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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 AoS7B:T;!  
    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! nqyD>>  
     
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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 _'=,c"  
    function z = zernfun(n,m,r,theta,nflag) 5;a*Xf%V  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. P%3pM*.  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N G|wtl(}3  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 0fsVbC  
    %   unit circle.  N is a vector of positive integers (including 0), and 4zoQe>v~  
    %   M is a vector with the same number of elements as N.  Each element NAR6q{c  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ~t6q-P  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 5n@YNaoIb  
    %   and THETA is a vector of angles.  R and THETA must have the same 2Rk}ovtD[  
    %   length.  The output Z is a matrix with one column for every (N,M) Yuv i{ 0  
    %   pair, and one row for every (R,THETA) pair. YF5}~M ymF  
    % !}&|a~U@`k  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike }Hg G<.H>  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), I|@+O#  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral gEh/m.L7  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, B~]Kqp7yU  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized }3(!kW  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. XM$ ~HG  
    % oZ'a}kF  
    %   The Zernike functions are an orthogonal basis on the unit circle. y* +y&  
    %   They are used in disciplines such as astronomy, optics, and /R# zu_i  
    %   optometry to describe functions on a circular domain. /"{d2  
    % 2\xw2VQ@P  
    %   The following table lists the first 15 Zernike functions. 4EB\R"rWXf  
    % @*6fEG{,q  
    %       n    m    Zernike function           Normalization :Jd7q.  
    %       -------------------------------------------------- \-\>JPO~<  
    %       0    0    1                                 1 8Y($ F2  
    %       1    1    r * cos(theta)                    2 l1+l@r\  
    %       1   -1    r * sin(theta)                    2 fUT[tkb/!  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) EZUaYp ~M  
    %       2    0    (2*r^2 - 1)                    sqrt(3) m:H^m/g  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 3lP;=* m.  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) /$~1e7 W  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) FQZ*i\G>>  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 7({)ou x  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) yaUtDC.|  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) !=[Y yh  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Y ;Ym=n'  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) _7Y h[I4  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1 .3#PdMR,  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 7)Toj  
    %       -------------------------------------------------- iU)I"#\l'k  
    % ?@64gdlwq  
    %   Example 1: W`>|OiuF  
    % Rh=" <'d  
    %       % Display the Zernike function Z(n=5,m=1) 6!<I'M'[e  
    %       x = -1:0.01:1; P>/:dt'GJ}  
    %       [X,Y] = meshgrid(x,x); s(,S~  
    %       [theta,r] = cart2pol(X,Y); ]J7qsMw  
    %       idx = r<=1; !cW rB9  
    %       z = nan(size(X)); _4S^'FDo  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); VPMu)1={:p  
    %       figure mqSVd^  
    %       pcolor(x,x,z), shading interp mF7 Ak&So^  
    %       axis square, colorbar CoN[Yf3\  
    %       title('Zernike function Z_5^1(r,\theta)') C=?S  
    % Sn=6[RQ>P  
    %   Example 2: MB]E[&Q!  
    % o_:v?Y>0  
    %       % Display the first 10 Zernike functions Ot=>~(u0  
    %       x = -1:0.01:1; E_,/)U8  
    %       [X,Y] = meshgrid(x,x); )G P;KUVae  
    %       [theta,r] = cart2pol(X,Y); &#p1ogf:  
    %       idx = r<=1; hx$]fvDevD  
    %       z = nan(size(X)); ~D52b1f  
    %       n = [0  1  1  2  2  2  3  3  3  3]; )V1XL   
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3];  s*u A3}j  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; rj4@  
    %       y = zernfun(n,m,r(idx),theta(idx)); E7uIur=g!  
    %       figure('Units','normalized') >* -I Io  
    %       for k = 1:10 'Ru(`" 1|  
    %           z(idx) = y(:,k); 1XGg0SC  
    %           subplot(4,7,Nplot(k)) ~k*]Z8Z  
    %           pcolor(x,x,z), shading interp .:S/x{~  
    %           set(gca,'XTick',[],'YTick',[]) :.:^\Q0  
    %           axis square ]kj^T?&n.  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) AL]gK)R  
    %       end 8Km&3nCv$Q  
    % K?.~}82c  
    %   See also ZERNPOL, ZERNFUN2. vs@d)$N  
    bZowc {!\  
    %   Paul Fricker 11/13/2006 !I7$e&Uz@  
    iI GK "}  
    x ;DoQx  
    % Check and prepare the inputs: |Ajd$+3  
    % ----------------------------- WK%cbFq(  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) x'|ty[87  
        error('zernfun:NMvectors','N and M must be vectors.') De$~ *2  
    end /T _M't@j  
    bT:u |/I  
    if length(n)~=length(m) (UkP AE  
        error('zernfun:NMlength','N and M must be the same length.') ~j!n`#.\  
    end tP'v;$)9F  
    u>>|ZPe  
    n = n(:); {&1L &f<  
    m = m(:); Wa;N(zw0h  
    if any(mod(n-m,2)) -`]9o3E7H  
        error('zernfun:NMmultiplesof2', ... ne#dEUD  
              'All N and M must differ by multiples of 2 (including 0).') f;E#CjlTL  
    end j0l{Mc5  
    jcCAXk055  
    if any(m>n) EX)&|2w  
        error('zernfun:MlessthanN', ... L>Y+}]~  
              'Each M must be less than or equal to its corresponding N.') ,%pCcM)  
    end l*ltS(?  
    1RAkqw<E  
    if any( r>1 | r<0 ) ]d*9@+Iu  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') }dc0ZRKgx  
    end Ca-"3aQkc  
    GcO2oq  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) >a"J);p  
        error('zernfun:RTHvector','R and THETA must be vectors.') @IG's-  
    end #`Su3~T=S  
    :WB uU  
    r = r(:); Z`TfS+O6  
    theta = theta(:); /^=1]+_!  
    length_r = length(r); IMM;LC%rD9  
    if length_r~=length(theta) ,_V V;P  
        error('zernfun:RTHlength', ... @eYpARF  
              'The number of R- and THETA-values must be equal.') a`wjZ"}'[  
    end Xi="gxp$%  
    9p_?t'&>q  
    % Check normalization: p?gm=b#  
    % -------------------- L;V 8c  
    if nargin==5 && ischar(nflag) n Bm ]?  
        isnorm = strcmpi(nflag,'norm'); ~RR!~q  
        if ~isnorm -Y_, .'ex  
            error('zernfun:normalization','Unrecognized normalization flag.') tLzLO#/n  
        end .`D'eS6b  
    else # ~<]z  
        isnorm = false; hBU)gP75  
    end %lCZ7z2o  
    &d6@ SQ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "7cty\  
    % Compute the Zernike Polynomials [Uup5+MCv  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Zc7;&cz  
    l>6tEOXt  
    % Determine the required powers of r: J[ }H^FR  
    % ----------------------------------- R3B+vLGX  
    m_abs = abs(m); oN032o?S  
    rpowers = []; '/O:@P5qY  
    for j = 1:length(n) %`]+sg[i  
        rpowers = [rpowers m_abs(j):2:n(j)]; x/,;:S  
    end Y j oe|  
    rpowers = unique(rpowers); oc1BOW z  
    dN2JOyS  
    % Pre-compute the values of r raised to the required powers, :^7w  
    % and compile them in a matrix: JxIJxhA>  
    % ----------------------------- ;!<}oZp{  
    if rpowers(1)==0 xXJ*xYn "}  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Ph3;;,v '  
        rpowern = cat(2,rpowern{:}); _xKn2?d8g  
        rpowern = [ones(length_r,1) rpowern]; uj.i(U s  
    else W )FxN,  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); sK2N3 B&6  
        rpowern = cat(2,rpowern{:}); UhH#> 2r_  
    end R4p Pt  
    4c5BlD  
    % Compute the values of the polynomials: --$* q"  
    % -------------------------------------- c~<;}ve^z  
    y = zeros(length_r,length(n)); +byOThuE  
    for j = 1:length(n) m?w_ ]  
        s = 0:(n(j)-m_abs(j))/2; O`Tz^Q /D  
        pows = n(j):-2:m_abs(j); ACyK#5E  
        for k = length(s):-1:1 Y4k2=w:D  
            p = (1-2*mod(s(k),2))* ... 9KVJk</:n  
                       prod(2:(n(j)-s(k)))/              ... |62` {+  
                       prod(2:s(k))/                     ... 4!dc/K  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... c}(H*VY2n  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); I=dG(?#7%  
            idx = (pows(k)==rpowers); xF8r+{_J)  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Znb={hh  
        end zu d_BOq{f  
         S;4:`?s=i  
        if isnorm (=j;rfvP  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); }$_@yt<{W@  
        end of B:7  
    end J ?o  
    % END: Compute the Zernike Polynomials wQSan&81Q  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t6"%u3W8M  
    wv9HiHz8gD  
    % Compute the Zernike functions: 7P1Pk?pxy  
    % ------------------------------ Qu|CXUk  
    idx_pos = m>0; 1_+ h"LE  
    idx_neg = m<0; ?tLApy^`?  
    O},}-%G  
    z = y; G4(R/<J,BQ  
    if any(idx_pos) v]k-x n|$j  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); r `PJb5^\|  
    end AR [m+E  
    if any(idx_neg) _,drOF|e  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); \V-N~_-H  
    end O,r;-t4vYU  
    R1zt6oY  
    % 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) ei5YxV6I  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ^.A*mMQ  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 3X gJZ  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive @} Ig*@  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, :-RB< Lj  
    %   and THETA is a vector of angles.  R and THETA must have the same pA!-spgX  
    %   length.  The output Z is a matrix with one column for every P-value, QXb2jWz  
    %   and one row for every (R,THETA) pair. c!\Gj|  
    % ]?}>D?5  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike @_do<'a  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) JVoC2Z<  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Jj=qC{]  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 Y:DopKRD  
    %   for all p. W]po RTJ:  
    % T]\1gs41  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 GxhE5f;  
    %   Zernike functions (order N<=7).  In some disciplines it is 'J?{/O^  
    %   traditional to label the first 36 functions using a single mode ,[;O'g?,g  
    %   number P instead of separate numbers for the order N and azimuthal w-Ph-L/  
    %   frequency M. %r|sb=(yT  
    % Q+Bl1xl  
    %   Example: $9YQ aN%  
    % 9Jwd*gevV  
    %       % Display the first 16 Zernike functions 3H,x4L5j  
    %       x = -1:0.01:1; wa[L[mw  
    %       [X,Y] = meshgrid(x,x); TdPd8ig8{  
    %       [theta,r] = cart2pol(X,Y); cM%?Ot,mK"  
    %       idx = r<=1; 9:2Bt <q  
    %       p = 0:15; `W x| 4  
    %       z = nan(size(X)); nL?P/ \  
    %       y = zernfun2(p,r(idx),theta(idx)); z@ `o(gh  
    %       figure('Units','normalized') % mQ&pk  
    %       for k = 1:length(p) iDDJJ>F26  
    %           z(idx) = y(:,k); #z P-, 2!r  
    %           subplot(4,4,k) ^Q#_  
    %           pcolor(x,x,z), shading interp Z'i@;^=A  
    %           set(gca,'XTick',[],'YTick',[]) <6(0ZO%,C!  
    %           axis square [l-o*@  
    %           title(['Z_{' num2str(p(k)) '}']) :aOR@])>o  
    %       end 2.Z#\6Vj  
    % K9@.l~n  
    %   See also ZERNPOL, ZERNFUN. )5@P|{FF  
    ovp/DM  
    %   Paul Fricker 11/13/2006 k@7#8(3  
    $6n J+  
    X2V+cre  
    % Check and prepare the inputs: O\Huj=  
    % ----------------------------- 'u.Dt*.Uq  
    if min(size(p))~=1 OP\jO DX  
        error('zernfun2:Pvector','Input P must be vector.') :|(YlNUv  
    end ug,AvHEnB  
    bo#xqSGQ  
    if any(p)>35 0f5 ag&  
        error('zernfun2:P36', ... ]0>  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... vEfj3+e  
               '(P = 0 to 35).']) z6w'XA1_+t  
    end +2{ f>KZ  
    B=]j=\o  
    % Get the order and frequency corresonding to the function number: 6 ZRc|ZQ  
    % ---------------------------------------------------------------- `i.fm1I]  
    p = p(:); eZ:iW#YF  
    n = ceil((-3+sqrt(9+8*p))/2); )<HvIr(xr  
    m = 2*p - n.*(n+2); a8TtItN  
    J299 mgB  
    % Pass the inputs to the function ZERNFUN: Vja 4WK*  
    % ---------------------------------------- v(;yy{>8"  
    switch nargin J%"5?)[z  
        case 3 NlF*/Rs  
            z = zernfun(n,m,r,theta); 4P#jMox  
        case 4 A _TaXl(  
            z = zernfun(n,m,r,theta,nflag); lq.:/_m0  
        otherwise 8`L]<Dm  
            error('zernfun2:nargin','Incorrect number of inputs.') uQ3sRJi  
    end At@0G\^  
    $Z;8@O3  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) / xv5we~  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 7W}%ralkg  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of lrc%GU):  
    %   order N and frequency M, evaluated at R.  N is a vector of UA'bE~i  
    %   positive integers (including 0), and M is a vector with the &FmTT8"l  
    %   same number of elements as N.  Each element k of M must be a wxB HlgK4z  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) lO\HchG zB  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is f-#:3k*7S  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix &M5v EPR  
    %   with one column for every (N,M) pair, and one row for every k-&<_ghT \  
    %   element in R. (su7*$wV  
    % ,62~u'hR5  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 1VYH:uGuAU  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is )7Hx <?P  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to KPy)%i  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1  R<1%Gdz  
    %   for all [n,m]. 9.xb-m7  
    % RUr ~u  
    %   The radial Zernike polynomials are the radial portion of the R/1e/t  
    %   Zernike functions, which are an orthogonal basis on the unit ,(oolx"Xa  
    %   circle.  The series representation of the radial Zernike rZ,3:x-:  
    %   polynomials is x]YzVJ=Y  
    % O: I]v@  
    %          (n-m)/2 #<Y3*^~5d  
    %            __ Ruq;:5u  
    %    m      \       s                                          n-2s , l!>+@  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 5Kd"W,  
    %    n      s=0 @G]*]rkKb  
    % vy2<'V*y}  
    %   The following table shows the first 12 polynomials. vg?(0Gasm*  
    % aVHID{Gf Z  
    %       n    m    Zernike polynomial    Normalization U}HSL5v  
    %       --------------------------------------------- 7 `~0j6FY  
    %       0    0    1                        sqrt(2) ^+%bh/2_W  
    %       1    1    r                           2  ]LsT  
    %       2    0    2*r^2 - 1                sqrt(6) /)v+|%U  
    %       2    2    r^2                      sqrt(6) a(IE8:yU`  
    %       3    1    3*r^3 - 2*r              sqrt(8) 0-OKbw5%=b  
    %       3    3    r^3                      sqrt(8) >l-u{([B  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) OS%[SHs  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) JkR%o #>5  
    %       4    4    r^4                      sqrt(10) #Qg)4[pMJ  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) U5ph4G  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) {pi_yr3  
    %       5    5    r^5                      sqrt(12) 7gLk~*  
    %       --------------------------------------------- 3Yx'/=]  
    % 3;b)pQ~6CJ  
    %   Example: _3u3b/%J?  
    % dVq9'{[3  
    %       % Display three example Zernike radial polynomials yS K81`  
    %       r = 0:0.01:1; ?.ObHV*k  
    %       n = [3 2 5]; `B&E?x  
    %       m = [1 2 1]; P$Y w'3v/  
    %       z = zernpol(n,m,r); > mCH!ey  
    %       figure |,KsJ2hD  
    %       plot(r,z) 0 -M i q  
    %       grid on b!MN QGs  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') d8 ~%(I9  
    % u*qI$?&  
    %   See also ZERNFUN, ZERNFUN2. =MJRQ V67  
    AzzHpfv,  
    % A note on the algorithm. DB|w&tygq  
    % ------------------------ G]xYQ]  
    % The radial Zernike polynomials are computed using the series Tw%1m  
    % representation shown in the Help section above. For many special o=7e8l  
    % functions, direct evaluation using the series representation can Dg~m}La  
    % produce poor numerical results (floating point errors), because AD!w:jT9  
    % the summation often involves computing small differences between D0 q42+5  
    % large successive terms in the series. (In such cases, the functions +p _?ekV\  
    % are often evaluated using alternative methods such as recurrence ORqqzy +  
    % relations: see the Legendre functions, for example). For the Zernike ]ZR` 6|"VO  
    % polynomials, however, this problem does not arise, because the r1.zURY  
    % polynomials are evaluated over the finite domain r = (0,1), and v:!TqfI  
    % because the coefficients for a given polynomial are generally all X%99@qv  
    % of similar magnitude. -#<{3BJTrz  
    % 7r3CO<fb  
    % ZERNPOL has been written using a vectorized implementation: multiple JSq3)o9?/  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 2>.b~q@  
    % values can be passed as inputs) for a vector of points R.  To achieve w|I5x}ZFG  
    % this vectorization most efficiently, the algorithm in ZERNPOL J 7dHD(R8  
    % involves pre-determining all the powers p of R that are required to 3KeY4b!h  
    % compute the outputs, and then compiling the {R^p} into a single qfAnMBM1@  
    % matrix.  This avoids any redundant computation of the R^p, and Pdh`Gu1:3  
    % minimizes the sizes of certain intermediate variables. g;q.vHvsc"  
    % 4a!%eBhX"K  
    %   Paul Fricker 11/13/2006 ,QA=)~;D  
    $\k)Y(&  
    W}7Uh b  
    % Check and prepare the inputs: q$H@W. f  
    % ----------------------------- li{<F{7  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7F2:'3SQ  
        error('zernpol:NMvectors','N and M must be vectors.') lpB:lRM  
    end A#Ga!a  
    C\Ob!sv%H  
    if length(n)~=length(m) @az<D7j2  
        error('zernpol:NMlength','N and M must be the same length.') d@8: f  
    end Y[VXx8"p  
    8Bc2?NI=   
    n = n(:); < )_#6)z:  
    m = m(:); fQ.S ,lMe  
    length_n = length(n); l  ~xXy<  
    2A*,9S|Y  
    if any(mod(n-m,2)) G&Yo2aADR  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') '%+LQ"Bp  
    end #;1RStb:zj  
    ~rq:I<5  
    if any(m<0) 7bGOE_r  
        error('zernpol:Mpositive','All M must be positive.') 9J_vvq`%`  
    end S<*1b 6%D  
    uHZjpMoM  
    if any(m>n) "-5FUKI-  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') <Vh5`-J  
    end SEu:31k{o  
    C=K{;.  
    if any( r>1 | r<0 ) )-iUUak  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 1Qjc*+JzO.  
    end " :[;}f;  
    3qV~C{ S  
    if ~any(size(r)==1) @QYCoEU8J  
        error('zernpol:Rvector','R must be a vector.') q+;lxR5D  
    end &P*r66  
    YXF^4||j.c  
    r = r(:); gH"a MEC  
    length_r = length(r); \O~WMN  
    U(~Nmo'  
    if nargin==4 +,T}x+D  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); kA2)T,s74  
        if ~isnorm $j!:ET'V  
            error('zernpol:normalization','Unrecognized normalization flag.')   LR4W  
        end ^"uD:f)  
    else Fy>g*3  
        isnorm = false; wXYT(R  
    end R(}!gv}s  
    =8]Ru(#Ig  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DU5rB\!.~  
    % Compute the Zernike Polynomials ;?-{Uk  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O|9Nl*rXz  
    xkkG#n)  
    % Determine the required powers of r: 96gaun J  
    % ----------------------------------- gxVJH'[V5  
    rpowers = []; ZY6%%7?1  
    for j = 1:length(n) B>"-8#B[4  
        rpowers = [rpowers m(j):2:n(j)]; ,[_)BM  
    end F"tM?V.|  
    rpowers = unique(rpowers);  ?f5||^7  
    6hFs{P7  
    % Pre-compute the values of r raised to the required powers, hig t(u  
    % and compile them in a matrix: UU#$Kt*frR  
    % ----------------------------- ,yfJjV*I  
    if rpowers(1)==0 5a&gdqg]  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); -H 5-6w$  
        rpowern = cat(2,rpowern{:}); D{+D.4\  
        rpowern = [ones(length_r,1) rpowern]; G+3uY25y  
    else Aq$o&t  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); #S|On[Q!  
        rpowern = cat(2,rpowern{:}); IJ{VCzi  
    end bvJ*REPL ?  
    Xi=4S[.4  
    % Compute the values of the polynomials: 7=A @P  
    % -------------------------------------- j{m{hVa  
    z = zeros(length_r,length_n); LH~ t5  
    for j = 1:length_n eW_EWVH  
        s = 0:(n(j)-m(j))/2; (d[JMO^@8  
        pows = n(j):-2:m(j); 6fT^t!<i  
        for k = length(s):-1:1 Lf Y[Z4  
            p = (1-2*mod(s(k),2))* ... ,`$2  
                       prod(2:(n(j)-s(k)))/          ... 2%pe.s tQ  
                       prod(2:s(k))/                 ... $BOIa  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... L[:M[,?=`  
                       prod(2:((n(j)+m(j))/2-s(k))); kPnuU!  
            idx = (pows(k)==rpowers); Z~"8C Kz  
            z(:,j) = z(:,j) + p*rpowern(:,idx); gg Hl{cl)  
        end 1fh6A`c  
         <9Ytv|t@0  
        if isnorm $bk_%R}s  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); uVw|jj  
        end b]WvKdq  
    end u3PM 7z!~  
    t\ 9Y)d  
    % 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)  %c,CfhEV%&  
    7!E7XP6,~>  
    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。 .(8eWc YK  
    v D4<G{  
    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)