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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 i uNBw]  
    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! -C>q,mDJZ  
     
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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 #Is/j =  
    function z = zernfun(n,m,r,theta,nflag) WN_i-A1G/h  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 6pKb!JJ  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N PN +<C7/  
    %   and angular frequency M, evaluated at positions (R,THETA) on the QIcg4\d%s  
    %   unit circle.  N is a vector of positive integers (including 0), and _kJ?mTk  
    %   M is a vector with the same number of elements as N.  Each element qXb{A*J  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ckZZ)lW`*  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 9AbSt&#  
    %   and THETA is a vector of angles.  R and THETA must have the same 3 E~d  
    %   length.  The output Z is a matrix with one column for every (N,M) )Q!3p={S*  
    %   pair, and one row for every (R,THETA) pair. b')Lj]%;k  
    % H=f'nm]dQ  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike p{sbf;-x}  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9qqzCMrI0e  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 7n_'2qY  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ub#>kCL9  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized HLP nbI-+  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. IO(Y_7  
    % E@f2hW2  
    %   The Zernike functions are an orthogonal basis on the unit circle. _;M46o%h  
    %   They are used in disciplines such as astronomy, optics, and AIx,c1G]K  
    %   optometry to describe functions on a circular domain. RCS91[  
    % Pdg%:aY  
    %   The following table lists the first 15 Zernike functions. !JkH$~  
    % j.5;0b_L^  
    %       n    m    Zernike function           Normalization Fp`MX>F  
    %       -------------------------------------------------- K)h\X~s  
    %       0    0    1                                 1 :*{>=BD  
    %       1    1    r * cos(theta)                    2 #kuk3}&  
    %       1   -1    r * sin(theta)                    2 0%m}tfQ5  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) '+ 8.nN  
    %       2    0    (2*r^2 - 1)                    sqrt(3) "DW; 6<m  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ?^# h|aUp.  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) !A6l\_  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) e^Ds|}{V  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) {O"?_6',  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) V&' :S{i  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) zeXMi:X  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Hko(@z  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) _>/T<Db  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) V?k"BU  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) /eoS$q  
    %       -------------------------------------------------- zW@OSKq4  
    % CD]2a@j {  
    %   Example 1: d^&F%)AT  
    % e|L$e0  
    %       % Display the Zernike function Z(n=5,m=1) )>! IY Q  
    %       x = -1:0.01:1; I3 %P_oW'  
    %       [X,Y] = meshgrid(x,x); W[dMf!(  
    %       [theta,r] = cart2pol(X,Y); Dm3/i |Y  
    %       idx = r<=1; is3nLm(  
    %       z = nan(size(X)); Wgh4DhAW  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); <Wn"_Ud=  
    %       figure yxECK&&P0#  
    %       pcolor(x,x,z), shading interp +3c!.] o;  
    %       axis square, colorbar wGqQR)a  
    %       title('Zernike function Z_5^1(r,\theta)') K|H&x"t  
    % $ljgFmR_  
    %   Example 2: U#B,Q6~  
    % I92c!`{  
    %       % Display the first 10 Zernike functions ,sAN,?eG~  
    %       x = -1:0.01:1; R|Oy/RGY$  
    %       [X,Y] = meshgrid(x,x); S;o U'KOY  
    %       [theta,r] = cart2pol(X,Y); %^L :K5V  
    %       idx = r<=1; 8Ee bWs*1  
    %       z = nan(size(X)); /12D >OK  
    %       n = [0  1  1  2  2  2  3  3  3  3]; "CEy r0h  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; W~1/vJ.*l  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ]~,V(K  
    %       y = zernfun(n,m,r(idx),theta(idx)); xHml" Y1  
    %       figure('Units','normalized') ~YIGOL"?  
    %       for k = 1:10 N.J;/!%!  
    %           z(idx) = y(:,k); @17hB h  
    %           subplot(4,7,Nplot(k)) AUloP?24  
    %           pcolor(x,x,z), shading interp CqXD z  
    %           set(gca,'XTick',[],'YTick',[]) 67I6]3[ Z  
    %           axis square u_aln[oIv  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Y$^x.^dT,  
    %       end 7]_lSYwrb  
    % ZCQ7xQD  
    %   See also ZERNPOL, ZERNFUN2. 7'[C+/:  
    HQ%-e5Q  
    %   Paul Fricker 11/13/2006 $*| :A  
    (D'Z4Y  
    TQ? D*&  
    % Check and prepare the inputs: )Oq N\  
    % ----------------------------- 4#5w^  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) i<g|+}I  
        error('zernfun:NMvectors','N and M must be vectors.') `_]Z#X&&h  
    end WUid5e2  
    U*Z P>Vv  
    if length(n)~=length(m) p[(VhbN  
        error('zernfun:NMlength','N and M must be the same length.') 8#%p[TLj  
    end ,L+tm>I  
    #@,39!;,:O  
    n = n(:); v>3)^l:=Y*  
    m = m(:); Sti)YCXH  
    if any(mod(n-m,2)) Q6y883>9  
        error('zernfun:NMmultiplesof2', ... W{Cc wq  
              'All N and M must differ by multiples of 2 (including 0).') ;lST@>  
    end %$j)?e  
    .>0e?A4,5?  
    if any(m>n) =2#a@D6Bl  
        error('zernfun:MlessthanN', ... O)MKEMuA  
              'Each M must be less than or equal to its corresponding N.') \ ?[#>L4  
    end _=Y]ZX`j  
     6h N~<  
    if any( r>1 | r<0 ) $Yt29AQ  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') #Zpp*S55  
    end 2}u hPW+  
    zCD?5*7  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) a z 7Vy-  
        error('zernfun:RTHvector','R and THETA must be vectors.') p6[a"~y  
    end 5y! 4ny _  
    w>T1D  
    r = r(:); rt%.IQdY  
    theta = theta(:); r)<]W@ Pr  
    length_r = length(r); 05:`(vl  
    if length_r~=length(theta) b r)oSw  
        error('zernfun:RTHlength', ... . m_y5J  
              'The number of R- and THETA-values must be equal.') 8NJ(l  
    end U">D_ 8  
    h0NM5   
    % Check normalization: OpY2Z7_  
    % -------------------- [~ bfM6Jw  
    if nargin==5 && ischar(nflag) @.fuR#  
        isnorm = strcmpi(nflag,'norm'); 4KE"r F  
        if ~isnorm 1)u 3  
            error('zernfun:normalization','Unrecognized normalization flag.') 2O {@W +Mt  
        end KyW6[WA9  
    else FG7}MUu  
        isnorm = false; ?eT^gWX  
    end /-<S FT`  
    fGJPZe  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #NVtZs!V/  
    % Compute the Zernike Polynomials M#on-[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \_FX}1Wc2.  
    cu|gM[  
    % Determine the required powers of r: < pI2}  
    % ----------------------------------- #M6@{R2_  
    m_abs = abs(m); cj-P&D[Ny[  
    rpowers = []; <CJua1l\  
    for j = 1:length(n) ,+P!R0PNH  
        rpowers = [rpowers m_abs(j):2:n(j)]; I,vy__ sZ  
    end )JE;#m0q  
    rpowers = unique(rpowers); .Vux~A  
    Lm\N`  
    % Pre-compute the values of r raised to the required powers, Z{`;Ys:zk  
    % and compile them in a matrix: ;rpjXP  
    % ----------------------------- T%K(opISc(  
    if rpowers(1)==0 VO>A+vx3M  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); #EAP<h  
        rpowern = cat(2,rpowern{:});  L5""  
        rpowern = [ones(length_r,1) rpowern]; 8Cz_LyL  
    else }pj>BK>  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Z}.N4 /  
        rpowern = cat(2,rpowern{:}); *. l,_68  
    end DDn@M|*$  
    KDgJ~T  
    % Compute the values of the polynomials: /j./  
    % -------------------------------------- Gvv~P3Dm  
    y = zeros(length_r,length(n)); aM?Xi6 U5  
    for j = 1:length(n) bLGgu#  
        s = 0:(n(j)-m_abs(j))/2; [=9-AG~}  
        pows = n(j):-2:m_abs(j); vmL% %7  
        for k = length(s):-1:1 >|!F.W  
            p = (1-2*mod(s(k),2))* ... KgX~PP>  
                       prod(2:(n(j)-s(k)))/              ... M~w =ZJ@  
                       prod(2:s(k))/                     ... 2}>jq8Y47  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ,xB&{ J  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); t>f<4~%MJ  
            idx = (pows(k)==rpowers); ,rc5r3  
            y(:,j) = y(:,j) + p*rpowern(:,idx); uQWJ7Xm  
        end lz@fXaZM  
         C_=! ( @`8  
        if isnorm EP&iG%(k  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); {<iIL3\mC  
        end {S(?E_id5b  
    end Z; Xg5  
    % END: Compute the Zernike Polynomials P %f],f  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% H1rge<  
    }9t$Cs%  
    % Compute the Zernike functions: (%"M% Qko  
    % ------------------------------ u_s  
    idx_pos = m>0; w-};\]I  
    idx_neg = m<0;  y$7Fq'  
    ;$l!mv 7  
    z = y; k'}}eu/ q  
    if any(idx_pos) T[B@7$Dp*  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 2]C`S,)  
    end |/C>xunzz  
    if any(idx_neg) 0[TZ$<v"  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Cq;t;qN,nQ  
    end gBUtv|(@>[  
    23@e?A=C  
    % 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) iq-n(Rfw~  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. P}N%**>`  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Ke,UwYG2~G  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Y>geP+ -  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, _ $PZID  
    %   and THETA is a vector of angles.  R and THETA must have the same JVf8KHDj  
    %   length.  The output Z is a matrix with one column for every P-value, k-xh-&  
    %   and one row for every (R,THETA) pair. 4_3Jpz*  
    % ]24aK_Uu  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike GLQ1rT  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) } *|_P  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 'A .c*<_  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 %sP C3L  
    %   for all p. st P~/}  
    % ]WR+>)ERb  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ;0O3b  
    %   Zernike functions (order N<=7).  In some disciplines it is dX{|-;6vm  
    %   traditional to label the first 36 functions using a single mode &Z/aM?  
    %   number P instead of separate numbers for the order N and azimuthal |8PUmax  
    %   frequency M. A-1Wn^,> *  
    % 4\4onCzuT  
    %   Example: @B %m,Mx  
    % ]N_(M   
    %       % Display the first 16 Zernike functions ~Wjm"|c  
    %       x = -1:0.01:1; UhYeyT  
    %       [X,Y] = meshgrid(x,x); SkBa- *MC  
    %       [theta,r] = cart2pol(X,Y); <:0649ZB  
    %       idx = r<=1; )9MmL-7K  
    %       p = 0:15; Kl GPu GL  
    %       z = nan(size(X)); 0?  (  
    %       y = zernfun2(p,r(idx),theta(idx)); uQazUFw  
    %       figure('Units','normalized') ]c]rIOTN  
    %       for k = 1:length(p) vBx*bZ  
    %           z(idx) = y(:,k); akHcN]sa2  
    %           subplot(4,4,k) eU 'DQp*  
    %           pcolor(x,x,z), shading interp 8M*[RlUJB  
    %           set(gca,'XTick',[],'YTick',[]) EQ'iyXhEe  
    %           axis square LvgNdVJDP|  
    %           title(['Z_{' num2str(p(k)) '}']) 1OK,r`   
    %       end Y,?s-AB  
    % @y3w_;P  
    %   See also ZERNPOL, ZERNFUN. G[n^SEY!  
    X> :@`}bq  
    %   Paul Fricker 11/13/2006 /uS(Z-@  
    \?7)oFNz  
    =)vmX0vL  
    % Check and prepare the inputs: #-dfG.*  
    % ----------------------------- i71 ,  
    if min(size(p))~=1 EKoAIC*?p  
        error('zernfun2:Pvector','Input P must be vector.') {i y[8eLg  
    end pV{MW#e  
    ,0%P3  
    if any(p)>35 l?v`kAMR  
        error('zernfun2:P36', ... ,}2yxo;i  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... eWzD'3h^  
               '(P = 0 to 35).']) Nq6~6Rr  
    end [T#5$J  
    / 1 lIV_Z  
    % Get the order and frequency corresonding to the function number: ?nJ7lLQA  
    % ---------------------------------------------------------------- O^ZOc0<  
    p = p(:); a3e<< <Z>R  
    n = ceil((-3+sqrt(9+8*p))/2); \PU3{_G]  
    m = 2*p - n.*(n+2); R+k-mbvnt  
    BoZ])Y6=  
    % Pass the inputs to the function ZERNFUN: bg;N BoZd  
    % ---------------------------------------- l G12Su/  
    switch nargin E;$)Oz  
        case 3 =xcA4"k  
            z = zernfun(n,m,r,theta); P.Pw .[:3  
        case 4 *5Upb,* *  
            z = zernfun(n,m,r,theta,nflag); Ry>c]\a]  
        otherwise P5/K?I~/So  
            error('zernfun2:nargin','Incorrect number of inputs.') 48dIh\TH"  
    end wJ@8-H 8}  
    wEL$QOu$  
    % 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) fTiqY72h  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. vw=OGjT_>m  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of u4TU"r("A  
    %   order N and frequency M, evaluated at R.  N is a vector of 9 2_F8y*D  
    %   positive integers (including 0), and M is a vector with the amq]&.M  
    %   same number of elements as N.  Each element k of M must be a !Cxo4Twg  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) hZ2!UW4'  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is "&?F 6Pi  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix bK;I:JK3  
    %   with one column for every (N,M) pair, and one row for every "3o{@TdU  
    %   element in R. h- .V[]<  
    % N),bhYS]  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ~$XbYR-  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is fP>_P# gZ  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to |_L\^T|6  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 $3>k/*=  
    %   for all [n,m]. vaL+@Kq~&  
    % 4 zuM?Dp  
    %   The radial Zernike polynomials are the radial portion of the [uK*=K/v  
    %   Zernike functions, which are an orthogonal basis on the unit wY|&qX,  
    %   circle.  The series representation of the radial Zernike %,f|H :+>u  
    %   polynomials is KrE:ilm#^Y  
    % )W9W8>Cc5_  
    %          (n-m)/2 i? 5jl&30  
    %            __ taOD,}c|$  
    %    m      \       s                                          n-2s YT\x'`>Q  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ,jJ&x7ra8  
    %    n      s=0 FEj{/  
    % B:S/ ?v  
    %   The following table shows the first 12 polynomials. C9zQ{G  
    % i5wXT  
    %       n    m    Zernike polynomial    Normalization ,l`4)@{G  
    %       --------------------------------------------- _j{^I^P  
    %       0    0    1                        sqrt(2) sv`+?hjG  
    %       1    1    r                           2 .;j}:<  
    %       2    0    2*r^2 - 1                sqrt(6) [=*c8  
    %       2    2    r^2                      sqrt(6) J J@O5  
    %       3    1    3*r^3 - 2*r              sqrt(8) P0O5CaR  
    %       3    3    r^3                      sqrt(8) 2mUq$kws  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) I;iJa@HWQ  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) '>dsROB->  
    %       4    4    r^4                      sqrt(10) S*;8z}5<\  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 1{@f:~v?  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) z5G<h  
    %       5    5    r^5                      sqrt(12) R2{y1b$l  
    %       --------------------------------------------- q\wT[W31@  
    % EIZSV>  
    %   Example: q#9JJWSs  
    % xo{3r\u?}  
    %       % Display three example Zernike radial polynomials uk%C:4T  
    %       r = 0:0.01:1; )>,; GVu"  
    %       n = [3 2 5]; 5bU[uT,`6  
    %       m = [1 2 1];  d(PS  
    %       z = zernpol(n,m,r); IG@.WsM_  
    %       figure P5 GM s  
    %       plot(r,z) R'R LF =  
    %       grid on qOaI4JP@  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') RC(fhqV  
    % 57r?`'#*  
    %   See also ZERNFUN, ZERNFUN2. r #H(kJu,  
    ^M?O  
    % A note on the algorithm. !ceT>i90h  
    % ------------------------ LASR*  
    % The radial Zernike polynomials are computed using the series cHN eiOF  
    % representation shown in the Help section above. For many special E}eu]2=nU}  
    % functions, direct evaluation using the series representation can g+>$_s  
    % produce poor numerical results (floating point errors), because 3^p<Wx  
    % the summation often involves computing small differences between dH4wyd`  
    % large successive terms in the series. (In such cases, the functions CZ2&9Vb9I  
    % are often evaluated using alternative methods such as recurrence Hkq""'Mx+w  
    % relations: see the Legendre functions, for example). For the Zernike 'r`#u@TTZ  
    % polynomials, however, this problem does not arise, because the p H&Tb4  
    % polynomials are evaluated over the finite domain r = (0,1), and GFM $1}  
    % because the coefficients for a given polynomial are generally all r&F(VF0 6  
    % of similar magnitude. 5 :O7cBr  
    %  L~F"  
    % ZERNPOL has been written using a vectorized implementation: multiple }.md$N_F  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] vLQ!kB^\W  
    % values can be passed as inputs) for a vector of points R.  To achieve ho*44=j  
    % this vectorization most efficiently, the algorithm in ZERNPOL Glz)-hjJ:n  
    % involves pre-determining all the powers p of R that are required to [k~V77w 14  
    % compute the outputs, and then compiling the {R^p} into a single U~{fbS3,  
    % matrix.  This avoids any redundant computation of the R^p, and 8@`"ZzM  
    % minimizes the sizes of certain intermediate variables. !uaV6K  
    % T\;7'  
    %   Paul Fricker 11/13/2006 _86pbr9  
    9qyA{ |3  
    r$3{1HXc  
    % Check and prepare the inputs: 1&{]jG{#  
    % ----------------------------- 9+'QH  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) z"4UObVs  
        error('zernpol:NMvectors','N and M must be vectors.') W)WL1@!Z  
    end s)_Xj`Q#  
    cYBv}ylw}R  
    if length(n)~=length(m) a.P7O!2Lp  
        error('zernpol:NMlength','N and M must be the same length.') 6Y!hz7D  
    end O{b.-<  
    JNY;;9o  
    n = n(:); i3C5"\y  
    m = m(:); ,E&PIbDL1  
    length_n = length(n); W i a%rm  
    h\+U+ ?u  
    if any(mod(n-m,2)) x13t@b  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') S`,(10Y  
    end J;Y=o B  
    {(mT,}`4  
    if any(m<0) bs-O3w  
        error('zernpol:Mpositive','All M must be positive.') 0bY}<x(;  
    end HsA4NRF'7  
    F8e]sa$K\  
    if any(m>n) ^[]G sF  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') g\sW2qXEw  
    end q}-q[p? 5  
    SM>V o+  
    if any( r>1 | r<0 ) Yh`P+L  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') U`gQ7  
    end /mMRV:pd  
    ~udi=J |  
    if ~any(size(r)==1) _D%aT6,G+(  
        error('zernpol:Rvector','R must be a vector.') z[kz [  
    end :W'Yt9v)  
    Z i-)PK^  
    r = r(:); aAHx^X^  
    length_r = length(r); .~#<>  
    /jJi`'{U  
    if nargin==4 D ==H{c1F  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); anwMG0  
        if ~isnorm }?f%cRT$  
            error('zernpol:normalization','Unrecognized normalization flag.') F+.:Ry FS  
        end !Pnvqgp/  
    else c_#\'yeW  
        isnorm = false; fmH"&>Loc  
    end \A gPkW  
    asT*Z"/Q!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ImJ2tz6  
    % Compute the Zernike Polynomials I[|5 DQ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x1['+!01  
    e1'<;;; L  
    % Determine the required powers of r: `<I+(8]Uz  
    % ----------------------------------- B+:'Ld](  
    rpowers = []; x5q5<-#  
    for j = 1:length(n) EsA)o 5  
        rpowers = [rpowers m(j):2:n(j)]; $~M#msK9  
    end _yje"  
    rpowers = unique(rpowers); }S{#DgZ@X  
    <0,c{e  
    % Pre-compute the values of r raised to the required powers, r+8%oWj  
    % and compile them in a matrix: ${"+bWG2G!  
    % ----------------------------- [}snKogp  
    if rpowers(1)==0 X}?`G?'  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ^8S'=Bk  
        rpowern = cat(2,rpowern{:}); ,DrE4")4  
        rpowern = [ones(length_r,1) rpowern]; l4c9.'6  
    else CBC0X}_`  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); &Q7vY  
        rpowern = cat(2,rpowern{:}); Y{P0?`  
    end ?#|Y'%a"  
    iU^KmM I  
    % Compute the values of the polynomials: `Q d_Gu,M  
    % -------------------------------------- Gi})*U]P|  
    z = zeros(length_r,length_n); mSSDV0Pfn  
    for j = 1:length_n CR$\$-  
        s = 0:(n(j)-m(j))/2; c8qsp n  
        pows = n(j):-2:m(j); SH# -3&$[  
        for k = length(s):-1:1 6 /8?:  
            p = (1-2*mod(s(k),2))* ... _~f&wkc  
                       prod(2:(n(j)-s(k)))/          ... @di mZsi1  
                       prod(2:s(k))/                 ... ]qZs^kQ  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... __Kn 1H{  
                       prod(2:((n(j)+m(j))/2-s(k))); 8^26g 3  
            idx = (pows(k)==rpowers); 7MXi_V;p<  
            z(:,j) = z(:,j) + p*rpowern(:,idx); tuzw% =Ey  
        end uveby:dh  
         Hk'D@(h S  
        if isnorm -LAYj:4  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); )H&rr(  
        end ?1\rf$l8  
    end }E626d}uA  
    =FXO1UZ!  
    % 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)  \DDR l{  
    @)i A V1r"  
    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。  xRTr@  
    $N@EH;{_0  
    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)