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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 ?7 \\e;j}  
    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! yhpz5[AuO  
     
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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 ;eL9{eF  
    function z = zernfun(n,m,r,theta,nflag) ememce,Np  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. p\p\q(S">  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Q`%R[#  
    %   and angular frequency M, evaluated at positions (R,THETA) on the S4CbyXW  
    %   unit circle.  N is a vector of positive integers (including 0), and Wg`AZ=t  
    %   M is a vector with the same number of elements as N.  Each element o>Er_r  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) &HW1mNF9  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, S6d`ioi-  
    %   and THETA is a vector of angles.  R and THETA must have the same 69K{+|  
    %   length.  The output Z is a matrix with one column for every (N,M) h#r^teui)  
    %   pair, and one row for every (R,THETA) pair. "=. t 36#  
    % V'Kgdj  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ;%C'FV e]  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), }x0- V8  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral /93l74.w  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, d[ >`")2)  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized H9)m^ *  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. }A=y=+4 j  
    % ?>c=}I#Ui-  
    %   The Zernike functions are an orthogonal basis on the unit circle. /cr}N%HZB  
    %   They are used in disciplines such as astronomy, optics, and ltB .Q  
    %   optometry to describe functions on a circular domain. M 3 '$[  
    % 6qWdd&1  
    %   The following table lists the first 15 Zernike functions. 0MMEo~dih  
    % D4Al3fe  
    %       n    m    Zernike function           Normalization :<Y}l-x  
    %       -------------------------------------------------- @`opDu!  
    %       0    0    1                                 1 o%{'UG  
    %       1    1    r * cos(theta)                    2 Zn=T#o  
    %       1   -1    r * sin(theta)                    2 %J:SO_6  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) &6 s) X  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Ho DVn/lr  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ;8WZx  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) (&|_quP7O  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Aa0b6?Jm  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) fkSO( C)  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) a,F&`Wg  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) W?yd#j  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?Xdak|?i  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) :@W.K5  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) g4`Kp; }&'  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 2?m.45`  
    %       -------------------------------------------------- }bN%u3mHws  
    % NK|?y  
    %   Example 1: c_aZ{S  
    % )"f N!9,F  
    %       % Display the Zernike function Z(n=5,m=1) 7Dnp'*H  
    %       x = -1:0.01:1; } VJfJ/  
    %       [X,Y] = meshgrid(x,x); b'MSkEiQG  
    %       [theta,r] = cart2pol(X,Y); o"wXIHUmV  
    %       idx = r<=1; u2oKH{/z  
    %       z = nan(size(X)); #HS]NA|e@  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ]E  =Iu  
    %       figure '$4O!YI9@  
    %       pcolor(x,x,z), shading interp x21dku<6K[  
    %       axis square, colorbar Z_\C*^  
    %       title('Zernike function Z_5^1(r,\theta)') i)#:qAtP*  
    % E_KCNn-f  
    %   Example 2: b jAnaya  
    % #%J5\+ua  
    %       % Display the first 10 Zernike functions $$:ZX  
    %       x = -1:0.01:1; ["\;kJ.  
    %       [X,Y] = meshgrid(x,x); l5l>d62  
    %       [theta,r] = cart2pol(X,Y); )54%HM_$k  
    %       idx = r<=1; 7~#:>OjW  
    %       z = nan(size(X)); Ax!+P\\2~  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ==i[w|  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; .]aF 1}AI  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; fgihy  
    %       y = zernfun(n,m,r(idx),theta(idx)); iC iZJ"  
    %       figure('Units','normalized') qfcYE=  
    %       for k = 1:10 Xl@cHO=i  
    %           z(idx) = y(:,k); L 8{\r$  
    %           subplot(4,7,Nplot(k)) ;n|^1S<[  
    %           pcolor(x,x,z), shading interp P{Q=mEQ  
    %           set(gca,'XTick',[],'YTick',[]) 5NJ4  
    %           axis square r\nx=  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) <F11m(  
    %       end DcSnia62f  
    % Scv#zuv_  
    %   See also ZERNPOL, ZERNFUN2. |yo\R{&6  
    j5@:a  
    %   Paul Fricker 11/13/2006 ?f/n0U4w  
    g/13~UM\  
    dg4vc][  
    % Check and prepare the inputs: _G1gtu]  
    % ----------------------------- nC{%quwh{  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) A)ipFB 6K  
        error('zernfun:NMvectors','N and M must be vectors.') 'l,V*5L  
    end =)|-?\[w  
    _HhbIU  
    if length(n)~=length(m) vFEQ7 qI  
        error('zernfun:NMlength','N and M must be the same length.') TIvRhbu  
    end O{ /q-~_  
    3]?#he  
    n = n(:); s`>[F@N7.o  
    m = m(:); [ Bl c^C{f  
    if any(mod(n-m,2)) 7t.!lh5G%  
        error('zernfun:NMmultiplesof2', ... ;2Q~0a|  
              'All N and M must differ by multiples of 2 (including 0).') sUPz/Z.h  
    end ?.Pg\ur  
    S;]*)i,v  
    if any(m>n) D-N8<:cA  
        error('zernfun:MlessthanN', ... Ks}Xgc\  
              'Each M must be less than or equal to its corresponding N.') } (!EuLL  
    end U!Ek'  
    zRPeNdX  
    if any( r>1 | r<0 ) p<:!)kt  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') NzNA>[$[  
    end Li Kxq=K  
    |\n_OS 7  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) V\6]n2  
        error('zernfun:RTHvector','R and THETA must be vectors.') } CfqG?)  
    end H<1WbM:w  
    "le>_Ze_>|  
    r = r(:); 9Vf1Xz  
    theta = theta(:); ;,]P=Ey  
    length_r = length(r); ]KJj6xn  
    if length_r~=length(theta) xp Og8u5  
        error('zernfun:RTHlength', ... @]ao"ui@/  
              'The number of R- and THETA-values must be equal.') Bm]8m=p  
    end @'GGm#<   
    uIZWO.OdU  
    % Check normalization: ?@V[#.  
    % -------------------- %AQIGBcgL  
    if nargin==5 && ischar(nflag) :kGU,>BN  
        isnorm = strcmpi(nflag,'norm'); [}&Sxgv  
        if ~isnorm QV\eMuNy  
            error('zernfun:normalization','Unrecognized normalization flag.') 9V5-%Iv  
        end k-}b{  
    else 5L!y-3  
        isnorm = false; @*sWu_ -Y%  
    end ~ yX2\i"  
    &%-73nYw  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% QqU!Najf  
    % Compute the Zernike Polynomials rEF0oJ.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V5rS T +  
    r9dyA5oD  
    % Determine the required powers of r: K>{T_){  
    % ----------------------------------- (P$H<FtH  
    m_abs = abs(m); q|),`.eh\  
    rpowers = []; }+sT4'Ah>  
    for j = 1:length(n) dN$Tf  
        rpowers = [rpowers m_abs(j):2:n(j)]; )KAEt.  
    end lg&t8FHa;  
    rpowers = unique(rpowers); m|G'K[8  
    q7E~+p(>(  
    % Pre-compute the values of r raised to the required powers, Z+=@<i''  
    % and compile them in a matrix: UNBH  
    % ----------------------------- pJtex^{!:  
    if rpowers(1)==0 1 9CK+;b  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ^cuc.g)c$?  
        rpowern = cat(2,rpowern{:}); =z /dcC$r  
        rpowern = [ones(length_r,1) rpowern]; &mx)~J^m  
    else .*)2SNH  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9_5ow  
        rpowern = cat(2,rpowern{:}); ;-qO'V:;  
    end aSnF KB  
    i,/0/?)*_  
    % Compute the values of the polynomials: B]l)++~  
    % -------------------------------------- HKUn`ng  
    y = zeros(length_r,length(n)); (P:<t6;+  
    for j = 1:length(n) 3(,?S$>  
        s = 0:(n(j)-m_abs(j))/2; ^\S~?0^m  
        pows = n(j):-2:m_abs(j); Nb'''W-iu  
        for k = length(s):-1:1 *vwbgJG! *  
            p = (1-2*mod(s(k),2))* ... e(<st r>  
                       prod(2:(n(j)-s(k)))/              ... \}|o1Xh2  
                       prod(2:s(k))/                     ... \r+8qC[,  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 71t* %  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); D1=((`v '  
            idx = (pows(k)==rpowers); pW J Fz-  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Rw0qcM\>|  
        end (O(}p~s  
          cHk)i  
        if isnorm lE(a%'36  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 2|*JSU.I  
        end B e0ND2oo  
    end %#xaA'? [  
    % END: Compute the Zernike Polynomials VL` z[|e @  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /k,-P  
    4?q <e*W  
    % Compute the Zernike functions: fO[+LR 'ax  
    % ------------------------------ _ =VqrK7T  
    idx_pos = m>0; 6"&6 `f  
    idx_neg = m<0; }%{LJ}\Px  
    DrY:9[LP  
    z = y; 2Tp1n8FV  
    if any(idx_pos) ?Yth0O6?sb  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Ay0U=#XP  
    end ,N]H dR  
    if any(idx_neg) r w2arx  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); sl`s_$J  
    end NRIG1v>  
    %1xb,g KO  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) y[)>yq y  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 26I_YL,S  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 2db3I:;E  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive NflD/q/ L  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, UU;(rS/  
    %   and THETA is a vector of angles.  R and THETA must have the same EIf5(/jo  
    %   length.  The output Z is a matrix with one column for every P-value, xSsa(b  
    %   and one row for every (R,THETA) pair. %In A+5s`  
    % $0>60<J  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Mh:L$f0A%O  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) i]YV {  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi)  #~2%)  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 G\Cp7:j}  
    %   for all p. z"-u95H  
    % irFMmIb  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 lm-dW'7&  
    %   Zernike functions (order N<=7).  In some disciplines it is CY s,`  
    %   traditional to label the first 36 functions using a single mode _VRpI)mu  
    %   number P instead of separate numbers for the order N and azimuthal m.# VYN`+A  
    %   frequency M. (_mnB W  
    % 8*#R]9  
    %   Example: F;4vPbH+  
    % pN:Kdi  
    %       % Display the first 16 Zernike functions (|:M&Cna]  
    %       x = -1:0.01:1; ~J wb`g.  
    %       [X,Y] = meshgrid(x,x); t{^*6XOcJ  
    %       [theta,r] = cart2pol(X,Y); Ai=s e2  
    %       idx = r<=1; r ~jm`y  
    %       p = 0:15; \r^qL^  
    %       z = nan(size(X)); JQ8fdP A  
    %       y = zernfun2(p,r(idx),theta(idx)); lT(WD}OS  
    %       figure('Units','normalized') /Y W>*?"N  
    %       for k = 1:length(p) =9O^p@Q#W  
    %           z(idx) = y(:,k); 4B =7:r  
    %           subplot(4,4,k) 42{Ew8  
    %           pcolor(x,x,z), shading interp , pDnRRJ!  
    %           set(gca,'XTick',[],'YTick',[]) z&#SPH*  
    %           axis square `w#Oih!6A|  
    %           title(['Z_{' num2str(p(k)) '}']) d6} r#\  
    %       end TJ_$vI  
    % 0=@?ob7  
    %   See also ZERNPOL, ZERNFUN. z1PBMSG  
    "CSsCA$/  
    %   Paul Fricker 11/13/2006 ddJQC|xR}  
    !;&p"E|b#  
    D.B.7-_8  
    % Check and prepare the inputs: cs,N <|  
    % ----------------------------- twL3\ }N/B  
    if min(size(p))~=1 dpAjR  
        error('zernfun2:Pvector','Input P must be vector.') j"ThEx0  
    end #C~+JL  
    GY6`JWk  
    if any(p)>35 mXRB7k  
        error('zernfun2:P36', ... [-65PC4aN  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... W98i[Q9A7  
               '(P = 0 to 35).']) <r .)hT"0  
    end \rx3aJl  
    / ;$#d}R  
    % Get the order and frequency corresonding to the function number: 1tEgl\u\  
    % ---------------------------------------------------------------- Fsmycr!R  
    p = p(:); 9_# >aOqL  
    n = ceil((-3+sqrt(9+8*p))/2); dsb`xw  
    m = 2*p - n.*(n+2); 6Z>FTz_  
    @K\~O__  
    % Pass the inputs to the function ZERNFUN: ^W`<gR  
    % ---------------------------------------- O,+1<.;+  
    switch nargin =e/9&993  
        case 3 WZ-s--n#  
            z = zernfun(n,m,r,theta); Jj>?GAir  
        case 4 iZ#!O* >  
            z = zernfun(n,m,r,theta,nflag); h;C5hU 4P  
        otherwise Eza`Z` ^el  
            error('zernfun2:nargin','Incorrect number of inputs.') p#ol*m5wE  
    end yQ_B)b  
    (|[2J3ZET  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) V@B__`y7  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. G3+.H  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of nim*/LC[:  
    %   order N and frequency M, evaluated at R.  N is a vector of HFKf kAl  
    %   positive integers (including 0), and M is a vector with the @E;=*9ek{u  
    %   same number of elements as N.  Each element k of M must be a =[H;orMr  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) -(~.6WnhS  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Vr)<\h  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 6DM$g=/ '  
    %   with one column for every (N,M) pair, and one row for every %b%-Ogz;4  
    %   element in R. q+Lr"&'Q  
    % Z8# (kmBdB  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly-  tR}MrM  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 8J=? 5  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to "8c@sHk(w  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 oo,uO;0G  
    %   for all [n,m]. pf%=h |  
    % ~rlB'8j(  
    %   The radial Zernike polynomials are the radial portion of the f*!j[U/r_  
    %   Zernike functions, which are an orthogonal basis on the unit  W,4QzcQR  
    %   circle.  The series representation of the radial Zernike yL%K4$z  
    %   polynomials is = 6tHsN23  
    % kjW+QT?T&  
    %          (n-m)/2 r#+d&.|  
    %            __ z8jk[5z  
    %    m      \       s                                          n-2s R%Qf7Q  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 2>|dF~"  
    %    n      s=0 VCu{&Sh*  
    % :j5n7s?&=y  
    %   The following table shows the first 12 polynomials. &+a9+y  
    % B268e  
    %       n    m    Zernike polynomial    Normalization 6FUw"|\u{  
    %       --------------------------------------------- Ipf|")*  
    %       0    0    1                        sqrt(2) Y&[1`:-~-  
    %       1    1    r                           2 +1Vjw'P  
    %       2    0    2*r^2 - 1                sqrt(6) @x9a?L.48  
    %       2    2    r^2                      sqrt(6) JIOh#VNU  
    %       3    1    3*r^3 - 2*r              sqrt(8) wmk *h-  
    %       3    3    r^3                      sqrt(8) E'v _#FLvR  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) __`*dL>*  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) RSr %n1  
    %       4    4    r^4                      sqrt(10) _.>QEh5"5  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 3t`P@nL0;  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) f_wvZ&  
    %       5    5    r^5                      sqrt(12) 9iXeBC  
    %       --------------------------------------------- Mx6@$tQ%  
    % - |kA)M[  
    %   Example: mYxuA0/k  
    % 5j:0Yt  
    %       % Display three example Zernike radial polynomials 4FEk5D  
    %       r = 0:0.01:1; IN4=YrM^  
    %       n = [3 2 5]; 9!f/aI  
    %       m = [1 2 1]; AcS|c:3MUy  
    %       z = zernpol(n,m,r); ie;]/v a  
    %       figure aY3^C q(r  
    %       plot(r,z) JCz@s~f\y  
    %       grid on sqla}~CiX  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') &yGaCq;0  
    % 8j Mk)-  
    %   See also ZERNFUN, ZERNFUN2. ~oI1 zNz/  
    8 ![|F:  
    % A note on the algorithm. IKABBW  
    % ------------------------ 2uG0/7  
    % The radial Zernike polynomials are computed using the series uJOJ-5}yt  
    % representation shown in the Help section above. For many special hD\rtW  
    % functions, direct evaluation using the series representation can MJ7Y#<u  
    % produce poor numerical results (floating point errors), because x6(~;J  
    % the summation often involves computing small differences between EzDk}uKY0R  
    % large successive terms in the series. (In such cases, the functions z8{a(nKP  
    % are often evaluated using alternative methods such as recurrence \x)n>{3C  
    % relations: see the Legendre functions, for example). For the Zernike >GQEqXs  
    % polynomials, however, this problem does not arise, because the ">~.$Jp_4  
    % polynomials are evaluated over the finite domain r = (0,1), and g$( V^  
    % because the coefficients for a given polynomial are generally all SPN5dE.@  
    % of similar magnitude. 8S>&WR%jH]  
    % 0zo?eI  
    % ZERNPOL has been written using a vectorized implementation: multiple {88)~  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 9U{a{~b  
    % values can be passed as inputs) for a vector of points R.  To achieve PH`9MXh  
    % this vectorization most efficiently, the algorithm in ZERNPOL @[5xq  
    % involves pre-determining all the powers p of R that are required to + hn+K1  
    % compute the outputs, and then compiling the {R^p} into a single RMX:9aQ3F  
    % matrix.  This avoids any redundant computation of the R^p, and e "/;7:J5\  
    % minimizes the sizes of certain intermediate variables. MHSs!^/g5  
    % >"gf3rioW  
    %   Paul Fricker 11/13/2006 W.z;B<  
    j]*j}%hz  
    OC1I&",Ai|  
    % Check and prepare the inputs: -M%_\;"de  
    % ----------------------------- n,=VQ Ou  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) )_{dWf1  
        error('zernpol:NMvectors','N and M must be vectors.') RMd[Yr2e  
    end @.G[s)x  
    ! vP[;6  
    if length(n)~=length(m) GN-mrQo  
        error('zernpol:NMlength','N and M must be the same length.') x\F,SEj  
    end VS9`{  
    5nv<^>[J  
    n = n(:); >2~+.WePu  
    m = m(:); 0dhF&*h|L  
    length_n = length(n); i-bJS6  
    MxSM@3v(  
    if any(mod(n-m,2)) )E6;-rD0^+  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') B+[A]dgS  
    end Q@ 2i~Qo[  
    CLeG<Hi ~  
    if any(m<0) @kk4]:,w  
        error('zernpol:Mpositive','All M must be positive.') /J04^ 6  
    end Q>/C*@  
    P8^hBv*  
    if any(m>n) zXv3:uRp.  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') +|bmT  
    end A]YV s  
    =Vy`J)z9  
    if any( r>1 | r<0 ) W5?yy>S6N  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') `9)t[7  
    end b8LoIY*  
    5O~HWBX.  
    if ~any(size(r)==1) 0@G")L Ue0  
        error('zernpol:Rvector','R must be a vector.') S|apw7C  
    end +x_Rfk$fb  
    BhM '@g*  
    r = r(:); RY c!~Wh~Y  
    length_r = length(r); Lg~B'd8m  
    f8M$45A'  
    if nargin==4 hF@Gn/  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); vk*=4}:  
        if ~isnorm ,Y|^^?'j Q  
            error('zernpol:normalization','Unrecognized normalization flag.') .q[SI$qO/  
        end X>NhZ5\  
    else c6nflk.l  
        isnorm = false; 8>X d2X  
    end }-~X4u#   
    %^I88,$&L  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% JNkwEZhHyg  
    % Compute the Zernike Polynomials [%9no B  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /%0<p,T  
    Bn61AFy`  
    % Determine the required powers of r: BT)X8>ct  
    % ----------------------------------- z;A>9vQ_J  
    rpowers = []; @O@GRq&V  
    for j = 1:length(n) 3 n'V\H vz  
        rpowers = [rpowers m(j):2:n(j)]; M7ers|&{  
    end w3=%*<  
    rpowers = unique(rpowers);  EI_  
    deM7fN4lTi  
    % Pre-compute the values of r raised to the required powers, ?[)}l9  
    % and compile them in a matrix: %g1,N k  
    % ----------------------------- TjHwjRa  
    if rpowers(1)==0 OCZ[D{i9@  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); L=q+|j1>  
        rpowern = cat(2,rpowern{:}); d?:=PH  
        rpowern = [ones(length_r,1) rpowern]; ,WvY$_#xW%  
    else %]I ZLJ  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); rS4@1`/R  
        rpowern = cat(2,rpowern{:});  .&9 i  
    end wjh=Q  
    VQ0fS!5'  
    % Compute the values of the polynomials: %{/%mJoX  
    % -------------------------------------- 3t<a $i  
    z = zeros(length_r,length_n); _~rI+lA  
    for j = 1:length_n /9zE^YcT  
        s = 0:(n(j)-m(j))/2; ?0YCpn  
        pows = n(j):-2:m(j); ~=KJzOS,S  
        for k = length(s):-1:1 *^RmjW1I  
            p = (1-2*mod(s(k),2))* ... v.:3"<ur}  
                       prod(2:(n(j)-s(k)))/          ... 2Fz|fW_  
                       prod(2:s(k))/                 ... A |3tI  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... - /(s#D  
                       prod(2:((n(j)+m(j))/2-s(k))); #% of;mJv  
            idx = (pows(k)==rpowers); wKi^C 8Z2  
            z(:,j) = z(:,j) + p*rpowern(:,idx); `.T}=j|  
        end -K rxMi  
         9*E7}b,  
        if isnorm Qt,M!i,  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); `5~ +,/Ys  
        end $Bj;D=d@V  
    end n+BJxu?  
    w.lAQ5)I%\  
    % 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)  ]APvp.Tw:  
    -[OXSaf6  
    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。 zq1mmFIO  
    g(O;{Q_  
    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)