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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 /be=u@KV  
    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! 8'_ 0g[s  
     
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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 20tO#{Li  
    function z = zernfun(n,m,r,theta,nflag) .WX,Nd3@  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~Y7dH Dn  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N T2=HG Z  
    %   and angular frequency M, evaluated at positions (R,THETA) on the @0NJ{  
    %   unit circle.  N is a vector of positive integers (including 0), and p=Y>i 'CG  
    %   M is a vector with the same number of elements as N.  Each element N|K4{Frm  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) (dqCa[  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ,DQjDMjrf  
    %   and THETA is a vector of angles.  R and THETA must have the same <jA105U"m>  
    %   length.  The output Z is a matrix with one column for every (N,M) [sy j#  
    %   pair, and one row for every (R,THETA) pair. j}f[W [2  
    % c5% 6Y2W0  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike wRvb8F 0  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ,<` )>2 'o  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral QkQ!Ep(  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ~F!,PM/  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ]Oeh=gq  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. BPv>$ m+.  
    % w0lT%CPx  
    %   The Zernike functions are an orthogonal basis on the unit circle. np9dM  
    %   They are used in disciplines such as astronomy, optics, and +ulagE|7  
    %   optometry to describe functions on a circular domain. vScjq5 "p  
    % -c*\o3)  
    %   The following table lists the first 15 Zernike functions. 5,)vJ,fs  
    % ZDt?j   
    %       n    m    Zernike function           Normalization G~,:2 o3  
    %       -------------------------------------------------- "ju'UOcS/  
    %       0    0    1                                 1 KP]"P*? ?  
    %       1    1    r * cos(theta)                    2 uLR<FpM  
    %       1   -1    r * sin(theta)                    2 (?0`d  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 'b&yrBFD  
    %       2    0    (2*r^2 - 1)                    sqrt(3) P8Qyhc  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) :-T*gqj|  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) o\VUD  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 3gEMRy*+  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) u]*0;-tz  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) UL$}{2N,_  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10)  #xh_  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) tc+WWDP#"  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) LeOP;#  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 88s/Q0l  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 49H+(*@v@  
    %       -------------------------------------------------- 4T6 {Y  
    % aB~S?.l  
    %   Example 1: qet>1<  
    % @(g_<@Jz  
    %       % Display the Zernike function Z(n=5,m=1) saf&dd  
    %       x = -1:0.01:1; W~1~k{A  
    %       [X,Y] = meshgrid(x,x); $'rG-g!f\  
    %       [theta,r] = cart2pol(X,Y); &ANP`=  
    %       idx = r<=1; >$WQxbwM(  
    %       z = nan(size(X)); tBbOY}.VD  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ]:M0Kj&h  
    %       figure 46@{5)Tq  
    %       pcolor(x,x,z), shading interp :Qt  
    %       axis square, colorbar D\dWt1n  
    %       title('Zernike function Z_5^1(r,\theta)') EOj"V'!  
    % Z<[<n0o1  
    %   Example 2: u$#Wv2|mk  
    % @mP]*$00  
    %       % Display the first 10 Zernike functions N["W I r  
    %       x = -1:0.01:1; S-Y=-"  
    %       [X,Y] = meshgrid(x,x); PXzsj.  
    %       [theta,r] = cart2pol(X,Y); E>'a,!QPv  
    %       idx = r<=1; 2Y\ d<.M  
    %       z = nan(size(X)); `]Fx.)C#  
    %       n = [0  1  1  2  2  2  3  3  3  3]; EP'h@zdz  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; X|f7K  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; fWfk[(M'9  
    %       y = zernfun(n,m,r(idx),theta(idx)); t7 n(Qkrv  
    %       figure('Units','normalized') Z>c3  
    %       for k = 1:10 wI]"U2L5  
    %           z(idx) = y(:,k); Un`^jw#_  
    %           subplot(4,7,Nplot(k)) R'EUV0KX>Y  
    %           pcolor(x,x,z), shading interp %,Sf1fUJ  
    %           set(gca,'XTick',[],'YTick',[]) c0B|F  
    %           axis square voP7"Dl[  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ('wY9kvL&  
    %       end <h%O?mkC  
    % poGc a1  
    %   See also ZERNPOL, ZERNFUN2. Nkxm m/Z  
    ;<yd^Xs  
    %   Paul Fricker 11/13/2006 m8'C_U^89  
    UcBe'r}G  
    aRG2@5  
    % Check and prepare the inputs: fkf1m:Ckh  
    % ----------------------------- \^ghdU  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) *.L81er5~  
        error('zernfun:NMvectors','N and M must be vectors.') 1) ta  
    end -F'b8:m  
    4wC+S9I#E^  
    if length(n)~=length(m) ?]D"k4  
        error('zernfun:NMlength','N and M must be the same length.') \fA{1  
    end d>;&9;)H  
    I}Nd$P)>  
    n = n(:); }ci#>  
    m = m(:); u[nyW3MZ  
    if any(mod(n-m,2)) (Yp+bS(PU*  
        error('zernfun:NMmultiplesof2', ... ? A(QyaKz  
              'All N and M must differ by multiples of 2 (including 0).') DXz} YIEC  
    end 'F>'(XWWQ  
    XGP6L0j  
    if any(m>n) t]7&\ihZi~  
        error('zernfun:MlessthanN', ... X[f=h=|  
              'Each M must be less than or equal to its corresponding N.') O@p]KSfk  
    end :Ad &$e g+  
    0a-:<zm  
    if any( r>1 | r<0 ) :_!8 WB  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') FQ g~l4WX  
    end `PY>Hgb  
    v) vkn/:  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) yAoe51h?  
        error('zernfun:RTHvector','R and THETA must be vectors.') A&nU]R8S  
    end 3p0LN'q]A  
    ]\7]%(  
    r = r(:); 035rPT7-2-  
    theta = theta(:); f=)2f =  
    length_r = length(r); ^f# F I&  
    if length_r~=length(theta) |SyMngIY  
        error('zernfun:RTHlength', ... L!=QR8?@E  
              'The number of R- and THETA-values must be equal.') 8 Y5  
    end kF/9-[]$g,  
    0v9rv.Y"  
    % Check normalization: ^tXJj:wtS  
    % -------------------- P2bZ65>3y  
    if nargin==5 && ischar(nflag) Yo[;W vu  
        isnorm = strcmpi(nflag,'norm'); =JJL[}a|  
        if ~isnorm )_U<7"~0l  
            error('zernfun:normalization','Unrecognized normalization flag.') lsJnI|  
        end Dk?\)lD`  
    else Nm |!#(L  
        isnorm = false; ki85!k=Q2  
    end o865 (<p  
    Th=eNL]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F7Zwh5W  
    % Compute the Zernike Polynomials ]\!?qsT3}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q[nEsYP  
    wXsmn1w9  
    % Determine the required powers of r: ^MVkZ{gtre  
    % ----------------------------------- ih7/}   
    m_abs = abs(m); tg{H9tU;  
    rpowers = []; j$+nKc$  
    for j = 1:length(n) y\ a1iy  
        rpowers = [rpowers m_abs(j):2:n(j)]; 3D2E?$dX  
    end 8 XU1 /i7N  
    rpowers = unique(rpowers); 2$Tj84'X  
    '_V2!?+RU+  
    % Pre-compute the values of r raised to the required powers, Y'ow  
    % and compile them in a matrix: ;UxP Kpl  
    % ----------------------------- p\<u6v ~J  
    if rpowers(1)==0 ,lyb!k8  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); X-wf:h?i  
        rpowern = cat(2,rpowern{:}); C+TI]{t  
        rpowern = [ones(length_r,1) rpowern]; VY3&  
    else XHK70: i  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); cJrmm2.0kD  
        rpowern = cat(2,rpowern{:}); l(02W  
    end +(h\fm7*-  
    ;72T|e  
    % Compute the values of the polynomials: dxmE3*b`  
    % -------------------------------------- 6Udov pl  
    y = zeros(length_r,length(n)); UX dUO@  
    for j = 1:length(n) >k'c' 7/  
        s = 0:(n(j)-m_abs(j))/2; #W|'1 OX4  
        pows = n(j):-2:m_abs(j); .,OVzW  
        for k = length(s):-1:1 ={z*akn,  
            p = (1-2*mod(s(k),2))* ... Z /9>  
                       prod(2:(n(j)-s(k)))/              ... u,UmrR  
                       prod(2:s(k))/                     ... %T~ig[GstX  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ...  Y*14v~\'  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); f\jLqZY  
            idx = (pows(k)==rpowers); #E&80#Z5  
            y(:,j) = y(:,j) + p*rpowern(:,idx); A -b [>} _  
        end ~x|F)~:0=  
         ,]d,-)KX8  
        if isnorm dUQ DO o  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 8@tPm$  
        end Ba!J"b]  
    end +1D+]*t_?[  
    % END: Compute the Zernike Polynomials L>3x9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F~P%AjAx'  
    ;S>])5<  
    % Compute the Zernike functions: >Vwc3d  
    % ------------------------------ jJ5W>Q1mK$  
    idx_pos = m>0; D/Mi^5H)  
    idx_neg = m<0; Gy6l<:;  
    WUh$^5W  
    z = y; fe9LEM8j  
    if any(idx_pos) _iir<}  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); eb=D/  
    end ]VD|xm:kj  
    if any(idx_neg) ayfFVTy1d  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); yp({>{u7  
    end /|D*w^ >  
    <x<"n t  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) njWL U!  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. jW  3c"  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated lx[oaCr  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive a+%6B_|\  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, C,v(:ZE$J7  
    %   and THETA is a vector of angles.  R and THETA must have the same /K. !sQ$  
    %   length.  The output Z is a matrix with one column for every P-value, eep1I :N  
    %   and one row for every (R,THETA) pair. ,f[>L|?e  
    % @ < Q|5  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 5nKj )RH7M  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) !Rhl f.x  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) XBp?w  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ]%IT|/;9Y  
    %   for all p. U G~ba  
    % LdL/399<  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Pc NkAo  
    %   Zernike functions (order N<=7).  In some disciplines it is {/ BT9|LI  
    %   traditional to label the first 36 functions using a single mode 5 4L\Jx  
    %   number P instead of separate numbers for the order N and azimuthal !& z(:d  
    %   frequency M. B>0]. CK`  
    % !'cl"\h  
    %   Example: Z2'Bk2 L  
    % mqSQL}vR  
    %       % Display the first 16 Zernike functions RT.D"WvT  
    %       x = -1:0.01:1; F*3j.lI  
    %       [X,Y] = meshgrid(x,x); 4UW_Do  
    %       [theta,r] = cart2pol(X,Y); ZHm7Isa1  
    %       idx = r<=1; >8qQK r\"  
    %       p = 0:15; U'<KC"f:'!  
    %       z = nan(size(X)); NbU[l  
    %       y = zernfun2(p,r(idx),theta(idx)); -T[lx\}  
    %       figure('Units','normalized') ^$'z!+QRM  
    %       for k = 1:length(p) Nw1#M%/!r!  
    %           z(idx) = y(:,k); aPm`^ q  
    %           subplot(4,4,k) Na?!;1]_  
    %           pcolor(x,x,z), shading interp {;:/-0s  
    %           set(gca,'XTick',[],'YTick',[]) 1ke g9]  
    %           axis square l@\#Ywz  
    %           title(['Z_{' num2str(p(k)) '}']) b"#WxgaF  
    %       end 4Dw@r{  
    % 4DL)rkO  
    %   See also ZERNPOL, ZERNFUN. 2gCX}4^3b  
    s#)5h0t#du  
    %   Paul Fricker 11/13/2006 Zf65`K3  
    S|]X'f  
    Zw ^kmSL"  
    % Check and prepare the inputs: q@nP}Pv&5  
    % ----------------------------- cM$P`{QrM  
    if min(size(p))~=1 lna}@]oR  
        error('zernfun2:Pvector','Input P must be vector.') T`;%TO*Y  
    end A8oo@z68n>  
    + EGD.S{  
    if any(p)>35 _U |>b>  
        error('zernfun2:P36', ... Q2F+?w;,  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... S'_-G;g.  
               '(P = 0 to 35).']) uFPF!Ern  
    end LRb{hUt=  
    =p=rg$?  
    % Get the order and frequency corresonding to the function number: /qy-qUh3h  
    % ---------------------------------------------------------------- @EnuJe  
    p = p(:); ;I}kQ!q  
    n = ceil((-3+sqrt(9+8*p))/2); *U]V@;XF  
    m = 2*p - n.*(n+2); e0T34x'  
    83Q 4On  
    % Pass the inputs to the function ZERNFUN: < F`>,Pm  
    % ---------------------------------------- ~,5gUl?Il  
    switch nargin 17G'jiY H  
        case 3 [N#, K02mk  
            z = zernfun(n,m,r,theta); 6u7?dG'4  
        case 4 C86J IC"  
            z = zernfun(n,m,r,theta,nflag); i5K[>5  
        otherwise :=\Hoz  
            error('zernfun2:nargin','Incorrect number of inputs.') Te}8!_ohyC  
    end Htgx`N|  
    & '}/f5s|  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) j&Wl0  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Nd cg/d  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of h_T7% #0  
    %   order N and frequency M, evaluated at R.  N is a vector of 8W#heW\-]  
    %   positive integers (including 0), and M is a vector with the jhg;%+KB  
    %   same number of elements as N.  Each element k of M must be a G6FEp`  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) L"j tf78  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is UM6(s@$  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix eK!V );  
    %   with one column for every (N,M) pair, and one row for every Y~(Md@!0S  
    %   element in R. M#=] k  
    % ?Vdia:  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- /O^RF}  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is t[oT-r  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 8_6Q~  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 3 "Q=Vl"  
    %   for all [n,m]. (;57Vw  
    % 70;Jl).\{  
    %   The radial Zernike polynomials are the radial portion of the Y5FbU  
    %   Zernike functions, which are an orthogonal basis on the unit `/ q|@B7  
    %   circle.  The series representation of the radial Zernike ;F~LqC$  
    %   polynomials is Bxfc}vC.  
    % yzLpK;  
    %          (n-m)/2 h}cy D7Wn  
    %            __ 'Ub g0"F(  
    %    m      \       s                                          n-2s KFvQ  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r NZ-\h  
    %    n      s=0 !H irhD N  
    % *EZHJt9  
    %   The following table shows the first 12 polynomials. U]vYV  
    % .G)(0z("s  
    %       n    m    Zernike polynomial    Normalization <B6&I$Wc+  
    %       --------------------------------------------- JA'h4AXk  
    %       0    0    1                        sqrt(2) 0;:.B j  
    %       1    1    r                           2 75T7+:p  
    %       2    0    2*r^2 - 1                sqrt(6) @g$Gti  
    %       2    2    r^2                      sqrt(6) :SGF45>B@  
    %       3    1    3*r^3 - 2*r              sqrt(8) %y|)=cm[  
    %       3    3    r^3                      sqrt(8) k=B] &F  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) t $xY #:  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) jlBanGs?  
    %       4    4    r^4                      sqrt(10) oNU0 qZ5  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) r]l!WRn  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) xg NJeQ  
    %       5    5    r^5                      sqrt(12) L?Qg#YSd ~  
    %       --------------------------------------------- ]) rrG/3  
    % w[ !^;#  
    %   Example: L.2/*H#  
    % W'.s\e?gh  
    %       % Display three example Zernike radial polynomials }f8Uc+  
    %       r = 0:0.01:1; J]G?Rc  
    %       n = [3 2 5]; A%D7bQ  
    %       m = [1 2 1]; w - Pk7I  
    %       z = zernpol(n,m,r); -Gw$#!  
    %       figure _-9@qe  
    %       plot(r,z) I{lT>go  
    %       grid on ni6{pK4Wqm  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ]"1`+q6i  
    % N\#MwLm  
    %   See also ZERNFUN, ZERNFUN2. z(fAnn T?  
    & M~`:R  
    % A note on the algorithm. Fx $Q;H!.  
    % ------------------------ ld^=#]g  
    % The radial Zernike polynomials are computed using the series qZh1`\G  
    % representation shown in the Help section above. For many special W0k0$\iX  
    % functions, direct evaluation using the series representation can |d*&y#kV  
    % produce poor numerical results (floating point errors), because 9XRZ$j}L  
    % the summation often involves computing small differences between kIGbG;"_  
    % large successive terms in the series. (In such cases, the functions `xywho%/Y  
    % are often evaluated using alternative methods such as recurrence phn9:{TI  
    % relations: see the Legendre functions, for example). For the Zernike eOXHQjuj  
    % polynomials, however, this problem does not arise, because the T.We: ,{  
    % polynomials are evaluated over the finite domain r = (0,1), and dd\n8f  
    % because the coefficients for a given polynomial are generally all VsN pHQG]  
    % of similar magnitude. =9z[[dQ|L  
    % /}PF\j9#4  
    % ZERNPOL has been written using a vectorized implementation: multiple lNL6M%e$Q  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 59%tXiO  
    % values can be passed as inputs) for a vector of points R.  To achieve s_> f5/i2  
    % this vectorization most efficiently, the algorithm in ZERNPOL \uXcLhXN  
    % involves pre-determining all the powers p of R that are required to e?Ho a$k  
    % compute the outputs, and then compiling the {R^p} into a single RheRe  
    % matrix.  This avoids any redundant computation of the R^p, and T[sDVkCbxf  
    % minimizes the sizes of certain intermediate variables. Pp| *J^U 4  
    % .9"Y_/0   
    %   Paul Fricker 11/13/2006 3nu^l'WQ  
    qWx][D"  
    @EDs~ lPv  
    % Check and prepare the inputs: RgGyoZ  
    % ----------------------------- }t ;(VynV)  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) x,otFp  
        error('zernpol:NMvectors','N and M must be vectors.') qR8u$2}NY  
    end .>mr%#p  
    :LQ5 u[g$\  
    if length(n)~=length(m) .'rW.'Ft  
        error('zernpol:NMlength','N and M must be the same length.') x)JOClLr  
    end >A<bBK#  
    :OkT? (i  
    n = n(:); <]T`3W9  
    m = m(:); [ e8x&{L-_  
    length_n = length(n); \igmv]G%  
    "d$m@c  
    if any(mod(n-m,2)) zt<WXw(  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') X5`AGyX  
    end N*`b%XGn3  
    TD9;kN1`  
    if any(m<0) n f.wCtf].  
        error('zernpol:Mpositive','All M must be positive.') v9D22,K-  
    end s-RQMK}H  
    }#qGqY*@LK  
    if any(m>n) ynB_"mg  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') %rF?dvb;?  
    end A^RR@D  
    p<&Xd}]"^W  
    if any( r>1 | r<0 ) js8{]04y  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') K^3co  
    end R-RDT9&<  
    d)[;e()  
    if ~any(size(r)==1) H> '>3]G  
        error('zernpol:Rvector','R must be a vector.') 9XHz-+bQ  
    end $F/EJ>  
    +4,2<\fX  
    r = r(:); 0UH*\<R  
    length_r = length(r); Z1h]  
    hx f'5uc  
    if nargin==4 u1~9{"P*  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); Zo}y(N1K}  
        if ~isnorm ErT{(t7  
            error('zernpol:normalization','Unrecognized normalization flag.') )' hH^(Yu  
        end {w8 NN-n  
    else 'Vr$MaO  
        isnorm = false; % ',F  
    end Hh4 n  
    2~*Ez!.3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C'!;J  
    % Compute the Zernike Polynomials 2TES>}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% { K _kPgKS  
    Cv*x2KF G  
    % Determine the required powers of r: ;),BW g  
    % ----------------------------------- t'~/$=9}  
    rpowers = []; N 6T{  
    for j = 1:length(n) m rJQ#  
        rpowers = [rpowers m(j):2:n(j)]; /{{UP-  
    end jr /lk  
    rpowers = unique(rpowers); H Y ynMP  
    T?p`)  
    % Pre-compute the values of r raised to the required powers, zi'Jr)n  
    % and compile them in a matrix: 3s:%2%jVK  
    % ----------------------------- 6ATtW+sN]  
    if rpowers(1)==0 #-Z8Z i"44  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); NJ MJ  
        rpowern = cat(2,rpowern{:}); @O}7XRJ_8  
        rpowern = [ones(length_r,1) rpowern]; /?6gdN  
    else 8*SP~q  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); JTqq0OD}  
        rpowern = cat(2,rpowern{:}); EQe5JFR  
    end m))<!3  
    }6-ZE9H-v  
    % Compute the values of the polynomials: Dw2Q 'E  
    % -------------------------------------- ^#):c`  
    z = zeros(length_r,length_n); P0i V<T4^  
    for j = 1:length_n ZCVl5R(mZ  
        s = 0:(n(j)-m(j))/2; SMf+qiM-E  
        pows = n(j):-2:m(j); vZ#!uU^a:  
        for k = length(s):-1:1  Q2p)7G  
            p = (1-2*mod(s(k),2))* ... W0zbxJKjd  
                       prod(2:(n(j)-s(k)))/          ... &48_2Q"{  
                       prod(2:s(k))/                 ... d"U(`E=H9  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... `aqrSH5^h  
                       prod(2:((n(j)+m(j))/2-s(k))); [Qdq}FYr  
            idx = (pows(k)==rpowers); gr-x |wK  
            z(:,j) = z(:,j) + p*rpowern(:,idx); @4!x>q$3  
        end %@R~DBS  
         <8 #ObdY!  
        if isnorm nd{R 9B  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); .9|u QEL  
        end l+y}4 k=/  
    end S*"u/b;  
    ~JuKV&&}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)  q5?L1  
    vpafru4  
    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。 %uEtQh[  
     ,F}r@  
    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)