ID  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `>&V_^y+  
    S0( ).2#  
    % Determine the required powers of r: U_ n1QU  
    % ----------------------------------- 9r.Os  
    m_abs = abs(m); }&A!h  
    rpowers = []; i"mN0%   
    for j = 1:length(n) ;0DT f  
        rpowers = [rpowers m_abs(j):2:n(j)]; 0dxEV]  
    end Q qGf*  
    rpowers = unique(rpowers); S]&f+g}&w  
    o))z8n?b  
    % Pre-compute the values of r raised to the required powers, 8qGK"%{ ~  
    % and compile them in a matrix: 1!_$HA  
    % ----------------------------- 5/Viz`hsz  
    if rpowers(1)==0 d-Vttxa6  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); S}6Ty2.\  
        rpowern = cat(2,rpowern{:}); +bpUb0.W  
        rpowern = [ones(length_r,1) rpowern]; Hhx"47:  
    else G"P@AOw  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); .4E5{F{~  
        rpowern = cat(2,rpowern{:}); \6o%gpUkD  
    end x"xl3dRu  
    " &2Kvsz  
    % Compute the values of the polynomials: y%%D="  
    % -------------------------------------- <QbD ;(%  
    y = zeros(length_r,length(n)); K ZoIjK]  
    for j = 1:length(n) G J"S*30  
        s = 0:(n(j)-m_abs(j))/2; l~P%mVC3m  
        pows = n(j):-2:m_abs(j); B].V|8h  
        for k = length(s):-1:1 Q@]~O-  
            p = (1-2*mod(s(k),2))* ... nvY3$ Ty  
                       prod(2:(n(j)-s(k)))/              ... (;DnL|"'8  
                       prod(2:s(k))/                     ... | 4/'~cYV  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ~q]@Jp  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); @@~OA>^  
            idx = (pows(k)==rpowers); N$>.V7H&  
            y(:,j) = y(:,j) + p*rpowern(:,idx); /)RyRS8c  
        end vbEAd)*S  
         }j<:hD QP  
        if isnorm SFhi]48&V  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); cV]c/*z A  
        end 1 ; _tu  
    end SSG57N-T  
    % END: Compute the Zernike Polynomials B(tLV9B3Q  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x\( @ v  
    7A:k  
    % Compute the Zernike functions: 7#/->Y  
    % ------------------------------ c;siMWw;  
    idx_pos = m>0; @bs YJ4-V  
    idx_neg = m<0; wW2b?b{*Z  
    XC 44]o4jx  
    z = y; |2RoDW  
    if any(idx_pos) \j C[|LM&  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 2J?ON|2M  
    end S-x'nu$u  
    if any(idx_neg) %f[0&)1!.v  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 581Jp'cje  
    end %L;z~C  
    h8 Wv t's  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) *[XVkt`H  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. iakqCjV  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 2=R}u-@6p  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive p7SX,kpt>  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ^7b[s pqE  
    %   and THETA is a vector of angles.  R and THETA must have the same 5&Y%N(  
    %   length.  The output Z is a matrix with one column for every P-value, h>0R!Rl8  
    %   and one row for every (R,THETA) pair. Y9}5&#  
    % dP[vXhc  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 1#nR$  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ]IF QD  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ?HD eiJ kX  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 TNi4H:\  
    %   for all p. sY|by\-c  
    % ajr);xd  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 #.Q3}[M  
    %   Zernike functions (order N<=7).  In some disciplines it is ucP"<,a  
    %   traditional to label the first 36 functions using a single mode cx+w_D9b!  
    %   number P instead of separate numbers for the order N and azimuthal rN$U%\.I  
    %   frequency M. aL)}S%5o?  
    % F+!w[}0  
    %   Example: L>trLD1pt  
    % `Zdeq.R]  
    %       % Display the first 16 Zernike functions adCTo  
    %       x = -1:0.01:1; *8I+D>x  
    %       [X,Y] = meshgrid(x,x); B|fh 4FNy  
    %       [theta,r] = cart2pol(X,Y); $m hIX A.  
    %       idx = r<=1; <R;t>~8x  
    %       p = 0:15; M53{e;.kN  
    %       z = nan(size(X)); W't?aj I|  
    %       y = zernfun2(p,r(idx),theta(idx)); YGdzA]3>  
    %       figure('Units','normalized') tVUC@M>'  
    %       for k = 1:length(p) Y5dt/8Jo  
    %           z(idx) = y(:,k); $Gy&  
    %           subplot(4,4,k) ?'xwr )v  
    %           pcolor(x,x,z), shading interp K3Sa6"U  
    %           set(gca,'XTick',[],'YTick',[]) o$4i{BL  
    %           axis square  L- '{   
    %           title(['Z_{' num2str(p(k)) '}'])  c6f=r  
    %       end \Fh#CI  
    % ce&Q}_  
    %   See also ZERNPOL, ZERNFUN. Q<c{$o  
    jV|j]m&t  
    %   Paul Fricker 11/13/2006 y^u9Ttf{  
    KfPYH\ 0  
    eb#yCDIC   
    % Check and prepare the inputs: _ArN[]Z  
    % ----------------------------- Y5/SbQYf1  
    if min(size(p))~=1 %&Fsk]T%:  
        error('zernfun2:Pvector','Input P must be vector.') hx.ln6=4  
    end Yl$R$u)  
    `SfBT1#5G  
    if any(p)>35 If*+yr|  
        error('zernfun2:P36', ... 7]8nW!h;  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... bb4 `s0  
               '(P = 0 to 35).']) n5NwiSE  
    end #/,WgsAC  
    Lu][0+-  
    % Get the order and frequency corresonding to the function number: w7d<Ky_C  
    % ---------------------------------------------------------------- Y[R;UJE`5  
    p = p(:); $bCN;yE  
    n = ceil((-3+sqrt(9+8*p))/2); rYK GBo8"  
    m = 2*p - n.*(n+2); zbL8 pp  
     Lw1aG;5  
    % Pass the inputs to the function ZERNFUN: m~f J_  
    % ---------------------------------------- >WZ_) `R  
    switch nargin f(*ygI  
        case 3 a5dc#f Kf  
            z = zernfun(n,m,r,theta); E9bc pup  
        case 4 AZ7m=Q97  
            z = zernfun(n,m,r,theta,nflag); b%TLvV 9F  
        otherwise `zP{E T_Y  
            error('zernfun2:nargin','Incorrect number of inputs.') W1!Nq`  
    end =>m x>R`S  
    !7>~=n_,L.  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) %NH{%K,  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. '2:HBJ  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of O?OG`{k  
    %   order N and frequency M, evaluated at R.  N is a vector of JU`5K}H<  
    %   positive integers (including 0), and M is a vector with the K)&oDwk  
    %   same number of elements as N.  Each element k of M must be a  K?]c  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) *Z}9S9YtN  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is [9H986=  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix nl/~7({  
    %   with one column for every (N,M) pair, and one row for every 7H$wpn Zln  
    %   element in R. ;Ob`B@!=b  
    % XNaiMpp'  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- QT)5-Jy  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is f2]O5rX p  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to =C4!h'hz  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 _!C M  
    %   for all [n,m]. P+gY LX8  
    % P>wTp)  
    %   The radial Zernike polynomials are the radial portion of the =;m;r!,K  
    %   Zernike functions, which are an orthogonal basis on the unit f}ES8 Hh[  
    %   circle.  The series representation of the radial Zernike "bmWr)  
    %   polynomials is S7kZpD $  
    % ;-"'sEu}  
    %          (n-m)/2 "yG*Kh7ur  
    %            __ ~fz9AhU8  
    %    m      \       s                                          n-2s NA{?DSP  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r .[:2M9Rx  
    %    n      s=0 <c_'(   
    % 9*CJWS;  
    %   The following table shows the first 12 polynomials. W</\F&  
    % !McRtxq?~  
    %       n    m    Zernike polynomial    Normalization )mh,F# "L  
    %       --------------------------------------------- ATkx_1]KM-  
    %       0    0    1                        sqrt(2) q6AL}9]9  
    %       1    1    r                           2 ?L(y8b}F(  
    %       2    0    2*r^2 - 1                sqrt(6) >pj)va[Q  
    %       2    2    r^2                      sqrt(6) Cp!Qd e  
    %       3    1    3*r^3 - 2*r              sqrt(8) x~n]r[!L  
    %       3    3    r^3                      sqrt(8) Y<)9TU:D!  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) KlSg0s  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) X{-901J1  
    %       4    4    r^4                      sqrt(10) BW5!@D2  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) -,pw[R  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) "Cn<x\E b  
    %       5    5    r^5                      sqrt(12) MgrJ ;?L  
    %       --------------------------------------------- h$_5)d~  
    % +I[Hxf~  
    %   Example: >>R,P Ow-  
    % .:A&5Y-   
    %       % Display three example Zernike radial polynomials ?D9>N'yH8  
    %       r = 0:0.01:1; N*6lyFcg  
    %       n = [3 2 5]; 4fgYO]  
    %       m = [1 2 1]; BQF7S<O+  
    %       z = zernpol(n,m,r); yd~}CF  
    %       figure XFJz\'{  
    %       plot(r,z) D\ kd6  
    %       grid on /~}<[6ZGCY  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') du_TiI  
    % *jrQ-'<T  
    %   See also ZERNFUN, ZERNFUN2. JzZ@Z8%a;  
    BU=;rz!;  
    % A note on the algorithm. ' "I-! +  
    % ------------------------ @*"H{xo.U  
    % The radial Zernike polynomials are computed using the series Cy2)M(RW  
    % representation shown in the Help section above. For many special p{W'[A{J .  
    % functions, direct evaluation using the series representation can C~q&  
    % produce poor numerical results (floating point errors), because )Nkf'&  
    % the summation often involves computing small differences between A#x_>fV  
    % large successive terms in the series. (In such cases, the functions Q zq3{%^x_  
    % are often evaluated using alternative methods such as recurrence L)-1( e<x  
    % relations: see the Legendre functions, for example). For the Zernike Cs?[   
    % polynomials, however, this problem does not arise, because the atYe$Db  
    % polynomials are evaluated over the finite domain r = (0,1), and Rt3/dw(p  
    % because the coefficients for a given polynomial are generally all Tt+E?C%Y  
    % of similar magnitude. #@8JYzMq%  
    % jPA^SxM  
    % ZERNPOL has been written using a vectorized implementation: multiple Ers8J V  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 0o_wy1O1,  
    % values can be passed as inputs) for a vector of points R.  To achieve o@ W:PmKW  
    % this vectorization most efficiently, the algorithm in ZERNPOL HqoCl  
    % involves pre-determining all the powers p of R that are required to CI+@G XY  
    % compute the outputs, and then compiling the {R^p} into a single ]mdO3P  
    % matrix.  This avoids any redundant computation of the R^p, and z=pGu_`2  
    % minimizes the sizes of certain intermediate variables. (^mpb  
    % 2YhtD A  
    %   Paul Fricker 11/13/2006 ,;k`N`#'  
    >A ?{cbJ  
    U?&&yynK  
    % Check and prepare the inputs: .V.ga2+  
    % ----------------------------- *e%(J$t  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /& wA$h  
        error('zernpol:NMvectors','N and M must be vectors.') XI7:y4M  
    end 1/<Z6 ?U  
    s8|F e_  
    if length(n)~=length(m) *7UDTgY  
        error('zernpol:NMlength','N and M must be the same length.') }KFM8CbS  
    end ]JvZ{fA%*  
    ObEp0-^?  
    n = n(:); 3w$Ib}7   
    m = m(:); tr-muhuK  
    length_n = length(n); ) ={ H  
    ,Uu#41ZOKL  
    if any(mod(n-m,2)) /6yH ,{(a  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') >@uFye$  
    end = @n`5g  
    FC }r~syqA  
    if any(m<0) /\ u1q<  
        error('zernpol:Mpositive','All M must be positive.') w i=&W  
    end ]B=2r^fn  
    l$DQkbOj  
    if any(m>n) [E6ZmMB&  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ? H7?>ZE  
    end p$1y8Zbor  
    4$MV]ldUI  
    if any( r>1 | r<0 ) idSc#n22  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') yYn7y1B  
    end z%~rQa./$  
    ;k5B@z/<S  
    if ~any(size(r)==1) z%44@TP  
        error('zernpol:Rvector','R must be a vector.') qJYEsI2M  
    end 5%;=(Oig  
    |7@@~|A  
    r = r(:); ;noZmPa  
    length_r = length(r); KxUO=v<u  
    x{I, gu|+  
    if nargin==4 IXof- I%8  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); Ah Rvyj  
        if ~isnorm r>Cv@4/j  
            error('zernpol:normalization','Unrecognized normalization flag.') Fd1jElt  
        end 2 3*OuY  
    else akV-|v_  
        isnorm = false; 4StoEgFS  
    end (Qj;B)  
    /j;HM[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #55_hY#  
    % Compute the Zernike Polynomials !G~`5?CvE  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7Kn}KO!Y8  
    L#Rj~&U  
    % Determine the required powers of r: prO ~g  
    % ----------------------------------- @nxpcHj  
    rpowers = []; `!lQd}W  
    for j = 1:length(n) &"mWi-Mpl  
        rpowers = [rpowers m(j):2:n(j)]; -AZ\u\xCB  
    end %1z`/B  
    rpowers = unique(rpowers); xY<*:&  
    0q_?<v_ 1  
    % Pre-compute the values of r raised to the required powers, Da&Brm   
    % and compile them in a matrix: VX]Ud\(  
    % ----------------------------- C1po]Ott*  
    if rpowers(1)==0 E<r<ObeRv`  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); zr^"zcfz&  
        rpowern = cat(2,rpowern{:}); )8 %lZ {  
        rpowern = [ones(length_r,1) rpowern]; %hN7K  
    else rB}2F*eT  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); [,Fu2j]  
        rpowern = cat(2,rpowern{:}); Y?xc#'  
    end eoxEnCU  
    qX:Y I3:,@  
    % Compute the values of the polynomials: \fJ _,  
    % -------------------------------------- yyc4'j+  
    z = zeros(length_r,length_n); oN&U@N/>aU  
    for j = 1:length_n Hd?#^X  
        s = 0:(n(j)-m(j))/2; U)] }EgpF  
        pows = n(j):-2:m(j); Wh^wKF~%  
        for k = length(s):-1:1 !MEA@^$#  
            p = (1-2*mod(s(k),2))* ...  %&pd`A/  
                       prod(2:(n(j)-s(k)))/          ... !;M5.Y1j&"  
                       prod(2:s(k))/                 ... Dgm"1+  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... O[15x H,  
                       prod(2:((n(j)+m(j))/2-s(k))); v{oHC4  
            idx = (pows(k)==rpowers); H/$oGhvl  
            z(:,j) = z(:,j) + p*rpowern(:,idx); UX@8  
        end |DLmMsS4  
         1!&m1  
        if isnorm F`M`c%  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); g^[BnP)I  
        end )Ag{S[yZ  
    end 8RjFp2) W  
    "J>8ZUP  
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
    发帖
    856
    光币
    846
    光券
    0
    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
    发帖
    4352
    光币
    5478
    光券
    1
    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  +P<#6<gR  
    ! =*k+gpF  
    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。 X=V2^zrt  
    F8|m i`f-  
    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)