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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 +V0uH pm  
    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! ",K6zALJ  
     
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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 sbG3,'i)  
    function z = zernfun(n,m,r,theta,nflag) Dzp9BRS 2f  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ?6a:!^eL  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N sKNN ahGjh  
    %   and angular frequency M, evaluated at positions (R,THETA) on the !,I}2,1%k  
    %   unit circle.  N is a vector of positive integers (including 0), and =>ignoeI  
    %   M is a vector with the same number of elements as N.  Each element *}LYMrP  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 7Xw #  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, \N!k)6\  
    %   and THETA is a vector of angles.  R and THETA must have the same &"25a[x{B  
    %   length.  The output Z is a matrix with one column for every (N,M) j'v2m6/  
    %   pair, and one row for every (R,THETA) pair. *)"`v]  
    % )<!y_;$A  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike |>d5 6  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ) |*HkdF`  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral l0]zZcpt  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, (?$}Vp  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ;i\i+:=  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. )=2iGEVW  
    % >/-<,,<\C  
    %   The Zernike functions are an orthogonal basis on the unit circle. P1)9OE  
    %   They are used in disciplines such as astronomy, optics, and #knpZ'  
    %   optometry to describe functions on a circular domain. r"k\G\,%  
    % eB5; wH  
    %   The following table lists the first 15 Zernike functions. mKn:EqA  
    % 0f1*#8-6  
    %       n    m    Zernike function           Normalization N^:)U"9*e  
    %       -------------------------------------------------- ECQ>VeP  
    %       0    0    1                                 1 Z^s&]  
    %       1    1    r * cos(theta)                    2 sJMT _yt;  
    %       1   -1    r * sin(theta)                    2 Fvl_5l  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) > u~ l_?  
    %       2    0    (2*r^2 - 1)                    sqrt(3) tP7l ;EX4  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 0~)cAKus  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Nx,.4CI  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) "1WwSh}Z  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) c]#F^(-A`  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) e")s1`  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) sBB>O@4  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6 [w_ /X"  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) <mi*AY  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) vm 1vX;  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 6f{Kj)  
    %       -------------------------------------------------- eG=Hyc  
    % w%KU@$  
    %   Example 1: Z;-=xp  
    % FK{Vnj0  
    %       % Display the Zernike function Z(n=5,m=1) %?@N-$j  
    %       x = -1:0.01:1; <"X\~  
    %       [X,Y] = meshgrid(x,x); Q6]SsV?x  
    %       [theta,r] = cart2pol(X,Y); w<*6pP y  
    %       idx = r<=1; T}M!A|   
    %       z = nan(size(X)); ^ yyL4{/  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); qwoF4_VN  
    %       figure s<h]2W  
    %       pcolor(x,x,z), shading interp STtjkZ6  
    %       axis square, colorbar  MV'q_{J  
    %       title('Zernike function Z_5^1(r,\theta)') D!^&*Ia?2  
    % Rm>AU=  
    %   Example 2: F^fL  
    % $oDc  
    %       % Display the first 10 Zernike functions Hyh$-iCa  
    %       x = -1:0.01:1; XOe)tz L  
    %       [X,Y] = meshgrid(x,x); Nb(c;|nV  
    %       [theta,r] = cart2pol(X,Y); A]c'`Nf  
    %       idx = r<=1; wxS.!9K  
    %       z = nan(size(X)); }%x2Z{VF  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 5%Hw,h   
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 14Y_ oH9  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; KP,#x$Bg  
    %       y = zernfun(n,m,r(idx),theta(idx)); DP_ ]\V<sT  
    %       figure('Units','normalized') Z8I  Y!d  
    %       for k = 1:10 # 3UrGom  
    %           z(idx) = y(:,k); Dc-v`jZ@)  
    %           subplot(4,7,Nplot(k)) KW`^uoY$  
    %           pcolor(x,x,z), shading interp @{n"/6t  
    %           set(gca,'XTick',[],'YTick',[]) e98f+,E/  
    %           axis square 4AW-'W  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) gc,%A'OR^<  
    %       end Kg?(Ax4  
    % v'=$K[_  
    %   See also ZERNPOL, ZERNFUN2. vLCyT=OB`  
    {8p<iY- %  
    %   Paul Fricker 11/13/2006 )09>#!*  
    uW;[FTcqy$  
    UZvF5Hoe+O  
    % Check and prepare the inputs: eO%w i.Q  
    % ----------------------------- @:s (L]  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) = j)5kY`  
        error('zernfun:NMvectors','N and M must be vectors.') (`'(`x#  
    end _?~EWT   
    ^`iqa-1  
    if length(n)~=length(m) &l M=>?  
        error('zernfun:NMlength','N and M must be the same length.') kZ5;Fe\*  
    end KJ (|skO  
    Y.yiUf/Q  
    n = n(:); }IJE%  
    m = m(:); D`c&Q4$:  
    if any(mod(n-m,2))  T&'p5h=l  
        error('zernfun:NMmultiplesof2', ... $Iz*W]B!  
              'All N and M must differ by multiples of 2 (including 0).') 7up~8e$_  
    end )>"|<h.2]  
    12]rfd   
    if any(m>n) kLE("I:7  
        error('zernfun:MlessthanN', ... "~2SHM@q  
              'Each M must be less than or equal to its corresponding N.') | -l9Z  
    end e92,@  
    W79Sz}):  
    if any( r>1 | r<0 ) t4d^DZDh!  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') F% < ZEVm  
    end "50 c<sZSB  
    2p %j@O  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) h[ cqa  
        error('zernfun:RTHvector','R and THETA must be vectors.') ~v>3lEGn*  
    end D/)E[Fv+  
    um,G^R   
    r = r(:); tNvjwgV\  
    theta = theta(:); >BWe"{;  
    length_r = length(r); b9R0"w!ml  
    if length_r~=length(theta) joA>-k04  
        error('zernfun:RTHlength', ... x1`4hB  
              'The number of R- and THETA-values must be equal.') e+~@"^|  
    end 4|/}~9/  
    J.(mg D  
    % Check normalization: LK|1[y^h  
    % -------------------- )k[{re  
    if nargin==5 && ischar(nflag) kxCN0e#_  
        isnorm = strcmpi(nflag,'norm'); fnJx$PD~  
        if ~isnorm Ak kth*p  
            error('zernfun:normalization','Unrecognized normalization flag.') {%Rntb  
        end g ySl.cxt  
    else l@:&0id4I  
        isnorm = false; laRn![[  
    end V}h <,E9  
    \_)[FC@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Nt,:`o |  
    % Compute the Zernike Polynomials \MDhm,H<  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  CH$K_\  
    9 lXnNK |]  
    % Determine the required powers of r: SuA  @S  
    % ----------------------------------- S&F[\4w5]  
    m_abs = abs(m); Y41b8.|P+  
    rpowers = []; /$d #9Uv  
    for j = 1:length(n) 9 K>~9Za  
        rpowers = [rpowers m_abs(j):2:n(j)]; Nd He::  
    end cTja<*W^xv  
    rpowers = unique(rpowers); l0r^LK$  
    2)Q%lEm`SP  
    % Pre-compute the values of r raised to the required powers,  eIj2(q9  
    % and compile them in a matrix: ]tNB^  
    % ----------------------------- KK?R|1VK9  
    if rpowers(1)==0 5mX"0a_Q  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); _"t"orD6  
        rpowern = cat(2,rpowern{:}); "^)$MAZ  
        rpowern = [ones(length_r,1) rpowern]; D}rnp wp{  
    else W0S\g#  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); +3 J5j+  
        rpowern = cat(2,rpowern{:}); 8dh ?JqX  
    end Am<){&XT ]  
    iU|X/>k?  
    % Compute the values of the polynomials: p^C$(}Yh  
    % -------------------------------------- yu jv^2/  
    y = zeros(length_r,length(n)); MKh}2B#S  
    for j = 1:length(n) by$S#e f  
        s = 0:(n(j)-m_abs(j))/2; TU1W!=Z  
        pows = n(j):-2:m_abs(j); Tdxc%'l  
        for k = length(s):-1:1 N97WI+`  
            p = (1-2*mod(s(k),2))* ... Bxf&gDwjgr  
                       prod(2:(n(j)-s(k)))/              ... RgD:"zeM  
                       prod(2:s(k))/                     ... *|,ye5"  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... WtlLqD!_D  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); sWq@E6,I  
            idx = (pows(k)==rpowers); yPf,GB"  
            y(:,j) = y(:,j) + p*rpowern(:,idx); m0*_  
        end O{Z bpa^  
         _=K\E0I.m  
        if isnorm Hv*+HUc(:  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); &r!jjT  
        end ?s]?2>p  
    end $e%m=@ga  
    % END: Compute the Zernike Polynomials >&JS-j Fg  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M y!;N1  
    t; @T~%  
    % Compute the Zernike functions: ' Uo|@tK  
    % ------------------------------ [M?&JA_$}  
    idx_pos = m>0; l M a||  
    idx_neg = m<0; pYj}  
    NkxW*w%}l  
    z = y; 2 U3WH.o  
    if any(idx_pos) #;\tgUQ  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); SpM Hq_MLM  
    end 0BN=>]V~j7  
    if any(idx_neg) >Ft:&N9L{  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); B7n1'?  
    end <%"CQT6g %  
    qJK6S4O]  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) 9U!#Y%*T  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. F"a31`L>H  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ~GjM:*  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 9]|G-cyt  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 2w:cdAv$  
    %   and THETA is a vector of angles.  R and THETA must have the same ETaLE[T%1  
    %   length.  The output Z is a matrix with one column for every P-value, #.<(/D+  
    %   and one row for every (R,THETA) pair. %loe8yt  
    % Gl5W4gW;&  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 88+J(^y>  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) (L"G,l  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) sX]ru^F3  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 Q".AmHn  
    %   for all p. Mh5 =]O+  
    % rY(^6[!  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ,IG?(CK|  
    %   Zernike functions (order N<=7).  In some disciplines it is ^/jALA9!  
    %   traditional to label the first 36 functions using a single mode K[i|OZWu  
    %   number P instead of separate numbers for the order N and azimuthal Z"a]AsG/Q#  
    %   frequency M. fg8U* 7  
    % x2z%J,z@4  
    %   Example: k&3'[&$I*,  
    % Sv03="&  
    %       % Display the first 16 Zernike functions G8F43!<  
    %       x = -1:0.01:1; +FYhDB~m  
    %       [X,Y] = meshgrid(x,x); }t.VH:02y  
    %       [theta,r] = cart2pol(X,Y); #zw 'H9l  
    %       idx = r<=1; }aa ~@K<A  
    %       p = 0:15; #'Lt_Yf!  
    %       z = nan(size(X)); +X7+:QQ }  
    %       y = zernfun2(p,r(idx),theta(idx)); IF.6sJg:  
    %       figure('Units','normalized') GA@Zfcg  
    %       for k = 1:length(p) ahm@ +/2  
    %           z(idx) = y(:,k); wQ/FJoB  
    %           subplot(4,4,k) /(skIvE|  
    %           pcolor(x,x,z), shading interp D[R<H((  
    %           set(gca,'XTick',[],'YTick',[]) 1R"ymWg"  
    %           axis square  IQCIc@5  
    %           title(['Z_{' num2str(p(k)) '}']) .+hM1OF`x  
    %       end Y7 `i~K;  
    % U)gr C8 C  
    %   See also ZERNPOL, ZERNFUN. j%OnLTZ  
    I4^}C;p0?  
    %   Paul Fricker 11/13/2006 6GtXM3qtS  
    C!aK5rqhv  
    9% AL f 9  
    % Check and prepare the inputs: \%N | X  
    % ----------------------------- 3re|=_ Hy  
    if min(size(p))~=1 5\$8"/H  
        error('zernfun2:Pvector','Input P must be vector.') o%\pI%  
    end j{u! /FD  
     mH*6Q>  
    if any(p)>35 VC@{cVT  
        error('zernfun2:P36', ... {9C+=v?  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ['rqz1DL5  
               '(P = 0 to 35).']) f,}]h~w\  
    end FGG 7;0(  
    y!?l;xMS  
    % Get the order and frequency corresonding to the function number: E>3fk  
    % ---------------------------------------------------------------- 1f^4J~{  
    p = p(:); 53cW`F  
    n = ceil((-3+sqrt(9+8*p))/2);  R)?zL;,x  
    m = 2*p - n.*(n+2);  ?[G!6  
    )R`w{V  
    % Pass the inputs to the function ZERNFUN: *PjW,   
    % ---------------------------------------- kM T73OI>_  
    switch nargin $!_]mz6*  
        case 3 px1{=~V/  
            z = zernfun(n,m,r,theta); ;/8oP ;X2  
        case 4 r&t)%R@q  
            z = zernfun(n,m,r,theta,nflag); <H)I06];  
        otherwise @fWmz,Ngl  
            error('zernfun2:nargin','Incorrect number of inputs.') dT9!gNvQ  
    end ?E?dg#yk  
    9ZEF%&58Y  
    % 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) <|s9@;(I  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. pA*cF!tq 7  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of bX:ARe O  
    %   order N and frequency M, evaluated at R.  N is a vector of DM*mOT  
    %   positive integers (including 0), and M is a vector with the 6uYCU|JsU  
    %   same number of elements as N.  Each element k of M must be a Zq--m/  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) &Oq& ikw  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is T/" 6iv\1  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix -VESe}c:nQ  
    %   with one column for every (N,M) pair, and one row for every }7Si2S  
    %   element in R. wPDA_ns~  
    % '[fo  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- aD~3C/?aW  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is L!7*U.+  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to uhFj|r$$  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 T3o}%wGW  
    %   for all [n,m]. eyZ /%4'q  
    % #L{QnV.3  
    %   The radial Zernike polynomials are the radial portion of the `":ch9rK  
    %   Zernike functions, which are an orthogonal basis on the unit @' DfNka  
    %   circle.  The series representation of the radial Zernike {ZH9W  
    %   polynomials is )POuH*j  
    % k=<,A'y-/  
    %          (n-m)/2 0)V<)"i  
    %            __ J(0.eD91v  
    %    m      \       s                                          n-2s xD;5z`A3  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r {|%N  
    %    n      s=0 ?L$ Dk5-W  
    % Vc3tKuMsiX  
    %   The following table shows the first 12 polynomials. ~O~c^fLH(B  
    % 2B7X~t>8a  
    %       n    m    Zernike polynomial    Normalization Z@=1-l  
    %       --------------------------------------------- }!\ZJoa  
    %       0    0    1                        sqrt(2) cjU*  
    %       1    1    r                           2 =Uta5$\a)  
    %       2    0    2*r^2 - 1                sqrt(6) hbhh m  
    %       2    2    r^2                      sqrt(6) 8? 4j-  
    %       3    1    3*r^3 - 2*r              sqrt(8) K+D`U6&  
    %       3    3    r^3                      sqrt(8) pq6}q($Rk  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 8oG0tX3i  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) +Eh1>m  
    %       4    4    r^4                      sqrt(10) , e{kC  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 2l#Ogn`k  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) SUXRWFl  
    %       5    5    r^5                      sqrt(12) "_JGe#=  
    %       --------------------------------------------- FW:x XK  
    % N.C<Mo  
    %   Example: .N8AkQ(Ok  
    % "w0>  
    %       % Display three example Zernike radial polynomials bR@ e6.<i  
    %       r = 0:0.01:1; `'[u%UE  
    %       n = [3 2 5]; S*6P=O*  
    %       m = [1 2 1]; _|xO4{X  
    %       z = zernpol(n,m,r); SIjdwr!+ZZ  
    %       figure yc2c{<Ya5  
    %       plot(r,z) A1:<-TF6^p  
    %       grid on D0tmNV@  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ;BqYhi  
    % OS6 l*S('  
    %   See also ZERNFUN, ZERNFUN2. V<AT"vU[  
    ua*k{0[  
    % A note on the algorithm. JS r& S[  
    % ------------------------ A^_BK(EY  
    % The radial Zernike polynomials are computed using the series kJqgY|  
    % representation shown in the Help section above. For many special u-3A6Q  
    % functions, direct evaluation using the series representation can rIg1]q  
    % produce poor numerical results (floating point errors), because ]'=]=o~4  
    % the summation often involves computing small differences between tK6z#)  
    % large successive terms in the series. (In such cases, the functions s<[%7 6Y!  
    % are often evaluated using alternative methods such as recurrence ozF>2`K }  
    % relations: see the Legendre functions, for example). For the Zernike S,Boutd  
    % polynomials, however, this problem does not arise, because the 7jT]J   
    % polynomials are evaluated over the finite domain r = (0,1), and N;7Xt9l  
    % because the coefficients for a given polynomial are generally all zlZ$t{[,  
    % of similar magnitude. Rz1&(_Ps  
    % wQ qI@  
    % ZERNPOL has been written using a vectorized implementation: multiple 7y:%^sl  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ~U#afGH$  
    % values can be passed as inputs) for a vector of points R.  To achieve *{8K b>D  
    % this vectorization most efficiently, the algorithm in ZERNPOL QWv+J a  
    % involves pre-determining all the powers p of R that are required to bB'iK4  
    % compute the outputs, and then compiling the {R^p} into a single @FKNB.>  
    % matrix.  This avoids any redundant computation of the R^p, and %geiJ z  
    % minimizes the sizes of certain intermediate variables. ~;k-/Z"  
    % NARW3\  
    %   Paul Fricker 11/13/2006 zE5%l`@|o  
    W/9dT^1y4'  
    a:Js i=  
    % Check and prepare the inputs: 4G0Er?D   
    % ----------------------------- "5u*C#T2$  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) .w\AyXp  
        error('zernpol:NMvectors','N and M must be vectors.') e5_a.c  
    end ~~k_A|&  
    6Y0k}+j|>E  
    if length(n)~=length(m) {^2``NYM_  
        error('zernpol:NMlength','N and M must be the same length.') PXG)?`^NX  
    end nIJ2*QJ  
    *,X;4?:,  
    n = n(:); yB\}e'J^  
    m = m(:); Tz3 L#0:j  
    length_n = length(n); 0gw0  
    :@{(^}N8u  
    if any(mod(n-m,2))  fy" q  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') DZ -5A  
    end FuiG=quY  
    Cqy84!Z<  
    if any(m<0) % 1ZJi}~  
        error('zernpol:Mpositive','All M must be positive.') U|. kAI*  
    end 1@sy:{ d`  
    Y3+DTR0|'  
    if any(m>n) =mxG[zDtQ  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') yEIM58l  
    end TZ{';oU  
    e/JbRbZX  
    if any( r>1 | r<0 ) ;QkUW<(  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') , f{<  
    end + X(@o  
    :["iBrFp  
    if ~any(size(r)==1) H)z}6[`  
        error('zernpol:Rvector','R must be a vector.') ]W39HL  
    end HZC^Q7]hy  
    p A7&  
    r = r(:); z ^t6VFM  
    length_r = length(r); U8Z(=*Z3  
    N|-M|1w96  
    if nargin==4 ekC 1wN l  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 8&<C.n KP  
        if ~isnorm I WKq_Zjkz  
            error('zernpol:normalization','Unrecognized normalization flag.') U Hh  
        end r:0F("},  
    else /ap3>xkt  
        isnorm = false; a)w *  
    end 5<ZE.'O  
    x9Um4!/t  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zLEl/yPE  
    % Compute the Zernike Polynomials ;Gixu9u'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jez=q  
    TLbnG$VQS  
    % Determine the required powers of r: oS3'q\  
    % ----------------------------------- MJ"ug8 N  
    rpowers = []; K",YAfJa  
    for j = 1:length(n) "IQ' (^-P  
        rpowers = [rpowers m(j):2:n(j)]; k|V%*BvY>  
    end e>z   
    rpowers = unique(rpowers); (%.[MilxPM  
    nk=+6r6  
    % Pre-compute the values of r raised to the required powers, MYNNeO  
    % and compile them in a matrix: -Uj)6PzGu  
    % ----------------------------- c"HB7  
    if rpowers(1)==0 8Ld{Xg  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); S&(MR%".  
        rpowern = cat(2,rpowern{:}); F[am2[/<A  
        rpowern = [ones(length_r,1) rpowern]; @GTkS!86  
    else C:z+8wt  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); wJc~AP)I%z  
        rpowern = cat(2,rpowern{:}); 6L5j  
    end 2tPW1"M.n  
    EHI %QT  
    % Compute the values of the polynomials: ,',  S  
    % -------------------------------------- O_=2{k~s0  
    z = zeros(length_r,length_n); y,&'nk}  
    for j = 1:length_n DzZEn]+zt  
        s = 0:(n(j)-m(j))/2; uBpnfIe  
        pows = n(j):-2:m(j); DpD19)ouy  
        for k = length(s):-1:1 Hwb+@'o  
            p = (1-2*mod(s(k),2))* ... 80J87\)  
                       prod(2:(n(j)-s(k)))/          ... 7Yv1et |  
                       prod(2:s(k))/                 ... YkWv*l  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... T0HNld  
                       prod(2:((n(j)+m(j))/2-s(k))); Oly"ll*K  
            idx = (pows(k)==rpowers); 287g 5  
            z(:,j) = z(:,j) + p*rpowern(:,idx);  9t$#!2z  
        end aMv?D(Meb  
         yV"k:_O{  
        if isnorm sr S2v\1:  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); <'T:9  
        end b"4'*<=au  
    end sF. oZ>  
    69q#Zw[,,  
    % 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)  qg2Vmj<H  
    W2&(:C8V@  
    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。 u}.mJDL  
    ?IG[W+M8  
    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)