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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 @ =x=dL(  
    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! -[OGZP`8  
     
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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 ],R rk]1  
    function z = zernfun(n,m,r,theta,nflag) yyxGVfr  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 1eI >Yy>}  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ^Qz8`1`;Z  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 'R8VCj  
    %   unit circle.  N is a vector of positive integers (including 0), and NZYtA7  
    %   M is a vector with the same number of elements as N.  Each element 3(%hHM7DM  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) sxJKu  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, \\ M2_mT  
    %   and THETA is a vector of angles.  R and THETA must have the same ?qYw9XQYL  
    %   length.  The output Z is a matrix with one column for every (N,M) j,eeQ KH  
    %   pair, and one row for every (R,THETA) pair. Ta?#o  
    % Y&`Vs(  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ~|@aV:k  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ;Avd$&::  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral O:Bfbna  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, N:[m,U9a  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized `zRgP#  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. K+Al8L?K_  
    % "cRc~4%K  
    %   The Zernike functions are an orthogonal basis on the unit circle. J7t5 B}}  
    %   They are used in disciplines such as astronomy, optics, and F%bv vw*(  
    %   optometry to describe functions on a circular domain. v>.nL(VLjP  
    % LslQZ]3MY  
    %   The following table lists the first 15 Zernike functions. g}|a-  
    % "R+ x  
    %       n    m    Zernike function           Normalization xZPSoxu  
    %       -------------------------------------------------- `23&vGk}  
    %       0    0    1                                 1 6 +^V  
    %       1    1    r * cos(theta)                    2 z|F>+6l"Y7  
    %       1   -1    r * sin(theta)                    2 e"hm|'  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) jJ?MT#v  
    %       2    0    (2*r^2 - 1)                    sqrt(3) nVw]0Yl  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) wKe^5|Rr  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) UP 1Y3  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) &D[dDUdHs  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) a+szA};  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) yEtI5Qk  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) m7z/@b[  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) rw8O<No4.o  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) t*zve,?}  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) cQzd0X  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) |OF<=GGO+  
    %       -------------------------------------------------- aoz+g,1 //  
    % @5\OM#WT~&  
    %   Example 1: Q{b ZD*  
    % B~-VGT 2o  
    %       % Display the Zernike function Z(n=5,m=1) -]~U_J]  
    %       x = -1:0.01:1; ;5ugnVXu  
    %       [X,Y] = meshgrid(x,x); 5&v'aiWK  
    %       [theta,r] = cart2pol(X,Y); )NRY9\H  
    %       idx = r<=1; {}N*e"<O  
    %       z = nan(size(X)); })g|r9=  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); jWiZ!dtUZ  
    %       figure (<s7X$(]e  
    %       pcolor(x,x,z), shading interp V%dMaX>^i  
    %       axis square, colorbar huWUd)Po%  
    %       title('Zernike function Z_5^1(r,\theta)') +VDwDJ)lG  
    % d"Y9go"Z  
    %   Example 2: -WE pBt7*  
    % m/=,O_  
    %       % Display the first 10 Zernike functions (k6=o';y  
    %       x = -1:0.01:1; 4o9#B:N]J  
    %       [X,Y] = meshgrid(x,x); 35) ]R`f  
    %       [theta,r] = cart2pol(X,Y); Hlp!6\gukp  
    %       idx = r<=1; eT[ ,k[#q  
    %       z = nan(size(X)); s!nFc{  
    %       n = [0  1  1  2  2  2  3  3  3  3]; :m_0WT  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ,[,+ _A  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; J*U,kyYF  
    %       y = zernfun(n,m,r(idx),theta(idx)); 3%{XJV   
    %       figure('Units','normalized') }h5pM`|1  
    %       for k = 1:10 zOLt)2-<  
    %           z(idx) = y(:,k); PDREwBX  
    %           subplot(4,7,Nplot(k)) /XEcA 5C<  
    %           pcolor(x,x,z), shading interp W>K2d  
    %           set(gca,'XTick',[],'YTick',[]) I"#jSazk  
    %           axis square W:4]-i?2  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Ag }hyIl  
    %       end Tcz67&c |W  
    % ppN96-]^0  
    %   See also ZERNPOL, ZERNFUN2. 1m|Oi%i4  
    8UwL%"?YB  
    %   Paul Fricker 11/13/2006 FgE6j;   
    PQWo<Uet  
    !lm^(SSv  
    % Check and prepare the inputs: g v&xC 6>  
    % ----------------------------- D2E~ c? V  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) #E5Sc\,  
        error('zernfun:NMvectors','N and M must be vectors.') @Rig@  
    end ]]d9\fw  
    G2ZF`WQ  
    if length(n)~=length(m) &?9p\oY[  
        error('zernfun:NMlength','N and M must be the same length.') `XP]y=  
    end %g5weiFM  
    (+4gq6b  
    n = n(:); {{ R/:-6?@  
    m = m(:); %.pX!jL  
    if any(mod(n-m,2)) 9j49#wG0"B  
        error('zernfun:NMmultiplesof2', ... wHWd~K_q  
              'All N and M must differ by multiples of 2 (including 0).') 2fO ~%!.G  
    end zbddn4bW9  
    E$ q/4  
    if any(m>n) '-D-H}%;}M  
        error('zernfun:MlessthanN', ... =9i:R!,W  
              'Each M must be less than or equal to its corresponding N.') ` R!0uRu  
    end ,'= Y  
    ]r$S{<  
    if any( r>1 | r<0 ) _{_LTy%[  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') UB|Nx(V s  
    end (jPN+yQ  
    KG'4;Z5J  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) x7L$x=8s  
        error('zernfun:RTHvector','R and THETA must be vectors.') 4Yt:PN2  
    end +VdYT6{p  
    tU!"CX  
    r = r(:); xh#ef=Bw  
    theta = theta(:); q_g'4VZv  
    length_r = length(r); pHsp]a  
    if length_r~=length(theta) |5V#&e\ES  
        error('zernfun:RTHlength', ... +&O[}%W  
              'The number of R- and THETA-values must be equal.') "}\z7^.W>  
    end }{ pNasAU  
    Um9!<G=;  
    % Check normalization: ! D'U:)  
    % -------------------- RB\>$D  
    if nargin==5 && ischar(nflag) yT-m9$^v  
        isnorm = strcmpi(nflag,'norm'); KB&t31aq  
        if ~isnorm xaoaZ3Ko  
            error('zernfun:normalization','Unrecognized normalization flag.') _q)`Y:2  
        end _ Eq:Qbw#  
    else /!eC;qp;[  
        isnorm = false; 67}y/C]<  
    end Fng":28o  
    I:]s/r7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b&*^\hY9b  
    % Compute the Zernike Polynomials A0oC*/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }dAb} 0XK.  
    5A7!Xd  
    % Determine the required powers of r: %ia/i :  
    % ----------------------------------- [LL"86D  
    m_abs = abs(m); y`mEsj  
    rpowers = []; QD+dP nZu  
    for j = 1:length(n) d7It}7@9  
        rpowers = [rpowers m_abs(j):2:n(j)]; fhLdM  
    end &%f y  
    rpowers = unique(rpowers); kzLj1Ix2  
    _Y|k \|'  
    % Pre-compute the values of r raised to the required powers, e|):%6#  
    % and compile them in a matrix: +TpM7QaL  
    % ----------------------------- Fu )V2[TY  
    if rpowers(1)==0 @-kzSm  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); G&/}P$  
        rpowern = cat(2,rpowern{:}); +_Fsiu_b  
        rpowern = [ones(length_r,1) rpowern]; q}ZZqYk  
    else (FH4\'t)  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9D(M>'Bh  
        rpowern = cat(2,rpowern{:}); OrPIvP<w@  
    end ?5$\8gZ  
    | (v/>t  
    % Compute the values of the polynomials: gO*cX&  
    % -------------------------------------- 89`AF1  
    y = zeros(length_r,length(n)); ^5 F-7R8Q  
    for j = 1:length(n) 8BE OE<  
        s = 0:(n(j)-m_abs(j))/2; 0Ny0#;P  
        pows = n(j):-2:m_abs(j); u<!!%C~+=  
        for k = length(s):-1:1 }s}b]v  
            p = (1-2*mod(s(k),2))* ... ]v rpr%K  
                       prod(2:(n(j)-s(k)))/              ... 7#MBT-ih  
                       prod(2:s(k))/                     ... "LaNXZ9  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... y"cK@sOo  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); gLl?e8[F  
            idx = (pows(k)==rpowers); 0AJ6g@ t[  
            y(:,j) = y(:,j) + p*rpowern(:,idx); u\^<V)  
        end m ~fqZK  
          7 g  
        if isnorm u5V<f;  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); `r_qvrC  
        end T"kaOy  
    end b1nw,(hLY  
    % END: Compute the Zernike Polynomials ;L(W'+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nP 2rN_:4  
    >^|\wy  
    % Compute the Zernike functions: JF: QQ\  
    % ------------------------------ ^w8H=UkP!+  
    idx_pos = m>0; :Q+ rEjw+  
    idx_neg = m<0; `q7I;w+g  
    F m h;d*IT  
    z = y; nLto=tNUO  
    if any(idx_pos) <g>_#fz"K  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); FLEf(  
    end Bwb3@vNA  
    if any(idx_neg) $aE %W? \  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); bxkp9o  
    end n3isLNvIp  
    %3fHitCikc  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) %pL ,A5M  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. e_k1pox]l  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated TL]2{rf~  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive COJqVC(#  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Nf-IDK  
    %   and THETA is a vector of angles.  R and THETA must have the same U>:CX XHRt  
    %   length.  The output Z is a matrix with one column for every P-value, qZKU=HM  
    %   and one row for every (R,THETA) pair. uO,90g[C/R  
    % qa`bR%eH  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike FK@rZP  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) bi#o1jR  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) :#d$[:r#  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 hd/5*C{s  
    %   for all p.  yZmQBh$  
    % mOGcv_L  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 JY9Hqf  
    %   Zernike functions (order N<=7).  In some disciplines it is [7|}h/  
    %   traditional to label the first 36 functions using a single mode c{|soc[#  
    %   number P instead of separate numbers for the order N and azimuthal  <gf:QX!  
    %   frequency M. F~W*"i+EZ  
    % <X|"5/h  
    %   Example: RX?Nv4-  
    % f+fF5Z\  
    %       % Display the first 16 Zernike functions xx!o]D-}  
    %       x = -1:0.01:1; d /Zt}{  
    %       [X,Y] = meshgrid(x,x); &vdGKYs 6  
    %       [theta,r] = cart2pol(X,Y); s) V7$D  
    %       idx = r<=1; #3kR}Amow  
    %       p = 0:15; =!{}:An1$  
    %       z = nan(size(X)); ?#pL\1"E  
    %       y = zernfun2(p,r(idx),theta(idx)); 'e;*V$+  
    %       figure('Units','normalized') 8 l}tYl`|  
    %       for k = 1:length(p) YCw^u  
    %           z(idx) = y(:,k); 47`{ e_YP0  
    %           subplot(4,4,k) akJ{-   
    %           pcolor(x,x,z), shading interp h-lMrI)U?h  
    %           set(gca,'XTick',[],'YTick',[]) 2ZIf@C{P.  
    %           axis square &rcr])jg[  
    %           title(['Z_{' num2str(p(k)) '}']) <adu^5BI  
    %       end o=;.RYi  
    % Fypqf|  
    %   See also ZERNPOL, ZERNFUN. ]$0{PBndW  
    ;)"r^M)):  
    %   Paul Fricker 11/13/2006 W%0-SR  
    }! zjj\g^  
    1hi^  
    % Check and prepare the inputs: nHyWb6  
    % ----------------------------- JXUO?9  
    if min(size(p))~=1 ;bP7|  
        error('zernfun2:Pvector','Input P must be vector.') -_>c P  
    end %b@>riR(y  
    4sNM#]%|  
    if any(p)>35 d 'x;]#S  
        error('zernfun2:P36', ... "pMXTRb  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 8Q#&=]W$  
               '(P = 0 to 35).']) uZ<Bfrc  
    end >tib21*  
    eA{,=, v)  
    % Get the order and frequency corresonding to the function number: m_\CK5T_  
    % ---------------------------------------------------------------- WYEvW<Hv  
    p = p(:); <XCH{Te1  
    n = ceil((-3+sqrt(9+8*p))/2); MW'z*r|,  
    m = 2*p - n.*(n+2); O!Mm~@MoA  
    ' nf"u  
    % Pass the inputs to the function ZERNFUN: gj6"U {D  
    % ---------------------------------------- E%H,Hk^  
    switch nargin nez5z:7F  
        case 3 [r^f5;Z  
            z = zernfun(n,m,r,theta); w$61+KHK  
        case 4 tet  
            z = zernfun(n,m,r,theta,nflag); O}#*U+j  
        otherwise >6jy d{  
            error('zernfun2:nargin','Incorrect number of inputs.') 2S!=2u+7  
    end ;#r tV;  
    mI0| lp 1$  
    % 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) I&1Mh4yu  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. :,GsbNKW  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ?;^_%XSQ*  
    %   order N and frequency M, evaluated at R.  N is a vector of QD\S E  
    %   positive integers (including 0), and M is a vector with the #-e3m/>  
    %   same number of elements as N.  Each element k of M must be a 9;h 1;9sC|  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) -#)xe W.d  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is joM98H@  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix &w/aQs~  
    %   with one column for every (N,M) pair, and one row for every ,:=E+sS  
    %   element in R. (">!vz  
    % li/O&@g`  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- nwO;>Qr  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is f$(w>B7..  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to G=~T)e  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 BT`/O D@  
    %   for all [n,m]. (CuaBHR  
    % 6pr}A  
    %   The radial Zernike polynomials are the radial portion of the {d^&$~  
    %   Zernike functions, which are an orthogonal basis on the unit 9D8el}uHf  
    %   circle.  The series representation of the radial Zernike p?Yovckm  
    %   polynomials is XPWK"t0 1  
    % tw*qlbFHv  
    %          (n-m)/2 0 w@~ynW[  
    %            __ kw=+"U   
    %    m      \       s                                          n-2s QdDdrR^&  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r u\=Nu4)Z F  
    %    n      s=0  $7|0{Dw  
    % }\l5|Ft[!  
    %   The following table shows the first 12 polynomials. 1j0yON  
    % tYmWze. j  
    %       n    m    Zernike polynomial    Normalization PX]A1Kt?  
    %       --------------------------------------------- [@>Kd`!'  
    %       0    0    1                        sqrt(2) 8?I(wn  
    %       1    1    r                           2 jL`S6E?7  
    %       2    0    2*r^2 - 1                sqrt(6) VQqEsnkz  
    %       2    2    r^2                      sqrt(6) j g$%WAEb  
    %       3    1    3*r^3 - 2*r              sqrt(8) B8w 0DJ  
    %       3    3    r^3                      sqrt(8) qx`*]lX  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) o{zo-:>Jp  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) lza'l  
    %       4    4    r^4                      sqrt(10) .&}}ro48  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 7CwG(c/5  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) N],A&}30  
    %       5    5    r^5                      sqrt(12) (Ptv#LSUX  
    %       --------------------------------------------- UyK|KL  
    % w6#hsRq[C  
    %   Example: B8B^@   
    % $!KV]]  
    %       % Display three example Zernike radial polynomials v*3ezf\  
    %       r = 0:0.01:1; _W?}%;  
    %       n = [3 2 5]; K*CO%:,-  
    %       m = [1 2 1]; P8;|>OLZ)  
    %       z = zernpol(n,m,r); C/ ;f)k<  
    %       figure q] ,&$d^@  
    %       plot(r,z) .sjM$#V=  
    %       grid on =I7#Vtd^K<  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') +J+]P\:  
    % m=j7 vb  
    %   See also ZERNFUN, ZERNFUN2. })T_D\2M  
    B6=8cf"i  
    % A note on the algorithm. CQ3;NY=o  
    % ------------------------ ]j_S2lt  
    % The radial Zernike polynomials are computed using the series >Qt#6X|  
    % representation shown in the Help section above. For many special Ybd){Je"z  
    % functions, direct evaluation using the series representation can X3RpJ#m"'  
    % produce poor numerical results (floating point errors), because n%Nf\z  
    % the summation often involves computing small differences between Pi)`[\{  
    % large successive terms in the series. (In such cases, the functions Pme`UcE3H  
    % are often evaluated using alternative methods such as recurrence  l R;<6  
    % relations: see the Legendre functions, for example). For the Zernike O!/J2SfuDH  
    % polynomials, however, this problem does not arise, because the E: XzX Fxx  
    % polynomials are evaluated over the finite domain r = (0,1), and 3- LO  
    % because the coefficients for a given polynomial are generally all {kW!|h&'  
    % of similar magnitude. 37 M7bB0  
    % `2S%l, >)#  
    % ZERNPOL has been written using a vectorized implementation: multiple m6'YFpf)V  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] VbA#D4;  
    % values can be passed as inputs) for a vector of points R.  To achieve 1z[WJ}$u  
    % this vectorization most efficiently, the algorithm in ZERNPOL p?8> 9  
    % involves pre-determining all the powers p of R that are required to Zf(ucAhL  
    % compute the outputs, and then compiling the {R^p} into a single Ig5J_Z^]b  
    % matrix.  This avoids any redundant computation of the R^p, and D~2,0K  
    % minimizes the sizes of certain intermediate variables. 1VJE+3  
    % 183'1Z$KA  
    %   Paul Fricker 11/13/2006 ^B]M- XG  
    }$g5:k!  
    W&Fa8  
    % Check and prepare the inputs: Vs9fAAXS4  
    % ----------------------------- Q"QrbU  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) S}*#$naK  
        error('zernpol:NMvectors','N and M must be vectors.') nLo:\I(  
    end KX`MX5?x  
    63F0Za}h  
    if length(n)~=length(m) 2R|2yAh  
        error('zernpol:NMlength','N and M must be the same length.') bumS>:  
    end FC vR  
    5+,&9;'Y^  
    n = n(:); Dr`A4LnqY  
    m = m(:); :/ "q NPJ  
    length_n = length(n); lc[\ S4  
    Z>Sv[Ec  
    if any(mod(n-m,2)) ?WUu@Z  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') G0a UZCw  
    end ]+fL6"OD/2  
    zb:p,T@5  
    if any(m<0) ~EX/IIa{  
        error('zernpol:Mpositive','All M must be positive.') ]9b*!n<z  
    end MPM_/dn-  
    }ZR3  
    if any(m>n) {?eD7xL:-  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') NV*t  
    end #gi&pR'$  
    bYow EzieF  
    if any( r>1 | r<0 ) t'R&$;z@b  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') `?`\!uP"  
    end ii{5z;I]X  
    Eepy%-\  
    if ~any(size(r)==1) U_oMR$/Z  
        error('zernpol:Rvector','R must be a vector.') 3%k@,Vvt  
    end :c<C;.  
    6VCw>x  
    r = r(:); `[Z?&'CRQ  
    length_r = length(r); 5b|_?Em7  
    njvmf*A?S  
    if nargin==4 +gK7`:v4O*  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ` YIpZ rB  
        if ~isnorm cl14FrpYu  
            error('zernpol:normalization','Unrecognized normalization flag.') e$Md ?Pq  
        end !K6:W1  
    else &eg]8kV  
        isnorm = false; rK)%n!Z  
    end =C5 [75z#+  
    5E}0 <&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MqXA8D  
    % Compute the Zernike Polynomials .>h|e_E  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CDR^xo5 dP  
    DF9Br D0{  
    % Determine the required powers of r: !"p,9  
    % ----------------------------------- /m9t2,KB  
    rpowers = []; D:%$a]_f  
    for j = 1:length(n) H6e ^" E  
        rpowers = [rpowers m(j):2:n(j)]; 85Ms*[g  
    end >TK`s@jdSV  
    rpowers = unique(rpowers); 1!BV]&,[  
    $]Y' [pE@  
    % Pre-compute the values of r raised to the required powers, 9;JU c0%  
    % and compile them in a matrix: RC\TPG/8!  
    % ----------------------------- ZtDHN L  
    if rpowers(1)==0 {s_0[>  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); X9zTz2 Fy  
        rpowern = cat(2,rpowern{:}); Y5Ey%M m6  
        rpowern = [ones(length_r,1) rpowern]; 5%,n[qj4IT  
    else y\)bxmC  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7.akp  
        rpowern = cat(2,rpowern{:}); %Sxy!gGz%%  
    end j+Wgjf  
    aLhTaB-va  
    % Compute the values of the polynomials: vT3LhN+1  
    % -------------------------------------- ~5]AXi'e~  
    z = zeros(length_r,length_n); b @5&<V;r2  
    for j = 1:length_n uodO^5"-  
        s = 0:(n(j)-m(j))/2; xI_WkoI  
        pows = n(j):-2:m(j); QT^( oog=  
        for k = length(s):-1:1 bCA3w%,kM  
            p = (1-2*mod(s(k),2))* ... mQ' ]0DS  
                       prod(2:(n(j)-s(k)))/          ... %l3RM*zb  
                       prod(2:s(k))/                 ... 2yeq2v   
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 4#4kfGoT  
                       prod(2:((n(j)+m(j))/2-s(k))); ]P]lG-  
            idx = (pows(k)==rpowers); 0#<_:E  
            z(:,j) = z(:,j) + p*rpowern(:,idx); OJkPlDym  
        end A+ Z3b:}~  
         69q8t*%O  
        if isnorm "ZT=[&2  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); }L:LcM  
        end *GD 1[:  
    end b*.)m  
    G|Q}.v  
    % 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)  ^//N-?Fx  
    (:[><-h.  
    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。 6^ /C+zuX  
    }i/{8Ou W  
    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)