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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 tuY= )?  
    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! 1 >2 /1>  
     
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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 6J$I8b#/  
    function z = zernfun(n,m,r,theta,nflag) P /q] u  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ]<Q&  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N XSh [#qJ  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ;W\?lGOs{  
    %   unit circle.  N is a vector of positive integers (including 0), and !g#y$  
    %   M is a vector with the same number of elements as N.  Each element ;!3: 3;  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) =xSf-\F  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Wk!<P" nHd  
    %   and THETA is a vector of angles.  R and THETA must have the same V <ilv<  
    %   length.  The output Z is a matrix with one column for every (N,M) zq3f@xOK  
    %   pair, and one row for every (R,THETA) pair. lJx5scN [  
    % EV|W:;Sg  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Ufor>  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^B7Ls{  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral w:R#F( 'B  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, )?6%d  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ~HKzqGQy >  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. I"KosSs  
    % s<3M_mt  
    %   The Zernike functions are an orthogonal basis on the unit circle. O+=}x]q*y  
    %   They are used in disciplines such as astronomy, optics, and Y'+K U/H  
    %   optometry to describe functions on a circular domain. `/B+  
    % -q? ,  
    %   The following table lists the first 15 Zernike functions. HTm`_}G9  
    % |U$ "GI  
    %       n    m    Zernike function           Normalization |PGTP#O<  
    %       -------------------------------------------------- 2gEF$?+q?  
    %       0    0    1                                 1 Tv~Ho&LS  
    %       1    1    r * cos(theta)                    2 dqFp"Xe"%  
    %       1   -1    r * sin(theta)                    2 )gAqWbkB  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) \,lIPA/L  
    %       2    0    (2*r^2 - 1)                    sqrt(3) K\mFb  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) q:vGGK^  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 4|4[3Ye7u:  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 4.~<|T8  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) PO:sF]5  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) N]\)Ok  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) LE?sAN  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) U % ?+N  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) )/2TU]//  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4jjo%N  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Eb5BJ-XeS^  
    %       -------------------------------------------------- ?t/\ ID  
    % >Dz8+y  
    %   Example 1: 15Jc PDV  
    % s E;2;2u"  
    %       % Display the Zernike function Z(n=5,m=1) X[SIk%{D  
    %       x = -1:0.01:1; -e0?1.A$  
    %       [X,Y] = meshgrid(x,x); l701$>>  
    %       [theta,r] = cart2pol(X,Y); (io[O?te  
    %       idx = r<=1; x]4>f[>*>  
    %       z = nan(size(X)); u Qg$hS  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); BE54L+$p  
    %       figure OgHqF,0MN  
    %       pcolor(x,x,z), shading interp g*w}m>O  
    %       axis square, colorbar VAe[x `  
    %       title('Zernike function Z_5^1(r,\theta)') jc,Q g2  
    % E;q+u[$  
    %   Example 2: q &S@\b  
    % pkTVQdtRG  
    %       % Display the first 10 Zernike functions d vo|9 >  
    %       x = -1:0.01:1; ^E~1%Md.  
    %       [X,Y] = meshgrid(x,x); 7c6- o"A  
    %       [theta,r] = cart2pol(X,Y); ^)aj, U[  
    %       idx = r<=1; a=6@} l1<  
    %       z = nan(size(X)); b7gN|Hw5 H  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 4i<GqG  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; $ P2*qpqy  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; $-s8tc(  
    %       y = zernfun(n,m,r(idx),theta(idx)); NiRb:F-  
    %       figure('Units','normalized') c}H}fyu%n  
    %       for k = 1:10 +k/=L9#e  
    %           z(idx) = y(:,k); r >sXvzv  
    %           subplot(4,7,Nplot(k)) JEP9!y9y  
    %           pcolor(x,x,z), shading interp [lu+"V,<LJ  
    %           set(gca,'XTick',[],'YTick',[]) w?Cho</Xu  
    %           axis square *Y!RU{w+Z  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ,4;'s  
    %       end ~3%aEj  
    % Y)#,6\=U  
    %   See also ZERNPOL, ZERNFUN2. Q:'r p  
    S@TfZ3Go|  
    %   Paul Fricker 11/13/2006 A-rj: k!  
    0sCWIGU W  
    $FZcvo3@*S  
    % Check and prepare the inputs: CdtCxy5  
    % ----------------------------- aN!,\D  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) NSq29#  
        error('zernfun:NMvectors','N and M must be vectors.') lwjA07 i  
    end 9hJ a K  
    =F5zU5`i  
    if length(n)~=length(m) /_yAd,^-+  
        error('zernfun:NMlength','N and M must be the same length.') ,|j\x  
    end S,a:H*Hf  
    EiyHZ  
    n = n(:); Z>dvth  
    m = m(:); \XfLTv  
    if any(mod(n-m,2)) D z[ ,;  
        error('zernfun:NMmultiplesof2', ... *qxv"PptX  
              'All N and M must differ by multiples of 2 (including 0).') ]LMtZUz  
    end >X5RRSo  
    S>Gb Jt(]  
    if any(m>n) zz8NBO  
        error('zernfun:MlessthanN', ... u(PUbxJ V  
              'Each M must be less than or equal to its corresponding N.') WmRu3O  
    end 1)f <  
     &'?Hh(  
    if any( r>1 | r<0 ) M'T[L%AP  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 42:,*4t(  
    end =Wz)(N  
    #RKd >ig%  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) e2pFX?  
        error('zernfun:RTHvector','R and THETA must be vectors.') Digx#'#jf  
    end 3FMYs&0r4  
    =Ew77  
    r = r(:); +WguWLO"  
    theta = theta(:); E `V?Io  
    length_r = length(r); aY DM)b}  
    if length_r~=length(theta) H|'n|\{lt  
        error('zernfun:RTHlength', ... N(O* "1b  
              'The number of R- and THETA-values must be equal.') ^+kymZ  
    end omT^jh  
    c_aj-`BKp  
    % Check normalization:  sHOBT,B  
    % -------------------- UMHFq-  
    if nargin==5 && ischar(nflag) _T;Kn'Gz(&  
        isnorm = strcmpi(nflag,'norm'); DU-dIq i  
        if ~isnorm +,)Iv_Xl$  
            error('zernfun:normalization','Unrecognized normalization flag.') D4?cnwU  
        end K 28s<i`  
    else 6zGeGW  
        isnorm = false; Ql,WKoj*  
    end *q@3yB}  
    OU*skc>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% w BoP&l  
    % Compute the Zernike Polynomials 6.a|w}C`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :w7?]y6~S  
    7dOpJjv?)  
    % Determine the required powers of r: we kb&?  
    % ----------------------------------- fVi[mH0=+  
    m_abs = abs(m); n- 1  
    rpowers = []; ViUx^e\  
    for j = 1:length(n) c2]h.G83  
        rpowers = [rpowers m_abs(j):2:n(j)]; M[e^Z}w.V  
    end W'e{2u  
    rpowers = unique(rpowers); hW\'EJ  
    7 4hRG~  
    % Pre-compute the values of r raised to the required powers, cb/$P!j7  
    % and compile them in a matrix: vorb?iVf>  
    % ----------------------------- Dw,LB>Eq,  
    if rpowers(1)==0 dki3(  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); kZfj"+p_S  
        rpowern = cat(2,rpowern{:}); f{|n/j;n=C  
        rpowern = [ones(length_r,1) rpowern]; pezfB{x?  
    else t&IWKu#  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); l vBcEg  
        rpowern = cat(2,rpowern{:}); ?q y*`  
    end _<?z-K_;I  
    /sqfw,h@  
    % Compute the values of the polynomials: K1o&(;l8G  
    % -------------------------------------- xFA`sAucr  
    y = zeros(length_r,length(n)); fe}RmnAC  
    for j = 1:length(n) kc2 8Q2  
        s = 0:(n(j)-m_abs(j))/2; ; NO#/  
        pows = n(j):-2:m_abs(j); rAD4}A_w  
        for k = length(s):-1:1 Yfy";C7X  
            p = (1-2*mod(s(k),2))* ... Ij9=J1c4  
                       prod(2:(n(j)-s(k)))/              ... FR\r/+n:t0  
                       prod(2:s(k))/                     ... @[Wf!8_  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... c57`mOe/b  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); %Siw>  
            idx = (pows(k)==rpowers); <Rz[G+0S=  
            y(:,j) = y(:,j) + p*rpowern(:,idx); X @7:FzU9  
        end @scSW5+  
         Q_*.1L  
        if isnorm _Ecs{'k  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); _6]tbni?v  
        end ZR8y9mx2"  
    end ]UZP dw1D  
    % END: Compute the Zernike Polynomials f+Fzpd?wS  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% aLwEz}-   
    'yh)6mid  
    % Compute the Zernike functions: IcNZUZGE  
    % ------------------------------ F'ez{ B\AX  
    idx_pos = m>0; y"H(F,(N  
    idx_neg = m<0; +KIBbXF7  
    <W*6=HZ'  
    z = y; m=w #l>!  
    if any(idx_pos) zJOyr"B'8  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ^xr & E  
    end ,,?XGx  
    if any(idx_neg) &C#?&AQ  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); tnq Zl S  
    end ifmX<'(9A  
    {H 3wL  
    % 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) 04a@  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Z0M|Bv9_  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated -8S Z}J  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 3RI %OCGF  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, `mquGk|)  
    %   and THETA is a vector of angles.  R and THETA must have the same zGP@!R`_  
    %   length.  The output Z is a matrix with one column for every P-value, {Ut,xi  
    %   and one row for every (R,THETA) pair. m;vm7]5  
    % *(sv5c!0M8  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Y*S(uqM  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) Ls&-8  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 5&]a8p{  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 _V3}F1?W  
    %   for all p. c7R6.T  
    % 0u I=8j  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 S AKIFNE  
    %   Zernike functions (order N<=7).  In some disciplines it is DyRU$U  
    %   traditional to label the first 36 functions using a single mode u>V~:q\X  
    %   number P instead of separate numbers for the order N and azimuthal 4u1au1c  
    %   frequency M. :=K+~?  
    % cY|@s?3NND  
    %   Example: :]8!G- Z  
    % Yr>7c1FZi  
    %       % Display the first 16 Zernike functions IkQ,#Bsb[  
    %       x = -1:0.01:1; WogCt,  
    %       [X,Y] = meshgrid(x,x); t;t;+M|W  
    %       [theta,r] = cart2pol(X,Y); Iz!]LW  
    %       idx = r<=1; Z jXn,W]~  
    %       p = 0:15; T~d_?UAw$  
    %       z = nan(size(X)); > v4+@o[~  
    %       y = zernfun2(p,r(idx),theta(idx)); 5zF$Q{3  
    %       figure('Units','normalized') ZD4:'m`T/  
    %       for k = 1:length(p) W'v o?  
    %           z(idx) = y(:,k); O 2+taB  
    %           subplot(4,4,k) nMBF/75  
    %           pcolor(x,x,z), shading interp ]'0}fuV  
    %           set(gca,'XTick',[],'YTick',[]) 2WB`+oWox  
    %           axis square J #;|P-pt  
    %           title(['Z_{' num2str(p(k)) '}']) -s 7a\H{~  
    %       end 3k:`7E.  
    % 12}!oS~_  
    %   See also ZERNPOL, ZERNFUN. OK \9`  
    c']m5q39'  
    %   Paul Fricker 11/13/2006 +]e) :J  
    UDlM?r:f  
    [u^~ND'  
    % Check and prepare the inputs: <4-g2.\  
    % ----------------------------- )vGxF}I3  
    if min(size(p))~=1 lXutZ<S[  
        error('zernfun2:Pvector','Input P must be vector.') ~b6c:db3  
    end WA#y&  
    w$jSlgUHy)  
    if any(p)>35 tSVS ogGd  
        error('zernfun2:P36', ... C-^8;xd  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... c7]0 >nU;  
               '(P = 0 to 35).']) ZSLvr-,D  
    end pwA~?$B1  
    K#R|GEwr  
    % Get the order and frequency corresonding to the function number: `X(H,Q}*;  
    % ---------------------------------------------------------------- /wi/i*;A  
    p = p(:); $?DEO[p.  
    n = ceil((-3+sqrt(9+8*p))/2); NOl/y@#  
    m = 2*p - n.*(n+2); D=M'g}l  
    D_BdvWSxj  
    % Pass the inputs to the function ZERNFUN: qU ,{jD$  
    % ---------------------------------------- RAA,%rRhu(  
    switch nargin 6|1*gl1_LD  
        case 3 D4T(Dce  
            z = zernfun(n,m,r,theta); m:cWnG  
        case 4 E*L5D4Kw  
            z = zernfun(n,m,r,theta,nflag); \cHF V  
        otherwise OUy} 1%HY  
            error('zernfun2:nargin','Incorrect number of inputs.') hcR^?  
    end *`t3z-L  
    -gv[u,R  
    % 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) ~oRT@E  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. '2SZ]   
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Sre:l'.  
    %   order N and frequency M, evaluated at R.  N is a vector of "P$')u wE  
    %   positive integers (including 0), and M is a vector with the ',I$`h  
    %   same number of elements as N.  Each element k of M must be a b[MKo7  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) G~/*!?&z  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is [>lQi X  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix d,o|>e$  
    %   with one column for every (N,M) pair, and one row for every ! )(To  
    %   element in R. e/$M6l$Q*4  
    % od*#)   
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- M[L@ej  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 0SJ(Ln`0K  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to |wuN`;gc"  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 9+~1# |  
    %   for all [n,m]. B. #-@  
    % a4",BDx  
    %   The radial Zernike polynomials are the radial portion of the "|/q4JN)7d  
    %   Zernike functions, which are an orthogonal basis on the unit e<"sZK  
    %   circle.  The series representation of the radial Zernike afjEN y1  
    %   polynomials is Iz&<rL;s  
    % W2A!BaH%  
    %          (n-m)/2 \psO$TxF=  
    %            __  9-y<= )  
    %    m      \       s                                          n-2s H`7T;`Yb  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ^O**ZndB/  
    %    n      s=0 9/O\769"'  
    % 9<5ii  
    %   The following table shows the first 12 polynomials. L tK,_j  
    % Hh%|}*f_,  
    %       n    m    Zernike polynomial    Normalization MF +F8h>/  
    %       --------------------------------------------- @ZtvpL}e  
    %       0    0    1                        sqrt(2) gKRlXVS  
    %       1    1    r                           2 .xtam 8@  
    %       2    0    2*r^2 - 1                sqrt(6) 'N/u< `)  
    %       2    2    r^2                      sqrt(6) y~ wN:  
    %       3    1    3*r^3 - 2*r              sqrt(8) N'?#g`*KW  
    %       3    3    r^3                      sqrt(8) 5w</Ga  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ~_ko$(;A  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 4cDe'9 LA  
    %       4    4    r^4                      sqrt(10) isz-MP$:K5  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) MFqb_q+  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) L"( {6H  
    %       5    5    r^5                      sqrt(12) ty8E;[ '  
    %       --------------------------------------------- J?f7!F:8  
    % A9LVS&52  
    %   Example: COA>y?  
    % hdYd2 j  
    %       % Display three example Zernike radial polynomials SI7r `'7A'  
    %       r = 0:0.01:1; \sS0@gnDI  
    %       n = [3 2 5]; U+ V yH4"  
    %       m = [1 2 1]; ?F|F~A8dr  
    %       z = zernpol(n,m,r); ex|h&Vma2V  
    %       figure ne=CN!=  
    %       plot(r,z) ~FnY'F<35  
    %       grid on c>wn e\(5H  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') [vxHsY3z  
    % KuMH,rXF  
    %   See also ZERNFUN, ZERNFUN2. 2FEi-m}  
    MK <\:g  
    % A note on the algorithm. 5]2 p>%G  
    % ------------------------ "FLiSz%ME  
    % The radial Zernike polynomials are computed using the series ccy q~  
    % representation shown in the Help section above. For many special _[N*k"  
    % functions, direct evaluation using the series representation can mH )i  
    % produce poor numerical results (floating point errors), because Z5[g[Q  
    % the summation often involves computing small differences between {}BAQ9|q  
    % large successive terms in the series. (In such cases, the functions B\+uRiD8w  
    % are often evaluated using alternative methods such as recurrence Eopb##o  
    % relations: see the Legendre functions, for example). For the Zernike 2 e&M/{  
    % polynomials, however, this problem does not arise, because the `{Fz  
    % polynomials are evaluated over the finite domain r = (0,1), and rg I Z  
    % because the coefficients for a given polynomial are generally all '>t'U?7w<  
    % of similar magnitude. ^O&&QRH~w  
    % RJdijj  
    % ZERNPOL has been written using a vectorized implementation: multiple Xl E0oN~{  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] '|G8yojz  
    % values can be passed as inputs) for a vector of points R.  To achieve J#\oc@  
    % this vectorization most efficiently, the algorithm in ZERNPOL [ic%ZoZ_  
    % involves pre-determining all the powers p of R that are required to 8I0G%hD  
    % compute the outputs, and then compiling the {R^p} into a single uz;eY D  
    % matrix.  This avoids any redundant computation of the R^p, and &a'LOq+r'  
    % minimizes the sizes of certain intermediate variables. c9+yU~(  
    % *C.Kdf3w  
    %   Paul Fricker 11/13/2006 [ZS.6{vr  
    ~gg&G~ ET  
    rv2;)3/*  
    % Check and prepare the inputs: imyfki $B  
    % ----------------------------- Nf}i /  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 8qoA5fW>  
        error('zernpol:NMvectors','N and M must be vectors.') 877Kv);  
    end T/jxsIt3  
     I^G6aw  
    if length(n)~=length(m) %I@ vMs^  
        error('zernpol:NMlength','N and M must be the same length.') ul!q)cPb{  
    end _.SpU`>/f  
    lz _ r  
    n = n(:); c!mMH~#  
    m = m(:); Bq tN=  
    length_n = length(n); kR{$&cE^  
    w^}* <q\  
    if any(mod(n-m,2)) dcfwUjp[  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') *pyC<4W  
    end o Va[  
    IH.EvierJ  
    if any(m<0) *?+2%zP  
        error('zernpol:Mpositive','All M must be positive.') (*\y  
    end i#PR Tbc  
    HstL'{&,-m  
    if any(m>n) GK#D R/OM  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') -jVg {f!  
    end "e/"$z'ca  
    0f9U:)1z  
    if any( r>1 | r<0 ) SBY0L.  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') \jAI~|3  
    end ;Hb"SB  
    T#HF! GH]  
    if ~any(size(r)==1) X7?j90tH  
        error('zernpol:Rvector','R must be a vector.') Cj J n  
    end 7**zO3 H  
    n;y[%H!g  
    r = r(:); S KGnx  
    length_r = length(r); #hXuGBZEI  
    M{p9b E[j  
    if nargin==4 ;HiaX<O!  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); WN o+%  
        if ~isnorm JvS ~.g1  
            error('zernpol:normalization','Unrecognized normalization flag.') _B\87e  
        end qJw\<7m  
    else %cASk>^i  
        isnorm = false; tZ:fOM  
    end s:I 8~Cc  
    GE\({V.W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]NKz5[9D  
    % Compute the Zernike Polynomials  1 K]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;|f]e/El  
    oUB9)C~  
    % Determine the required powers of r: T7N\b]?j@Y  
    % ----------------------------------- `R*!GHro  
    rpowers = []; 8DFq eY0S  
    for j = 1:length(n) Z1wfy\9c8  
        rpowers = [rpowers m(j):2:n(j)]; OOYdrv,  
    end 6L2Wv5C  
    rpowers = unique(rpowers); A[f `xE  
    ZL9|/ PY  
    % Pre-compute the values of r raised to the required powers, N8X)/W  
    % and compile them in a matrix: B9'2$s+Z;  
    % ----------------------------- mOFp!(  
    if rpowers(1)==0 <iM}p^jX9  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ZQmg;L&7  
        rpowern = cat(2,rpowern{:}); D c]J3r  
        rpowern = [ones(length_r,1) rpowern]; 2-^ ['R  
    else x_= 3 !)  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ='(;!3ZH  
        rpowern = cat(2,rpowern{:}); Z*'_/Grv?  
    end \*c=bz&l  
    Z-aB[hE  
    % Compute the values of the polynomials: d%oHcn  
    % -------------------------------------- u2*."W\  
    z = zeros(length_r,length_n); 1119YeL  
    for j = 1:length_n K:Z|# i-  
        s = 0:(n(j)-m(j))/2; 6>h"Lsww  
        pows = n(j):-2:m(j); ^;@!\Rc  
        for k = length(s):-1:1 Nl{on"il  
            p = (1-2*mod(s(k),2))* ... G ahY+$L,  
                       prod(2:(n(j)-s(k)))/          ... )XYCr<s2"  
                       prod(2:s(k))/                 ... -U@ycx|r  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... axv-U dE;  
                       prod(2:((n(j)+m(j))/2-s(k))); RMAbu*D0  
            idx = (pows(k)==rpowers); \G6V-W  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 3GZrVhU?m  
        end E,[v%Xw   
         $ccCI \  
        if isnorm Bhe0z|&  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); hxGo~<. :  
        end u<HJFGLzI  
    end M,SIs 3  
    7Ur'@wr  
    % 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)  G9i?yd4n=B  
    YkbZ 2J*-  
    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。 by%k*y  
    ^\zf8kPti  
    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)