切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11486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 k$H%.l;E  
    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! 66.5QD0  
     
    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 ^| /](  
    function z = zernfun(n,m,r,theta,nflag) }~"hC3w  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle.  ?p(/_@  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N lW(px^&IN  
    %   and angular frequency M, evaluated at positions (R,THETA) on the QHWBAGA  
    %   unit circle.  N is a vector of positive integers (including 0), and [8Qro8  
    %   M is a vector with the same number of elements as N.  Each element #]#sGmW/L  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) wMdal:n^  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Wm);C~Le  
    %   and THETA is a vector of angles.  R and THETA must have the same -S$1Yn  
    %   length.  The output Z is a matrix with one column for every (N,M) c%[#~;E  
    %   pair, and one row for every (R,THETA) pair. K]j0_~3s  
    % +V{7")px6  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /F4pb]U!*  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), _UT$,0u_i  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral n+BJxu?  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, w.lAQ5)I%\  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized UN%Vg:=  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. !2z?YZhu  
    % >~`r:0',  
    %   The Zernike functions are an orthogonal basis on the unit circle. "Ae@lINn[y  
    %   They are used in disciplines such as astronomy, optics, and $uap8nN  
    %   optometry to describe functions on a circular domain. ^':!1  
    % N.4q.  
    %   The following table lists the first 15 Zernike functions. .[Ap=UYI>  
    % V^hE}`>z&  
    %       n    m    Zernike function           Normalization +<}0|Xl&  
    %       -------------------------------------------------- 9elga"4:'  
    %       0    0    1                                 1 p|Q*5TO  
    %       1    1    r * cos(theta)                    2 fm(e3]  
    %       1   -1    r * sin(theta)                    2 vk>b#%1{  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) fx@j?*Qb  
    %       2    0    (2*r^2 - 1)                    sqrt(3) zO V=9"~{  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 2MATpV#BT  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ?x+Z)`w_  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 6<N5_1  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) LY[~Os W  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) xB@|LtdO9;  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 4n %?YQ[t  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 3d-%>?-ee  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) H*bs31i{  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?%VI{[y#>  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) M;0]u.D*=  
    %       -------------------------------------------------- @x eAc0.^  
    %  Y!WG)u5  
    %   Example 1: #U*_1P0h  
    % Wm H~m k"  
    %       % Display the Zernike function Z(n=5,m=1) _{Sm k [  
    %       x = -1:0.01:1; / }Rz=&  
    %       [X,Y] = meshgrid(x,x); Cn>ADWpT&  
    %       [theta,r] = cart2pol(X,Y); $5v0m#[^  
    %       idx = r<=1; ]c&<zeX,  
    %       z = nan(size(X)); N`E-+9L)  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); $ ''9K  
    %       figure !r`,=jK"  
    %       pcolor(x,x,z), shading interp ifo7%XPcg  
    %       axis square, colorbar 9}c8Xt^&  
    %       title('Zernike function Z_5^1(r,\theta)') 3:{yJdpg  
    % R/^u/~<  
    %   Example 2: V97,1`  
    % CiR%Ujf  
    %       % Display the first 10 Zernike functions h?-#9<A  
    %       x = -1:0.01:1; xr7+$:>a  
    %       [X,Y] = meshgrid(x,x);  PlYm&  
    %       [theta,r] = cart2pol(X,Y); -!0_:m3  
    %       idx = r<=1; 0<PR+Iv*i  
    %       z = nan(size(X)); jqH3J2L  
    %       n = [0  1  1  2  2  2  3  3  3  3]; @:tj<\G]  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; y7S4d~&  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; .XkMk|t8  
    %       y = zernfun(n,m,r(idx),theta(idx)); % aUsOB-RV  
    %       figure('Units','normalized') k<RZKwQc  
    %       for k = 1:10 j F-v% ?  
    %           z(idx) = y(:,k); -k(CJ5H9  
    %           subplot(4,7,Nplot(k)) Cda!Mk:  
    %           pcolor(x,x,z), shading interp SlSM+F  
    %           set(gca,'XTick',[],'YTick',[]) Mc-)OtmG[  
    %           axis square BYY RoE[P  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) cEe? *\G  
    %       end *jMk/9oa<N  
    % XE3'`D !  
    %   See also ZERNPOL, ZERNFUN2. kz"3ZDR  
    J(#mtj>v_  
    %   Paul Fricker 11/13/2006 V:/7f*n7  
    #{9G sD  
    "lNzGi-H  
    % Check and prepare the inputs: 5'w^@Rs5  
    % ----------------------------- QQe;1O  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) `VQb-V  
        error('zernfun:NMvectors','N and M must be vectors.') 9'x)M?{8  
    end )2DQ>cm  
    \@}#Gez  
    if length(n)~=length(m) Xnuzr" 4u  
        error('zernfun:NMlength','N and M must be the same length.') GHF_R,7  
    end ]APvp.Tw:  
    - O"i3>C  
    n = n(:); F?m?UQS'u  
    m = m(:); T@%m7|P  
    if any(mod(n-m,2)) N~pIC2Woo  
        error('zernfun:NMmultiplesof2', ... }X;U|]d  
              'All N and M must differ by multiples of 2 (including 0).') +%N KQ'49I  
    end Pv<FLo%u<  
    o{*ay$vA]  
    if any(m>n) *2}O-e  
        error('zernfun:MlessthanN', ... M[~{Vd  
              'Each M must be less than or equal to its corresponding N.') `]$?uQ  
    end yMLOUUWa8x  
    mL~z~w*s  
    if any( r>1 | r<0 ) 8hA^`Y  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 0Q593F  
    end p.fF}B  
    h{ lDxOH*  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) WxbsD S;  
        error('zernfun:RTHvector','R and THETA must be vectors.') 9kKnAf4Z  
    end Sd IX-k.  
    6zIgQ4Bp24  
    r = r(:); 69kJC/1+l  
    theta = theta(:); <B /5J:o<  
    length_r = length(r); ,jy*1Hjd  
    if length_r~=length(theta) Ip}Vb6}  
        error('zernfun:RTHlength', ... 4z:#I;  
              'The number of R- and THETA-values must be equal.') Sx ] T/xq  
    end Lc<eRVNd,  
    +Ra3bjl  
    % Check normalization: RA a[t :|  
    % -------------------- %;z((3F  
    if nargin==5 && ischar(nflag) ~un%4]U  
        isnorm = strcmpi(nflag,'norm'); J NC  
        if ~isnorm :f'&z47  
            error('zernfun:normalization','Unrecognized normalization flag.') &"uV~AM  
        end 1u]P4Gf=  
    else K#K\-TR|$  
        isnorm = false; 4ZT A>   
    end L6 6-LMkH  
    }tST)=M`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =QV ::/  
    % Compute the Zernike Polynomials 0"xPX#Cvj  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% km:nE: |  
    yy2Ie  
    % Determine the required powers of r: FM^9}*  
    % ----------------------------------- Gie@JX  
    m_abs = abs(m); i}TwOy<4s  
    rpowers = []; ]*%+H|l  
    for j = 1:length(n) Em13dem  
        rpowers = [rpowers m_abs(j):2:n(j)]; t~K%.|'0  
    end K.>wQA&  
    rpowers = unique(rpowers); ;n#%G^!H  
    a0Oe:]mo\  
    % Pre-compute the values of r raised to the required powers, E@QA".  
    % and compile them in a matrix: FE5Q?*Ea  
    % ----------------------------- H D/5!d  
    if rpowers(1)==0 OCyG_DLT$5  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ,*,sw:=2  
        rpowern = cat(2,rpowern{:}); #P2;K dDO  
        rpowern = [ones(length_r,1) rpowern]; /k:$l9C[  
    else ~el-*=<m  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ;~zNqdlH  
        rpowern = cat(2,rpowern{:}); xc'vS>&  
    end ;Fl<v@9  
    NVIWWX9?  
    % Compute the values of the polynomials: o96:4j4  
    % -------------------------------------- WXUkuO  
    y = zeros(length_r,length(n)); `U`#I,Ln[  
    for j = 1:length(n) 0=U70nKr  
        s = 0:(n(j)-m_abs(j))/2; 8KjRCm,I  
        pows = n(j):-2:m_abs(j); eS!C3xC;J]  
        for k = length(s):-1:1 fu\s`W6f&  
            p = (1-2*mod(s(k),2))* ... l\q} |o  
                       prod(2:(n(j)-s(k)))/              ... PjqeE,5  
                       prod(2:s(k))/                     ... }HZ{(?  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... HD# r0)  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 2P~)I)3V  
            idx = (pows(k)==rpowers); hCc0sRp  
            y(:,j) = y(:,j) + p*rpowern(:,idx); |w)5;uQ&\  
        end k&s; {|!  
         -6EK#!+  
        if isnorm [ x>  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); $Tl<V/  
        end }Zl"9A#K  
    end CJ w$j`k  
    % END: Compute the Zernike Polynomials ]EL\)xCr  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v|+5:jFOqb  
    ZCiY,;c  
    % Compute the Zernike functions: $iMC/Kym  
    % ------------------------------ o)]FtL:mm  
    idx_pos = m>0; WfVMdwz=  
    idx_neg = m<0; Y)p4]>lT+8  
    r+g jc?Ol  
    z = y; Lar r}o=  
    if any(idx_pos) hLuJWjCV  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); (r F?If  
    end emWGIo  
    if any(idx_neg) !EFBI+?&  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); M9"Sgb`g  
    end obGWxI%a  
    T_ ^C#>  
    % 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) _t.FL@3e  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. (T;9us0  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated W8* 2;F]  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Bcaw~WD  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 5P\N"Yjx'  
    %   and THETA is a vector of angles.  R and THETA must have the same wgZrrq/W|  
    %   length.  The output Z is a matrix with one column for every P-value, Mo|yv[(K ,  
    %   and one row for every (R,THETA) pair. )0|):g   
    % $c9=mjwH  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike l\aUresm  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) FfXZ|o$;  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ak2dn]]D  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 csvO g[  
    %   for all p. 41 'EA \V  
    % _80ns&q  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 +S0u=u65  
    %   Zernike functions (order N<=7).  In some disciplines it is C1ZyB"{  
    %   traditional to label the first 36 functions using a single mode ZY Kd  
    %   number P instead of separate numbers for the order N and azimuthal ?#ihJt,  
    %   frequency M. u:5IjOb2^  
    % sY^lQN  
    %   Example: *H*\gaSh  
    % A;t zRe  
    %       % Display the first 16 Zernike functions ,RN|d0dE  
    %       x = -1:0.01:1; T/Q==Q{W:  
    %       [X,Y] = meshgrid(x,x); L]>4Nd  
    %       [theta,r] = cart2pol(X,Y); 3{q[q#"  
    %       idx = r<=1; "OJr*B  
    %       p = 0:15; AA.Ys89V  
    %       z = nan(size(X)); L5C2ng>  
    %       y = zernfun2(p,r(idx),theta(idx)); 4tnjXP8  
    %       figure('Units','normalized') :p$EiR  
    %       for k = 1:length(p) TK %< a/  
    %           z(idx) = y(:,k); &%:*\_2s  
    %           subplot(4,4,k) -fQX4'3R  
    %           pcolor(x,x,z), shading interp 3.~h6r5-  
    %           set(gca,'XTick',[],'YTick',[]) x Ty7lfSe  
    %           axis square N1s.3`  
    %           title(['Z_{' num2str(p(k)) '}']) #'iPDRYy  
    %       end c.-cpFk^L&  
    % oB}K[3uB:t  
    %   See also ZERNPOL, ZERNFUN. '2xcce#  
    3B -NY Ja  
    %   Paul Fricker 11/13/2006 -;<>tq'3`  
    kU(kU2u%9  
    26}u4W$  
    % Check and prepare the inputs: v*XkWH5  
    % ----------------------------- NkoofhZ  
    if min(size(p))~=1 QA!#s\  
        error('zernfun2:Pvector','Input P must be vector.') ^f6 {0  
    end lT3|D?sF  
     )Oo2<:"  
    if any(p)>35 9PCa*,  
        error('zernfun2:P36', ... p4y6R4kyT  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... >F~ITk5`Oo  
               '(P = 0 to 35).']) ;9vIa7L&  
    end A$N+9n\  
    ]qMH=>pOsj  
    % Get the order and frequency corresonding to the function number: OMi02tSm  
    % ---------------------------------------------------------------- qz87iJp&  
    p = p(:); +#9xA6,AE  
    n = ceil((-3+sqrt(9+8*p))/2); e6o/q)9#  
    m = 2*p - n.*(n+2); ' #KA+?@  
    {9Db9K^  
    % Pass the inputs to the function ZERNFUN: D|[/>x  
    % ---------------------------------------- )ph30B  
    switch nargin tr5'dX4]  
        case 3 q _19&;&  
            z = zernfun(n,m,r,theta); ` %l&zwj>  
        case 4 ),M U+*`  
            z = zernfun(n,m,r,theta,nflag); {clC n  
        otherwise 'Z59<Ya&x  
            error('zernfun2:nargin','Incorrect number of inputs.') 98h :X%  
    end 7t`E@dm  
    8B_0!U& ]  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) Bl=nj.g  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. v^<<[I2 C  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of =jsx (3V   
    %   order N and frequency M, evaluated at R.  N is a vector of YGfA qI y  
    %   positive integers (including 0), and M is a vector with the h\/^Aa0  
    %   same number of elements as N.  Each element k of M must be a (_s;aK  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) .mC~Ry+t  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ~wa%fM  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix BN_!Y)F l  
    %   with one column for every (N,M) pair, and one row for every <zfO1~^  
    %   element in R. b=V)?"e-  
    % jkZ_c!  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- W]} #\\$z  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is L-`(!j  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Z`^ K%P=  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 f&=K]:WDe  
    %   for all [n,m]. v!nm &"  
    % _e;N'DZ  
    %   The radial Zernike polynomials are the radial portion of the H<v c\r  
    %   Zernike functions, which are an orthogonal basis on the unit -[G/2F'  
    %   circle.  The series representation of the radial Zernike yW%&_s0  
    %   polynomials is j@%K*Gb`  
    % 5wT' ,U"+  
    %          (n-m)/2 7^n,Ti g  
    %            __ Z'voCWCd  
    %    m      \       s                                          n-2s ;%v%K+}r  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Tbe_x s^  
    %    n      s=0 ac2}3 $u  
    % C}x4#bNK  
    %   The following table shows the first 12 polynomials. ^nG1/}  
    % QWU5-p9e8  
    %       n    m    Zernike polynomial    Normalization hdo+Qezu:  
    %       --------------------------------------------- pA*D/P-  
    %       0    0    1                        sqrt(2) 71K\.[ =-  
    %       1    1    r                           2 jXc5fXO N  
    %       2    0    2*r^2 - 1                sqrt(6) _Cu[s?,kS  
    %       2    2    r^2                      sqrt(6) }T?i%l  
    %       3    1    3*r^3 - 2*r              sqrt(8)  Bq~AU#  
    %       3    3    r^3                      sqrt(8) z4 4  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) $xKg }cO  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) v]LFZI5  
    %       4    4    r^4                      sqrt(10) ~Da >{zHt  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) _Ym&UY.u#  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) jrDz7AfA  
    %       5    5    r^5                      sqrt(12) D<+ bzC  
    %       --------------------------------------------- DR.3 J`?K  
    % S/#) :,YS  
    %   Example: TKj/6Jz|  
    % a7QlU=\  
    %       % Display three example Zernike radial polynomials >=.ch5h3J)  
    %       r = 0:0.01:1; 44Seq  
    %       n = [3 2 5]; F?yh23&_4  
    %       m = [1 2 1]; 31 KDeFg  
    %       z = zernpol(n,m,r); v*vub#wP  
    %       figure N[|by}@n  
    %       plot(r,z) C=xo&I7  
    %       grid on umq$4}T '$  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') z|t.y.JX  
    % {Sd@u$&  
    %   See also ZERNFUN, ZERNFUN2. Hl4vLx@  
    =RCfibT!C  
    % A note on the algorithm. {[(W4NAlH  
    % ------------------------ +lY\r +;  
    % The radial Zernike polynomials are computed using the series b;&Yw-\nZ;  
    % representation shown in the Help section above. For many special ONg<  
    % functions, direct evaluation using the series representation can B1 jH.(  
    % produce poor numerical results (floating point errors), because ^*$WZMMJ1  
    % the summation often involves computing small differences between 1Ud t9$~T  
    % large successive terms in the series. (In such cases, the functions jk WBw.(  
    % are often evaluated using alternative methods such as recurrence yKX:Z4I/  
    % relations: see the Legendre functions, for example). For the Zernike Rx_,J%0Fq  
    % polynomials, however, this problem does not arise, because the VNOK>+  
    % polynomials are evaluated over the finite domain r = (0,1), and W#oEF/G  
    % because the coefficients for a given polynomial are generally all )[^:]}%r  
    % of similar magnitude. V!yp@%D  
    % .{-iq(3  
    % ZERNPOL has been written using a vectorized implementation: multiple Gc3PN  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ( S C7m /  
    % values can be passed as inputs) for a vector of points R.  To achieve g*"J10hyP  
    % this vectorization most efficiently, the algorithm in ZERNPOL AR[M8RA  
    % involves pre-determining all the powers p of R that are required to  ^qSf  
    % compute the outputs, and then compiling the {R^p} into a single .q'FSEkMJ  
    % matrix.  This avoids any redundant computation of the R^p, and &L[8Mju6  
    % minimizes the sizes of certain intermediate variables. x r+E  
    % z~A(IQO  
    %   Paul Fricker 11/13/2006 )nbyV a  
    MO(5-R`  
    //4p1^%  
    % Check and prepare the inputs:   t`&s  
    % ----------------------------- \a~;8):q=i  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <try%p|f  
        error('zernpol:NMvectors','N and M must be vectors.') `qYc#_ELv  
    end +@<^i?ale  
    G%W03c  
    if length(n)~=length(m) e-T9HM&%P  
        error('zernpol:NMlength','N and M must be the same length.') ,rvZW}=  
    end U`vt/#j 1  
    ~k:>Xo[|O  
    n = n(:); 2-B8>-   
    m = m(:); I{X@<o}  
    length_n = length(n); Q%b46"  
    CsQ}P)  
    if any(mod(n-m,2)) 'DB({s  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 7u&H*e7  
    end F0o18k_"  
    YRT}fd>R&  
    if any(m<0) (vYf?+Kb  
        error('zernpol:Mpositive','All M must be positive.') "p_[A  
    end 5Dh&ez`oR'  
    qkyX*_}  
    if any(m>n) k+>p!1  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') m<VL19o>R  
    end ~A{[=v  
    l<+,(E=  
    if any( r>1 | r<0 ) 'rcsK  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 0^tJX1L  
    end [+[fD  
    %%-Tjw o  
    if ~any(size(r)==1) Bg 8t'dw?K  
        error('zernpol:Rvector','R must be a vector.') F\$}8,9  
    end S3[oA&  
    <i`K%+<WO  
    r = r(:); v(WL 3[y;  
    length_r = length(r); Y_ u7 0@`  
    k? _$h<Y  
    if nargin==4 I[YfF  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ^gVbVz[17  
        if ~isnorm 8B(Q7Qj  
            error('zernpol:normalization','Unrecognized normalization flag.') (j\UoKLRt  
        end X wn|.  
    else OTr!?xi  
        isnorm = false; r;s3(@[,@  
    end i_Q4bhVj  
    b9!J}hto,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pz z`4VS:  
    % Compute the Zernike Polynomials EC&19  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ql!6I(  
    'G By^hj?  
    % Determine the required powers of r: p RfHbPV?  
    % ----------------------------------- dYttse'  
    rpowers = []; B?>#cpW j  
    for j = 1:length(n) 7 5cr!+  
        rpowers = [rpowers m(j):2:n(j)]; enO=-#  
    end 7B>cmi  
    rpowers = unique(rpowers); I5 7<0  
    jZgnt{  
    % Pre-compute the values of r raised to the required powers, $2Tty 7  
    % and compile them in a matrix: doUqUak  
    % ----------------------------- kA$;vbm  
    if rpowers(1)==0 LHGK!zI  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5L'@WB|{4u  
        rpowern = cat(2,rpowern{:}); X([n>w  
        rpowern = [ones(length_r,1) rpowern]; ?>Ci`XlLr  
    else U8@*I>vA  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); SOY#, Zu  
        rpowern = cat(2,rpowern{:}); {d5ur@G1  
    end `rFGSq$9  
    oA^ ]x>  
    % Compute the values of the polynomials: x[<#mt  
    % -------------------------------------- D}C*8s bC}  
    z = zeros(length_r,length_n); c;fyUi  
    for j = 1:length_n m_W.r+s~C4  
        s = 0:(n(j)-m(j))/2; 4zvU"np  
        pows = n(j):-2:m(j); 6O?Sr,  
        for k = length(s):-1:1 '48|f`8$  
            p = (1-2*mod(s(k),2))* ... BJ;cF"Kp  
                       prod(2:(n(j)-s(k)))/          ... Q14;G<l-  
                       prod(2:s(k))/                 ... _p^ "!  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... e(5Px!B  
                       prod(2:((n(j)+m(j))/2-s(k))); 3B]+]e~  
            idx = (pows(k)==rpowers); LGue=Hkp  
            z(:,j) = z(:,j) + p*rpowern(:,idx); )HiTYV)]'  
        end -|UX}t*  
         [UrS%]OSR  
        if isnorm 3). c [F^l  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); UmMYe4LQR  
        end )Syf5I  
    end "U~@o4u;  
    iV$75Atk  
    % 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)  Is[0ri   
    79_MP  
    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。 7j8_O@_  
    D.?gV_  
    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)