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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 ec/>LJDX7  
    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! O;i0xWUh  
     
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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 erbk (  
    function z = zernfun(n,m,r,theta,nflag) p\F%Nj,  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. [ p~,;%  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N c#"t.j<E}  
    %   and angular frequency M, evaluated at positions (R,THETA) on the K)se$vb6  
    %   unit circle.  N is a vector of positive integers (including 0), and F?05+  
    %   M is a vector with the same number of elements as N.  Each element Kop(+]Q&n  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) %''L7o.#a  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, -`eB4j'7  
    %   and THETA is a vector of angles.  R and THETA must have the same B2P@9u|9  
    %   length.  The output Z is a matrix with one column for every (N,M) ,W|-?b?   
    %   pair, and one row for every (R,THETA) pair. ah_ >:x  
    % <Z<meB[g  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike )wCNLi>4  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), _ZFEo< `'  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral +xU({/  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, vJ=Q{_D=\  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized W]7/ e  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. lw[c+F7  
    % <F(2D<d{;)  
    %   The Zernike functions are an orthogonal basis on the unit circle. YURMXbj  
    %   They are used in disciplines such as astronomy, optics, and GGr82)E  
    %   optometry to describe functions on a circular domain. e0(aRN{W  
    % +egwZ$5I  
    %   The following table lists the first 15 Zernike functions. m%apGp'=1  
    % 6hv.;n};  
    %       n    m    Zernike function           Normalization u:2Ll[ eo  
    %       -------------------------------------------------- zBTW&  
    %       0    0    1                                 1 3\Q9>>  
    %       1    1    r * cos(theta)                    2 qy)~OBY  
    %       1   -1    r * sin(theta)                    2 mfaU_Vo&  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) _p+E(i 9  
    %       2    0    (2*r^2 - 1)                    sqrt(3) %)?jaE}[  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) kaB4[u  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) X~c?C-fV  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 3Cc#{X-+  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) :S_]!'H  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) %dg[ho  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 1O NkmVtL  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )X[2~E  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) _/noWwVu  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) p/VVb%  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) |g)>6+?]W  
    %       -------------------------------------------------- $*iovam>^]  
    % vno/V#e$WX  
    %   Example 1: O^row1D_  
    % rf:H$\yw  
    %       % Display the Zernike function Z(n=5,m=1) B5|\<CF  
    %       x = -1:0.01:1; Cp"7R&s  
    %       [X,Y] = meshgrid(x,x); ,&WwADZ-s  
    %       [theta,r] = cart2pol(X,Y); Cd"{7<OyM4  
    %       idx = r<=1; Y.]$T8  
    %       z = nan(size(X)); 7g(Z @  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 6`@J=Q?  
    %       figure PBCGC^0{  
    %       pcolor(x,x,z), shading interp lYJSg70P  
    %       axis square, colorbar U|%}B(  
    %       title('Zernike function Z_5^1(r,\theta)') W Eif&<Y  
    % & rab,I"  
    %   Example 2: VDbbA\  
    % tMX$8W0 c  
    %       % Display the first 10 Zernike functions y"q>}5  
    %       x = -1:0.01:1; vBl:&99[/  
    %       [X,Y] = meshgrid(x,x); 60u_,@rV  
    %       [theta,r] = cart2pol(X,Y); 7\,9Gcv1  
    %       idx = r<=1; [%N?D#;  
    %       z = nan(size(X)); iP"sw0V8  
    %       n = [0  1  1  2  2  2  3  3  3  3]; dM^Z,; u  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; DJ:'<"zH7  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; DI{*E  
    %       y = zernfun(n,m,r(idx),theta(idx)); Q'jw=w!|g  
    %       figure('Units','normalized') t'Wv? ,  
    %       for k = 1:10 {XmCG%%L  
    %           z(idx) = y(:,k); \>- M&C  
    %           subplot(4,7,Nplot(k)) u/u(Z&  
    %           pcolor(x,x,z), shading interp Bso#+v5  
    %           set(gca,'XTick',[],'YTick',[]) ()?83Xj[c  
    %           axis square N8dxgh!,  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) MkPQ@so  
    %       end ;: 2U}p^-  
    % h&$h<zL[  
    %   See also ZERNPOL, ZERNFUN2. C'#)mo_@t  
    BA]$Fi.Mw  
    %   Paul Fricker 11/13/2006 lbBWOx/|  
    gYc]z5`  
    -PE_qZ^  
    % Check and prepare the inputs: ?!U[~Gq  
    % ----------------------------- *c94'Tcl  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) S-7&$n  
        error('zernfun:NMvectors','N and M must be vectors.') .PUp3X-  
    end j fY7ich  
    /q]rA  
    if length(n)~=length(m) 2H)4}5H  
        error('zernfun:NMlength','N and M must be the same length.') *(?Wzanh  
    end +SH{`7r  
    mOsp~|d  
    n = n(:); MxIa,M <  
    m = m(:); (O5Yd 6u  
    if any(mod(n-m,2)) 4\Y5RfLB_  
        error('zernfun:NMmultiplesof2', ... <ukBAux,D  
              'All N and M must differ by multiples of 2 (including 0).') YOD.y!.zq7  
    end Zp9. ~&4o-  
    %'=*utOxy  
    if any(m>n) i.vH$  
        error('zernfun:MlessthanN', ...  S=(O6+U  
              'Each M must be less than or equal to its corresponding N.') 00QJ596  
    end P9 <U+\z  
    k||t<&`Ze  
    if any( r>1 | r<0 ) +nDy b  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') vt|R)[,  
    end qq| 5[I.?  
    MIrx,d  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 27e!KG[&  
        error('zernfun:RTHvector','R and THETA must be vectors.') N7+L@CC6T  
    end _5jT}I<k  
    _F;v3|`D@<  
    r = r(:); 0D s3wNz  
    theta = theta(:); ) CP  
    length_r = length(r); {arqcILr  
    if length_r~=length(theta)  <OMwi9  
        error('zernfun:RTHlength', ...  8s0+6{vW  
              'The number of R- and THETA-values must be equal.') f<Hi=Qpm  
    end +(3_V$|Dv  
    Rm} ym9  
    % Check normalization: 6}"c4 ^k6  
    % -------------------- }X&rJV  
    if nargin==5 && ischar(nflag) U#` e~d t<  
        isnorm = strcmpi(nflag,'norm'); `t~jHe4!Y  
        if ~isnorm ;.A}c)b  
            error('zernfun:normalization','Unrecognized normalization flag.') s<9g3Gh  
        end m+TAaK  
    else 'r?ULft1  
        isnorm = false; -zR<m  
    end \H fAKBT  
    Iux3f+H  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ')y2W1  
    % Compute the Zernike Polynomials FE~D:)Xj'?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $.SBW=^V  
    L8VOiK=,  
    % Determine the required powers of r: ZSC*{dD$E  
    % ----------------------------------- ZEP?~zV\A  
    m_abs = abs(m); ,&P 4%N"  
    rpowers = []; ->sxz/L  
    for j = 1:length(n) mlnF,+s  
        rpowers = [rpowers m_abs(j):2:n(j)]; `^bP9X_a  
    end R6+)&:Ab{R  
    rpowers = unique(rpowers); gq7tSkH@  
    v ,8;: sD  
    % Pre-compute the values of r raised to the required powers, c|&3e84U  
    % and compile them in a matrix: r;#"j%z  
    % ----------------------------- S /hx\TzC  
    if rpowers(1)==0 {M]_]L{&7  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); sdFHr4  
        rpowern = cat(2,rpowern{:}); x< A-Ws{^V  
        rpowern = [ones(length_r,1) rpowern]; 1 /. BP  
    else ;tjOEmIiU  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ^4dE8Ve"@  
        rpowern = cat(2,rpowern{:}); :<QknU}dwy  
    end {213/@,  
    t#k]K]  
    % Compute the values of the polynomials: p5G'})x  
    % -------------------------------------- (2g a: }K  
    y = zeros(length_r,length(n)); VW-qQe  
    for j = 1:length(n) H+v&4}f  
        s = 0:(n(j)-m_abs(j))/2; NJUKH1lIhR  
        pows = n(j):-2:m_abs(j); <J/ =$u/  
        for k = length(s):-1:1 mq`/nAmt  
            p = (1-2*mod(s(k),2))* ... Y6` xb`  
                       prod(2:(n(j)-s(k)))/              ... I |Oco?Q"  
                       prod(2:s(k))/                     ... *_(X$qfoW  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... S,#1^S  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); oz) [ -  
            idx = (pows(k)==rpowers); yPN'@{ 5#  
            y(:,j) = y(:,j) + p*rpowern(:,idx); o`bch? ]  
        end uO%0rKW  
         1Cr&6't  
        if isnorm po| Ux`u  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); bJ ~H  
        end (Ou%0 KW  
    end n(: <pz  
    % END: Compute the Zernike Polynomials lSxb:$g  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [&)]-2w2  
    %Bs. XW,  
    % Compute the Zernike functions: nV' 1 $L#  
    % ------------------------------ BEdCA]T  
    idx_pos = m>0; Pvxb6\G&d  
    idx_neg = m<0; =rjU=3!&(  
    #N|\7(#~u  
    z = y; m'o dVZ7  
    if any(idx_pos) yW_yHSx;  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); u`pTFy  
    end %yRXOt2(  
    if any(idx_neg) #}`sfaT  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); dWAt#xII  
    end c;l!i-  
    Q:}]-lJg  
    % 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) I2 [U#4n  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. K~W(ZmB  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated X/5m}-6d]  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive D#;7S'C  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, %OQdUH4x  
    %   and THETA is a vector of angles.  R and THETA must have the same JEUU~L;  
    %   length.  The output Z is a matrix with one column for every P-value, |iM,bs  
    %   and one row for every (R,THETA) pair. #{i*9'  
    % X8C7d6ca  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Xw H>F7HPe  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) Kz HYh  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) `fv5U%  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 S?~0)EXj(  
    %   for all p. e3I""D{)[=  
    % 6v`3/o  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 RGW@@  
    %   Zernike functions (order N<=7).  In some disciplines it is rXx#<7`  
    %   traditional to label the first 36 functions using a single mode {j2V k)\[i  
    %   number P instead of separate numbers for the order N and azimuthal dCC*|b8h  
    %   frequency M. e~)[I!n  
    % \}Q=q$)  
    %   Example: f"6W ;b2L.  
    % y`I>|5[ `  
    %       % Display the first 16 Zernike functions \Y P,}_ ~  
    %       x = -1:0.01:1; (W1 $+X  
    %       [X,Y] = meshgrid(x,x); 4Aj~mA  
    %       [theta,r] = cart2pol(X,Y); MN?aPpr>  
    %       idx = r<=1; '$ei3  
    %       p = 0:15; @16GF!.  
    %       z = nan(size(X)); /\mtCa.O  
    %       y = zernfun2(p,r(idx),theta(idx)); ) Sn0Y B  
    %       figure('Units','normalized') g=Xf&}&=x  
    %       for k = 1:length(p) f$I=o N  
    %           z(idx) = y(:,k); atL<mhRz  
    %           subplot(4,4,k) X[BP0:`t  
    %           pcolor(x,x,z), shading interp O( ^h_  
    %           set(gca,'XTick',[],'YTick',[]) #asg5 }  
    %           axis square =?5)M_6)  
    %           title(['Z_{' num2str(p(k)) '}']) * EWWN?d  
    %       end K%q5:9m  
    % E&U_1D9=L<  
    %   See also ZERNPOL, ZERNFUN. |{9<%Ok4P  
    .}IW!$ dq  
    %   Paul Fricker 11/13/2006 eL3 _Lz  
    [LoQYDku  
    pz%s_g'  
    % Check and prepare the inputs: ;(C<gt,r}  
    % ----------------------------- *,\v|]fc  
    if min(size(p))~=1 !M^O\C)  
        error('zernfun2:Pvector','Input P must be vector.') +_GS@)L`%  
    end ?I+L  
    b RAD_  
    if any(p)>35 gAAC>{Wh  
        error('zernfun2:P36', ... }gbLWx'iG  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... v,w af`)J  
               '(P = 0 to 35).']) "*d6E}wG  
    end <KMCNCU\+  
    T$;S   
    % Get the order and frequency corresonding to the function number: Q'OtXs 80  
    % ---------------------------------------------------------------- AC RuDY  
    p = p(:); n`,  <g  
    n = ceil((-3+sqrt(9+8*p))/2); { 4J.  
    m = 2*p - n.*(n+2); V}y]<  
    Qs1p  
    % Pass the inputs to the function ZERNFUN: |A&;m}(Mt  
    % ---------------------------------------- :nx+(xgw  
    switch nargin  ?eS;Yc  
        case 3 b-u@?G|<  
            z = zernfun(n,m,r,theta); yqN`R\d  
        case 4 =B}IsBn'J  
            z = zernfun(n,m,r,theta,nflag); ~?\U];l  
        otherwise f,G*e367:  
            error('zernfun2:nargin','Incorrect number of inputs.') }0'LKwIR  
    end jQ{ @ol}n  
    o/Ismg-p  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) !v8R(  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. FX!KX/OE)  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of tVG;A&\,6  
    %   order N and frequency M, evaluated at R.  N is a vector of h*Fv~j'p  
    %   positive integers (including 0), and M is a vector with the x?L0R{?WW  
    %   same number of elements as N.  Each element k of M must be a VeQGdyhY  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) OBWb0t5H?  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is i:s=  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix e]~p:  
    %   with one column for every (N,M) pair, and one row for every in>+D|q c  
    %   element in R. )U~|QdZ  
    % pS$9mzY  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- cKTjQJ#  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is "z9C@T  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 3t-STk?  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 }H ~-oYMu  
    %   for all [n,m]. d88A.Z3w  
    % (\ab%M   
    %   The radial Zernike polynomials are the radial portion of the )0/9 L  
    %   Zernike functions, which are an orthogonal basis on the unit }u;K<<h:  
    %   circle.  The series representation of the radial Zernike Jl_W6gY"Z  
    %   polynomials is bMK X9`*o  
    % 0OO[@Ht  
    %          (n-m)/2 t=B1yvE "  
    %            __ v~ >Bbe  
    %    m      \       s                                          n-2s F^GNOD3J  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r P*KIk~J  
    %    n      s=0 b-ss^UL  
    % 3)WfBvG  
    %   The following table shows the first 12 polynomials. 4EM+Ye  
    % 'vNju1sfk  
    %       n    m    Zernike polynomial    Normalization V 1'otQH2l  
    %       --------------------------------------------- Jg$<2CR&  
    %       0    0    1                        sqrt(2) /Bt!xSI  
    %       1    1    r                           2 ~u&gU1}  
    %       2    0    2*r^2 - 1                sqrt(6) {&+M.Xn  
    %       2    2    r^2                      sqrt(6) NF0_D1Goi  
    %       3    1    3*r^3 - 2*r              sqrt(8) 'I>USl3hI  
    %       3    3    r^3                      sqrt(8) T +vo)9w  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10)  x+cL(R  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) d.? }>jl  
    %       4    4    r^4                      sqrt(10) NK qI x  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) m""+ $  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ]EKg)E  
    %       5    5    r^5                      sqrt(12) glLVT i  
    %       --------------------------------------------- [mzed{p]]  
    % h/n(  
    %   Example: ) A:h  
    % UN'n~d @~  
    %       % Display three example Zernike radial polynomials OKh0m_ )7  
    %       r = 0:0.01:1; Lf(( zk:pt  
    %       n = [3 2 5]; a,Pw2Gcid  
    %       m = [1 2 1]; ~B|m"qY{i  
    %       z = zernpol(n,m,r); nF'YG+;|@  
    %       figure Ry >y  
    %       plot(r,z) Gko"iO#  
    %       grid on X$5  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') nG<oae6z"  
    % *k7BE_&*0Z  
    %   See also ZERNFUN, ZERNFUN2. X&WP.n)  
    bkd`7(r  
    % A note on the algorithm. :^ywc O   
    % ------------------------ &%rM|  
    % The radial Zernike polynomials are computed using the series hdDT'+  
    % representation shown in the Help section above. For many special " AUSgVE+h  
    % functions, direct evaluation using the series representation can `h@fW- r  
    % produce poor numerical results (floating point errors), because 'Ou C[$Z  
    % the summation often involves computing small differences between R `ViRJh  
    % large successive terms in the series. (In such cases, the functions O[VY|.MEk  
    % are often evaluated using alternative methods such as recurrence _Z(t**Zh6y  
    % relations: see the Legendre functions, for example). For the Zernike Wh i#Ii~  
    % polynomials, however, this problem does not arise, because the >OaD7  
    % polynomials are evaluated over the finite domain r = (0,1), and 6C2~0b   
    % because the coefficients for a given polynomial are generally all |'z8>1  
    % of similar magnitude. WGz)-IB!PE  
    % Imv#7{ndq  
    % ZERNPOL has been written using a vectorized implementation: multiple %rb$tKk  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] "`ftcJUd  
    % values can be passed as inputs) for a vector of points R.  To achieve )I&.6l!#  
    % this vectorization most efficiently, the algorithm in ZERNPOL "jAd.x?X7e  
    % involves pre-determining all the powers p of R that are required to ~1+6gG  
    % compute the outputs, and then compiling the {R^p} into a single }gQ2\6o2g  
    % matrix.  This avoids any redundant computation of the R^p, and }sH[_%)  
    % minimizes the sizes of certain intermediate variables. Kkp dcc  
    % T [$-])iK  
    %   Paul Fricker 11/13/2006 Ms|c" ?se  
     p?f\/  
    _CHzwNU  
    % Check and prepare the inputs: 3q'AgiW  
    % ----------------------------- ;~<To9O  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) [eD0L7 1[  
        error('zernpol:NMvectors','N and M must be vectors.') GCJ[xn(_  
    end _f0AV;S:vd  
    0S4BV%7F  
    if length(n)~=length(m) Wa|V~PL+T  
        error('zernpol:NMlength','N and M must be the same length.') aG]>{(~cL  
    end /-p!|T}w  
    FL{?W(M  
    n = n(:); +7b8ye  
    m = m(:); (|BY<Ac3  
    length_n = length(n); Jx5`0?  
    jn5xYKv  
    if any(mod(n-m,2)) nx'c=gp  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') d[_26.  
    end zzZ EX  
    gQr+ ~O  
    if any(m<0) bq E'9GI  
        error('zernpol:Mpositive','All M must be positive.') ^;_~ mq.  
    end ."+lij=56  
    6|NH*#s  
    if any(m>n) !vnC-&G  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') wS}c \!@<,  
    end n- 2X?<_Z  
    #`u}#(  
    if any( r>1 | r<0 ) ,Iyc0  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') bKpy?5&>  
    end ~`AB-0t.u  
    dQ8RrD=$&  
    if ~any(size(r)==1) hty'L61\z  
        error('zernpol:Rvector','R must be a vector.') |H t5a.  
    end ~^obf(N`  
    59oTU  
    r = r(:); NVb}uH*i  
    length_r = length(r); w||t3!M+n  
     _BP%@o  
    if nargin==4 SxY z)aF~  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); K+`GVmD  
        if ~isnorm jg?UwR&  
            error('zernpol:normalization','Unrecognized normalization flag.') aLh(8;$  
        end )5b_>Uy  
    else X_2N9$},  
        isnorm = false; 2V@5:tf  
    end \< .BN;t{  
    .hW>#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %k#+nad  
    % Compute the Zernike Polynomials q8$t4_pF  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j~N*TXkC  
    yF)J7a:U  
    % Determine the required powers of r: |1%% c %  
    % ----------------------------------- :Tpf8  
    rpowers = []; sLA.bp.O  
    for j = 1:length(n) CC=I|/mBM  
        rpowers = [rpowers m(j):2:n(j)]; "zcAYg^U  
    end 7 {92_xRL  
    rpowers = unique(rpowers); U:*rlA@_.  
    TAUl{??,  
    % Pre-compute the values of r raised to the required powers, +DRt2a #  
    % and compile them in a matrix: FXr^ 4B}  
    % ----------------------------- [k$GUU,jY  
    if rpowers(1)==0 %:~Ah6R1  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); gg`{kN^r.a  
        rpowern = cat(2,rpowern{:}); %d+Fq=<  
        rpowern = [ones(length_r,1) rpowern]; DoczQc-U+  
    else 'b.jKkW7  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); TIJH} Ri  
        rpowern = cat(2,rpowern{:}); \uTlwS  
    end g}hUCx(  
    \r IOnZ.WK  
    % Compute the values of the polynomials: ,&,%B|gT]  
    % -------------------------------------- h^(U:M=A  
    z = zeros(length_r,length_n); e&x)g;bn  
    for j = 1:length_n wxHd^b  
        s = 0:(n(j)-m(j))/2; #+o$Tg  
        pows = n(j):-2:m(j); _AF$E"f@  
        for k = length(s):-1:1 Q qF<HCO  
            p = (1-2*mod(s(k),2))* ... ]lA}5  
                       prod(2:(n(j)-s(k)))/          ... IrZjlnht  
                       prod(2:s(k))/                 ... j(y<oxh  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... s#5#WNzP  
                       prod(2:((n(j)+m(j))/2-s(k))); r#WqXh_uk  
            idx = (pows(k)==rpowers); fL| 9/sojz  
            z(:,j) = z(:,j) + p*rpowern(:,idx); <zqIq9}r  
        end er_6PV  
         5{yg  
        if isnorm K-]) RIM  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); L&+k`b  
        end _kBmKE  
    end yreH/$Ou 8  
    #!_4ZX  
    % 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)  Gy6x.GX  
    pFBK'NE  
    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。 {EVy.F  
    (6 RWI#  
    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)