切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11429阅读
    • 9回复

    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

    上一主题 下一主题
    离线niuhelen
     
    发帖
    19
    光币
    28
    光券
    0
    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 E#2k|TpH4  
    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! J_[[BJ&}x  
     
    分享到
    离线phility
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    19
    光币
    28
    光券
    0
    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 DD$P r&~=  
    function z = zernfun(n,m,r,theta,nflag) )zt4'b\)v  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. S=amjcC  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N u&_U CJCf  
    %   and angular frequency M, evaluated at positions (R,THETA) on the [gdPHXs  
    %   unit circle.  N is a vector of positive integers (including 0), and })SdaZ  
    %   M is a vector with the same number of elements as N.  Each element L.:QI<n  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 5_C#_=E  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, sfPN\^k2  
    %   and THETA is a vector of angles.  R and THETA must have the same / lM~K:  
    %   length.  The output Z is a matrix with one column for every (N,M) Ib8{+j  
    %   pair, and one row for every (R,THETA) pair. 'I>#0VRr  
    % 4bzn^  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike OwIy(ukTI  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Jo$Dxa z  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral []3}(8yxGb  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, rPpAg  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized +mOtYf W  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. O:p649A  
    % bCe-0!Q  
    %   The Zernike functions are an orthogonal basis on the unit circle. V@'S#K#  
    %   They are used in disciplines such as astronomy, optics, and }Y ];ccT  
    %   optometry to describe functions on a circular domain. -86:PL(I"  
    % k[)@I;m  
    %   The following table lists the first 15 Zernike functions. R./6Q1  
    % h:sG23@=  
    %       n    m    Zernike function           Normalization `80Hxp@  
    %       -------------------------------------------------- y]4 `d  
    %       0    0    1                                 1 "$pg mf2  
    %       1    1    r * cos(theta)                    2 Ht^2)~e~:  
    %       1   -1    r * sin(theta)                    2 5w{pX1z1  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) *Y0,d`  
    %       2    0    (2*r^2 - 1)                    sqrt(3) <1.mm_pw  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ucPMT0k  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) $QBUnLOek&  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) `2+e\%f/0  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) g9Gy3zk=  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) '\\Cpc_g  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) BQ0\+  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ka\b_P&  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) xG/qDc  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) AK?j1Pk  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) }3y\cv0ct  
    %       -------------------------------------------------- :]Qx T8B  
    % NWK_(=n  
    %   Example 1: :?k=Yr  
    % Q 9<_:3  
    %       % Display the Zernike function Z(n=5,m=1) 3F!+c 8e  
    %       x = -1:0.01:1; iRHQRdij  
    %       [X,Y] = meshgrid(x,x); + aqo8'a  
    %       [theta,r] = cart2pol(X,Y); T["(YFCByg  
    %       idx = r<=1; !r0P\  
    %       z = nan(size(X)); 695ppiKU  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ++"PPbOe&D  
    %       figure ?} tQaj  
    %       pcolor(x,x,z), shading interp p;=(-4\V}  
    %       axis square, colorbar 9'h^59  
    %       title('Zernike function Z_5^1(r,\theta)') Asu"#sd  
    % hAyPaS#  
    %   Example 2: <t37DnCgI  
    % V/}8+Xq  
    %       % Display the first 10 Zernike functions AI;=k  
    %       x = -1:0.01:1; TJ:Lz]l >  
    %       [X,Y] = meshgrid(x,x); !I_4GE,  
    %       [theta,r] = cart2pol(X,Y); f"^tOgGH  
    %       idx = r<=1; $7d"9s\$"  
    %       z = nan(size(X)); <5~>.DuE  
    %       n = [0  1  1  2  2  2  3  3  3  3]; @ RBwT  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; X-F HJ4  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; nB0 ol-<  
    %       y = zernfun(n,m,r(idx),theta(idx)); 0+pJv0u  
    %       figure('Units','normalized') jMbK7 1K%  
    %       for k = 1:10 V1A3l{>L  
    %           z(idx) = y(:,k); .y+U7 "?s*  
    %           subplot(4,7,Nplot(k)) a"aV&t  
    %           pcolor(x,x,z), shading interp w,9F riW  
    %           set(gca,'XTick',[],'YTick',[])  c @fc7  
    %           axis square Q2?qvNZ  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 3/FB>w gt  
    %       end ;D:T ^4  
    % o7zfD94I  
    %   See also ZERNPOL, ZERNFUN2. p]4 sN  
    GK&Dd"v  
    %   Paul Fricker 11/13/2006 n\Ixv  
    HXI}f\6x  
    m@~x*+Iz  
    % Check and prepare the inputs: )zo ;r!eP  
    % ----------------------------- !d(V7`8  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) `f]O  
        error('zernfun:NMvectors','N and M must be vectors.') ]EQ/*ct  
    end T 1=M6iJ  
    q3`t0eLZ  
    if length(n)~=length(m) >k|[U[@  
        error('zernfun:NMlength','N and M must be the same length.') e.V){}{V  
    end {A UEVt  
    H #_Z6J  
    n = n(:); (xL=X%6a  
    m = m(:); |=s3a5sl  
    if any(mod(n-m,2)) :f;|^(]"  
        error('zernfun:NMmultiplesof2', ... aDuanGC/V  
              'All N and M must differ by multiples of 2 (including 0).') gzF&7trN  
    end za7wNe(s  
    {wI0 =U  
    if any(m>n) n} {cs  
        error('zernfun:MlessthanN', ... l1WVt}  
              'Each M must be less than or equal to its corresponding N.') {'!~j!1'j  
    end 3yN1cd"#?  
    .U_=LV]C  
    if any( r>1 | r<0 ) 9lv 2  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') if>] )g2lr  
    end &bQ^J%\  
    Bx F  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) \`C3;}o:"P  
        error('zernfun:RTHvector','R and THETA must be vectors.') v(`$%V.  
    end ,dBI=D'  
    uk,f}Xc  
    r = r(:); M_K&x-H0  
    theta = theta(:); 2lRZ/xaF%P  
    length_r = length(r); 7f>n`nq?  
    if length_r~=length(theta)  >pKI'  
        error('zernfun:RTHlength', ... D$HxPfDZ  
              'The number of R- and THETA-values must be equal.') J++D\x#@  
    end A7H=#L+C  
    AI2CfH#:C  
    % Check normalization: 71_N9ub@z  
    % -------------------- 0W> ",2|z  
    if nargin==5 && ischar(nflag) A\`Uu&  
        isnorm = strcmpi(nflag,'norm'); )1/O_N6C  
        if ~isnorm Lst5  
            error('zernfun:normalization','Unrecognized normalization flag.') _wBPn6gg`  
        end ^d,d<Uc  
    else J3=jC5=J4  
        isnorm = false; w]_a0{Uh  
    end ?=/l@d  
    ',f[y:v;  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% lgl/| ^ Uw  
    % Compute the Zernike Polynomials eo!z>9#.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% eC?N>wHH  
    n" sGI  
    % Determine the required powers of r: bTj,5,8 i  
    % ----------------------------------- "TPMSx&Ei  
    m_abs = abs(m); Mtu8zm  
    rpowers = []; H,'c&  
    for j = 1:length(n) lI9 3{!+>  
        rpowers = [rpowers m_abs(j):2:n(j)]; c!zu0\[Id  
    end WVZ\4y  
    rpowers = unique(rpowers); E%TvGe;#  
    i> ;G4  
    % Pre-compute the values of r raised to the required powers, sMZ \6  
    % and compile them in a matrix: [eImP V]  
    % ----------------------------- zC7;Zj*k  
    if rpowers(1)==0 ^#+9v  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 3iB8QO;pp  
        rpowern = cat(2,rpowern{:}); nP.d5%E  
        rpowern = [ones(length_r,1) rpowern]; 79\ =)m}$Q  
    else d<]/,BY'  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ]Sh&8 #  
        rpowern = cat(2,rpowern{:}); AK[c!mzx  
    end ;k>{I8L~  
    E)Dik`Ccl  
    % Compute the values of the polynomials: ~34$D],D  
    % -------------------------------------- T"O!  
    y = zeros(length_r,length(n)); @I%m}>4Jm  
    for j = 1:length(n) \>+gZc]an  
        s = 0:(n(j)-m_abs(j))/2; =3FXU{"Qi4  
        pows = n(j):-2:m_abs(j); PqfH}d0l  
        for k = length(s):-1:1 Epx.0TA=t  
            p = (1-2*mod(s(k),2))* ... d97wiE/i<  
                       prod(2:(n(j)-s(k)))/              ... il: ""x7^y  
                       prod(2:s(k))/                     ... 4WLB,<b}  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... =uHTpHR  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); h<?Vzl  
            idx = (pows(k)==rpowers); ak%8|'}  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Gb"PMai  
        end PWTAy\  
         #VLTx!5o  
        if isnorm T+I|2HYqOj  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Ba"Z^(:  
        end B|!Re4`0  
    end Xs4`bbap  
    % END: Compute the Zernike Polynomials Ox58L>:0m  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uJi|@{V  
    b(wiJ&t  
    % Compute the Zernike functions: W)KV"A3C  
    % ------------------------------ \hg12],#:@  
    idx_pos = m>0; ur;8uv2o  
    idx_neg = m<0; STO6cNi  
    ~#wq sm  
    z = y; /MA4Er r  
    if any(idx_pos) nfc&.(6x<  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Rt+s\MC^r  
    end -q[?,h  
    if any(idx_neg) %N2=:;f  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ZZ.GpB.  
    end 0 j6/H?OT  
    l/SbJrM*  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) 4~D?F'o  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. EiSS_Lc  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ~qs 97'  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive p;g$D=2  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ]"^U  
    %   and THETA is a vector of angles.  R and THETA must have the same Ue! &Vm  
    %   length.  The output Z is a matrix with one column for every P-value, 0m!+gZ@  
    %   and one row for every (R,THETA) pair. q'[5h>Pa  
    % s~,Ypo?  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike >A#]60w.  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) Yz4Q!tL  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) @a+1Ri`)  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 "d9"Md0k  
    %   for all p. aH*)W'N?  
    % }!x\qpA  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 A?=g!(wB  
    %   Zernike functions (order N<=7).  In some disciplines it is Ovh[qm?Z  
    %   traditional to label the first 36 functions using a single mode 3 cu`U`  
    %   number P instead of separate numbers for the order N and azimuthal (i1 ]+.  
    %   frequency M. YRqIC -_  
    % ckS.j)@.c  
    %   Example: }[k~JXt  
    % d[J+):aW  
    %       % Display the first 16 Zernike functions ,!Gw40t  
    %       x = -1:0.01:1; hvkLcpE  
    %       [X,Y] = meshgrid(x,x); /{6PwlP5  
    %       [theta,r] = cart2pol(X,Y); ihdN{Mx<2  
    %       idx = r<=1; <`}Oi 5nW  
    %       p = 0:15; j@ lHgis  
    %       z = nan(size(X)); D:4Iex9$F"  
    %       y = zernfun2(p,r(idx),theta(idx)); R_`i=>Z-  
    %       figure('Units','normalized') To.CY^M  
    %       for k = 1:length(p) B|zJrz0q3  
    %           z(idx) = y(:,k); )%I2#Q"Nt-  
    %           subplot(4,4,k) 1:(qoA:  
    %           pcolor(x,x,z), shading interp !`JaYUL[e  
    %           set(gca,'XTick',[],'YTick',[]) ]yy10Pk[!  
    %           axis square KEEHb2q  
    %           title(['Z_{' num2str(p(k)) '}']) Dyyf%'\M  
    %       end ],V_"\ATD  
    % &'Pwz  
    %   See also ZERNPOL, ZERNFUN. hCS|(8g  
    3 - Nwg9 U  
    %   Paul Fricker 11/13/2006 .5 Sw  
    R7pdwKD  
    MOi.bHCQJP  
    % Check and prepare the inputs: xM"k qRZ  
    % -----------------------------  rl"$6{Z}  
    if min(size(p))~=1 p~Di\AQ/  
        error('zernfun2:Pvector','Input P must be vector.') yhxen  
    end I&%{%*y  
    4>x]v!d  
    if any(p)>35 ?NkweT(  
        error('zernfun2:P36', ... e=e^;K4  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... /%fBkA#n  
               '(P = 0 to 35).']) Jr+~'  
    end Myaj81  
    M$iDaEu-  
    % Get the order and frequency corresonding to the function number: CobMagPhr  
    % ---------------------------------------------------------------- o}O"  
    p = p(:); P@lDhzd  
    n = ceil((-3+sqrt(9+8*p))/2); HGM? ?=  
    m = 2*p - n.*(n+2); }ya@*jH  
    >ka*-8?  
    % Pass the inputs to the function ZERNFUN: 4IfOvAN%  
    % ---------------------------------------- `< _A#@  
    switch nargin P5-1z&9O  
        case 3 $v5)d J  
            z = zernfun(n,m,r,theta); [&y="6No  
        case 4 YD>5zV%!D  
            z = zernfun(n,m,r,theta,nflag); NX.%Rj*  
        otherwise ;J [ed>v;3  
            error('zernfun2:nargin','Incorrect number of inputs.') uzG{jc^  
    end /6S% h-#\  
    G4O $gg  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) !W\Zq+^^J3  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. n{Ce%gy  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of s0D,n1x  
    %   order N and frequency M, evaluated at R.  N is a vector of ppYIVI  
    %   positive integers (including 0), and M is a vector with the Ebk9[=  
    %   same number of elements as N.  Each element k of M must be a WxE^S ??|  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k)  x&^>|'H  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is oY NIJXln  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 6>  L)  
    %   with one column for every (N,M) pair, and one row for every XHN*'@ 77;  
    %   element in R. _Fc :<Ym?  
    % /kZ{+4M  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- #k}x} rn<'  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is Nj5V" c  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to %1JN%  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 1UHlA8w7 Q  
    %   for all [n,m]. 322)r$!"  
    % TK! D=M  
    %   The radial Zernike polynomials are the radial portion of the <q}w,XU  
    %   Zernike functions, which are an orthogonal basis on the unit _R/^P>Q?  
    %   circle.  The series representation of the radial Zernike Nd;)V  
    %   polynomials is 27"M]17)  
    % KzgW+6*G  
    %          (n-m)/2 76'@}wNnw  
    %            __ sLHUQ(S!  
    %    m      \       s                                          n-2s 9>QGsf.3  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r PQ0l<]Y  
    %    n      s=0 LvM;ZfAEv  
    % }Cs. Hm0P  
    %   The following table shows the first 12 polynomials. 5u:{lcC.X  
    % JXhHitUD  
    %       n    m    Zernike polynomial    Normalization [c`u   
    %       --------------------------------------------- !EwL"4pPw  
    %       0    0    1                        sqrt(2) GS*Mv{JJ  
    %       1    1    r                           2 %)t9b@c!}  
    %       2    0    2*r^2 - 1                sqrt(6) jsp)e=  
    %       2    2    r^2                      sqrt(6) O,D/& 0  
    %       3    1    3*r^3 - 2*r              sqrt(8) *X%dg$VcV  
    %       3    3    r^3                      sqrt(8) xPcH]Gs^b  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) DQ08dP((v  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 3oo Tn-`{  
    %       4    4    r^4                      sqrt(10) M~!DQ1u  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) :c?}~a~JO(  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 5eL_iNqJM  
    %       5    5    r^5                      sqrt(12) 7,v}Ap]Pa  
    %       --------------------------------------------- S&q(PI_"  
    % Yw!(]8PYdU  
    %   Example: RJ63"F $  
    % PV(TDb:0  
    %       % Display three example Zernike radial polynomials /c4@QbB  
    %       r = 0:0.01:1; )@hG#KMK  
    %       n = [3 2 5]; QBD\2VR  
    %       m = [1 2 1]; }#bX{?f  
    %       z = zernpol(n,m,r); \9Yc2$dY  
    %       figure $qp,7RW  
    %       plot(r,z) Qzh`x-S  
    %       grid on Shag4-*@hi  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 9)~Ha iVB  
    % .29y3}[PO  
    %   See also ZERNFUN, ZERNFUN2. ", Ge:\TR=  
    -~HyzX\cZB  
    % A note on the algorithm. ;zpSyyp@  
    % ------------------------ $%GW~|S\C  
    % The radial Zernike polynomials are computed using the series z>&|:VGG  
    % representation shown in the Help section above. For many special jb'A Os  
    % functions, direct evaluation using the series representation can q\I2lZ  
    % produce poor numerical results (floating point errors), because L2WH-XP=  
    % the summation often involves computing small differences between +<TnE+>j  
    % large successive terms in the series. (In such cases, the functions qiyX{J7Z  
    % are often evaluated using alternative methods such as recurrence zEJZ,<  
    % relations: see the Legendre functions, for example). For the Zernike U%qE=u-  
    % polynomials, however, this problem does not arise, because the [m+):q^  
    % polynomials are evaluated over the finite domain r = (0,1), and FVo_=O)  
    % because the coefficients for a given polynomial are generally all %9HL "  
    % of similar magnitude. ;5.S"  
    % ]N#%exBVo  
    % ZERNPOL has been written using a vectorized implementation: multiple 4r+s" |  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ch-.+p3  
    % values can be passed as inputs) for a vector of points R.  To achieve -0G/a&ss  
    % this vectorization most efficiently, the algorithm in ZERNPOL pI]tv@>:f  
    % involves pre-determining all the powers p of R that are required to B{dR/q3;@  
    % compute the outputs, and then compiling the {R^p} into a single c 0/vB  
    % matrix.  This avoids any redundant computation of the R^p, and ~L55l2u7  
    % minimizes the sizes of certain intermediate variables. 6$*\%  
    % WKDa]({k%  
    %   Paul Fricker 11/13/2006 Yg<4}l."  
    '^# =,+ A  
    QGkMT +A  
    % Check and prepare the inputs: +T,Yf/^Fn  
    % ----------------------------- Q"VS;uh.v  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) G Ch]5\  
        error('zernpol:NMvectors','N and M must be vectors.') J =j6rD  
    end Oh]RIWL  
    9irT}e  
    if length(n)~=length(m) #J_+ SL[  
        error('zernpol:NMlength','N and M must be the same length.') hALg5.E{T  
    end ob(S/t  
    J6s@}@R1  
    n = n(:); dF#`_!4pbf  
    m = m(:); (h $[g"8  
    length_n = length(n); X 8#Uk}/  
    xJemc3]2  
    if any(mod(n-m,2)) K|Kc.   
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ("!P_Q#  
    end O S%  
    Zp'q;h_  
    if any(m<0) J}M_Ka  
        error('zernpol:Mpositive','All M must be positive.') uNoP8U%*  
    end ]@G$ L,3  
    W}0cM9 g  
    if any(m>n) =j&qat  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') gfU@`A_N"  
    end 5+yT{,(5  
    -]$=.0 l  
    if any( r>1 | r<0 ) 6U!zc]>  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') qy$1+>f1  
    end ]\ DIJ>JZ  
    9~Ve}NB#z&  
    if ~any(size(r)==1) P"k`h=>!4  
        error('zernpol:Rvector','R must be a vector.') {S*:pG:+q  
    end '}pe$=  
    7~H.\4HB  
    r = r(:); <JkmJ/X  
    length_r = length(r); Q(0eq_X|6  
    zh6 0b{  
    if nargin==4 [e.@Yx_}  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); tg|7\Z7i  
        if ~isnorm j7u\.xu9  
            error('zernpol:normalization','Unrecognized normalization flag.') >^|( AzS  
        end miv)R  
    else g$a 5  
        isnorm = false; l+n0=^ Z  
    end  ~d\>f  
    Sb,lY<=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p+7ZGB  
    % Compute the Zernike Polynomials {DVu* %|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% iM'rl0  
    UX!)\5-  
    % Determine the required powers of r: PEIf)**0N  
    % ----------------------------------- s^6"qhTa  
    rpowers = []; oe,37xa4  
    for j = 1:length(n) gT8%?U:  
        rpowers = [rpowers m(j):2:n(j)]; -!JnyD   
    end VHlo}Ek<#  
    rpowers = unique(rpowers); /~nPPC  
    ?jy6%Y#,i  
    % Pre-compute the values of r raised to the required powers,  XeRbn  
    % and compile them in a matrix: DuzJQ Sv  
    % ----------------------------- ,LpGE>s  
    if rpowers(1)==0 ZlEH3-Zv  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); eT<T[; m  
        rpowern = cat(2,rpowern{:}); tuWJj^  
        rpowern = [ones(length_r,1) rpowern]; l$mfsm|{:  
    else m c q!_#{y  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); >ngP\&\  
        rpowern = cat(2,rpowern{:}); L kA_M'G  
    end [t.x cO  
    s J~WzQ  
    % Compute the values of the polynomials: HAOl&\)7"_  
    % -------------------------------------- X@cO`P  
    z = zeros(length_r,length_n); 8&2W^f5  
    for j = 1:length_n G j9WUv[P  
        s = 0:(n(j)-m(j))/2; G;k#06  
        pows = n(j):-2:m(j); gxf{/EjH  
        for k = length(s):-1:1 `zmj iC  
            p = (1-2*mod(s(k),2))* ... i *9Bu;  
                       prod(2:(n(j)-s(k)))/          ... E%tGwbi7  
                       prod(2:s(k))/                 ... fH6mv0  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... $<QOMfY>  
                       prod(2:((n(j)+m(j))/2-s(k))); %M KZ':m  
            idx = (pows(k)==rpowers); lf?dTPrD  
            z(:,j) = z(:,j) + p*rpowern(:,idx); "PhP1;A9,  
        end ^;[|,:8f7L  
         F9\T <  
        if isnorm O>)Fl42IeD  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 1NI%J B  
        end GR ^d/  
    end jXCSD@?]K  
    BGO!c[-  
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    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
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
    发帖
    856
    光币
    846
    光券
    0
    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
    发帖
    4352
    光币
    5478
    光券
    1
    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 qR8uL>  
    q!5 *) nw"  
    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。 pjh o#yP  
    0VOj,)K=  
    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)