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    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

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    离线niuhelen
     
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    只看楼主 倒序阅读 楼主  发表于: 2011-03-12
    小弟不是学光学的,所以想请各位大侠指点啊!谢谢啦 l9"0Wu@_x  
    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! kB {  
     
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    离线phility
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    只看该作者 1楼 发表于: 2011-03-12
    可以用matlab编程,用zernike多项式进行波面拟合,求出zernike多项式的系数,拟合的算法有很多种,最简单的是最小二乘法,你可以查下相关资料,挺简单的
    离线phility
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    只看该作者 2楼 发表于: 2011-03-12
    泽尼克多项式的前9项对应象差的
    离线niuhelen
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    只看该作者 3楼 发表于: 2011-03-12
    回 2楼(phility) 的帖子
    非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 3fPd|F.kF  
    function z = zernfun(n,m,r,theta,nflag) Xmr|k:z  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. I,;@\  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N )@+lfIE(l  
    %   and angular frequency M, evaluated at positions (R,THETA) on the vFKX@wV S  
    %   unit circle.  N is a vector of positive integers (including 0), and /{@^h#4M1  
    %   M is a vector with the same number of elements as N.  Each element QP/%+[E.  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) u"eO&Vc  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, +@*}_%^l"  
    %   and THETA is a vector of angles.  R and THETA must have the same ,ab_u@  
    %   length.  The output Z is a matrix with one column for every (N,M) qYo"-D*  
    %   pair, and one row for every (R,THETA) pair. C+ibLS4i  
    % !kCMw%[  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike *FhD%><  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), xuBXOr4"P  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral V6l~Aj}/  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, GP=i6I6C  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized l{q$[/J~)  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. v`&  
    % M]9oSi  
    %   The Zernike functions are an orthogonal basis on the unit circle. s}"5uDfn1F  
    %   They are used in disciplines such as astronomy, optics, and ;VM',40  
    %   optometry to describe functions on a circular domain. Zx$q,Zo<  
    % d'j8P  
    %   The following table lists the first 15 Zernike functions. YdsY2  
    % `"~s<+  
    %       n    m    Zernike function           Normalization kkWqP20q  
    %       -------------------------------------------------- xW|^2k  
    %       0    0    1                                 1 ~{69&T}9  
    %       1    1    r * cos(theta)                    2 "s-e)svB  
    %       1   -1    r * sin(theta)                    2 >6 p <n  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) +!_?f'kv`  
    %       2    0    (2*r^2 - 1)                    sqrt(3) R}~p1=D  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) zx)^!dEMM  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ?}f+PP,  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ]LGp3)T-  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) RSL%<  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Q2^~^'Y k  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) e|Ip7`  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) e| AA7  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) >R|*FYam  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) aJh=4j~.  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) *Nfn6lVB  
    %       -------------------------------------------------- _PTo !aJL  
    % b1X.#pz7F  
    %   Example 1: .-kqt^Gc  
    % $#Mew:J  
    %       % Display the Zernike function Z(n=5,m=1)  }qf9ra  
    %       x = -1:0.01:1;  $^&SEz  
    %       [X,Y] = meshgrid(x,x); 'Na \9b(  
    %       [theta,r] = cart2pol(X,Y); XD1 x*#  
    %       idx = r<=1; /t "p^9!^  
    %       z = nan(size(X)); 6 yIl)5/=  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); *~p~IX{  
    %       figure p)  x.Y  
    %       pcolor(x,x,z), shading interp >)Ih[0~M  
    %       axis square, colorbar ]>utLi5dX  
    %       title('Zernike function Z_5^1(r,\theta)') iU)-YFO  
    % R'E8>ee; ^  
    %   Example 2: m~K[+P  
    % c[=%v]j:u  
    %       % Display the first 10 Zernike functions Bjg 21bw^  
    %       x = -1:0.01:1; mtfyhFk  
    %       [X,Y] = meshgrid(x,x); Sr7+DCr  
    %       [theta,r] = cart2pol(X,Y); [V#"7O vl  
    %       idx = r<=1; OtopA)  
    %       z = nan(size(X)); 9JF*xXd>Q  
    %       n = [0  1  1  2  2  2  3  3  3  3]; kvU0$1  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; eYL7G-3  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; uj.~/W1,!  
    %       y = zernfun(n,m,r(idx),theta(idx)); K;2]c3T  
    %       figure('Units','normalized') +MQvq\%tG  
    %       for k = 1:10 Q]*YIb~D  
    %           z(idx) = y(:,k); K#"@nVWJ.m  
    %           subplot(4,7,Nplot(k)) uO$ujbWZ  
    %           pcolor(x,x,z), shading interp @5gZK[?|I  
    %           set(gca,'XTick',[],'YTick',[]) $s2-O!P?  
    %           axis square l+'1>T.I  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) %iPu51+=  
    %       end N0s)Nao4  
    % qa![oMKc  
    %   See also ZERNPOL, ZERNFUN2. {GF>HHQb  
    2|k*rv}l  
    %   Paul Fricker 11/13/2006 c$f|a$$b   
    i '!M<>7  
    W7N Hr5RC  
    % Check and prepare the inputs: ^H+j;K{5,  
    % ----------------------------- bw*@0;  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) $ Z;HE/ 3  
        error('zernfun:NMvectors','N and M must be vectors.') ?`F")y  
    end (4V1%0  
    LvpHR#K)F5  
    if length(n)~=length(m) {nQ}t }B  
        error('zernfun:NMlength','N and M must be the same length.') _ED1".&#f  
    end H+zn:j@~L  
    *jWU8.W  
    n = n(:); ADX}  
    m = m(:); Q}jbk9gM5  
    if any(mod(n-m,2)) hMJ \a  
        error('zernfun:NMmultiplesof2', ... vg5zsR0u  
              'All N and M must differ by multiples of 2 (including 0).') T[)) ful  
    end TJY$<:  
    T4 SByX9  
    if any(m>n) Fga9  
        error('zernfun:MlessthanN', ... k?Jzy  
              'Each M must be less than or equal to its corresponding N.') '4sT+q  
    end F *; +-e  
    !W:QLOe6F  
    if any( r>1 | r<0 ) whNRUOK:  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ;J\{r$q  
    end 8O{]ML  
    'D(Hqdr;:  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 7kn=j6I  
        error('zernfun:RTHvector','R and THETA must be vectors.') \Y9=d E}  
    end 9[N' HpQ3  
    SU# S'  
    r = r(:); @n(=#Q3  
    theta = theta(:); 1jmhh !,  
    length_r = length(r); [v-?MS  
    if length_r~=length(theta) IJ, ,aCj4g  
        error('zernfun:RTHlength', ... r"fu{4aX  
              'The number of R- and THETA-values must be equal.') MC#bo{Bq3-  
    end  1 ,PFz  
    mQ%kGqs  
    % Check normalization: (I.uQP~H  
    % -------------------- svpWABO  
    if nargin==5 && ischar(nflag) H@IX$+;z  
        isnorm = strcmpi(nflag,'norm'); n E-=7S L  
        if ~isnorm @=wAk5[IN  
            error('zernfun:normalization','Unrecognized normalization flag.') B_cn[?M  
        end ^e>v{AE%  
    else =< CH(4!  
        isnorm = false; ?5mVC]W?]  
    end =|3 L'cDC  
    QHs=Zh;"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N83RsL "}_  
    % Compute the Zernike Polynomials ]VJcV.7`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% '%RMpyK~  
    s* 9tWSd  
    % Determine the required powers of r: mA^>Y_:  
    % ----------------------------------- !W$Br\<  
    m_abs = abs(m); /HzhgMV3  
    rpowers = []; YSrFHVq  
    for j = 1:length(n) l}Xmm^@)  
        rpowers = [rpowers m_abs(j):2:n(j)]; `MTOe 1  
    end !y] Y'j  
    rpowers = unique(rpowers); Xkv>@7ec  
    1}jE?{V*  
    % Pre-compute the values of r raised to the required powers, X<9DE!/)  
    % and compile them in a matrix: I:6xDDpZG`  
    % ----------------------------- 6AQ;P  
    if rpowers(1)==0 g"Ii'JZ?  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); *R~oA`  
        rpowern = cat(2,rpowern{:}); CKBi-q FH  
        rpowern = [ones(length_r,1) rpowern]; oub4/0tN,~  
    else Y"l!3^   
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); It_yh #s  
        rpowern = cat(2,rpowern{:}); 8%xtb6#7M  
    end zV80r+y  
    V GvOwd)E  
    % Compute the values of the polynomials: ] lO$oO  
    % -------------------------------------- rz7yAm  
    y = zeros(length_r,length(n)); [\.>BK  
    for j = 1:length(n)  %ANPv=  
        s = 0:(n(j)-m_abs(j))/2; SiBbz4  
        pows = n(j):-2:m_abs(j); JnsXEkM)  
        for k = length(s):-1:1 15eHddd  
            p = (1-2*mod(s(k),2))* ... Mvcl9  
                       prod(2:(n(j)-s(k)))/              ... g<lX Xj2  
                       prod(2:s(k))/                     ... d?{2A84S  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &0C!P=-p  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); LA wS8t',  
            idx = (pows(k)==rpowers); qJ!oH&/cD  
            y(:,j) = y(:,j) + p*rpowern(:,idx); {(MG: B  
        end Y-{spTI  
         blPC"3}3Vd  
        if isnorm ud#8`/!mq  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); r=[}7N  
        end uBMNkN8  
    end 9E#(iP  
    % END: Compute the Zernike Polynomials QV 'y6m\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ./g#<  
    L%8"d6  
    % Compute the Zernike functions: w`v\/a_  
    % ------------------------------ Q?;ntzi  
    idx_pos = m>0; z"vgwOP su  
    idx_neg = m<0; <?7~,#AK  
    jXDo!a| 4y  
    z = y; K*}j1A  
    if any(idx_pos) vVf!XZF  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); V9bLm,DtT  
    end ^R$dG[Qf  
    if any(idx_neg) enr mjA&3  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); .R"L$V$RU.  
    end $.cGRz  
    3gh^a;uC  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) {ULnQ 6@  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 7L6M#B[)e5  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated # 0 (\s@r.  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Uwk|M?94  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, aKy|$ {RC  
    %   and THETA is a vector of angles.  R and THETA must have the same T{M:)}V  
    %   length.  The output Z is a matrix with one column for every P-value, /km3L7L%R  
    %   and one row for every (R,THETA) pair.  f#nmr5F  
    % X9j+$X \j  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike !;a<E:  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 5b'S~Qj#r$  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) m t^1[  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 Uf<vw3  
    %   for all p. *)1z-rH`  
    % iE`aGoA  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 "lZ<bG  
    %   Zernike functions (order N<=7).  In some disciplines it is n58jB:XR(  
    %   traditional to label the first 36 functions using a single mode yw<xv-Q=i  
    %   number P instead of separate numbers for the order N and azimuthal g < o;\\  
    %   frequency M. ~] 2R+  
    % ( -@>  
    %   Example: M[{:o/]<  
    % J5T#}!f  
    %       % Display the first 16 Zernike functions aB)DX  
    %       x = -1:0.01:1; A{%;Hd`0/  
    %       [X,Y] = meshgrid(x,x); >>D i  
    %       [theta,r] = cart2pol(X,Y); Fm':sd)'X  
    %       idx = r<=1; SI9hS4<j  
    %       p = 0:15; <k6xScy$}  
    %       z = nan(size(X)); bYc qscW  
    %       y = zernfun2(p,r(idx),theta(idx)); Se`N5hQ  
    %       figure('Units','normalized') z-G (!]:  
    %       for k = 1:length(p) R.T-Ptene  
    %           z(idx) = y(:,k); i#=X#_ +El  
    %           subplot(4,4,k) J.l%H U  
    %           pcolor(x,x,z), shading interp %5gJ6>@6Z  
    %           set(gca,'XTick',[],'YTick',[]) M(uB ;Te  
    %           axis square L#Y;a 5b  
    %           title(['Z_{' num2str(p(k)) '}']) 9(WC#-,  
    %       end |Ze}bM=N  
    % %#a%Luq  
    %   See also ZERNPOL, ZERNFUN. '=.Uz3D'0  
    NN'<-0~  
    %   Paul Fricker 11/13/2006 n #I}!x>2  
     &7&*As  
    z:5ROlk0  
    % Check and prepare the inputs: u_8 22Z  
    % ----------------------------- oZ[ w  
    if min(size(p))~=1 SGd.z6"H  
        error('zernfun2:Pvector','Input P must be vector.') A#: c  
    end *XO KH+_u  
    -RQQ|:O$  
    if any(p)>35 #U D  
        error('zernfun2:P36', ... ?/MXcI(  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... )d u{ZWr  
               '(P = 0 to 35).']) );DIrA  
    end U(4_X[qD  
    @|Bp'`j%J  
    % Get the order and frequency corresonding to the function number: FF~4y>R7u  
    % ---------------------------------------------------------------- m0\}Cc  
    p = p(:); {~g  
    n = ceil((-3+sqrt(9+8*p))/2); \#,#_  
    m = 2*p - n.*(n+2); {VG[m@  
    2z# @:Q  
    % Pass the inputs to the function ZERNFUN: L.[uMuUa  
    % ---------------------------------------- r.^X>?  
    switch nargin [#'_@zZz  
        case 3 )#~fS28j  
            z = zernfun(n,m,r,theta); d}cJ5 !d  
        case 4 no6]{qn=6  
            z = zernfun(n,m,r,theta,nflag); M~F2cX W  
        otherwise rxp9B>~  
            error('zernfun2:nargin','Incorrect number of inputs.') 'TsZuZW]  
    end WCTW#<izm  
    wg_CI,Kq  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) DLVs>?Y  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. T4Gw\Z%  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of y"L`bl A9}  
    %   order N and frequency M, evaluated at R.  N is a vector of OrJlHMz  
    %   positive integers (including 0), and M is a vector with the lT!$\E$1   
    %   same number of elements as N.  Each element k of M must be a 0QH3,Ps1C  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) )u/ ^aK53^  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is `Mp7 })  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix L{%a4 Ip  
    %   with one column for every (N,M) pair, and one row for every ,W8Iabi^  
    %   element in R. y{I[}$k  
    % _JIUds5  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ^*ez j1  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is fy>And*  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to nEcd+7(  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 jy@i(@Z  
    %   for all [n,m]. N z3%}6F:  
    % o``>sBZOq  
    %   The radial Zernike polynomials are the radial portion of the )^>XZ*eK  
    %   Zernike functions, which are an orthogonal basis on the unit !v4j`A;%  
    %   circle.  The series representation of the radial Zernike AD%D ,l  
    %   polynomials is  {%~4RZA  
    % #Q7x:,f  
    %          (n-m)/2 OPt;G,$ta  
    %            __ agU!D[M_G  
    %    m      \       s                                          n-2s u#@{%kPW  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r hd ;S>K/C  
    %    n      s=0 .A!0.M|  
    % $gl<{{  
    %   The following table shows the first 12 polynomials. O:=|b]t  
    % |}p}`Mb)a  
    %       n    m    Zernike polynomial    Normalization +\8krA  
    %       --------------------------------------------- ._MAHBx+G  
    %       0    0    1                        sqrt(2) , 64t  
    %       1    1    r                           2 ;, v L  
    %       2    0    2*r^2 - 1                sqrt(6) x gT~b9  
    %       2    2    r^2                      sqrt(6) 27 145  
    %       3    1    3*r^3 - 2*r              sqrt(8) zPh\3B  
    %       3    3    r^3                      sqrt(8) {+ 6D-rDw  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) mV*/zWh_  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) :{ WrS  
    %       4    4    r^4                      sqrt(10) LQ(5D_yG.  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) -6~y$c&c  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) " $IXZ  
    %       5    5    r^5                      sqrt(12) =Wk/q_.  
    %       --------------------------------------------- #W%)$k c  
    % 6;[/ 9  
    %   Example: N>pmhskN?  
    % %V(N U_o  
    %       % Display three example Zernike radial polynomials u|OzW}xb7j  
    %       r = 0:0.01:1; z(` }:t  
    %       n = [3 2 5]; D_n}p8blT  
    %       m = [1 2 1]; \9(- /rE  
    %       z = zernpol(n,m,r); b~r{J5x@  
    %       figure p;u 1{  
    %       plot(r,z) xBRh !w  
    %       grid on 'LbeL1ca  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') GKyG #Fl  
    % 4OOn,09  
    %   See also ZERNFUN, ZERNFUN2. Q-8'?S  
    t.E4Tqzc>  
    % A note on the algorithm. B?#kW!wj  
    % ------------------------ 9XoQO9*Q  
    % The radial Zernike polynomials are computed using the series S'!q}|7X 3  
    % representation shown in the Help section above. For many special &`yOIX-H_  
    % functions, direct evaluation using the series representation can GT'7,+<?N  
    % produce poor numerical results (floating point errors), because 45+%K@@x  
    % the summation often involves computing small differences between hY=w|b=Y  
    % large successive terms in the series. (In such cases, the functions F?8BS*r_  
    % are often evaluated using alternative methods such as recurrence _}e7L7B7g  
    % relations: see the Legendre functions, for example). For the Zernike BU9J_rCIv  
    % polynomials, however, this problem does not arise, because the ) Ab6!"'  
    % polynomials are evaluated over the finite domain r = (0,1), and cZgMA8 F  
    % because the coefficients for a given polynomial are generally all 2sqm7th  
    % of similar magnitude. Y8%0;!T  
    % i>Fvmw  
    % ZERNPOL has been written using a vectorized implementation: multiple Lvi[*une|  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] gE@$~Q>M  
    % values can be passed as inputs) for a vector of points R.  To achieve /BMtcCPG!  
    % this vectorization most efficiently, the algorithm in ZERNPOL Ts(t:^  
    % involves pre-determining all the powers p of R that are required to }U%^3r-  
    % compute the outputs, and then compiling the {R^p} into a single y7JZKtsFA  
    % matrix.  This avoids any redundant computation of the R^p, and `k(u:yGK  
    % minimizes the sizes of certain intermediate variables. 3a#PA4Ql  
    % Sk/#J!T8{  
    %   Paul Fricker 11/13/2006 \T`["<  
    U!c]_q  
    g@@&sB-A"  
    % Check and prepare the inputs: F<Hqo>G  
    % ----------------------------- WHUT/:?f  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 0%s3Mp6H  
        error('zernpol:NMvectors','N and M must be vectors.') IV{FH&t^T"  
    end P3iA(3I24<  
    2yln7[a  
    if length(n)~=length(m) %M{k.FE(  
        error('zernpol:NMlength','N and M must be the same length.') ~n[b^b  
    end *O@sh  
    A3AP51 !  
    n = n(:); v@8S5KJ  
    m = m(:); B(j02<-  
    length_n = length(n); #>8T*B  
    {~"7vkc+  
    if any(mod(n-m,2)) tu\mFHvlg  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') iOT)0@f'  
    end r^$\t0h(U8  
    [kbC'Eh*  
    if any(m<0) D@8jGcz62  
        error('zernpol:Mpositive','All M must be positive.') VpkD'<G  
    end Y4mC_4EU  
    \\ jIl3Z  
    if any(m>n) 6*ZU}xT  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') Fr-[UZ~V  
    end U~aWG\h#X  
    [tUv*jw%  
    if any( r>1 | r<0 ) -$U@By<SJ  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ]o-Fi$h!  
    end wms1IV%;  
    6W5d7`A  
    if ~any(size(r)==1) U1l0Uke  
        error('zernpol:Rvector','R must be a vector.') I-xwJi9?,  
    end cDCJ]iDs  
     ]}Pl%.  
    r = r(:); $`|5/,M%QN  
    length_r = length(r); z@zD .  
    ~}BJ0P(VMc  
    if nargin==4 }wG,BB%N  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); Aj,]n>{  
        if ~isnorm (]2<?x*  
            error('zernpol:normalization','Unrecognized normalization flag.') tqK=\{U  
        end |K%}}g[<e;  
    else 1e/L\Y=m  
        isnorm = false; TMK'(6dH  
    end Vu}806kB  
    qgtn5] A  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PdT83vOCE  
    % Compute the Zernike Polynomials @0$}? 2  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% rJu[ N(2k  
    C1d 04Q  
    % Determine the required powers of r: jZRhKT  
    % ----------------------------------- *vYn_wE  
    rpowers = []; 8Jr1_a  
    for j = 1:length(n) ~;[&K%n  
        rpowers = [rpowers m(j):2:n(j)]; 8x9Rm  
    end "3SWO3-x  
    rpowers = unique(rpowers); YgEM:'1f  
    jo)6 %w]  
    % Pre-compute the values of r raised to the required powers, \^LWCp,C"  
    % and compile them in a matrix: tw=K&/@^O  
    % ----------------------------- y_*n9 )Ct  
    if rpowers(1)==0 !i^]UN   
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); >8|+%pK8<  
        rpowern = cat(2,rpowern{:}); 402x<H  
        rpowern = [ones(length_r,1) rpowern]; jeq:  
    else (3VGaUlx  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); .]P2}w)x?  
        rpowern = cat(2,rpowern{:}); g5y;?fqJ  
    end e2c1pgs&+  
    tjj^O%SV<  
    % Compute the values of the polynomials: #|Oj]bd(=  
    % -------------------------------------- g`n;R  
    z = zeros(length_r,length_n); ,^ 7 CP  
    for j = 1:length_n [SkKz>rC  
        s = 0:(n(j)-m(j))/2; sK&,):"]R  
        pows = n(j):-2:m(j); yyP'Z~0  
        for k = length(s):-1:1 Rn-G @}f  
            p = (1-2*mod(s(k),2))* ... @u/H8\.l  
                       prod(2:(n(j)-s(k)))/          ... M;KA]fmc  
                       prod(2:s(k))/                 ... 9${Xer'  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 3;-^YG  
                       prod(2:((n(j)+m(j))/2-s(k))); 78 d_io}w  
            idx = (pows(k)==rpowers); V@%  
            z(:,j) = z(:,j) + p*rpowern(:,idx); h*3{6X#(/  
        end ;#&fgj  
         4PWAGuN^  
        if isnorm QL97WK\$  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); MS*G-C  
        end ` H XEZ|  
    end Ly7!R$X  
    K"\MU  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  [U@; \V$  
    ``VW;l{  
    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。 /4\wn?f  
    +HT1ct+dI  
    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)