切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11332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 V Ku|=m2vB  
    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! q U^`fIa  
     
    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 a>eg H og  
    function z = zernfun(n,m,r,theta,nflag) gCaxZ~o  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. :)kWQQ+,  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N B dxV [SF  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 6o~CX  
    %   unit circle.  N is a vector of positive integers (including 0), and #4F0o@Z  
    %   M is a vector with the same number of elements as N.  Each element dyt.( 2  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Q6d>tqWhq  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, B+[L/C}=;  
    %   and THETA is a vector of angles.  R and THETA must have the same 66HxwY3a  
    %   length.  The output Z is a matrix with one column for every (N,M) j!K{1s[.y  
    %   pair, and one row for every (R,THETA) pair. V(F1i%9lg  
    % >uJU25)|  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike kI,O9z7A7  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 3H`ES_JL  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ) -@Dh6F  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Z"E2ZSa0  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized rXVR X#Lh  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 9Qn*frdY,  
    % =]P|!$!}0  
    %   The Zernike functions are an orthogonal basis on the unit circle. Fr1OzS^&(  
    %   They are used in disciplines such as astronomy, optics, and I1W~;2cK  
    %   optometry to describe functions on a circular domain. r-5xo.J'  
    % }PzHtA,V  
    %   The following table lists the first 15 Zernike functions. 3j w4#GW  
    % ]%[.>mR  
    %       n    m    Zernike function           Normalization `,Y/!(:;  
    %       -------------------------------------------------- 1V ; ,ZGI*  
    %       0    0    1                                 1 1_z~<d @?;  
    %       1    1    r * cos(theta)                    2 V0y_c^x  
    %       1   -1    r * sin(theta)                    2 2Y%E.){  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Hf9F:yH  
    %       2    0    (2*r^2 - 1)                    sqrt(3) z}2  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) D>K=D"  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) qIk( ei  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) [wcp2g3Px  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) w>&g'  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) )<_:%oB  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) _tfi6UQ&lY  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !Z%pdqo`.  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 4?l:.\fB:  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) KHoDD=O  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 3| 0OW Jk  
    %       -------------------------------------------------- JvM:xy9  
    % k Hh0&~ (  
    %   Example 1: [\.@,Y0j  
    % idNg&'   
    %       % Display the Zernike function Z(n=5,m=1) n hGh5,  
    %       x = -1:0.01:1; pt~b=+bBm  
    %       [X,Y] = meshgrid(x,x); U Kf0cU  
    %       [theta,r] = cart2pol(X,Y); cB}6{c$_sW  
    %       idx = r<=1; ;a`I8Fj  
    %       z = nan(size(X)); !p(N DQm  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); vp`s< ;CA  
    %       figure _hi8m o  
    %       pcolor(x,x,z), shading interp >\Ml \CyL  
    %       axis square, colorbar 2w>yW]  
    %       title('Zernike function Z_5^1(r,\theta)') "SU O2-Gj  
    % Kl w9  
    %   Example 2:  +D|E8sz8  
    % ~P!%i9e_  
    %       % Display the first 10 Zernike functions b!z kQ?h  
    %       x = -1:0.01:1; B S+=*3J  
    %       [X,Y] = meshgrid(x,x); fk(h*L|sI  
    %       [theta,r] = cart2pol(X,Y); X!f` !tZ:{  
    %       idx = r<=1; N m@UM*D  
    %       z = nan(size(X)); @xN)mi  
    %       n = [0  1  1  2  2  2  3  3  3  3]; >jpk R  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; z460a[Wl  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; l6< bV#_qe  
    %       y = zernfun(n,m,r(idx),theta(idx)); 9v(k<('_  
    %       figure('Units','normalized') 5VGr<i&A  
    %       for k = 1:10 <CGJ:% AY  
    %           z(idx) = y(:,k); U].3vju`c  
    %           subplot(4,7,Nplot(k)) F'^?s= QX  
    %           pcolor(x,x,z), shading interp 48n7<M;I  
    %           set(gca,'XTick',[],'YTick',[]) JI|MR#_u  
    %           axis square ~3qt<"  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) }Z8DVTpX}  
    %       end v42Z&PO   
    % "$PX [:  
    %   See also ZERNPOL, ZERNFUN2. nBGcf(BE.$  
    S/xCX!  
    %   Paul Fricker 11/13/2006 JG=z~STz  
    NnqAr ,  
    wZKEUJpQ  
    % Check and prepare the inputs: .X{U\{c|a  
    % ----------------------------- 2G)q?_Q4S  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) #iP5@:!Wm~  
        error('zernfun:NMvectors','N and M must be vectors.') +X!QH/ 8  
    end 6Wc'5t3  
    (GbZt{.  
    if length(n)~=length(m) JId|LHf*P  
        error('zernfun:NMlength','N and M must be the same length.') TV*@h2C"i  
    end eT Z2f  
    QZamf lk  
    n = n(:); v_NL2eQ~  
    m = m(:); ,w0Io   
    if any(mod(n-m,2)) S~Yu;  
        error('zernfun:NMmultiplesof2', ... 6G]hs gro  
              'All N and M must differ by multiples of 2 (including 0).') Vv3:x1S  
    end d^^EfWU  
    0M 5m8  
    if any(m>n) fkJElO-F  
        error('zernfun:MlessthanN', ... 4?.L+wL  
              'Each M must be less than or equal to its corresponding N.') AMc`qh  
    end yf2$HF  
    Gc{s?rB_  
    if any( r>1 | r<0 ) HR$;QHl~F  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') |oV_7%mlu  
    end C,nU.0  
    n+:}p D  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) *#Hw6N0#   
        error('zernfun:RTHvector','R and THETA must be vectors.') |ZJ<N\\h-  
    end 8 ;o*c6+  
    [?nM)4d  
    r = r(:); x`VA3nE9  
    theta = theta(:); d^Zr I\AJ  
    length_r = length(r); Kld#C51X f  
    if length_r~=length(theta) zM!2JC  
        error('zernfun:RTHlength', ... ]c\d][R N  
              'The number of R- and THETA-values must be equal.') )@a_|q@V  
    end FFpG>+*3  
    1cY,)Z%l #  
    % Check normalization: &~#y-o"  
    % -------------------- #Hi$squJ  
    if nargin==5 && ischar(nflag) N Ah^2X  
        isnorm = strcmpi(nflag,'norm'); X^9eCj;c  
        if ~isnorm eGQ -Ht,N  
            error('zernfun:normalization','Unrecognized normalization flag.') "*Gp@  
        end N=~aj7B%  
    else ) |j?aVqZ  
        isnorm = false; hLF;MH@  
    end jC_m0Iwc  
    klSAY  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?"L ^ 0%  
    % Compute the Zernike Polynomials *g!7PzJ'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )l[bu6bM  
    =uD^#AX  
    % Determine the required powers of r: mk]8}+^.  
    % ----------------------------------- ^@OdY& 5^  
    m_abs = abs(m); R;`C;Rbf  
    rpowers = []; Q+a"Z^Z|  
    for j = 1:length(n) se&Q\!&M  
        rpowers = [rpowers m_abs(j):2:n(j)]; -mY,nMDb  
    end @tg4rl  
    rpowers = unique(rpowers); ] 8dzTEjk  
    .vWwYG  
    % Pre-compute the values of r raised to the required powers, MyaJhA6c  
    % and compile them in a matrix: 1AAOg+Y@U"  
    % ----------------------------- ^b^}6L'Z  
    if rpowers(1)==0 j-TRa,4bN  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); h"t\x}8qq  
        rpowern = cat(2,rpowern{:}); %hCd*[Z}j  
        rpowern = [ones(length_r,1) rpowern]; G*%:"qleT$  
    else JUd Q Q  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); E8"$vl&c]  
        rpowern = cat(2,rpowern{:}); Lf+"Gp  
    end ^vha4<'-qG  
    3V%ts7:a  
    % Compute the values of the polynomials: V?&P).5)  
    % -------------------------------------- |ZtNCB5{^j  
    y = zeros(length_r,length(n)); 'mO>hD`V  
    for j = 1:length(n) J/B`c(  
        s = 0:(n(j)-m_abs(j))/2; +a0` ,Jc  
        pows = n(j):-2:m_abs(j); #dDM "s  
        for k = length(s):-1:1 U6F1QLSLz  
            p = (1-2*mod(s(k),2))* ... 6o<(,\ad [  
                       prod(2:(n(j)-s(k)))/              ... OU'm0Jlk  
                       prod(2:s(k))/                     ... t$g@+1p4  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... v:?l C<,  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); D-4{9[  
            idx = (pows(k)==rpowers); y7| 3]>Z  
            y(:,j) = y(:,j) + p*rpowern(:,idx); y85R"d  
        end ,)ZI&BL5  
         Kjt\A]R%  
        if isnorm do:IkjU~  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); }No8to  
        end #Fz/}lO  
    end /X%+z5  
    % END: Compute the Zernike Polynomials _)[UartKx  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "NtY[sT{V  
    ,-[z?dvO  
    % Compute the Zernike functions: 0t7vg#v|  
    % ------------------------------ 0sI7UK`m  
    idx_pos = m>0; a!B"WNb+  
    idx_neg = m<0; ziC%Q8  
    8p_6RvG  
    z = y; V6Y0#sTU  
    if any(idx_pos) i>e?$H,/  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); bX>R9i$  
    end vwAtX($  
    if any(idx_neg) a'(B}B=h  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); YO9;NA{sH  
    end oS^KC}X  
    Ug\$Ob5=q  
    % 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) ~*x 2IPi H  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. @qEUp7W.?  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated .wB'"z8L  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive c(aykIVOo  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ]kd:p*U6P  
    %   and THETA is a vector of angles.  R and THETA must have the same SEVB.;  
    %   length.  The output Z is a matrix with one column for every P-value, $O%"[w  
    %   and one row for every (R,THETA) pair. b0Dco0U(  
    % [iZH[7&j  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike RL3*fRlb  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 4w)>}  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ;cB3D3fR.  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 sNM ]bei  
    %   for all p. `aTw!QBfG  
    % x#gZC 1$Y  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 =#=}|Q}  
    %   Zernike functions (order N<=7).  In some disciplines it is @S:T8 *~}  
    %   traditional to label the first 36 functions using a single mode xkv%4H>  
    %   number P instead of separate numbers for the order N and azimuthal )FNn  
    %   frequency M. p=odyf1hK  
    % HLOr Dlj7  
    %   Example: [>t;P ,  
    % GUZ.Pw  
    %       % Display the first 16 Zernike functions 4}s'xMT!  
    %       x = -1:0.01:1; ka6E s~  
    %       [X,Y] = meshgrid(x,x); ) J.xQ}g  
    %       [theta,r] = cart2pol(X,Y); *V4%&&{  
    %       idx = r<=1; D|ra ;d  
    %       p = 0:15; 4EmdQn  
    %       z = nan(size(X)); z%#-2&i  
    %       y = zernfun2(p,r(idx),theta(idx)); g9fYt&  
    %       figure('Units','normalized') T<"Bb[kH  
    %       for k = 1:length(p) (T%?@'\  
    %           z(idx) = y(:,k); {2YqEX-I*  
    %           subplot(4,4,k) ~(8fUob  
    %           pcolor(x,x,z), shading interp UI"UBZZ$  
    %           set(gca,'XTick',[],'YTick',[]) #:By/9}-  
    %           axis square V:>r6  
    %           title(['Z_{' num2str(p(k)) '}']) un4fnoc  
    %       end 6 Ia HaV+P  
    % ]YtN6Rq/  
    %   See also ZERNPOL, ZERNFUN. 7e\Jg/FU  
    :)i,K>y3i  
    %   Paul Fricker 11/13/2006 l i)6^f#  
    J]#rh5um  
    :9av]Yv&  
    % Check and prepare the inputs: kJpO0k9?eY  
    % ----------------------------- <b .p/uA  
    if min(size(p))~=1 Hqs!L`oW)  
        error('zernfun2:Pvector','Input P must be vector.') ' )0@J`  
    end *hru);OJr  
    $}{[_2  
    if any(p)>35 9!(%Vf>  
        error('zernfun2:P36', ... S3l^h4  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Km $o@  
               '(P = 0 to 35).']) ge(,>xB  
    end 57EX#:a  
    Zp^O1&\SK?  
    % Get the order and frequency corresonding to the function number: (WJ)!  
    % ---------------------------------------------------------------- ?_d6 ;  
    p = p(:); T.3{}230<  
    n = ceil((-3+sqrt(9+8*p))/2); 9 :Oz-b  
    m = 2*p - n.*(n+2); vi *A 5  
    n1{[CCee@  
    % Pass the inputs to the function ZERNFUN: PPH;'!>s"  
    % ---------------------------------------- r5N TTc  
    switch nargin ?&;_>0P  
        case 3 Ak,JPz T  
            z = zernfun(n,m,r,theta); (Hj[9[=  
        case 4 A&)2m  
            z = zernfun(n,m,r,theta,nflag); +Wg/ O -  
        otherwise M:GpyE%  
            error('zernfun2:nargin','Incorrect number of inputs.') U 7.kYu  
    end @fYVlHT%E  
    )ds]fvMW]N  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) s$isDG#Sr  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. e)n ,Y  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 5RN!"YLI3  
    %   order N and frequency M, evaluated at R.  N is a vector of n 5R9<A^  
    %   positive integers (including 0), and M is a vector with the #p@GhI!6  
    %   same number of elements as N.  Each element k of M must be a %]&$VVVh  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) Lo1ySLo$G  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is I2WP/  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ^J#*sn  
    %   with one column for every (N,M) pair, and one row for every H" `'d  
    %   element in R. U%s@np  
    % dh7`eAMY   
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- #| _VN %!  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is KCyV |,+n  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to BP@tI|  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ` = O  
    %   for all [n,m]. =yZq]g6Q  
    % Bh2l3J4X  
    %   The radial Zernike polynomials are the radial portion of the rhbz|Uq  
    %   Zernike functions, which are an orthogonal basis on the unit 0>Snps3*Z  
    %   circle.  The series representation of the radial Zernike > v%.q]E6n  
    %   polynomials is kEnGr6e  
    % dEtjcId  
    %          (n-m)/2 H?];8wq$G  
    %            __ 'wk,t^)  
    %    m      \       s                                          n-2s ih".y3  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r @!fUp b  
    %    n      s=0 bpwA|H%{M  
    % qx5`lm~L  
    %   The following table shows the first 12 polynomials. / S]RP>cQ  
    % MSQ^ovph  
    %       n    m    Zernike polynomial    Normalization P-Y_$Nv0g  
    %       --------------------------------------------- ]6^<VC`5D  
    %       0    0    1                        sqrt(2) bPxL+ +  
    %       1    1    r                           2 YUEyGhkMV{  
    %       2    0    2*r^2 - 1                sqrt(6) 1;$XX#7o  
    %       2    2    r^2                      sqrt(6) s6 g"uF>k  
    %       3    1    3*r^3 - 2*r              sqrt(8) }8x+F2i  
    %       3    3    r^3                      sqrt(8) sh_;98^  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ]##aAh-P4&  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) w-pgtO|Us  
    %       4    4    r^4                      sqrt(10) s) ]j X  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ^qR|lA@=\  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 4-.K<-T%D  
    %       5    5    r^5                      sqrt(12) .@,t}:lD  
    %       --------------------------------------------- q-<DYVG+  
    % 7x6 M]1F  
    %   Example: kP%hgZ  
    % *I(6hB  
    %       % Display three example Zernike radial polynomials "5V;~}=S  
    %       r = 0:0.01:1; W]oD(eZ  
    %       n = [3 2 5]; Sk|e#{  
    %       m = [1 2 1]; \~hrS/$[$  
    %       z = zernpol(n,m,r); 89LD:+p/  
    %       figure pr#%VM[':R  
    %       plot(r,z) %, psUOY  
    %       grid on +Umsr  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') nXRa_M(z8  
    % =8T!ldVxES  
    %   See also ZERNFUN, ZERNFUN2. mF;mJq<d  
    9Y,JYc#  
    % A note on the algorithm. cpjwc@UMe  
    % ------------------------ M4C8K{}  
    % The radial Zernike polynomials are computed using the series ?.VKVTX^  
    % representation shown in the Help section above. For many special F<I*?${[  
    % functions, direct evaluation using the series representation can e3; &  
    % produce poor numerical results (floating point errors), because GJ*IH9YR  
    % the summation often involves computing small differences between L?[m$l!T}  
    % large successive terms in the series. (In such cases, the functions &|k=mxox\  
    % are often evaluated using alternative methods such as recurrence xx`YBn~"  
    % relations: see the Legendre functions, for example). For the Zernike {1Ra |,;  
    % polynomials, however, this problem does not arise, because the GGuU(sL*  
    % polynomials are evaluated over the finite domain r = (0,1), and vdq=F|&  
    % because the coefficients for a given polynomial are generally all  8${n}}  
    % of similar magnitude. =PRQ3/?5  
    % l/G +Xj4M  
    % ZERNPOL has been written using a vectorized implementation: multiple S/`#6  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] )*BZo>"  
    % values can be passed as inputs) for a vector of points R.  To achieve f(|k0$EIu  
    % this vectorization most efficiently, the algorithm in ZERNPOL .#QE*<T)]  
    % involves pre-determining all the powers p of R that are required to dBXiLrEbs  
    % compute the outputs, and then compiling the {R^p} into a single B"&-) (  
    % matrix.  This avoids any redundant computation of the R^p, and rC-E+%y  
    % minimizes the sizes of certain intermediate variables. |eu8;~A  
    % fY00  
    %   Paul Fricker 11/13/2006 W Ej{2+  
    G]ek-[-  
    A2 + %  
    % Check and prepare the inputs: 64?HqO 6(  
    % ----------------------------- H d|p@$I  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) _c>ww<*3  
        error('zernpol:NMvectors','N and M must be vectors.') F\D iT|?}  
    end :01d9|#  
    yI: ;+K  
    if length(n)~=length(m) r/sSkF F  
        error('zernpol:NMlength','N and M must be the same length.') `}?;Ow&2CY  
    end O6G\0o  
    m%[e_eS  
    n = n(:); \AwkK3  
    m = m(:); dkw.o.e  
    length_n = length(n); >_2~uF@pb  
    DPT6]pl"y  
    if any(mod(n-m,2)) K+}0:W=P  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') zTa5 N  
    end R 0RxcB tG  
    7%  D4  
    if any(m<0) ^`kwSC  
        error('zernpol:Mpositive','All M must be positive.') QR&e~rks  
    end Q7aPW\-  
    V5K/)\#  
    if any(m>n) c7XBZ%D  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 4 Im>2 )  
    end B.; qvuM~  
    9A"s7iJ)  
    if any( r>1 | r<0 ) v.(dOIrX  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') \\k=N(n  
    end eNd&47lJ  
    *tUOTA 3L  
    if ~any(size(r)==1) f'=u`*(b7  
        error('zernpol:Rvector','R must be a vector.') %LrOGr  
    end O t)}:oG  
    Y%?S:&GH  
    r = r(:); qofAA!3z  
    length_r = length(r); }b\hRy~=r  
    k' st^1T  
    if nargin==4 tDRR3=9pX  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); )h}IZSm  
        if ~isnorm fbh,V%t7  
            error('zernpol:normalization','Unrecognized normalization flag.') {U;yW)  
        end t5+p]7  
    else ,-Hj  
        isnorm = false; mXz*Gi  
    end EX8+3>)  
    !h.hJt  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U823q-x  
    % Compute the Zernike Polynomials yJI~{VmU7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y > =Y  
    vaB ql(?'2  
    % Determine the required powers of r: GeP={lj  
    % ----------------------------------- Wq4<9D  
    rpowers = []; :IZAdlz[@  
    for j = 1:length(n) ; dzL9P9IU  
        rpowers = [rpowers m(j):2:n(j)]; (\F9_y,6*\  
    end #Nh'1@@  
    rpowers = unique(rpowers); (F&LN!Hn>p  
    bA)nWWSg=  
    % Pre-compute the values of r raised to the required powers, m#'eDO:  
    % and compile them in a matrix: Y!L-5|G  
    % ----------------------------- osXEzr(  
    if rpowers(1)==0 f8;?WSGyD2  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); PZ|I3z  
        rpowern = cat(2,rpowern{:}); |*c1S -#  
        rpowern = [ones(length_r,1) rpowern];  @/s|<*  
    else j,i9,oF6]  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);  v%:deaF  
        rpowern = cat(2,rpowern{:}); 3 2z4G =l  
    end GURiW42  
    (X "J)x aQ  
    % Compute the values of the polynomials: V*@aE  
    % -------------------------------------- RB %+|@c  
    z = zeros(length_r,length_n); 9295:Y| w1  
    for j = 1:length_n 4  eLZ  
        s = 0:(n(j)-m(j))/2; 6Hnez@d  
        pows = n(j):-2:m(j); ye.6tlW  
        for k = length(s):-1:1 @*y4uI6&  
            p = (1-2*mod(s(k),2))* ... |#_p0yPy  
                       prod(2:(n(j)-s(k)))/          ... BaQyn 6B  
                       prod(2:s(k))/                 ... \x-2qlZ  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... gkd4)\9  
                       prod(2:((n(j)+m(j))/2-s(k))); ~3.*b% ,  
            idx = (pows(k)==rpowers); RvAgv[8  
            z(:,j) = z(:,j) + p*rpowern(:,idx); A^,E~Z!x  
        end )jOa!E"  
         `$/a-K}  
        if isnorm f- XUto  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); RxqNgun@  
        end v7"VH90`!  
    end /Z6lnm7wJ  
    N)"8CvQL  
    % 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)  5{(4%  
    3ux7^au  
    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。  lha;|  
    Fk49~z   
    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)