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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 (|F} B  
    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! c~1X/,biA  
     
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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 cmzu @zq  
    function z = zernfun(n,m,r,theta,nflag) LEq"g7YH  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. W;Rx(o>  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N {M7`z,,[  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 'E4`qq  
    %   unit circle.  N is a vector of positive integers (including 0), and (6aSDx Sc  
    %   M is a vector with the same number of elements as N.  Each element \k#|5W  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 4>uy+"8PO  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, b.`<T "y  
    %   and THETA is a vector of angles.  R and THETA must have the same },"T,t#  
    %   length.  The output Z is a matrix with one column for every (N,M) SNV[KdvP*  
    %   pair, and one row for every (R,THETA) pair. ,ZpcvK/S  
    % G2bZl% ,D  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike !J5k?J&{=  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), cB;:}Q08#  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral n~1'M/wh  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +0w~Skd,  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized {*=+g>R gD  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. \(nb >K  
    % h6*&1r  
    %   The Zernike functions are an orthogonal basis on the unit circle. hmA$gR_  
    %   They are used in disciplines such as astronomy, optics, and ?e`4 s f_~  
    %   optometry to describe functions on a circular domain. )yV|vn  
    % %:v59:i}  
    %   The following table lists the first 15 Zernike functions. hPC t-  
    % ){AtV&{$  
    %       n    m    Zernike function           Normalization x>>#<hOz[  
    %       -------------------------------------------------- *4i)aj  
    %       0    0    1                                 1 L[]*vj   
    %       1    1    r * cos(theta)                    2 vhw"Nl  
    %       1   -1    r * sin(theta)                    2 0XrB+nt  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) *V\z]Dy-[  
    %       2    0    (2*r^2 - 1)                    sqrt(3) rjmKe*_1V  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) [79 eq=  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) e}x}Fj</(  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) (xp<@-  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) DFgr,~  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) >m}U|#;W  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Yy 4EM  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `1cGb*b/  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) AL%gqt]  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^2gDhoO_  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) [0_JS2KE  
    %       -------------------------------------------------- 6sRe. ct<  
    % ? m$uqi  
    %   Example 1: ]L'FYOfrpx  
    % qf/1a CQiP  
    %       % Display the Zernike function Z(n=5,m=1) \9U4V>p  
    %       x = -1:0.01:1; 9;Z2.P"w  
    %       [X,Y] = meshgrid(x,x); }PZz(Ms  
    %       [theta,r] = cart2pol(X,Y); @%4MFc0`!  
    %       idx = r<=1; M*DFtp<  
    %       z = nan(size(X)); \JJ>y  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); b-/ztZ@u  
    %       figure =j)y.x(  
    %       pcolor(x,x,z), shading interp T6X%.tR>`  
    %       axis square, colorbar j3Ng] @N  
    %       title('Zernike function Z_5^1(r,\theta)') ]gN]Cw\L  
    % wzw`9^B  
    %   Example 2: 64Ot`=A"  
    % Hpo/CY/  
    %       % Display the first 10 Zernike functions ]dXHjOpA  
    %       x = -1:0.01:1; omxBd#;F$  
    %       [X,Y] = meshgrid(x,x); A),nkw0X  
    %       [theta,r] = cart2pol(X,Y); 2<d l23  
    %       idx = r<=1; br!:g]Vh  
    %       z = nan(size(X)); r ]XXN2[jO  
    %       n = [0  1  1  2  2  2  3  3  3  3]; T5mdC  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ;Nw.  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; mhp&; Q9  
    %       y = zernfun(n,m,r(idx),theta(idx)); }3tbqFiH  
    %       figure('Units','normalized') ?/mkFDN  
    %       for k = 1:10 ryz [A:^G  
    %           z(idx) = y(:,k); OSQt:58K  
    %           subplot(4,7,Nplot(k)) _1z|QC  
    %           pcolor(x,x,z), shading interp L*ZC` .h  
    %           set(gca,'XTick',[],'YTick',[]) ];bl;BP  
    %           axis square rm7$i9DH2  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ; Q-f6)+&  
    %       end )P6n,\  
    % gTI!b  
    %   See also ZERNPOL, ZERNFUN2. b\/:-][  
    )4d)G5{  
    %   Paul Fricker 11/13/2006 3Lxk7D>0c  
    O[p;IG`  
    G)(\!0pNZ  
    % Check and prepare the inputs: ],*^wQ   
    % ----------------------------- _":yUa0D  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Cdjh/+!f  
        error('zernfun:NMvectors','N and M must be vectors.') > ,L'A;c}  
    end Mr}]P(4h  
    cD-.thHO  
    if length(n)~=length(m) Luxo,Ve  
        error('zernfun:NMlength','N and M must be the same length.') 9N9dQ}[:g  
    end \NYtxGV[Z  
    1Aq*|JSk(  
    n = n(:); F+;{s(wx  
    m = m(:); *}9i@DP1,  
    if any(mod(n-m,2)) yVThbL_YJ  
        error('zernfun:NMmultiplesof2', ... o E+s8Q  
              'All N and M must differ by multiples of 2 (including 0).') Mis t,H7  
    end =<-tD<  
    |Rr^K5hmD  
    if any(m>n) zcrLd={  
        error('zernfun:MlessthanN', ... 0B(<I?a/  
              'Each M must be less than or equal to its corresponding N.') %0]vW;Q5  
    end (wmMHo|  
    WA-` *m$v  
    if any( r>1 | r<0 ) I*e8 5wef  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') @l9qH1  
    end k^q}F%UV  
    Jji~MiMn  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) e&dE>m  
        error('zernfun:RTHvector','R and THETA must be vectors.') ; 6Wlu3I  
    end [Rh[Z# 6  
    9I>+Q&   
    r = r(:); :$&%Pxm  
    theta = theta(:); qC9$xIWq  
    length_r = length(r); |]a =He;  
    if length_r~=length(theta) q# W|*kL3  
        error('zernfun:RTHlength', ... L&1VPli  
              'The number of R- and THETA-values must be equal.') QDlEby m  
    end !g /&ws&  
    EG5'kYw2  
    % Check normalization: q<>  
    % -------------------- `nc cRy< l  
    if nargin==5 && ischar(nflag) wiWpzJz  
        isnorm = strcmpi(nflag,'norm'); Hz$l)g}U  
        if ~isnorm _8C0z=hz  
            error('zernfun:normalization','Unrecognized normalization flag.') = GirUW D  
        end `fEB,0j^  
    else \oF79   
        isnorm = false; @;}bBHQz{p  
    end :+ef|,:`/  
    (: IUg   
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jsS xjf;O  
    % Compute the Zernike Polynomials :>to?~Z1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @sly-2{e1  
    - |mWi  
    % Determine the required powers of r: &H!3]  
    % ----------------------------------- P  F!S  
    m_abs = abs(m); f[3DKA  
    rpowers = []; $ WAFr  
    for j = 1:length(n) .$+]N[-=  
        rpowers = [rpowers m_abs(j):2:n(j)]; OKfJ  
    end Ec| Gom?  
    rpowers = unique(rpowers); u-Pa:wm0-  
    orn9;|8q  
    % Pre-compute the values of r raised to the required powers, wZVY h  
    % and compile them in a matrix: ZdHfZ3)dB  
    % ----------------------------- PL/as3O^A  
    if rpowers(1)==0 3vPb}  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); #q1Qa_LXc  
        rpowern = cat(2,rpowern{:}); uR{HCZ-  
        rpowern = [ones(length_r,1) rpowern]; #%k!`?^fbK  
    else 2"lDKjj  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )xiiTkJd5  
        rpowern = cat(2,rpowern{:}); E4RvVfA0F  
    end V6"<lK8"  
    a'w~7y!}  
    % Compute the values of the polynomials: M}NmA  
    % -------------------------------------- ?Y2ZqI  
    y = zeros(length_r,length(n)); `e]L.P_e?  
    for j = 1:length(n) O(;K ]8  
        s = 0:(n(j)-m_abs(j))/2; Y - 6 ?x  
        pows = n(j):-2:m_abs(j); D.o|pTZ  
        for k = length(s):-1:1 Vh^fbv`?  
            p = (1-2*mod(s(k),2))* ... kM5N#|!  
                       prod(2:(n(j)-s(k)))/              ... 2?ac\c6"  
                       prod(2:s(k))/                     ... Z<ozANbk  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... J@Eqqyf"  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); vJDK]p<}  
            idx = (pows(k)==rpowers); %pgie"k   
            y(:,j) = y(:,j) + p*rpowern(:,idx); ~U`oew  
        end 2yR*<yj  
         /2-S/,a  
        if isnorm / <WB%O  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); <b>@'\w9  
        end 8<M'~G%CEq  
    end . $uvQpyh  
    % END: Compute the Zernike Polynomials LAeJz_9U  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AEx I!  
    }$3eRu +  
    % Compute the Zernike functions: x/Se /C  
    % ------------------------------ #+HLb  
    idx_pos = m>0; xRYL{+  
    idx_neg = m<0; Xu`c_  
    $j:$ `  
    z = y; xy$73K6  
    if any(idx_pos) 'gk.J  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ,ii*[{X?  
    end Sj;B1&  
    if any(idx_neg) %"PG/avo  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ?D-1xnxep  
    end \UM9cAX`  
    >k,|N4(  
    % 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) sS;6QkI"y  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. `<!Nk^2ap  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated G'IqAKJ  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive jY%&G#4  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, /!;oO_U:#  
    %   and THETA is a vector of angles.  R and THETA must have the same h\\fb[``  
    %   length.  The output Z is a matrix with one column for every P-value, Z"PPXv-<jY  
    %   and one row for every (R,THETA) pair. k `JP  
    % B.CUk.  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Pe6}y  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) !E!i`yF  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) y\K r@;q0w  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 IWcgh`8  
    %   for all p. YXI_ '  
    % .zQ:u{FT  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 IvGQ7 VLr  
    %   Zernike functions (order N<=7).  In some disciplines it is wBZ=IMDu\  
    %   traditional to label the first 36 functions using a single mode |N_tVE  
    %   number P instead of separate numbers for the order N and azimuthal 2g5i3C.q$  
    %   frequency M. 4-;"w;  
    % Fw5|_@&k  
    %   Example: 0JzH dz  
    % %@ UH,Ew  
    %       % Display the first 16 Zernike functions Q^X  
    %       x = -1:0.01:1; ap=m5h27  
    %       [X,Y] = meshgrid(x,x); `i5U&K. 7  
    %       [theta,r] = cart2pol(X,Y); WLl_;BgN  
    %       idx = r<=1; TI4#A E  
    %       p = 0:15; ~!UC:&UKo  
    %       z = nan(size(X)); `G*7y7  
    %       y = zernfun2(p,r(idx),theta(idx)); (5- w>(  
    %       figure('Units','normalized') ]&6# {I-  
    %       for k = 1:length(p) _5TSI'@.4  
    %           z(idx) = y(:,k); ,Y *unk<S  
    %           subplot(4,4,k) FjRt'  
    %           pcolor(x,x,z), shading interp <=.0 P/N  
    %           set(gca,'XTick',[],'YTick',[]) Aq' yr,  
    %           axis square ^mxOQc !  
    %           title(['Z_{' num2str(p(k)) '}']) Zjqa n  
    %       end x` T  
    % xCN6?  
    %   See also ZERNPOL, ZERNFUN. TNe,'S,%  
    S_; 5mb+b  
    %   Paul Fricker 11/13/2006 oSVo~F  
    8K+(CS>xvO  
    $A~UA  
    % Check and prepare the inputs: 8B#;ffkmN  
    % ----------------------------- 9V&} %  
    if min(size(p))~=1 Z] ?Tx2|7  
        error('zernfun2:Pvector','Input P must be vector.') O/g|E47  
    end PWeCk2xH  
    ZK:dhwer  
    if any(p)>35 iMG)zPj  
        error('zernfun2:P36', ... od~^''/b  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... y cYT1Sg 8  
               '(P = 0 to 35).']) s18o,Zs'  
    end \H<gKZquR  
    W P7RX|7  
    % Get the order and frequency corresonding to the function number: 'Cv,:Q  
    % ---------------------------------------------------------------- +m?;,JGt  
    p = p(:); =&+]>g{T  
    n = ceil((-3+sqrt(9+8*p))/2); oh*Hzb  
    m = 2*p - n.*(n+2); HIiMq'H^  
    Br/qOO:n$}  
    % Pass the inputs to the function ZERNFUN: \s_lB~"P!3  
    % ---------------------------------------- & gF*p  
    switch nargin s5/5>a V  
        case 3 PJd7t% m;  
            z = zernfun(n,m,r,theta); t][U`1>i  
        case 4 A8,9^cQ]  
            z = zernfun(n,m,r,theta,nflag); $cl[Qcw  
        otherwise K:!){a[  
            error('zernfun2:nargin','Incorrect number of inputs.') -CV_yySc  
    end Xjb 4dip  
    s|er+-'  
    % 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) Za,MzKd=  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. I+JWDYk  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ku2g FO  
    %   order N and frequency M, evaluated at R.  N is a vector of oJ\)-qSf  
    %   positive integers (including 0), and M is a vector with the TcB^Sctf  
    %   same number of elements as N.  Each element k of M must be a \.0cA4)[$  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) m(2(Caz{  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is NO$n-<ag  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Ufx^@%v  
    %   with one column for every (N,M) pair, and one row for every 2bJqZ,@  
    %   element in R. K)-Gv|*t  
    % N=2BrKb)o  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ! z!lQ~  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 01N]|F:  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to O $dcy!  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 )gX7qQ  
    %   for all [n,m]. @B.;V=8wJ  
    % ? PIq/[tk  
    %   The radial Zernike polynomials are the radial portion of the ';H"Ye:D=7  
    %   Zernike functions, which are an orthogonal basis on the unit ~* R:UTBtw  
    %   circle.  The series representation of the radial Zernike ^Rel-=Z$B  
    %   polynomials is yQP!Vt^  
    % !yUn|v>&p  
    %          (n-m)/2  uj8G6'm%  
    %            __ J A=9EnTU  
    %    m      \       s                                          n-2s 72i ]`   
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r R,Gr{"H  
    %    n      s=0 Sn o7Ru2  
    % ;HKb  
    %   The following table shows the first 12 polynomials. v~l_6V}  
    % 33R1<dRk  
    %       n    m    Zernike polynomial    Normalization 1\'?.  
    %       --------------------------------------------- 3Jt7IM!9[  
    %       0    0    1                        sqrt(2) WA'&0i4  
    %       1    1    r                           2 m=s aUhI*9  
    %       2    0    2*r^2 - 1                sqrt(6) th!$R  
    %       2    2    r^2                      sqrt(6) ZQL4<fy'E  
    %       3    1    3*r^3 - 2*r              sqrt(8) "ITC P<+  
    %       3    3    r^3                      sqrt(8) y15 MWZ  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) K;n2mXYGM  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ^Vbx9UN/  
    %       4    4    r^4                      sqrt(10) L&gC  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) mbf'xGO  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) i146@<\G{P  
    %       5    5    r^5                      sqrt(12) &1=Je$,  
    %       --------------------------------------------- d65fkz==A)  
    % }Q }&3m~g  
    %   Example: bCV3h3<  
    % ]q,5'[=~4h  
    %       % Display three example Zernike radial polynomials J 8!D."'Q0  
    %       r = 0:0.01:1; 'zRi ;:UHA  
    %       n = [3 2 5]; o {W4@:Ib  
    %       m = [1 2 1]; /mi9 q  
    %       z = zernpol(n,m,r); Z.4 vKO[<  
    %       figure uTWij4)a  
    %       plot(r,z) n]G_# ;  
    %       grid on 9s#Q[\B!  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') iRbTH}4i  
    % z%4E~u10  
    %   See also ZERNFUN, ZERNFUN2. /w!!jj^  
    )td?t.4  
    % A note on the algorithm. N5ph70#y3  
    % ------------------------ )aV\=a |A  
    % The radial Zernike polynomials are computed using the series 5s5GBJ?  
    % representation shown in the Help section above. For many special AQUl:0!  
    % functions, direct evaluation using the series representation can P#0U[`ltK  
    % produce poor numerical results (floating point errors), because Z+gG.|"k  
    % the summation often involves computing small differences between %^`b)   
    % large successive terms in the series. (In such cases, the functions *A^j>lV  
    % are often evaluated using alternative methods such as recurrence 34D7qR  
    % relations: see the Legendre functions, for example). For the Zernike v$WH#;(\  
    % polynomials, however, this problem does not arise, because the ].TAZ-4s  
    % polynomials are evaluated over the finite domain r = (0,1), and bNH72gX2Yh  
    % because the coefficients for a given polynomial are generally all +vDEDOS1  
    % of similar magnitude. 4 6yq F  
    % uVSc1 MS1  
    % ZERNPOL has been written using a vectorized implementation: multiple 0mb|JoE(  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] _`oP*g =  
    % values can be passed as inputs) for a vector of points R.  To achieve B}:/2?gQ  
    % this vectorization most efficiently, the algorithm in ZERNPOL 6~oo.6bA  
    % involves pre-determining all the powers p of R that are required to $] gwaJ:  
    % compute the outputs, and then compiling the {R^p} into a single =\QKzQ'BC  
    % matrix.  This avoids any redundant computation of the R^p, and +i\ +bR  
    % minimizes the sizes of certain intermediate variables. lc:dKGF6  
    % 3 L:s5  
    %   Paul Fricker 11/13/2006 wt@Qjbqd8  
    `<?{%ja  
    J~e%EjN5e  
    % Check and prepare the inputs: 5xS ze;  
    % ----------------------------- `wMHjcUP  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) "X[sW%# F  
        error('zernpol:NMvectors','N and M must be vectors.') V(?PKb-w)  
    end 5PcN$r"P  
    <n+]\a97*  
    if length(n)~=length(m) ]* #k|>Fl  
        error('zernpol:NMlength','N and M must be the same length.') S-5|t]LV  
    end 9s.x%m,  
    Pse1NMK9 [  
    n = n(:); 0\y{/P?I$  
    m = m(:); hja;d1yH  
    length_n = length(n); <[oPh(!V  
    \{GBaMwG~  
    if any(mod(n-m,2)) x;w^&<hQ\  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') tM% f#O  
    end l DWg%pI+  
    =V[ey  
    if any(m<0) l%f &vOcd  
        error('zernpol:Mpositive','All M must be positive.') I,nW~;OV0  
    end nt5x[xa  
    %k"qpu  
    if any(m>n) pA%Sybw+  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 1DTA Dh0  
    end pBbfU2p  
    TwaK>t96[  
    if any( r>1 | r<0 ) l>kREfHq!{  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 6m\MYay  
    end 6-+q3#e  
    <mk'n6B  
    if ~any(size(r)==1) AB:JXMyK  
        error('zernpol:Rvector','R must be a vector.') O^2@9 w  
    end 3j h: K   
    @[=K`n:n_  
    r = r(:); Eq\PSa=gz  
    length_r = length(r); D,c53B6M  
    `w;8xD(  
    if nargin==4 v90)G8|q  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); K:cZ q3F  
        if ~isnorm y$Y*%D^w  
            error('zernpol:normalization','Unrecognized normalization flag.') Twi7g3}/jB  
        end $ Ith8p~  
    else &yabxl_  
        isnorm = false; Ld9YbL:  
    end A><q-`bw  
    p-S&Wq  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :g+5cs  
    % Compute the Zernike Polynomials nG~#o  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'LyEdlC]  
    70MSP;^  
    % Determine the required powers of r: }j_2K1NS{  
    % ----------------------------------- p!_3j^"{  
    rpowers = []; Ocp`6Fj  
    for j = 1:length(n) C-:lM1  
        rpowers = [rpowers m(j):2:n(j)]; q; n  
    end RP9jZRDbZ  
    rpowers = unique(rpowers); ) u(Gf*t  
    %Ums'<xJ  
    % Pre-compute the values of r raised to the required powers, !,0%ZG}]7  
    % and compile them in a matrix: e*Gt%'  
    % ----------------------------- vUNmN2pRJ  
    if rpowers(1)==0 ->rr4xaKC  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); iL6Yk @  
        rpowern = cat(2,rpowern{:}); ~ZuFMVR  
        rpowern = [ones(length_r,1) rpowern]; 2x<A7l)6  
    else M#CYDEB  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);  <j<V{Wc  
        rpowern = cat(2,rpowern{:}); bgNN0,+8  
    end ~M !9E])  
    |RS(QU<QE  
    % Compute the values of the polynomials: ~B2,edkM  
    % -------------------------------------- |3:e$  
    z = zeros(length_r,length_n); brpsZU  
    for j = 1:length_n x&4gy%b  
        s = 0:(n(j)-m(j))/2; pFW^   
        pows = n(j):-2:m(j); (]}52%~  
        for k = length(s):-1:1 u;/5@ADW  
            p = (1-2*mod(s(k),2))* ... tF&g3)D:NV  
                       prod(2:(n(j)-s(k)))/          ... K K_  
                       prod(2:s(k))/                 ... 6oA2"!u^w  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ,'%wadOo  
                       prod(2:((n(j)+m(j))/2-s(k))); "pa2,-&  
            idx = (pows(k)==rpowers); _ mJP=+i  
            z(:,j) = z(:,j) + p*rpowern(:,idx); x,rK4L7U  
        end j YVR"D;  
         !C3ozZ<  
        if isnorm &9ZrZ"]  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); `m_ ('N  
        end Gdu5 &]H#6  
    end E8LZ% N#  
    L8WYxJ k  
    % 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)  U(:Di]>{  
    9GtVcucN  
    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。 Cs'<;|r(  
    a`6R}|ZB  
    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)