切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11387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 8!SiTOzR?  
    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! mD)O\.uA  
     
    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 oK:P@V6!  
    function z = zernfun(n,m,r,theta,nflag) zn1Rou]6  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. f\U&M,L\ '  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ;;hyjFGq%  
    %   and angular frequency M, evaluated at positions (R,THETA) on the }k0-?_Z=1  
    %   unit circle.  N is a vector of positive integers (including 0), and eSNSnh]'  
    %   M is a vector with the same number of elements as N.  Each element 5qkuK F  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) _I-VWDCk  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, jZT :-w  
    %   and THETA is a vector of angles.  R and THETA must have the same .]s( c!{y  
    %   length.  The output Z is a matrix with one column for every (N,M) 1 3 `0d  
    %   pair, and one row for every (R,THETA) pair.  0(/D|  
    % yPh2P5}H>  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike >04>rn#},,  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), L2.`1Aag  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral UW[{d/.wC  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, D *I;|.=u  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized T) tZU?  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Df:7P>  
    % 56SS >b  
    %   The Zernike functions are an orthogonal basis on the unit circle. )QCM2  
    %   They are used in disciplines such as astronomy, optics, and l()MYuLNV  
    %   optometry to describe functions on a circular domain. qJXsf M6  
    % pNE\@U|4E  
    %   The following table lists the first 15 Zernike functions. k7|z$=zY  
    % q6JW@GT  
    %       n    m    Zernike function           Normalization (S)E|;f%C  
    %       -------------------------------------------------- Oqpl2Y"/  
    %       0    0    1                                 1 R 4$Q3vcH  
    %       1    1    r * cos(theta)                    2 ,' r L'Ys  
    %       1   -1    r * sin(theta)                    2 dEd]U49u  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) t)gi.Ed1"L  
    %       2    0    (2*r^2 - 1)                    sqrt(3) \btR^;_\A  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ,mjfZ*N  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) h>Uid &:?  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ca/o#9:N`:  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) hQ}7Z&O  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) }{wTlR.]  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ,)rZAI  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?(/j<,m^  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) yOUX E>-  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) iQ|,&K0d]  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Ly)(_Tp@+  
    %       -------------------------------------------------- 1 73<x){  
    % N`<4:v[P  
    %   Example 1: &H4uvJ_<  
    % gJ3OK!/  
    %       % Display the Zernike function Z(n=5,m=1) \YlF>{LVe  
    %       x = -1:0.01:1; I51oG:6fR?  
    %       [X,Y] = meshgrid(x,x); !<=%;+  
    %       [theta,r] = cart2pol(X,Y); VqClM  
    %       idx = r<=1; JU'WiR bcb  
    %       z = nan(size(X)); ?VZ11?u  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Dpdn%8+Z  
    %       figure yD[zzEuQ  
    %       pcolor(x,x,z), shading interp xdL/0 N3  
    %       axis square, colorbar ,zN3? /7  
    %       title('Zernike function Z_5^1(r,\theta)') jKj=#O  
    % "s>fV9YyZ  
    %   Example 2: %|*nmIPq(  
    % C,{F0-D  
    %       % Display the first 10 Zernike functions y^ X\^Kq  
    %       x = -1:0.01:1; r}oURy,5  
    %       [X,Y] = meshgrid(x,x); -OrY{^F  
    %       [theta,r] = cart2pol(X,Y); &N"'7bK6n  
    %       idx = r<=1; nxyjL)!)0  
    %       z = nan(size(X)); %wt2F-u  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ` y^zM/Ib  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ){+[$@9  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; #ox9&  
    %       y = zernfun(n,m,r(idx),theta(idx)); [;?"R-V"z  
    %       figure('Units','normalized') msc 1^2  
    %       for k = 1:10 C{UF~  
    %           z(idx) = y(:,k); 0~+NB-L}  
    %           subplot(4,7,Nplot(k)) ShWHHU(QQ  
    %           pcolor(x,x,z), shading interp selP=Q!  
    %           set(gca,'XTick',[],'YTick',[]) I(OAEIz  
    %           axis square TsaW5ho<p  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) GSz @rDGY  
    %       end y Y>-MoF/t  
    % 83KfM!w  
    %   See also ZERNPOL, ZERNFUN2. *.m{jgi1X  
    ]{IR&{EI-  
    %   Paul Fricker 11/13/2006 ~LawF_]6  
    %bIsrQ~B  
    Y&vHOA  
    % Check and prepare the inputs: y)3~]h\a  
    % ----------------------------- x7 "z(rKl  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) [3j$ 4rP  
        error('zernfun:NMvectors','N and M must be vectors.') L!;^ #g  
    end 8W~lU~-  
    brg":V1a  
    if length(n)~=length(m)  r=fE8[,  
        error('zernfun:NMlength','N and M must be the same length.') 8yE!7$Mj  
    end >j50 ;</  
    7$(_j<o`  
    n = n(:); jrm0@K+<IA  
    m = m(:); bK3B3r#$  
    if any(mod(n-m,2)) ?^LG hdR  
        error('zernfun:NMmultiplesof2', ... { EA2   
              'All N and M must differ by multiples of 2 (including 0).') w$gS j/  
    end 94Xjz(  
    i{gDW+N  
    if any(m>n) [O=W>l  
        error('zernfun:MlessthanN', ... X_D6eYF  
              'Each M must be less than or equal to its corresponding N.') OuB2 x=B  
    end L~*u4  
    3YR* ^  
    if any( r>1 | r<0 ) xME(B@j  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 3PsxOb+  
    end a*Rz<08  
    fO*)LPen.z  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) XjX 2[*l  
        error('zernfun:RTHvector','R and THETA must be vectors.') c  Qld$  
    end k_]\(myq  
    X(IyvfC  
    r = r(:); y k?SD1hj  
    theta = theta(:); ,# ]+HS^B  
    length_r = length(r); YVoao#!  
    if length_r~=length(theta) F4Rr26M  
        error('zernfun:RTHlength', ... f, |QAj=a  
              'The number of R- and THETA-values must be equal.') >f>V5L%1  
    end V {p*z  
    i wUv`>l&  
    % Check normalization: ]de\i=?|  
    % -------------------- $u:<x  
    if nargin==5 && ischar(nflag) &9RH}zv6  
        isnorm = strcmpi(nflag,'norm'); (I[s3EnhS  
        if ~isnorm Qe_+r(3)k  
            error('zernfun:normalization','Unrecognized normalization flag.') 6VC-KY  
        end gt'*B5F(  
    else 7m\vRMK  
        isnorm = false; [~COYjp  
    end }7%9}2}Iw  
    >E, Q  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f_rp<R>Uu  
    % Compute the Zernike Polynomials ((qGh>*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F'1k<V?  
    p+$+MeBz  
    % Determine the required powers of r: 0 <g{ V  
    % ----------------------------------- \Dfm(R  
    m_abs = abs(m); guU=NQZ  
    rpowers = []; t ^m~  
    for j = 1:length(n) sds}bo  
        rpowers = [rpowers m_abs(j):2:n(j)]; / $_M@>  
    end <KX&zi<L)  
    rpowers = unique(rpowers); ul$,q05nb  
    SyAo, )j  
    % Pre-compute the values of r raised to the required powers,  c-5Ysg  
    % and compile them in a matrix: 19p8B&  
    % ----------------------------- Ls1B \Aw_  
    if rpowers(1)==0 > VP5vkv=  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 6x/s|RWL1  
        rpowern = cat(2,rpowern{:}); 9p4y>3  
        rpowern = [ones(length_r,1) rpowern]; Hs$'0:  
    else KU]ok '  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); g rspt}  
        rpowern = cat(2,rpowern{:}); 1 DqX:WM6  
    end 4@h;5   
    "TNVD"RLY  
    % Compute the values of the polynomials: hCAZ{+`z  
    % -------------------------------------- W&YU^&`Yr  
    y = zeros(length_r,length(n)); FIS "Z(  
    for j = 1:length(n) DHv2&zH  
        s = 0:(n(j)-m_abs(j))/2; *GJ:+U&m[  
        pows = n(j):-2:m_abs(j); f0DK>L  
        for k = length(s):-1:1 &qKig kLd  
            p = (1-2*mod(s(k),2))* ... E=]]b;u-n  
                       prod(2:(n(j)-s(k)))/              ... 6WeM rWx  
                       prod(2:s(k))/                     ... q_sEw~~@!  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &?y7I Pp  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); x#r<,uNn,  
            idx = (pows(k)==rpowers); /OG zt  
            y(:,j) = y(:,j) + p*rpowern(:,idx); gfN2/TDC]P  
        end t"|DWC*  
         45<y{8  
        if isnorm w"~<h;  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); k"0;D-lTZ>  
        end s6n`?,vw  
    end pawl|Z'Ez  
    % END: Compute the Zernike Polynomials @PX\{6&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nxfoWy  
    [Gtb+'8  
    % Compute the Zernike functions: Xb,T{.3@  
    % ------------------------------ oL-2qtv  
    idx_pos = m>0; \f%.n]>  
    idx_neg = m<0; \k; n20\u  
    MA* :<l  
    z = y; RV;!05^<  
    if any(idx_pos) "VTF}#Uo  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 2+Yb 7 uI,  
    end )%F5t&lum  
    if any(idx_neg) ! %Ny0JkO  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Ryv_1gR!  
    end +qy6d7^  
    p!DP`Ouc3\  
    % 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) Fr [7  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. U>Gg0`>  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated .zkP~xQ~  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive <[i}n55  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, G5Ykbw#  
    %   and THETA is a vector of angles.  R and THETA must have the same 6gU{(H   
    %   length.  The output Z is a matrix with one column for every P-value, c^9tYNn  
    %   and one row for every (R,THETA) pair. *9 D!A  
    % /.Q4~Hw%}  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike G%{0i20_  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) D$q'FZH  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ~ap2m  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 4 b,N8  
    %   for all p. 93o;n1rS  
    % <]d LX}C)  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 : ]II-$/8  
    %   Zernike functions (order N<=7).  In some disciplines it is Z^ar.boc  
    %   traditional to label the first 36 functions using a single mode Or+p%K}-7  
    %   number P instead of separate numbers for the order N and azimuthal {' 5qv@3  
    %   frequency M. L:R<e#kgS  
    % >*1}1~uU`'  
    %   Example: @_yoX(.E&  
    % /,tAoa~FA  
    %       % Display the first 16 Zernike functions tef^ShF]  
    %       x = -1:0.01:1; N#k61x  
    %       [X,Y] = meshgrid(x,x); |J>WC}g@n  
    %       [theta,r] = cart2pol(X,Y); 0XU}B\'<  
    %       idx = r<=1; 7~UR!T9  
    %       p = 0:15; h{'t5&yY  
    %       z = nan(size(X)); Qa4MZj ;$K  
    %       y = zernfun2(p,r(idx),theta(idx)); dh -,E  
    %       figure('Units','normalized') `I;F$`\  
    %       for k = 1:length(p) |i7a@'0)  
    %           z(idx) = y(:,k); 55DE\<r  
    %           subplot(4,4,k) 'Jj=RAV`  
    %           pcolor(x,x,z), shading interp  $xgBKD  
    %           set(gca,'XTick',[],'YTick',[]) TqAPAHg  
    %           axis square 7Y( 5]A9=  
    %           title(['Z_{' num2str(p(k)) '}']) Da1aI]{I  
    %       end Xm!-~n@-m7  
    % diT=x52  
    %   See also ZERNPOL, ZERNFUN. n/Dp"4H%q  
    I4c!m_sr  
    %   Paul Fricker 11/13/2006 WO*9+\[v  
    \}"m'(\c  
    Acm<-de  
    % Check and prepare the inputs: A\sI<WrH  
    % ----------------------------- ~r*P]*51x  
    if min(size(p))~=1 EbQa?  
        error('zernfun2:Pvector','Input P must be vector.') {2KFD\i\  
    end N{Qxq>6 G  
    9pSUIl9|j  
    if any(p)>35 $)Bg JDr  
        error('zernfun2:P36', ... Ym8}ZW-  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... rofNZ;nu  
               '(P = 0 to 35).'])  IDFFc&  
    end @{HrJ/4%:&  
    =:I+6PlF@  
    % Get the order and frequency corresonding to the function number: dK9Zg,DZL  
    % ---------------------------------------------------------------- sM2MLh'D  
    p = p(:); R9o-`Wz  
    n = ceil((-3+sqrt(9+8*p))/2); Gh( A%x)  
    m = 2*p - n.*(n+2); HIvZQQW|  
    F5T3E?_  
    % Pass the inputs to the function ZERNFUN: gzn^#3b  
    % ---------------------------------------- ^QX bJJ  
    switch nargin lS5ny  
        case 3 !cX[-}Q  
            z = zernfun(n,m,r,theta); ~/#1G.H  
        case 4 D-p.kA3MJ  
            z = zernfun(n,m,r,theta,nflag); Ctu?o+^;z  
        otherwise 7<\C ?`q"  
            error('zernfun2:nargin','Incorrect number of inputs.') (P|pRVO  
    end ;{Ux_JEg  
    t*S." q  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) 6sfwlT  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. aoW6U{\  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Fj p.T;  
    %   order N and frequency M, evaluated at R.  N is a vector of L@Nu/(pB=  
    %   positive integers (including 0), and M is a vector with the afG{lWE)  
    %   same number of elements as N.  Each element k of M must be a kAYb!h[`  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) )X+mV  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is DvXHK  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ;3'NMk  
    %   with one column for every (N,M) pair, and one row for every ^%T7.1'x  
    %   element in R.  vb{i  
    % \%jVg\4 '  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- i'/m4 !>h  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is n$L51#'  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to E+95WF|4k"  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 uzr\oj+>  
    %   for all [n,m]. V&{MQWy  
    % WN]<q`.  
    %   The radial Zernike polynomials are the radial portion of the "f.Z}AbP  
    %   Zernike functions, which are an orthogonal basis on the unit kma?v B  
    %   circle.  The series representation of the radial Zernike YPDf Y<?v  
    %   polynomials is Av J4\  
    % r),PtI0X  
    %          (n-m)/2 uq3{h B#  
    %            __ 7*o*6,/  
    %    m      \       s                                          n-2s &]6) LFm  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r : esg(  
    %    n      s=0 $^/0<i$   
    % 6aft$A}XnD  
    %   The following table shows the first 12 polynomials. )eeN1G`rDE  
    % ] ,etZ%z&  
    %       n    m    Zernike polynomial    Normalization ~EiH-z4U  
    %       --------------------------------------------- Dr<='Ux[5  
    %       0    0    1                        sqrt(2) \*T"M*;  
    %       1    1    r                           2 jyS=!ydn+  
    %       2    0    2*r^2 - 1                sqrt(6) )=pD%$iq  
    %       2    2    r^2                      sqrt(6) E$s/]wnr[  
    %       3    1    3*r^3 - 2*r              sqrt(8) M)-6T{[IT  
    %       3    3    r^3                      sqrt(8) alMYk  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) Xf'=+f2p  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) "Y: /= Gx  
    %       4    4    r^4                      sqrt(10) ;Y9=!.Ak0y  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) DPgm%Xq9(!  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Ol /\t  
    %       5    5    r^5                      sqrt(12) 3L>IX8_   
    %       --------------------------------------------- 9Ru;`  
    % f7urJ'!V  
    %   Example: iO w3MfO  
    % RF}X ER  
    %       % Display three example Zernike radial polynomials R{Z-m2La  
    %       r = 0:0.01:1; V)M1YZV{  
    %       n = [3 2 5]; vYmSKS  
    %       m = [1 2 1]; RSfM]w}Hq#  
    %       z = zernpol(n,m,r); y8Xv~4qQW  
    %       figure q(o/yx{bm  
    %       plot(r,z) g:ErZ;[  
    %       grid on ~!iQ6N?PY  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') I_)*)d44_  
    % G#`\(NW  
    %   See also ZERNFUN, ZERNFUN2. #^#Kcg  
    `|O yRU"EK  
    % A note on the algorithm. | $^;wP  
    % ------------------------ kfb/n)b'  
    % The radial Zernike polynomials are computed using the series shC;hR&;  
    % representation shown in the Help section above. For many special 5MTgK=c  
    % functions, direct evaluation using the series representation can VaZn{z  
    % produce poor numerical results (floating point errors), because R,2=&+ e  
    % the summation often involves computing small differences between &[R8Q|1 j  
    % large successive terms in the series. (In such cases, the functions 2RtHg_d_l  
    % are often evaluated using alternative methods such as recurrence hn)a@  
    % relations: see the Legendre functions, for example). For the Zernike S0/usC[r  
    % polynomials, however, this problem does not arise, because the )emOKS  
    % polynomials are evaluated over the finite domain r = (0,1), and q0mOG^  
    % because the coefficients for a given polynomial are generally all H!IshZfktn  
    % of similar magnitude. 5AeQQU  
    % p0p4Xh1 e  
    % ZERNPOL has been written using a vectorized implementation: multiple z2c5m  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] +t)n;JHN  
    % values can be passed as inputs) for a vector of points R.  To achieve _W!p8cB  
    % this vectorization most efficiently, the algorithm in ZERNPOL '(+<UpG_Q}  
    % involves pre-determining all the powers p of R that are required to Zi$ziDz&  
    % compute the outputs, and then compiling the {R^p} into a single a~LC+8|JW  
    % matrix.  This avoids any redundant computation of the R^p, and qOV[TP,  
    % minimizes the sizes of certain intermediate variables. .aOnGp  
    % Rf %HIAVE  
    %   Paul Fricker 11/13/2006 HjNxqaljt  
    B6P|Z%E;D6  
    hqSJ(gs{  
    % Check and prepare the inputs: |aToUi.Q%  
    % ----------------------------- Y$8JM  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) uYG^Pc^v  
        error('zernpol:NMvectors','N and M must be vectors.') U!5)5c}G  
    end dj6*6qX0'^  
    S]3Ev#>  
    if length(n)~=length(m) )U<Y0bZA!  
        error('zernpol:NMlength','N and M must be the same length.') a?5[k}\  
    end X~.f7Ao[  
    9a$56GnW1  
    n = n(:); g&/p*c_  
    m = m(:); " K*  
    length_n = length(n); SF ]@|  
    +?D6T!)  
    if any(mod(n-m,2)) th5g\h%j*  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 8euZTfK9e  
    end C_:k8?  
    E N^Uki`  
    if any(m<0) $gle8Z-  
        error('zernpol:Mpositive','All M must be positive.') u0`o A  
    end !|?e7u7  
    6~meM@  
    if any(m>n) [|`U6 8}u  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') M&Y .;  
    end wRNroQ  
    8t"~Om5sG  
    if any( r>1 | r<0 ) Y]`.InG@  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') >"3>s%  
    end *DI)?  
    \g)Xt?w0Wo  
    if ~any(size(r)==1) PG5- ;i/  
        error('zernpol:Rvector','R must be a vector.') p^m5`{1]x  
    end eEc4bVQa  
    _+f+`]iM  
    r = r(:); =;~I_)Pg1  
    length_r = length(r); J<n+\F-s  
    Wk;5/  
    if nargin==4 f,i5iSYf  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); mZk0@C&:6  
        if ~isnorm YOyX[&oi  
            error('zernpol:normalization','Unrecognized normalization flag.') t6N*6ld2b  
        end v *hRz;  
    else +m\|e{G  
        isnorm = false; |tMn={  
    end JwnAW}=  
    J<j&;:IRd  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zRl~^~sY  
    % Compute the Zernike Polynomials /BKe+]dS*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )w~Fo,   
    {43yb_B(  
    % Determine the required powers of r: =(AtfW^H  
    % ----------------------------------- m8]?hJY 3l  
    rpowers = []; DsDzkwJE  
    for j = 1:length(n) 2+8#H.  
        rpowers = [rpowers m(j):2:n(j)]; 0O!cN_l|  
    end yTM{|D]$(  
    rpowers = unique(rpowers); FXKF\1`( H  
    ~o3Hdd_#}N  
    % Pre-compute the values of r raised to the required powers, )8gGv  
    % and compile them in a matrix: d4[(8} x$/  
    % ----------------------------- D6D1S/:ij'  
    if rpowers(1)==0 Q<tu)Qo  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 1nj(h g  
        rpowern = cat(2,rpowern{:}); $*[{J+t_  
        rpowern = [ones(length_r,1) rpowern]; CCijf]+  
    else Sywu=b  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); >PKBo  
        rpowern = cat(2,rpowern{:}); &Jc_Fc(M  
    end wy1X\PJjH  
    X##1! ad  
    % Compute the values of the polynomials: >/f_F6ay#  
    % -------------------------------------- |Q~cX!;  
    z = zeros(length_r,length_n); HYr}wG  
    for j = 1:length_n p(SRjQt  
        s = 0:(n(j)-m(j))/2; c2l_$p  
        pows = n(j):-2:m(j); H2gj=krK  
        for k = length(s):-1:1 +y+"Fyl  
            p = (1-2*mod(s(k),2))* ... I 1d0iU  
                       prod(2:(n(j)-s(k)))/          ... UQ Co}vM  
                       prod(2:s(k))/                 ... T4e\0.If  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... _Yb _D/  
                       prod(2:((n(j)+m(j))/2-s(k))); Q }k.JS~#  
            idx = (pows(k)==rpowers); ~iBgw&Y  
            z(:,j) = z(:,j) + p*rpowern(:,idx); a[bBT@f  
        end Q3W#`6jpF  
         }'"Gr%jf(  
        if isnorm ,"-Rf<q/  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); YEu1#N  
        end F7m?xy  
    end "tit\a6\(  
    qMBR *f  
    % 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)  %t=kdc0=_  
    l5%G'1w#,j  
    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。 rVvR!"//yH  
    MfO:m[s  
    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)