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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 &U <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! o0SQJ1.a$  
     
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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 Kr;7~`$[  
    function z = zernfun(n,m,r,theta,nflag)  G{4~{{tI  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. S`N_},  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N RU r0K#]  
    %   and angular frequency M, evaluated at positions (R,THETA) on the f76bEe/B9  
    %   unit circle.  N is a vector of positive integers (including 0), and Ds}ctL{6"  
    %   M is a vector with the same number of elements as N.  Each element KN41 kkN  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) f;Cu@z{b  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 47(/K2  
    %   and THETA is a vector of angles.  R and THETA must have the same +x?_\?&Ks  
    %   length.  The output Z is a matrix with one column for every (N,M) fF~3"!1#\I  
    %   pair, and one row for every (R,THETA) pair. wF@mHv  
    % \&|zD"*  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike xKo l  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ?AL;m.X-@  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral fJjtrvNy)  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, /.?m9O^ F  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized > `uk2QdC  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. {e>E4(  
    % #5Zf6w  
    %   The Zernike functions are an orthogonal basis on the unit circle. ]GSs{'Uh B  
    %   They are used in disciplines such as astronomy, optics, and s :4<wmu4=  
    %   optometry to describe functions on a circular domain. `63?FzT y  
    % X?RnP3t~  
    %   The following table lists the first 15 Zernike functions. &n|S:"B  
    % 4sj:%% UE  
    %       n    m    Zernike function           Normalization Wa/&H$d\u@  
    %       -------------------------------------------------- "q-,140_  
    %       0    0    1                                 1 %Pz'D6 /  
    %       1    1    r * cos(theta)                    2 aP%& -W$D|  
    %       1   -1    r * sin(theta)                    2 N[(ovr  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 3]*_*<D  
    %       2    0    (2*r^2 - 1)                    sqrt(3) "cK@Yo  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 4V$DV!dPQ}  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) EPY64 {  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 8SG*7[T7  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) K >-)O=$s  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 3IrmDT  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) zsQhydTR  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) _~^JRC[q  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ka3(sctZ5  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) %J_`-\)"{~  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 2g)W-M  
    %       -------------------------------------------------- %B;e 7 UJ  
    % sz5&P )X  
    %   Example 1: ~ jR:oN  
    % OZHQnvZ  
    %       % Display the Zernike function Z(n=5,m=1) jz\LI  
    %       x = -1:0.01:1; E"EBj7<s  
    %       [X,Y] = meshgrid(x,x); 0K0[mC}ZwM  
    %       [theta,r] = cart2pol(X,Y); [sM~B  
    %       idx = r<=1; ~@3X&E0S  
    %       z = nan(size(X)); QasUgZ  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); d[b(+sHp a  
    %       figure 0st)/\  
    %       pcolor(x,x,z), shading interp [&Kn&bdKW  
    %       axis square, colorbar ?5%0zMC  
    %       title('Zernike function Z_5^1(r,\theta)') O Oa}+^-j  
    % 4 Ar\`{c>  
    %   Example 2: B/*`u  
    % :HDl-8]Lw  
    %       % Display the first 10 Zernike functions dkz79G}e  
    %       x = -1:0.01:1; LI>tN R~  
    %       [X,Y] = meshgrid(x,x); Dm,*G`Js  
    %       [theta,r] = cart2pol(X,Y); kfod[*3  
    %       idx = r<=1; mOLP77(o  
    %       z = nan(size(X)); H;QE',a9+i  
    %       n = [0  1  1  2  2  2  3  3  3  3]; &&N]u e@>  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; R'#[}s  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; _x.!, g{  
    %       y = zernfun(n,m,r(idx),theta(idx)); ur'a{BI2R  
    %       figure('Units','normalized') L_>j SP  
    %       for k = 1:10 ^Fy{Q*p`(  
    %           z(idx) = y(:,k); kc0YWW Q-:  
    %           subplot(4,7,Nplot(k)) ;P` z ?>J:  
    %           pcolor(x,x,z), shading interp $)L=MEdx  
    %           set(gca,'XTick',[],'YTick',[]) ZfzUvN&!  
    %           axis square e}Y|' bG  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ?m)3n0Uh  
    %       end <f l-P  
    % |.A#wjF9  
    %   See also ZERNPOL, ZERNFUN2. @KM !g,f  
    G0Q8"]  
    %   Paul Fricker 11/13/2006 2#sJ`pdQ  
    <X7x  
    &^R0kCF`  
    % Check and prepare the inputs: ryd*Ha">I  
    % ----------------------------- {8NnRnzU  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) )G7")I J/X  
        error('zernfun:NMvectors','N and M must be vectors.') D ^ mfWJS  
    end <2cl1Fb  
    r!qr'Ht<  
    if length(n)~=length(m) mL!)(Bb  
        error('zernfun:NMlength','N and M must be the same length.') 'USol<  
    end 3SRz14/W_R  
    29]T:I1d[  
    n = n(:); oW:p6d  
    m = m(:); u$7o d$&S  
    if any(mod(n-m,2)) k79" xyXX  
        error('zernfun:NMmultiplesof2', ... %R%e0|a  
              'All N and M must differ by multiples of 2 (including 0).') p'lL2 n$E  
    end 1^G*)Qn5Df  
    .xR J )9q  
    if any(m>n) K{]!hm,[3  
        error('zernfun:MlessthanN', ... YlI/~J  
              'Each M must be less than or equal to its corresponding N.') W'Wr8~{h  
    end LwpO_/qV  
    g]^@bxdg  
    if any( r>1 | r<0 ) Z.a`S~U  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') kaSy 9Y{  
    end S#IlWU  
    b' 1%g}  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) [.M<h^xrB  
        error('zernfun:RTHvector','R and THETA must be vectors.') >t-9yO1XQq  
    end VdrqbZ   
    d!+8  
    r = r(:); [:cy.K!Uo%  
    theta = theta(:); h J*2q"  
    length_r = length(r); dLV>FpA\  
    if length_r~=length(theta) 9oOr-9t3  
        error('zernfun:RTHlength', ... #0K122oY  
              'The number of R- and THETA-values must be equal.') !Cq2<[K#  
    end [TUy><Z  
    dQD YN_  
    % Check normalization: u:~2:3B  
    % -------------------- [LDV*79Z  
    if nargin==5 && ischar(nflag) 0 K T.@P  
        isnorm = strcmpi(nflag,'norm'); Z=VAjJ;i[  
        if ~isnorm ZPrL)']  
            error('zernfun:normalization','Unrecognized normalization flag.') ~j%g?;#*  
        end 8lG@8tbW^  
    else E$B7E@(U  
        isnorm = false; EbEQ@6t  
    end rkdf htpI  
    ElJM. a  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MeD}S@H  
    % Compute the Zernike Polynomials ^gP pmb<x  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Y?cdm}:Ou  
    8y9oj9 ;E]  
    % Determine the required powers of r:  T06BrX  
    % ----------------------------------- >HvgU_  
    m_abs = abs(m); q)Qd+:a7{  
    rpowers = []; V`F]L^m=L  
    for j = 1:length(n) PL;PId<9w  
        rpowers = [rpowers m_abs(j):2:n(j)]; wR)U&da`@  
    end 6Fp}U  
    rpowers = unique(rpowers); QWqEe|}6  
    i98>=y~  
    % Pre-compute the values of r raised to the required powers, B=E<</i  
    % and compile them in a matrix: O=2"t%Gc  
    % ----------------------------- 6Vr:?TI7  
    if rpowers(1)==0 8SV.giG;  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); uB;\nj5'D  
        rpowern = cat(2,rpowern{:}); ^[]q/v'3m!  
        rpowern = [ones(length_r,1) rpowern]; ;+d2qbGd  
    else " 3ryp A  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); s L;  
        rpowern = cat(2,rpowern{:}); ]r]=Q"/5  
    end ~ ZkSYW<  
    \Y37wy4  
    % Compute the values of the polynomials: F+%6?2 J  
    % -------------------------------------- HF(pC7/a:  
    y = zeros(length_r,length(n)); b FV+|0  
    for j = 1:length(n) 6V[ce4a%  
        s = 0:(n(j)-m_abs(j))/2; wH?r522`c  
        pows = n(j):-2:m_abs(j); }6U`/"RfcO  
        for k = length(s):-1:1 pDw^~5P  
            p = (1-2*mod(s(k),2))* ... c34s(>AC  
                       prod(2:(n(j)-s(k)))/              ... WA~PE` U  
                       prod(2:s(k))/                     ... {jnfe}]  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Me*woCos'  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); eSAB :L,K  
            idx = (pows(k)==rpowers); /UwB6s(  
            y(:,j) = y(:,j) + p*rpowern(:,idx); l1<]pdLTR  
        end \FE  
         # Uc0 W  
        if isnorm #3fS_;G  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); w6b\l1Z  
        end #*J+4a w3  
    end `5J`<BPs  
    % END: Compute the Zernike Polynomials u 2)#Ml  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% OI@;ffHSW  
    G@Jl4iHug"  
    % Compute the Zernike functions: @;^7kt  
    % ------------------------------ C rA7lu'  
    idx_pos = m>0; u~JCMM$  
    idx_neg = m<0; !(%^Tg=  
    p\>im+0oh  
    z = y; z8MKGM  
    if any(idx_pos) bcVzl]9  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ZvQ~K(3  
    end khXp}p!Zm  
    if any(idx_neg) f( %r)%  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 7v{X?86&  
    end `W& :*  
    } `X.^}oe  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) E0SP  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. yf lt2 R  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated lZ\Si  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive (toN? ?r  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ~F{u4p7{N  
    %   and THETA is a vector of angles.  R and THETA must have the same KS9 e V  
    %   length.  The output Z is a matrix with one column for every P-value, vX9B^W||x  
    %   and one row for every (R,THETA) pair. 5O7 x4bY  
    % Boi?Bt  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike |aaoi4OJ  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 31FQ=(K  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Pc{0Js5VzE  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 A_:YpQ07@  
    %   for all p. C>A*L4c]F  
    % r@|{mQOxa  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 c@uNA0 p  
    %   Zernike functions (order N<=7).  In some disciplines it is .lcI"%>  
    %   traditional to label the first 36 functions using a single mode MDyPwv\  
    %   number P instead of separate numbers for the order N and azimuthal P S$6`6G  
    %   frequency M. SK>*tKY  
    % i&%/]Nq  
    %   Example: t V]BcDp  
    %  e>FK5rz  
    %       % Display the first 16 Zernike functions ,hggmzA~  
    %       x = -1:0.01:1; [6$n  
    %       [X,Y] = meshgrid(x,x); GfG!CG^ %  
    %       [theta,r] = cart2pol(X,Y); qh40nqS;9  
    %       idx = r<=1; O<H5W|cM  
    %       p = 0:15; wM2[i  
    %       z = nan(size(X)); >f !  
    %       y = zernfun2(p,r(idx),theta(idx)); *j`{ K  
    %       figure('Units','normalized') Fq-A vU  
    %       for k = 1:length(p) oD@~wcMIT0  
    %           z(idx) = y(:,k); bPe|/wp  
    %           subplot(4,4,k) ^hMJNy&R  
    %           pcolor(x,x,z), shading interp pOe"S  
    %           set(gca,'XTick',[],'YTick',[]) mvCH$}w8&  
    %           axis square RKt#2%FFO  
    %           title(['Z_{' num2str(p(k)) '}'])  hxedQvW  
    %       end aYmC LLj  
    % pyf/%9R:d  
    %   See also ZERNPOL, ZERNFUN. NI1jJfH|l  
    &B;M.sz~C4  
    %   Paul Fricker 11/13/2006 figCeJ!W4  
    8}Qmhm`_j=  
    @77%15_Jz  
    % Check and prepare the inputs: `Tt;)D  
    % ----------------------------- t/3t69\x  
    if min(size(p))~=1 t:SME'~.P  
        error('zernfun2:Pvector','Input P must be vector.') k9'`<82Y  
    end NJe^5>4`  
    aj$#8l |zu  
    if any(p)>35 '5*8'.4Sy  
        error('zernfun2:P36', ... sXpA^pT"T  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... <z=d5g{n  
               '(P = 0 to 35).']) F/QRgXV  
    end #cZ<[K q6  
    +ROwk  
    % Get the order and frequency corresonding to the function number: JIA'3"C  
    % ---------------------------------------------------------------- l-} );zH74  
    p = p(:); :'F7^N3;H  
    n = ceil((-3+sqrt(9+8*p))/2); 7a<-}>sU  
    m = 2*p - n.*(n+2); 8,l~e8&  
    zS6oz=  
    % Pass the inputs to the function ZERNFUN: ]{/1F:bcQ  
    % ---------------------------------------- uxKj7!(#  
    switch nargin j~ 'a %P  
        case 3 C.& R,$  
            z = zernfun(n,m,r,theta); 0+vt LDq@P  
        case 4 Y >83G`*}b  
            z = zernfun(n,m,r,theta,nflag); y\M Kd[G7  
        otherwise +W8L^Wl  
            error('zernfun2:nargin','Incorrect number of inputs.') j\uh]8N3<  
    end `F^~*FnR,B  
    :O~*}7G  
    % 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) ^qro0]"LD  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. $1F$3"k  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of lO>9Q]S<  
    %   order N and frequency M, evaluated at R.  N is a vector of 3utv  
    %   positive integers (including 0), and M is a vector with the 6N<v&7cSB  
    %   same number of elements as N.  Each element k of M must be a ],3#[n[ m  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 8q5 `A Gl  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is JSAbh\Mq6  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix s[}4Q|s%  
    %   with one column for every (N,M) pair, and one row for every bh~"LQS1  
    %   element in R. )yj:P  
    % }=fVO<R v  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- N Qdz]o  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is _?YP0GpU  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to NO%x 2dx0  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 I8s%wY9  
    %   for all [n,m]. z5|m`$gy  
    % *>#mI/#}  
    %   The radial Zernike polynomials are the radial portion of the )^)j=xs  
    %   Zernike functions, which are an orthogonal basis on the unit e-`=?tct  
    %   circle.  The series representation of the radial Zernike ! /qQ:k-.  
    %   polynomials is Ul`~d !3zH  
    % 'PBuf:9lN  
    %          (n-m)/2 0&@pD`K e  
    %            __ :=J^"c  
    %    m      \       s                                          n-2s uYu/0fQD  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Jj :Bi&C  
    %    n      s=0 UgBD| ~zu  
    % 0YApaL+jt  
    %   The following table shows the first 12 polynomials. _x&fK$Y)B  
    % 6bacU#0o  
    %       n    m    Zernike polynomial    Normalization "{lw;AA5F  
    %       --------------------------------------------- it\U+xu  
    %       0    0    1                        sqrt(2) E;>Bc Pt5  
    %       1    1    r                           2 l?rT_uO4  
    %       2    0    2*r^2 - 1                sqrt(6) ku&m)'  
    %       2    2    r^2                      sqrt(6) j/Dc';,d.(  
    %       3    1    3*r^3 - 2*r              sqrt(8) '/[9Xwh9  
    %       3    3    r^3                      sqrt(8) v Wt{kg;  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) kR1dk4I4  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) uh8+Y%V p  
    %       4    4    r^4                      sqrt(10) .R<Ke\y/  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 5'mpd  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) j0eGg::  
    %       5    5    r^5                      sqrt(12) p`CVq`k  
    %       --------------------------------------------- 1/l;4~p7'  
    % d ~`_;.z  
    %   Example: %{sL/H_  
    % ;#>,eD2u  
    %       % Display three example Zernike radial polynomials )= :gO`"D  
    %       r = 0:0.01:1; &AS<2hB  
    %       n = [3 2 5]; K5ywO8_6`  
    %       m = [1 2 1]; j&qJK,~  
    %       z = zernpol(n,m,r); @=0O' XM  
    %       figure `Qrrnq  
    %       plot(r,z) UojHlTg#bT  
    %       grid on -y+u0,=p.  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 4NN81~v 4  
    % 2^TJ_xG~  
    %   See also ZERNFUN, ZERNFUN2. [:MpOl-KIz  
    `"#0\Wh  
    % A note on the algorithm. Bp.z6x4  
    % ------------------------ Y unY'xY  
    % The radial Zernike polynomials are computed using the series !6 k{]v  
    % representation shown in the Help section above. For many special {Yp;R  
    % functions, direct evaluation using the series representation can '~Z#h  P  
    % produce poor numerical results (floating point errors), because MUs~ZF  
    % the summation often involves computing small differences between DOzJ-uww1  
    % large successive terms in the series. (In such cases, the functions R06zca  
    % are often evaluated using alternative methods such as recurrence Kr*s]O  
    % relations: see the Legendre functions, for example). For the Zernike OG C|elSM  
    % polynomials, however, this problem does not arise, because the R\+O.vX  
    % polynomials are evaluated over the finite domain r = (0,1), and _&~y{;)S  
    % because the coefficients for a given polynomial are generally all `B4Px|3  
    % of similar magnitude. G|"`kAa  
    % c/g"/ICs  
    % ZERNPOL has been written using a vectorized implementation: multiple cHG>iW9C  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] @6~OQN  
    % values can be passed as inputs) for a vector of points R.  To achieve ~Xf&<&5d T  
    % this vectorization most efficiently, the algorithm in ZERNPOL `c-(1 ;Jb  
    % involves pre-determining all the powers p of R that are required to o (OC3  
    % compute the outputs, and then compiling the {R^p} into a single 6kc/  
    % matrix.  This avoids any redundant computation of the R^p, and %NI'PXpI  
    % minimizes the sizes of certain intermediate variables. 0aF&5Lk`y  
    % wU|Y`wJmF  
    %   Paul Fricker 11/13/2006 !{L6 4qI  
    lYz$~/sd  
    x!<?/I)X  
    % Check and prepare the inputs: r$7D;>*O{  
    % ----------------------------- NVFgRJ&  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ke(LjRS  
        error('zernpol:NMvectors','N and M must be vectors.') SLiQHWw*J  
    end O 0lQ1<=  
    W9$mgs=S`E  
    if length(n)~=length(m) |0wUOs*5  
        error('zernpol:NMlength','N and M must be the same length.') >H ,t^i}@  
    end >#MGGCGL  
    Ef}rMkv  
    n = n(:); -ty_<m]  
    m = m(:); |c]Y1WwDx  
    length_n = length(n); t-vH\m  
    &f\ng{  
    if any(mod(n-m,2)) Xu1tN9:oE  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') f y|Ae  
    end 05<MsxB"w  
    qX(sx2TK  
    if any(m<0) bB^SD] }C  
        error('zernpol:Mpositive','All M must be positive.') ^c9~~m16+  
    end \\qw"w9  
    D;V FM P  
    if any(m>n) U 9?!|h;7  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') *(~=L%s  
    end iCouGd}  
    XG5mfKMt+  
    if any( r>1 | r<0 ) 8: KlU(J  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') #nL&x3  
    end UeVRd  
    r[:)-`]b  
    if ~any(size(r)==1) sQT0y(FW  
        error('zernpol:Rvector','R must be a vector.') C?Sy90f  
    end ]i=\5FH e  
    S*o%#ZJN  
    r = r(:); (wNL,<%~  
    length_r = length(r); r9/PmZo4x  
    0<+=Ew5Z  
    if nargin==4 OG_2k3v  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); x)h p3&L  
        if ~isnorm &b 2Vt  
            error('zernpol:normalization','Unrecognized normalization flag.') aF:_1. LC  
        end <B;l).[6  
    else uC>X;<^   
        isnorm = false; 3B(6^iS  
    end o)P'H"Ki  
    > dJvl|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S-"&#OfWg<  
    % Compute the Zernike Polynomials G<C[A  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Tsez&R$k  
    "%aJ 'l2  
    % Determine the required powers of r: -Fodqq@,  
    % ----------------------------------- +jFcq:`#UG  
    rpowers = []; zR'lQ<u  
    for j = 1:length(n) bBkF,`/f$  
        rpowers = [rpowers m(j):2:n(j)]; 9L}=xX`>?  
    end |pv:'']J  
    rpowers = unique(rpowers); hsVf/%  
    ;}b.gpG  
    % Pre-compute the values of r raised to the required powers, 9PA\Eo|Yb  
    % and compile them in a matrix: /q4<ZS#  
    % ----------------------------- v1yNVs \}  
    if rpowers(1)==0 Z-RgN  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); slV+2b  
        rpowern = cat(2,rpowern{:}); 'AX/?Srd  
        rpowern = [ones(length_r,1) rpowern]; Uhc2`r#q  
    else -{i;!XE$SR  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 6@-VLO))O  
        rpowern = cat(2,rpowern{:}); Y"&&=M#  
    end QvK-3w;=  
    %aU4d e^  
    % Compute the values of the polynomials: ">}l8MA  
    % -------------------------------------- W6t"n_%?"  
    z = zeros(length_r,length_n); Kb~s'cTxIO  
    for j = 1:length_n )+c4n]  
        s = 0:(n(j)-m(j))/2; RLN>*X  
        pows = n(j):-2:m(j); )miY>7K  
        for k = length(s):-1:1 GZ# 6}/;b  
            p = (1-2*mod(s(k),2))* ... gG0P &9xz  
                       prod(2:(n(j)-s(k)))/          ... k=j--`$8k  
                       prod(2:s(k))/                 ... k&4@$;Ap  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... n$Z@7r  
                       prod(2:((n(j)+m(j))/2-s(k))); TY[1jW~{r  
            idx = (pows(k)==rpowers); %D|27gh  
            z(:,j) = z(:,j) + p*rpowern(:,idx); R9o3T)9V  
        end F#KO!\iA+  
         D!kv+<+  
        if isnorm 0wV!mC  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); O1pBr=+j+{  
        end pOlo_na}[  
    end P8DY*B k  
    l@Vl^f~P  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  QP f*!E  
    heJI5t,  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 S~L$sqt  
    [:<CgU9C  
    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)