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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 5!wjYQt3  
    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! ?=1i:h  
     
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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 &" 5Yt&{  
    function z = zernfun(n,m,r,theta,nflag) ?5^DQ|Hg ^  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 3qDbfO[  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N )c 79&S  
    %   and angular frequency M, evaluated at positions (R,THETA) on the m( %PZ*s  
    %   unit circle.  N is a vector of positive integers (including 0), and +D[C.is>]}  
    %   M is a vector with the same number of elements as N.  Each element b2j ~"9  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) I]pz3!On4,  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, obv_?i1  
    %   and THETA is a vector of angles.  R and THETA must have the same X`-o0HG  
    %   length.  The output Z is a matrix with one column for every (N,M) k! x`cp  
    %   pair, and one row for every (R,THETA) pair. ixoN#'y<"  
    % p;D {?H/  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 'F:Tv[qx  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), w4&\-S#  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral i[z#5;x+<  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Bt1v7M  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized /^gu&xnS  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. <K>qK]|C  
    % A6E~GJa  
    %   The Zernike functions are an orthogonal basis on the unit circle. 0HQTe>!  
    %   They are used in disciplines such as astronomy, optics, and o{l]n*  
    %   optometry to describe functions on a circular domain. 8%a ^j\L  
    % -q nOq[  
    %   The following table lists the first 15 Zernike functions. tWQ$`<h  
    % .ezZ+@LI+#  
    %       n    m    Zernike function           Normalization \J;]g\&I"  
    %       -------------------------------------------------- m%.[|sZ3EM  
    %       0    0    1                                 1 5Q8s{WQ  
    %       1    1    r * cos(theta)                    2 ^ ]+vtk  
    %       1   -1    r * sin(theta)                    2 pwB>$7(_h  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) %F}d'TPx  
    %       2    0    (2*r^2 - 1)                    sqrt(3) nyOmNvZf  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 6uk}4bdvq  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) THgEHR0,}[  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) :KGPQ@:O  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) f|3LeOyz  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Mp[2Auf  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) @~&^1%37)  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Q~rE+?n9 F  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ?V(+Cc  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 8KKhD$  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) )M"xCO3a  
    %       -------------------------------------------------- !-&;t7R  
    % 5{vuN)K3  
    %   Example 1: J: I@kM  
    % O3#eQs  
    %       % Display the Zernike function Z(n=5,m=1) UA*Kuad  
    %       x = -1:0.01:1; SDk^fTV8x  
    %       [X,Y] = meshgrid(x,x); kQn}lD  
    %       [theta,r] = cart2pol(X,Y); 9oG)\M.6w  
    %       idx = r<=1; VtGZB3  
    %       z = nan(size(X)); p9S>H  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); IABF_GwF  
    %       figure ,pVe@d'  
    %       pcolor(x,x,z), shading interp ft4hzmuzM  
    %       axis square, colorbar ~]'yUd1gSZ  
    %       title('Zernike function Z_5^1(r,\theta)') g yT0h?xDt  
    % C 5e;U  
    %   Example 2: L@ejFXQg  
    % +%K~HYN  
    %       % Display the first 10 Zernike functions WSGho(\  
    %       x = -1:0.01:1; VssWtL  
    %       [X,Y] = meshgrid(x,x); _g'x=VJF  
    %       [theta,r] = cart2pol(X,Y); 2h)Qz+|7  
    %       idx = r<=1; ktp<o.f[  
    %       z = nan(size(X)); yW"[}L h4  
    %       n = [0  1  1  2  2  2  3  3  3  3]; >Pvz5Hf/wW  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; _N0N #L4M  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; @3S:W2k  
    %       y = zernfun(n,m,r(idx),theta(idx)); <|w(Sn  
    %       figure('Units','normalized') rFp>A`TJ  
    %       for k = 1:10 QUh`kt(E  
    %           z(idx) = y(:,k); uH[:R vC0  
    %           subplot(4,7,Nplot(k)) v I,T1%llu  
    %           pcolor(x,x,z), shading interp @Qp#Tg<'  
    %           set(gca,'XTick',[],'YTick',[]) ViG>gMGv  
    %           axis square ?},RN  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) k~, k@mR  
    %       end /!`xqG#  
    % U"~W3vwJ  
    %   See also ZERNPOL, ZERNFUN2. jX^_(Kg  
    MT$)A:"  
    %   Paul Fricker 11/13/2006 fVdu9 l  
    \^jRMIM==  
    a|4Q6Ycu  
    % Check and prepare the inputs: su3Wk,MLP  
    % ----------------------------- p%K(dA  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) O=^/58(m  
        error('zernfun:NMvectors','N and M must be vectors.') g}L>k}I?!W  
    end ~qK/w0=j  
    aK 3'u   
    if length(n)~=length(m) Ch:EL-L  
        error('zernfun:NMlength','N and M must be the same length.') <d >!%  
    end F07X9s44E  
    }]JHY P\  
    n = n(:); ;WgUhA ;q  
    m = m(:); ~R50-O  
    if any(mod(n-m,2)) {<?8Y  
        error('zernfun:NMmultiplesof2', ... wN :"(mQ  
              'All N and M must differ by multiples of 2 (including 0).') bR8`Y(=F9b  
    end ExeZj8U  
    <Y$( l szT  
    if any(m>n) R'" c  
        error('zernfun:MlessthanN', ... 7+qKA1t^  
              'Each M must be less than or equal to its corresponding N.') |"+Uf w^  
    end 9[sOh<W  
    [1O{yPV3s  
    if any( r>1 | r<0 ) A~ _2"  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') o^m?w0 \  
    end !e*T. 1Kz  
    tBX71d T  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 5}c8v2R:B  
        error('zernfun:RTHvector','R and THETA must be vectors.') 0N$FIw2  
    end cLw|[!5:  
    II!~"-WH  
    r = r(:); l@ (:Q!Sk  
    theta = theta(:); Y*S:/b~y  
    length_r = length(r); 1Kd6tnX  
    if length_r~=length(theta) =itQ@ ``r  
        error('zernfun:RTHlength', ... t[@>u'YKt  
              'The number of R- and THETA-values must be equal.') 0m"Ni:KEf  
    end n9n)eI)R  
    A7|L|+ ?  
    % Check normalization: z,4 D'F&  
    % -------------------- sx}S,aIU  
    if nargin==5 && ischar(nflag) _uXb>V*8  
        isnorm = strcmpi(nflag,'norm'); e `OQ6|.k8  
        if ~isnorm bdG@%K',  
            error('zernfun:normalization','Unrecognized normalization flag.') d ez4g  
        end =%7s0l3z  
    else P,9Pn)M|  
        isnorm = false; S>S7\b'  
    end Aa4Tq2G  
    ?~!9\dek,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >?rMMR+A  
    % Compute the Zernike Polynomials To5hVL<Ex"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $*T?}r>  
    UGj |)/  
    % Determine the required powers of r: 5t"FNL <(M  
    % ----------------------------------- .{} 8mFi1  
    m_abs = abs(m); R=F_U  
    rpowers = []; aB?usVoS  
    for j = 1:length(n) j<k6z   
        rpowers = [rpowers m_abs(j):2:n(j)]; xwi6#>  
    end v(!:HK0oeT  
    rpowers = unique(rpowers); / *PHX@  
    zn7)>cQ905  
    % Pre-compute the values of r raised to the required powers, 32j}ep.*  
    % and compile them in a matrix: 7 )r L<+  
    % ----------------------------- E)ZL+(  
    if rpowers(1)==0 q b/}&J7+  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); H-U_  
        rpowern = cat(2,rpowern{:}); eZN"t~\rX  
        rpowern = [ones(length_r,1) rpowern]; Y#tur`N  
    else D79:L:  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 5j6`W?|q  
        rpowern = cat(2,rpowern{:}); PP>6  
    end j49Uj}:j  
    d7 H*F  
    % Compute the values of the polynomials: R&J?X Q  
    % -------------------------------------- :dAd5v2f  
    y = zeros(length_r,length(n)); x3Y)l1gh  
    for j = 1:length(n) ,"XiI$Le  
        s = 0:(n(j)-m_abs(j))/2; ?Rx(@  
        pows = n(j):-2:m_abs(j); upL3M`  
        for k = length(s):-1:1 'A3skznX{  
            p = (1-2*mod(s(k),2))* ... VqpC@C$  
                       prod(2:(n(j)-s(k)))/              ... v{fcQb  
                       prod(2:s(k))/                     ... . R/y`:1:W  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... -!:5jfT"  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ne/JC(  
            idx = (pows(k)==rpowers); 0FgF,  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ]| +M0:2?  
        end L/V^#$  
         ]L7A$sTUQ  
        if isnorm ;'= cNj  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); nGkSS_X  
        end =4a:)g'  
    end S!.&#sc  
    % END: Compute the Zernike Polynomials !W9:)5^X  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u0 t lf  
    Cl]?qH*:  
    % Compute the Zernike functions: O6R)>Y4  
    % ------------------------------ Qop,~yK  
    idx_pos = m>0; rUj\F9*5#  
    idx_neg = m<0; }: HG)V  
    kzDN(_<1  
    z = y; )J}v.8   
    if any(idx_pos) Oo}h:3?  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); O'mcN*  
    end bYnq,JRA  
    if any(idx_neg) ,T<JNd'  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); DylO;+  
    end "J1A9|  
    L ,dh$F  
    % 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) l< f9$l^U  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Z~~6y6p  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated _SAM8!q4,  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive * 9^8NY]  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, si]VM_w6  
    %   and THETA is a vector of angles.  R and THETA must have the same @MES.g  
    %   length.  The output Z is a matrix with one column for every P-value,  Sfz1p  
    %   and one row for every (R,THETA) pair. g Ed A hfx  
    % tDX& ~1s  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike zjQ746<&)i  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) @M5+12FYt  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) -3{Q`@F  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 rx1u*L  
    %   for all p. CUu Owx6%  
    % hv|a8=U!R  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ""0 Y^M2I  
    %   Zernike functions (order N<=7).  In some disciplines it is QxYm3x5  
    %   traditional to label the first 36 functions using a single mode $r/$aq=K  
    %   number P instead of separate numbers for the order N and azimuthal V`^*Z}d9  
    %   frequency M. da7"Q{f+  
    % '[ t.  
    %   Example: SK}sf9gTv  
    % 8tx*z"2S  
    %       % Display the first 16 Zernike functions bC `<A  
    %       x = -1:0.01:1; .~f )4'T 9  
    %       [X,Y] = meshgrid(x,x); `Nx@MPo  
    %       [theta,r] = cart2pol(X,Y); Qr$'Q7  
    %       idx = r<=1; )QE6X67i  
    %       p = 0:15; wk|+[Rl;L  
    %       z = nan(size(X)); tO M$'0u  
    %       y = zernfun2(p,r(idx),theta(idx)); J3eud}w  
    %       figure('Units','normalized') 3K &637  
    %       for k = 1:length(p) ys9:";X;}  
    %           z(idx) = y(:,k); 4YfM.~ 6  
    %           subplot(4,4,k) \sNgs#{7E7  
    %           pcolor(x,x,z), shading interp }dkXRce*  
    %           set(gca,'XTick',[],'YTick',[]) ~ WWhCRq  
    %           axis square 6!\V|  
    %           title(['Z_{' num2str(p(k)) '}']) lVvcrU  
    %       end D S U`(`  
    % ip-X r|Bq  
    %   See also ZERNPOL, ZERNFUN. ^Arv6kD,  
    q/EX`%U  
    %   Paul Fricker 11/13/2006 8^UF0>`'  
    )U %`7(bN  
    m!FuC=e  
    % Check and prepare the inputs: /wJ#-DZ  
    % ----------------------------- & kC  
    if min(size(p))~=1 c4fH/-  
        error('zernfun2:Pvector','Input P must be vector.') qp})4XTv  
    end \CjJa(vV  
    )'+[,z ;s  
    if any(p)>35 Cbff:IP  
        error('zernfun2:P36', ... R-Edht|{  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... .LDZqWr-  
               '(P = 0 to 35).']) pJHdY)Cz  
    end eFiG:LS7  
    ]}L'jK 0  
    % Get the order and frequency corresonding to the function number: V4,Gt ]4  
    % ---------------------------------------------------------------- IC cr  
    p = p(:); Kv@P Uzu  
    n = ceil((-3+sqrt(9+8*p))/2); h#YO;m2wd  
    m = 2*p - n.*(n+2); v@\S$qU2  
     0s;~9>  
    % Pass the inputs to the function ZERNFUN: Y$JVxly  
    % ---------------------------------------- Ae>+Fcv  
    switch nargin b/S:&%E  
        case 3 s<YN*~  
            z = zernfun(n,m,r,theta); Ey=2 zo^F  
        case 4 #*iUZo  
            z = zernfun(n,m,r,theta,nflag); r&LZH.$oh  
        otherwise lh;fqn`  
            error('zernfun2:nargin','Incorrect number of inputs.') hz:7W8  
    end 'zUV(K?2]  
    m9[ 7"I  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) ]aPf-O*  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. og";mC  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 6_`Bo%  
    %   order N and frequency M, evaluated at R.  N is a vector of Qz@_"wm[  
    %   positive integers (including 0), and M is a vector with the GN_L"|#)=  
    %   same number of elements as N.  Each element k of M must be a yr%[IX]R  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) %IO*(5f  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ;^N lq3N  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix daSe0:daJ  
    %   with one column for every (N,M) pair, and one row for every gAqK/9;  
    %   element in R. O:0{vu9AQ  
    % iy~h|YK;  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- h3`}{ w  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is kP`#zwp'Ci  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ^SpQtW118  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 rO#w(]   
    %   for all [n,m]. {Z c8,jm  
    % y]Nk^ga:U6  
    %   The radial Zernike polynomials are the radial portion of the r)gK5Mv  
    %   Zernike functions, which are an orthogonal basis on the unit JU)^b V_  
    %   circle.  The series representation of the radial Zernike |Ahf 01  
    %   polynomials is a#]V|1*O  
    % ^iONC&r  
    %          (n-m)/2 `t/j6 e]  
    %            __ C+' -TLeu  
    %    m      \       s                                          n-2s u[**,.Ecg  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ec ;  
    %    n      s=0 3(oMASf  
    % 3@" :&  
    %   The following table shows the first 12 polynomials. O+W<l:|$  
    % g|Lbe4?  
    %       n    m    Zernike polynomial    Normalization Pm%xX~H  
    %       --------------------------------------------- Fv]6 a n.  
    %       0    0    1                        sqrt(2) {@2+oOuYfN  
    %       1    1    r                           2 ]$ d ;P  
    %       2    0    2*r^2 - 1                sqrt(6) 'xta/@Sq  
    %       2    2    r^2                      sqrt(6) 9\EW~OgTu  
    %       3    1    3*r^3 - 2*r              sqrt(8) i+14!LlI  
    %       3    3    r^3                      sqrt(8) ?t%{2a<X  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) Dn)yBA%  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) },d^y:m  
    %       4    4    r^4                      sqrt(10) [;wJM|Z J0  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ;B@#,6t/  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) _&]7  
    %       5    5    r^5                      sqrt(12) :fj>JF\[  
    %       --------------------------------------------- PE5*]+lW.  
    % '1D $ ;  
    %   Example: *IOrv)  
    % MiZ<v/L2  
    %       % Display three example Zernike radial polynomials 6CFnE7TQf  
    %       r = 0:0.01:1; ^mL X}E]  
    %       n = [3 2 5]; 7G+!9^  
    %       m = [1 2 1]; Gy \ ]j  
    %       z = zernpol(n,m,r); e.vt"eRB  
    %       figure ZP~H!  
    %       plot(r,z) `<g]p-=":  
    %       grid on )5( jx  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') no8\Oees  
    % !-)!UQ~|8  
    %   See also ZERNFUN, ZERNFUN2. $9?:P}$v  
    MH#Tp#RG  
    % A note on the algorithm. ]r#b:W\  
    % ------------------------ oaQW~R`_  
    % The radial Zernike polynomials are computed using the series 'dWUE-  
    % representation shown in the Help section above. For many special ;SE*En  
    % functions, direct evaluation using the series representation can aB6/-T+ u  
    % produce poor numerical results (floating point errors), because oh-EEo4,  
    % the summation often involves computing small differences between }vh <x6  
    % large successive terms in the series. (In such cases, the functions [s$x"Ex  
    % are often evaluated using alternative methods such as recurrence 7C'@g)@^/  
    % relations: see the Legendre functions, for example). For the Zernike j1`<+YT<#  
    % polynomials, however, this problem does not arise, because the (W#CDw<ja  
    % polynomials are evaluated over the finite domain r = (0,1), and 4L,wBce;,t  
    % because the coefficients for a given polynomial are generally all D]_6OlIE#'  
    % of similar magnitude. ]A}ZaXd  
    % f?:=@35  
    % ZERNPOL has been written using a vectorized implementation: multiple q6pHL  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] g$NUu  
    % values can be passed as inputs) for a vector of points R.  To achieve ?5CE<[  
    % this vectorization most efficiently, the algorithm in ZERNPOL ?#GTD?3d  
    % involves pre-determining all the powers p of R that are required to {X<g93  
    % compute the outputs, and then compiling the {R^p} into a single YZ<z lU  
    % matrix.  This avoids any redundant computation of the R^p, and 8o+:|V~X  
    % minimizes the sizes of certain intermediate variables. 2T}>9X  
    % 4!Radl3`  
    %   Paul Fricker 11/13/2006 {J)%6eL?  
    JkN*hm?  
    <.Zh{"$qo  
    % Check and prepare the inputs: 7 q!==P=  
    % ----------------------------- O [= L#wi  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +%j27~ R>D  
        error('zernpol:NMvectors','N and M must be vectors.') AmC9qk8Q  
    end c/ImK`:)4a  
    ~ S<aIk0l  
    if length(n)~=length(m) A{4,ih"5  
        error('zernpol:NMlength','N and M must be the same length.') q]yw",muT  
    end QOK,-  
    |J4sQ!%K  
    n = n(:); QuEX|h,F  
    m = m(:); OD7^*j(p`  
    length_n = length(n); Y=|p}>.}  
    Q9 AvNj>X  
    if any(mod(n-m,2)) x-c5iahp'  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') <[n:Ij  
    end D N)o|p  
    =8#.=J[/  
    if any(m<0) U2?R&c;b  
        error('zernpol:Mpositive','All M must be positive.') q#AIN`H  
    end 9[JUJ,#X'0  
    =r/8~~=  
    if any(m>n) |hj!NhBe  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') a T(]  
    end 4Cu\|"5)  
    'm`}XGUBS  
    if any( r>1 | r<0 ) 7w2$?k',-  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') R~iv%+  
    end B-_b.4ND)  
    &xB*Shp,B  
    if ~any(size(r)==1) LI@BB:)[  
        error('zernpol:Rvector','R must be a vector.') Wk7E&?-:6  
    end fZ &  
    $ c-O+~  
    r = r(:); Z8Ig,  
    length_r = length(r); O >+=cg  
    n= 4  
    if nargin==4 B<L7`xL  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); *<S>PbqLw  
        if ~isnorm 2DPv7\fW  
            error('zernpol:normalization','Unrecognized normalization flag.') MG=8`J-`  
        end Nc(CGl:  
    else ms5?^kS2O  
        isnorm = false; Y!oLNGY  
    end vE^tdzAG  
    JDR_k  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q\?p' i  
    % Compute the Zernike Polynomials J;Z2<x/H  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G\ F>*  
    OFtf)cGE  
    % Determine the required powers of r: dq.U#Rhrx  
    % ----------------------------------- 17?YN<  
    rpowers = []; d/yF}%0QI  
    for j = 1:length(n) ^wWbW&<Tg  
        rpowers = [rpowers m(j):2:n(j)]; W;=Ae~  
    end 2<B'PR-??y  
    rpowers = unique(rpowers); 3%5YUG@  
    hT1JEu  
    % Pre-compute the values of r raised to the required powers, %H\J@{f  
    % and compile them in a matrix: DFWO5Y_  
    % ----------------------------- Wgh@XB  
    if rpowers(1)==0 5\z<xpJ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 04dz ?`HuB  
        rpowern = cat(2,rpowern{:}); =MQ/z#:-P  
        rpowern = [ones(length_r,1) rpowern]; nyi!D   
    else ou-UR5  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ':;k<(<-  
        rpowern = cat(2,rpowern{:}); 4zS0kk;+  
    end ;DK%!."%  
    K [DpH&  
    % Compute the values of the polynomials: }r@dZ Bp:  
    % -------------------------------------- & V>rq'~;  
    z = zeros(length_r,length_n); y& yf&p  
    for j = 1:length_n zsJ# CDm  
        s = 0:(n(j)-m(j))/2; *'{-!Y  
        pows = n(j):-2:m(j); G*+^b'7  
        for k = length(s):-1:1 T%)E!:}v  
            p = (1-2*mod(s(k),2))* ... lvWwr!w  
                       prod(2:(n(j)-s(k)))/          ... exhU!p8  
                       prod(2:s(k))/                 ... !\4B.  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 1X5g(B  
                       prod(2:((n(j)+m(j))/2-s(k))); PhC3F4  
            idx = (pows(k)==rpowers); mF\!~ag|  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 1V1I[CxlX  
        end Cty#|6 k  
         Tp;W4]'a*:  
        if isnorm A_9^S!  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); $!>.h*np  
        end 3U>-~-DS  
    end {V6pC  
    To>,8E+GAb  
    % 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)  ( ay AP  
    6}4})B2  
    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。 tb3V qFx  
    pO` KtagL  
    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)