切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11317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 A>HCX 4i  
    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! -lAY*2Jg  
     
    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 Dk8@x8  
    function z = zernfun(n,m,r,theta,nflag) 4mpcI  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. dxtG3  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N XV`8Vb  
    %   and angular frequency M, evaluated at positions (R,THETA) on the "}H2dn2n  
    %   unit circle.  N is a vector of positive integers (including 0), and >B*zzj  
    %   M is a vector with the same number of elements as N.  Each element 02T'B&&~  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 5P 5Tgk  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 6E^9>  
    %   and THETA is a vector of angles.  R and THETA must have the same V)ag ss w?  
    %   length.  The output Z is a matrix with one column for every (N,M) FP*kA_z$  
    %   pair, and one row for every (R,THETA) pair. &FYv4J  
    % b^P\Q s*m  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 3a=\$x@  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), #YK3Ogb,  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral bqx2lQf,_  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, BlcsDB =ka  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 8LXK3D}?3  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. yUO%@;  
    % b@K1;A! S  
    %   The Zernike functions are an orthogonal basis on the unit circle. R|wS*xd,  
    %   They are used in disciplines such as astronomy, optics, and l0g+OMt  
    %   optometry to describe functions on a circular domain. t -fmA?\  
    % >RpMw!NT  
    %   The following table lists the first 15 Zernike functions. 2+ g'ul`  
    % \$F#bIjC  
    %       n    m    Zernike function           Normalization 'Z#>K*  
    %       -------------------------------------------------- Fzy#!^9Nu  
    %       0    0    1                                 1 P4|A\|t  
    %       1    1    r * cos(theta)                    2 =ReSlt  
    %       1   -1    r * sin(theta)                    2 40dwp*/!  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 2pP"dX  
    %       2    0    (2*r^2 - 1)                    sqrt(3) qG g29  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) %mzDmrzq  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) >}JEX]V  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) *m`x/_y+  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) %P(2uesd  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) HYY+Fv5  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) q]SH'Wd  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) i<=2 L?[.I  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) :()K2<E  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) LZE9]Gd  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) kV!1k<f  
    %       -------------------------------------------------- 0(&Rm R  
    % s%6L94\t  
    %   Example 1: 2t>>08T  
    % 78?cCj{e  
    %       % Display the Zernike function Z(n=5,m=1) Wc;N;K52   
    %       x = -1:0.01:1; :lmimAMt  
    %       [X,Y] = meshgrid(x,x); =5YbK1Q^  
    %       [theta,r] = cart2pol(X,Y); c+8 Y|GB  
    %       idx = r<=1; HsT6 #K  
    %       z = nan(size(X)); w"O;: `|n  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Rz6kwh=q  
    %       figure ApplWa3  
    %       pcolor(x,x,z), shading interp M8y|Lm}o  
    %       axis square, colorbar 9F~5Ht  
    %       title('Zernike function Z_5^1(r,\theta)') wjT#D|soI  
    % \]\h,Y8  
    %   Example 2: WH fl|e  
    % Y/pK  
    %       % Display the first 10 Zernike functions $~?)E;S  
    %       x = -1:0.01:1; Fx)><+-  
    %       [X,Y] = meshgrid(x,x); yC4%z) t&R  
    %       [theta,r] = cart2pol(X,Y); C+mPl+}w  
    %       idx = r<=1; {BJH}vV1)  
    %       z = nan(size(X)); t~!ag#3['.  
    %       n = [0  1  1  2  2  2  3  3  3  3]; q^<;B Y  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; V!e*J,g  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; WE-+WC!!:  
    %       y = zernfun(n,m,r(idx),theta(idx)); ,jD-fL/:  
    %       figure('Units','normalized') Qp2~ `hD  
    %       for k = 1:10 k ,r*xt  
    %           z(idx) = y(:,k); fWF!%|L  
    %           subplot(4,7,Nplot(k)) 'RNj5r  
    %           pcolor(x,x,z), shading interp ~L> &p  
    %           set(gca,'XTick',[],'YTick',[]) h96<9L  
    %           axis square ^W^Y"0y9`  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) t_(S e  
    %       end >N}+O<Fc  
    % 0TiDQ4}i[  
    %   See also ZERNPOL, ZERNFUN2. ?,[$8V  
    JxM32?Rm*w  
    %   Paul Fricker 11/13/2006 'gsO}xj  
    A-$ C6q   
    -Q ];o~  
    % Check and prepare the inputs: RL/5 o"  
    % ----------------------------- [DTe  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 1 7 iw`@  
        error('zernfun:NMvectors','N and M must be vectors.') y\dx \  
    end HPo><u  
    2&AX_#P  
    if length(n)~=length(m) ~i>'3j0@k  
        error('zernfun:NMlength','N and M must be the same length.') ,I,Zl.5  
    end gx C`Ml  
    vH E:TQo4  
    n = n(:); V_|HzYJJ5  
    m = m(:); "ZmxHMf  
    if any(mod(n-m,2)) &iy7It  
        error('zernfun:NMmultiplesof2', ... +]hc!s8  
              'All N and M must differ by multiples of 2 (including 0).') ^lK!tOeO  
    end 2t=&h|6EW  
    IQmlmu  
    if any(m>n) X6?Gxf,  
        error('zernfun:MlessthanN', ... (?.h<v1}  
              'Each M must be less than or equal to its corresponding N.') $ylxl"Y  
    end I6S>*V  
    ?~]mOv>  
    if any( r>1 | r<0 ) n~i^+pD@  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Ku3NE-)  
    end i/C0 (!  
    DnF|wS  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ^{E_fQJX  
        error('zernfun:RTHvector','R and THETA must be vectors.') V@1,((,l  
    end ?b]f$ 2  
    ;BHIss7  
    r = r(:); ZMK1V)ohn  
    theta = theta(:); S@4bpnhK  
    length_r = length(r); bF +d_t  
    if length_r~=length(theta) W)Yo-%  
        error('zernfun:RTHlength', ... s>TC~d82  
              'The number of R- and THETA-values must be equal.') 4!?4Tc!X  
    end 5?E;Yy A  
    o+S?j*mv@  
    % Check normalization: \PmM856=ms  
    % -------------------- dcE(uf  
    if nargin==5 && ischar(nflag) 9HlM0qE5b  
        isnorm = strcmpi(nflag,'norm'); eNm Wul  
        if ~isnorm MA7&fNjB  
            error('zernfun:normalization','Unrecognized normalization flag.') %XXjQ5p  
        end q+lCA#Sx  
    else Ti#x62X{  
        isnorm = false; !VvM  
    end dmMrZ1u2  
    s-l3_210  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F@ZB6~T~.  
    % Compute the Zernike Polynomials @,pn/[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >vuR:4B  
    W9A F}  
    % Determine the required powers of r: }</"~Kw!  
    % ----------------------------------- Vz y )jf  
    m_abs = abs(m); i6^-fl  
    rpowers = []; 1_G+sDw$  
    for j = 1:length(n) 48 mTL+*  
        rpowers = [rpowers m_abs(j):2:n(j)]; YD5mJ[1t"2  
    end N,ZmGzNP)  
    rpowers = unique(rpowers);  b|Eo\l2  
    cs]3Rp^g  
    % Pre-compute the values of r raised to the required powers, pq]>Ep  
    % and compile them in a matrix: 2y9$ k\<xV  
    % ----------------------------- W{kTM4  
    if rpowers(1)==0 1EliR uJ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); qqu ]r  
        rpowern = cat(2,rpowern{:}); )fc+B_  
        rpowern = [ones(length_r,1) rpowern]; IXR%IggJA  
    else `Z (`  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);  t&G #%  
        rpowern = cat(2,rpowern{:}); XBTjb  
    end Z&GjG6t  
    ?"p.Gy)  
    % Compute the values of the polynomials: _P=L| U#C  
    % -------------------------------------- //^{u[lr  
    y = zeros(length_r,length(n)); XeAH.i<  
    for j = 1:length(n) Qgl5Jr.  
        s = 0:(n(j)-m_abs(j))/2; _2<d6@}  
        pows = n(j):-2:m_abs(j); B)&z% +  
        for k = length(s):-1:1 tLGNYW!K  
            p = (1-2*mod(s(k),2))* ... 9]a!1  
                       prod(2:(n(j)-s(k)))/              ... HU-#xK  
                       prod(2:s(k))/                     ... j|y"Lcq  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 5>h# hcL  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); OUm,;WNLf  
            idx = (pows(k)==rpowers); WAb@d=H{+>  
            y(:,j) = y(:,j) + p*rpowern(:,idx); AD"L>7  
        end H$)otDOE  
         stOD5yi  
        if isnorm d-#yN:}0  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); s&6/fa  
        end G5$YXNV  
    end (KphAA8  
    % END: Compute the Zernike Polynomials 51!#m|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -p20UP 1I  
    Jrx]/CM  
    % Compute the Zernike functions: WL<f!   
    % ------------------------------ bm(.(0MI  
    idx_pos = m>0; ZJ |&t  
    idx_neg = m<0; b!z=:  
    u%nhQ%  
    z = y; hKN/&P^  
    if any(idx_pos) uBo~PiJ2"  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); oMF[<Xf  
    end j$khGR!  
    if any(idx_neg) ljk,R G  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); -VZ? c  
    end qk!,:T  
    -W)8Z.  
    % 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) !FbW3p f  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. h2ou ]  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated O|0}m  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive *uvE`4V^Jg  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, MF4B 2d  
    %   and THETA is a vector of angles.  R and THETA must have the same BPC>  
    %   length.  The output Z is a matrix with one column for every P-value, $ Q*^c"&  
    %   and one row for every (R,THETA) pair. J8PZVeWx  
    % F!ra$5u  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike fBct%M 3  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) p|'Rm ]&jb  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 4>te>[  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 P<IZ%eS3B  
    %   for all p. "o$)z'q  
    % B3V+/o6  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 bODyJ7=[  
    %   Zernike functions (order N<=7).  In some disciplines it is ~DUOL ~E  
    %   traditional to label the first 36 functions using a single mode {$)pkhJ  
    %   number P instead of separate numbers for the order N and azimuthal O ftjm X_  
    %   frequency M. U7d05y'  
    % )r:gDd#/X  
    %   Example: 'Rw*WK  
    % R ;3!?`  
    %       % Display the first 16 Zernike functions RV, cQ K  
    %       x = -1:0.01:1; p"=8{LrO  
    %       [X,Y] = meshgrid(x,x); &8QkGUbS<  
    %       [theta,r] = cart2pol(X,Y); Pxe7 \e  
    %       idx = r<=1; $D(q  
    %       p = 0:15; %scQP{%aD  
    %       z = nan(size(X)); <5sP%Fs)  
    %       y = zernfun2(p,r(idx),theta(idx)); f&`yiy_  
    %       figure('Units','normalized') pDG>9P#mO  
    %       for k = 1:length(p) Ky6 d{|H  
    %           z(idx) = y(:,k); -Oc  
    %           subplot(4,4,k) h v/+  
    %           pcolor(x,x,z), shading interp .0#{ ?R,  
    %           set(gca,'XTick',[],'YTick',[]) _&/2-3]\B  
    %           axis square "!_,N@\t  
    %           title(['Z_{' num2str(p(k)) '}']) 5D`!Tu3  
    %       end yo"!C?82=  
    % o.KE=zp&z  
    %   See also ZERNPOL, ZERNFUN. !eGUiE=  
    </)QCl'd  
    %   Paul Fricker 11/13/2006 |bv7N@?e  
    .Sjg  
    %pr}Xs(-f  
    % Check and prepare the inputs: L QA6iZBP  
    % ----------------------------- ed4`n!3  
    if min(size(p))~=1 HWi: CDgm  
        error('zernfun2:Pvector','Input P must be vector.') P/doNv}iG  
    end t Ai?Bjo  
    BZAF;j  
    if any(p)>35 X16r$~Pb  
        error('zernfun2:P36', ... }R2afTn[;  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... q OX=M  
               '(P = 0 to 35).']) QVPJ$~x  
    end [,|;rt\o>  
    P_%kYcX'  
    % Get the order and frequency corresonding to the function number: oAxCI/  
    % ---------------------------------------------------------------- 5WU ? Km  
    p = p(:); 95hdQ<W  
    n = ceil((-3+sqrt(9+8*p))/2); yc3i> w`  
    m = 2*p - n.*(n+2); K8g9IZ*lT  
    ()%;s2>F  
    % Pass the inputs to the function ZERNFUN: Xo~kB)|,  
    % ---------------------------------------- I<LIw8LI  
    switch nargin `Fs-z  
        case 3 %~^:[@xa*  
            z = zernfun(n,m,r,theta); Ur2) ];WZ  
        case 4 C#D8 E.W  
            z = zernfun(n,m,r,theta,nflag); >19j_[n@VC  
        otherwise gtw?u b  
            error('zernfun2:nargin','Incorrect number of inputs.') K:8. Dvn  
    end *d$r`.9j  
    g,d'&r"JWt  
    % 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) Nr:%yvk%s  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. GP;UuQz  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of L#9g ~>~  
    %   order N and frequency M, evaluated at R.  N is a vector of U# 7K^(E9  
    %   positive integers (including 0), and M is a vector with the d+158qQOh]  
    %   same number of elements as N.  Each element k of M must be a {2MS,Ua{  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) JWo).  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is %r >Y)@$Vt  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix );oE^3]f  
    %   with one column for every (N,M) pair, and one row for every [3fmhc  
    %   element in R. }D7} %P]  
    % |tqYRWn0  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- j]bNOC2.L  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is _1kcz]]F  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to w-?|6I}T  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 l"app]uVZ  
    %   for all [n,m]. U<jAZU[L  
    % ?)L X4GY  
    %   The radial Zernike polynomials are the radial portion of the G+zIh}9  
    %   Zernike functions, which are an orthogonal basis on the unit +je{%,*  
    %   circle.  The series representation of the radial Zernike JPGEE1!B{b  
    %   polynomials is kH}HFl  
    % Fv T;8ik:3  
    %          (n-m)/2 d<+@cf_9  
    %            __ 68, (+vkB  
    %    m      \       s                                          n-2s m(eR Wx&pZ  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r * J|]E(  
    %    n      s=0 J'#R9NO<  
    % mqk tM6  
    %   The following table shows the first 12 polynomials. 6;{E-y  
    % mdy+ >e <  
    %       n    m    Zernike polynomial    Normalization *Ph@XkhU  
    %       --------------------------------------------- x3]y*6  
    %       0    0    1                        sqrt(2) M&~cU{9c  
    %       1    1    r                           2 FvaelB  
    %       2    0    2*r^2 - 1                sqrt(6) Xl\yOMfp  
    %       2    2    r^2                      sqrt(6) 7zEpuw  
    %       3    1    3*r^3 - 2*r              sqrt(8) w6FVSU]sY  
    %       3    3    r^3                      sqrt(8) WSV[)-=:  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) cNvh2JI  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) E!zAUEVQm[  
    %       4    4    r^4                      sqrt(10) :Co+haW  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) t o2y#4'.  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ?Y | *EH  
    %       5    5    r^5                      sqrt(12) TB(!*t  
    %       --------------------------------------------- (X(c.Jj  
    % !:|[?M.`  
    %   Example: (3fU2{sm  
    % :Yy8Ie#  
    %       % Display three example Zernike radial polynomials 1H]E:Bq  
    %       r = 0:0.01:1; f+W8Gszi  
    %       n = [3 2 5]; tj;<EaM  
    %       m = [1 2 1]; <N}*|z7=b  
    %       z = zernpol(n,m,r); (D <o=Q  
    %       figure ]/y69ou  
    %       plot(r,z) \r7gubD  
    %       grid on JBxizJBP  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') hD! 9[Gb  
    % 4,P!D3SH  
    %   See also ZERNFUN, ZERNFUN2. \B1<fF2  
    7<p? E7  
    % A note on the algorithm. 2<GN+W v[#  
    % ------------------------ K}1eQS&$a  
    % The radial Zernike polynomials are computed using the series &nX,)"  
    % representation shown in the Help section above. For many special RRBBz7:~  
    % functions, direct evaluation using the series representation can T_1p1Sg  
    % produce poor numerical results (floating point errors), because gP 6`q  
    % the summation often involves computing small differences between g{%2*{;i  
    % large successive terms in the series. (In such cases, the functions 3PU'd^  
    % are often evaluated using alternative methods such as recurrence aB+B1YdY"  
    % relations: see the Legendre functions, for example). For the Zernike 5'hQ6i8  
    % polynomials, however, this problem does not arise, because the Eh*t;J=O  
    % polynomials are evaluated over the finite domain r = (0,1), and iz pFl@WS  
    % because the coefficients for a given polynomial are generally all 96.Vm*/7  
    % of similar magnitude. I7/X6^/}  
    % 6K 6uB ~  
    % ZERNPOL has been written using a vectorized implementation: multiple kx_PMpc  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] EU@XLm6  
    % values can be passed as inputs) for a vector of points R.  To achieve xPJ kadu  
    % this vectorization most efficiently, the algorithm in ZERNPOL n`af2I2  
    % involves pre-determining all the powers p of R that are required to 8 y+Nl&"V  
    % compute the outputs, and then compiling the {R^p} into a single wM#BQe3t#  
    % matrix.  This avoids any redundant computation of the R^p, and 1[Ffl^\ARp  
    % minimizes the sizes of certain intermediate variables. .ugQH<B  
    % }2-p= Y:6  
    %   Paul Fricker 11/13/2006 >i IUS  
    b/ dyH  
    ;UB$Uqs6  
    % Check and prepare the inputs:  ;%tu;  
    % ----------------------------- Vn1hr;i]  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) v'zj<|2  
        error('zernpol:NMvectors','N and M must be vectors.') \r+8}8  
    end 2r~&+0sBP  
    SXI3y  
    if length(n)~=length(m) Ap [}[:U  
        error('zernpol:NMlength','N and M must be the same length.') Jxy94y*  
    end )-4xI4  
    A"8"e*  
    n = n(:); OK"B`*  
    m = m(:); rJ UXA<:2  
    length_n = length(n); Q]]5\C.  
    K@PQLL#yJp  
    if any(mod(n-m,2)) rtM!|apr  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') d|8iD`sZz  
    end /2\%X`]<  
    zz+p6`   
    if any(m<0) z nc'  
        error('zernpol:Mpositive','All M must be positive.') w 9mi2=  
    end -n`igC  
    [# '38  
    if any(m>n) <_tkd3t#W  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') g, %xGQ4+  
    end <Y9xHn&  
     `=h`:`  
    if any( r>1 | r<0 ) iS"(  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') & M wvj  
    end 1L=6Z2*fB4  
    u0(PWCi2  
    if ~any(size(r)==1) Z$? Ql@M  
        error('zernpol:Rvector','R must be a vector.') 0` {6~p  
    end @ !:~gQ  
    7.o:(P1??g  
    r = r(:); V~uH)IMkh7  
    length_r = length(r); domaD"C  
    PmGW\E[ni  
    if nargin==4 .kO;9z\B  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); >7VO ytc  
        if ~isnorm c_fx,; ;  
            error('zernpol:normalization','Unrecognized normalization flag.') 1px8af]  
        end ZO2$Aan  
    else %`P6a38j  
        isnorm = false; L]goHs  
    end w< hw>e^.  
    9aR-kcvJIJ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Qv%"iSe~J  
    % Compute the Zernike Polynomials 7X$[E*kd  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mOb*VH  
    YFVNkB O%  
    % Determine the required powers of r: 4X@ <PX5  
    % ----------------------------------- V #0F2GV<,  
    rpowers = []; DB'd9<  
    for j = 1:length(n) 3f2%+2Zjt,  
        rpowers = [rpowers m(j):2:n(j)]; c7[<X<yk  
    end 1jJ>(S  
    rpowers = unique(rpowers); k|$08EK $  
    gJ_{V;R  
    % Pre-compute the values of r raised to the required powers, vap,)kILF  
    % and compile them in a matrix: I]"wT2@T;7  
    % ----------------------------- P9)E1]Dc$  
    if rpowers(1)==0 zqDIwfW  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 'E{n1[b  
        rpowern = cat(2,rpowern{:}); Zxm Mw  
        rpowern = [ones(length_r,1) rpowern]; p9;Oe,Il  
    else 9y(75Bn9  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); F_9eju^|  
        rpowern = cat(2,rpowern{:}); Q2c|sK8  
    end .a%D:4GYR  
    k !S0-/ h  
    % Compute the values of the polynomials: 8IYn9<L  
    % -------------------------------------- +rw?k/  
    z = zeros(length_r,length_n); /Q4TQ\:  
    for j = 1:length_n 8\?7k  
        s = 0:(n(j)-m(j))/2; sw.cw}1  
        pows = n(j):-2:m(j); 2NHkK_B1P  
        for k = length(s):-1:1 wo($7'.@  
            p = (1-2*mod(s(k),2))* ... e6C;A]T2E  
                       prod(2:(n(j)-s(k)))/          ... v=A ]#O%  
                       prod(2:s(k))/                 ... ^tS{a*Yn  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... G x;U 3iV  
                       prod(2:((n(j)+m(j))/2-s(k))); O,`#h*{N  
            idx = (pows(k)==rpowers); 'u6T^YS  
            z(:,j) = z(:,j) + p*rpowern(:,idx); >hkmL](^  
        end $4^cbk  
         '?Mt*%J@=$  
        if isnorm }Ut*Y*  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); CdCo+U5z{  
        end kiLwN nq  
    end OOzk@j^  
    '- >%b  
    % 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)  O:% ,.??<%  
    . r?URC  
    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。 R!+_mPb=Q*  
    9Sb[5_Q  
    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)