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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 Y;I(6`,Y  
    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! jW+VUF-t  
     
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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 }T*xT>p^3  
    function z = zernfun(n,m,r,theta,nflag) R8W4 4I*R:  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. LkbvA  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N RlPByG5K  
    %   and angular frequency M, evaluated at positions (R,THETA) on the g1!L. On  
    %   unit circle.  N is a vector of positive integers (including 0), and CzsY=DBH=  
    %   M is a vector with the same number of elements as N.  Each element oP`M\KXau  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) N %/DN  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, _w,0wn9N$  
    %   and THETA is a vector of angles.  R and THETA must have the same \rnG 1o  
    %   length.  The output Z is a matrix with one column for every (N,M) 50hh0!1  
    %   pair, and one row for every (R,THETA) pair. />I8nS}T  
    % 59J$SE  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 5 nIlG  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9PfU'm|h  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral o 0 #]EMr  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, . t%Vx  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Oqe.t;E 0}  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. T-8nUo}i  
    % E&tmWOMj>  
    %   The Zernike functions are an orthogonal basis on the unit circle. "}aM*(l+\  
    %   They are used in disciplines such as astronomy, optics, and B]}V$*$ \?  
    %   optometry to describe functions on a circular domain. imq(3?  
    % Q>c6ouuJ  
    %   The following table lists the first 15 Zernike functions. !l~aRj-WZ  
    % 7?WBzo!!L  
    %       n    m    Zernike function           Normalization kxf=%<l  
    %       -------------------------------------------------- TFA  
    %       0    0    1                                 1 g-gBg\y{v  
    %       1    1    r * cos(theta)                    2 %~(i[Ur;  
    %       1   -1    r * sin(theta)                    2 {hP&P  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) =v=!x  
    %       2    0    (2*r^2 - 1)                    sqrt(3) *pUV-^uo  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) +( (31l  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) =9@yJ9c-  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) "fJ|DE&@<i  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) AFUl   
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 5VoiDM=\c  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) A+E@OOw*~  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {Y TF]J $  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) nv Gd:]Z  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0\^2HjsJ  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) fzG1<Gem  
    %       -------------------------------------------------- fR;_6?p*B  
    % YEoT_>A$dB  
    %   Example 1: ;!sGfrs 0$  
    % ~,-O  
    %       % Display the Zernike function Z(n=5,m=1) C2i..iD  
    %       x = -1:0.01:1; {S(T1ua  
    %       [X,Y] = meshgrid(x,x); <s3(   
    %       [theta,r] = cart2pol(X,Y); DA@hf  
    %       idx = r<=1; jn Y3G  
    %       z = nan(size(X)); ^{bEq\5&  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ^}\R]})w"  
    %       figure K 8c#/o  
    %       pcolor(x,x,z), shading interp ^i1:PlW]  
    %       axis square, colorbar bj{f[nZ d  
    %       title('Zernike function Z_5^1(r,\theta)')  IomJo  
    % A6.'1OD  
    %   Example 2: 6^u(PzlA|~  
    % T^G<)IX`c  
    %       % Display the first 10 Zernike functions HNT8~s.2  
    %       x = -1:0.01:1; N)Kr4GC  
    %       [X,Y] = meshgrid(x,x); aC 0Jfo  
    %       [theta,r] = cart2pol(X,Y); 2MeavTr  
    %       idx = r<=1; U# B  
    %       z = nan(size(X)); VbR.tz  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Z`t?kXDNoI  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; W RaO.3Q@.  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Jz'+@q6h  
    %       y = zernfun(n,m,r(idx),theta(idx)); Mp=+*I[  
    %       figure('Units','normalized') ~-i?=  
    %       for k = 1:10 XePBA J  
    %           z(idx) = y(:,k); qNL~m'  
    %           subplot(4,7,Nplot(k)) !,"G/}'^;  
    %           pcolor(x,x,z), shading interp 5 Vqvb|  
    %           set(gca,'XTick',[],'YTick',[]) s$6#3%h  
    %           axis square _,~zy9{,  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) bf(&N-"A  
    %       end e[!>ezaIY  
    % MEUqQ4/Gl  
    %   See also ZERNPOL, ZERNFUN2. :nEV/"#F  
    Gzt5efygKt  
    %   Paul Fricker 11/13/2006 L9)&9 /f  
    |;yb *  
    lsi8?91  
    % Check and prepare the inputs: .#|pje^  
    % ----------------------------- :[3\jLrc  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) P s;:g0  
        error('zernfun:NMvectors','N and M must be vectors.') v%[mt` I  
    end t57b)5{FM  
    VRt*!v<")  
    if length(n)~=length(m) )`-]nMc  
        error('zernfun:NMlength','N and M must be the same length.') 4[q * 7m  
    end =T]OYk  
    < .!3yy  
    n = n(:); 0f1#T gX  
    m = m(:); A?zW!'  
    if any(mod(n-m,2)) }Jfo(j  
        error('zernfun:NMmultiplesof2', ... )`^:G3w  
              'All N and M must differ by multiples of 2 (including 0).') kpu^:N &  
    end jFfki.H  
    |?kH]Trr  
    if any(m>n) uX[ "w|  
        error('zernfun:MlessthanN', ... d]]qy  
              'Each M must be less than or equal to its corresponding N.') 'CX KphlWs  
    end Jhc S  
    rge/jE,^~Z  
    if any( r>1 | r<0 ) ,}0pK\Y>$  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') M<Mr (z  
    end +|;IIwo  
    b&1@rE-  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Zpmy)W]1  
        error('zernfun:RTHvector','R and THETA must be vectors.') #UQ[8e  
    end X c^~|%+  
    k|5nu-B0v  
    r = r(:); 7Go!W(8  
    theta = theta(:); icmDPq  
    length_r = length(r); 0"N %Vm  
    if length_r~=length(theta) /rW{rf^  
        error('zernfun:RTHlength', ... NL 37Y{b  
              'The number of R- and THETA-values must be equal.') 4SYN$?.Mp  
    end MR}\fw$(.  
    RAC-;~$WB  
    % Check normalization: KJiwM(o  
    % -------------------- V|)>  
    if nargin==5 && ischar(nflag) /L.a:Er$  
        isnorm = strcmpi(nflag,'norm'); X#yl8k_  
        if ~isnorm '<Gqu_-  
            error('zernfun:normalization','Unrecognized normalization flag.') Ar==@777j  
        end BlUY9`VWh@  
    else fVM%.`  
        isnorm = false; &ly[mBP~  
    end 8~i@7~ J  
    1;W>ceN"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uOQ5.S+  
    % Compute the Zernike Polynomials 5 Jhl4p}w  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |1D`v9  
    Ogb_WO;)  
    % Determine the required powers of r: [H6>]&  
    % ----------------------------------- <Yc:,CU  
    m_abs = abs(m); ~&x%;cnv_  
    rpowers = []; 5+UiAc$  
    for j = 1:length(n) u2t<auE9^  
        rpowers = [rpowers m_abs(j):2:n(j)]; 2Y+*vNs3  
    end i]nE86.;  
    rpowers = unique(rpowers); \&H%k   
    CbZ1<r" /  
    % Pre-compute the values of r raised to the required powers, fp7Qb $-A  
    % and compile them in a matrix: r!#3>F;B  
    % ----------------------------- .\VjS^o&Z&  
    if rpowers(1)==0 1}6pq 2  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ew(6;}+^/  
        rpowern = cat(2,rpowern{:}); &eg,*K}'  
        rpowern = [ones(length_r,1) rpowern]; S;])Nt'X'  
    else 6]Jv3Re'(I  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ^6*? a9jO>  
        rpowern = cat(2,rpowern{:}); o$-P hl  
    end $3L7R  
    &l Q j?]  
    % Compute the values of the polynomials: tT7$2 9  
    % -------------------------------------- 4Qd g t*  
    y = zeros(length_r,length(n)); &[YG\8sxWa  
    for j = 1:length(n) 7v-C-u[E`  
        s = 0:(n(j)-m_abs(j))/2; 6-3l6q  
        pows = n(j):-2:m_abs(j); "rXGXQu  
        for k = length(s):-1:1 Cn,jLy  
            p = (1-2*mod(s(k),2))* ... ct  ZW7  
                       prod(2:(n(j)-s(k)))/              ... 9K49<u0O  
                       prod(2:s(k))/                     ... $H#&.IjY  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... vl#/8]0!  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ;[xDc>&("Q  
            idx = (pows(k)==rpowers); P ,i)A  
            y(:,j) = y(:,j) + p*rpowern(:,idx); U0rz 4fxc  
        end pQp}HD!-  
         J.-#:OZ  
        if isnorm 3 !,%;Vz=  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ZD,l 2DQ?  
        end "%Jx,L\f{  
    end t~AesHZpk  
    % END: Compute the Zernike Polynomials 1)r1/0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% IOA{l N6  
    OD i)#  
    % Compute the Zernike functions: HV sIbQS  
    % ------------------------------ h*d,AJz &.  
    idx_pos = m>0; Xm*Dh#H  
    idx_neg = m<0; WV8<gx`Q  
    9J?j2!D  
    z = y; #zXDh3%]a  
    if any(idx_pos) \z_@.Jw{  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ;7hf'k  
    end +z4NxR   
    if any(idx_neg) {5to;\.  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); tly:$;K  
    end $exu}%  
    hE=cgO`QU  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) B^6P 6,  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 9u:MF0:W  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated GxvVh71zP  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive , vky  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, wHAh6lm  
    %   and THETA is a vector of angles.  R and THETA must have the same >V]> h&`  
    %   length.  The output Z is a matrix with one column for every P-value, vj#gY2qZ  
    %   and one row for every (R,THETA) pair. b~\![HoCMM  
    % J)R2O4OEd  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike im&| H-  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 3{:d$- y  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) !0w'S>e  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 6 Fm.^9@  
    %   for all p. Edjh*  
    % <cl$?].RE!  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 9 Iw+g]`y*  
    %   Zernike functions (order N<=7).  In some disciplines it is I\[*vgjm3G  
    %   traditional to label the first 36 functions using a single mode E=LaPjEIj  
    %   number P instead of separate numbers for the order N and azimuthal H(0d(c1s  
    %   frequency M. J +9D/VT  
    % |5O%@  
    %   Example: 2bCa|HTv  
    % lRO4- y  
    %       % Display the first 16 Zernike functions x.d9mjLN8m  
    %       x = -1:0.01:1; ncWASw`  
    %       [X,Y] = meshgrid(x,x);  fBQZ=zh  
    %       [theta,r] = cart2pol(X,Y); i4->XvC  
    %       idx = r<=1; |C5i3?  
    %       p = 0:15; w("jyvV[C  
    %       z = nan(size(X)); T *$uc,  
    %       y = zernfun2(p,r(idx),theta(idx)); Q,s,EooIx  
    %       figure('Units','normalized') !{SEm"J^  
    %       for k = 1:length(p) 0a(*/u  
    %           z(idx) = y(:,k); vK6bpzI 3  
    %           subplot(4,4,k) C#gQJ=!B  
    %           pcolor(x,x,z), shading interp D]4?UL  
    %           set(gca,'XTick',[],'YTick',[]) +[cm  
    %           axis square hwexv 9""  
    %           title(['Z_{' num2str(p(k)) '}']) b?r0n]  
    %       end bi,%QZZ  
    % & ??)gMM[  
    %   See also ZERNPOL, ZERNFUN. I{M2nQi  
    F9d][ P@@  
    %   Paul Fricker 11/13/2006 $i =-A  
    E fqa*,k  
    EK#w: "  
    % Check and prepare the inputs: xE+Go  
    % ----------------------------- ysL8w"t  
    if min(size(p))~=1 l ='lV]  
        error('zernfun2:Pvector','Input P must be vector.') .%*.nq  
    end XbHcd8N T  
    S?D2`b  
    if any(p)>35 .}Xkr+ +]  
        error('zernfun2:P36', ... o]jo R3  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... xNjA>S\]W5  
               '(P = 0 to 35).']) Q>X ;7nt0  
    end 2*Gl|@~N  
    ."3 J;j  
    % Get the order and frequency corresonding to the function number: 4$_8#w B1&  
    % ---------------------------------------------------------------- KnbP@!+c  
    p = p(:); Q9rE_} Z  
    n = ceil((-3+sqrt(9+8*p))/2); gAR];(*  
    m = 2*p - n.*(n+2); !WbQ`]uN/#  
    YP#OI 6u  
    % Pass the inputs to the function ZERNFUN: Wmp\J3  
    % ---------------------------------------- F*Qw%  
    switch nargin L5U>`lx6$  
        case 3 Z5NuLB'  
            z = zernfun(n,m,r,theta); Z3[,Xw  
        case 4 a z`5{hK  
            z = zernfun(n,m,r,theta,nflag); 76c}Rk^  
        otherwise R4{}ZT  
            error('zernfun2:nargin','Incorrect number of inputs.') 's*UU:R  
    end %zY3,4~  
    &M<431y  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) UY>{e>/H9  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. =niU6Q}  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of  E?%k  
    %   order N and frequency M, evaluated at R.  N is a vector of M8 ++JI  
    %   positive integers (including 0), and M is a vector with the &0Nd9%>  
    %   same number of elements as N.  Each element k of M must be a ab 2 V.S  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) F[~qgS*;  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 6~D:O?2  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Xr':/Qjf  
    %   with one column for every (N,M) pair, and one row for every M~3(4,  
    %   element in R. t$s)S>  
    % x37r{$2  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- J&h 3,  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is |"l g4S%  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to $k}+,tHtJO  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 {>5c,L$  
    %   for all [n,m]. ]_#[o S  
    % mb?yG:L=0b  
    %   The radial Zernike polynomials are the radial portion of the EMJ}tvL0Tp  
    %   Zernike functions, which are an orthogonal basis on the unit UlQ}   
    %   circle.  The series representation of the radial Zernike tjYe82  
    %   polynomials is E6BW&Xp  
    % o'R_kadN[T  
    %          (n-m)/2 ?jb7Oq#[  
    %            __ yUBic~S  
    %    m      \       s                                          n-2s @-Gf+*GZys  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 6eQrupa  
    %    n      s=0 QULrE+@  
    % /&vUi7'  
    %   The following table shows the first 12 polynomials. mo <g'|0  
    % !'n+0  
    %       n    m    Zernike polynomial    Normalization MQp1j:CK  
    %       --------------------------------------------- 3",6 E(  
    %       0    0    1                        sqrt(2) - !7QH'  
    %       1    1    r                           2 |h8C}P&Z  
    %       2    0    2*r^2 - 1                sqrt(6) |1rBK.8  
    %       2    2    r^2                      sqrt(6) CYG'WFvZZ  
    %       3    1    3*r^3 - 2*r              sqrt(8) uy7)9w  
    %       3    3    r^3                      sqrt(8) "<bL-k*H)  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) DlTV1X-^1  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) `s@1'IG;R_  
    %       4    4    r^4                      sqrt(10) EYMwg_  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) r+\it&cW+  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 4dl?US[-  
    %       5    5    r^5                      sqrt(12) R"K{@8b  
    %       --------------------------------------------- 33-=Z9|r  
    % DR^mT$  
    %   Example: 4 YI,:  
    % |yw-H2k1  
    %       % Display three example Zernike radial polynomials 7;c{lQOj}  
    %       r = 0:0.01:1; Fx)]AJ~[t  
    %       n = [3 2 5]; _MnMT9  
    %       m = [1 2 1]; b(K.p?bt  
    %       z = zernpol(n,m,r); qo4AQ}0 <  
    %       figure {.eC"  
    %       plot(r,z) ; N!K/[p=  
    %       grid on NIQa{R/H  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') >P+V!-%#  
    % #P18vK5  
    %   See also ZERNFUN, ZERNFUN2. #S_LKc  
    ;I]TM#qGF  
    % A note on the algorithm. }?8KFe7U  
    % ------------------------ V?5QpBK I  
    % The radial Zernike polynomials are computed using the series &<k )W  
    % representation shown in the Help section above. For many special 5+giT5K*h  
    % functions, direct evaluation using the series representation can vg *+>lbA  
    % produce poor numerical results (floating point errors), because 9JP{F  
    % the summation often involves computing small differences between !=I:Uc-Y  
    % large successive terms in the series. (In such cases, the functions SO8Ej)m  
    % are often evaluated using alternative methods such as recurrence UV@<55)K  
    % relations: see the Legendre functions, for example). For the Zernike B% BO  
    % polynomials, however, this problem does not arise, because the v]Pw]m5=U  
    % polynomials are evaluated over the finite domain r = (0,1), and K\=bpc"Fy  
    % because the coefficients for a given polynomial are generally all Ab8~'<F$B  
    % of similar magnitude. ]X@/0  
    % < _c84,[V  
    % ZERNPOL has been written using a vectorized implementation: multiple 2-UZ|y  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] G8 f7N; D  
    % values can be passed as inputs) for a vector of points R.  To achieve ?Q:se  
    % this vectorization most efficiently, the algorithm in ZERNPOL d ID] {  
    % involves pre-determining all the powers p of R that are required to :IbrV@gN{@  
    % compute the outputs, and then compiling the {R^p} into a single `hI1  
    % matrix.  This avoids any redundant computation of the R^p, and jAN(r>zVL  
    % minimizes the sizes of certain intermediate variables. xLq+n jH E  
    % dax|4R  
    %   Paul Fricker 11/13/2006 ~d){7OG  
    irgjq/&d  
    [uZU p*.V  
    % Check and prepare the inputs: q>!T*BQ  
    % ----------------------------- 9]7+fu  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) DlfXzKn;  
        error('zernpol:NMvectors','N and M must be vectors.') &> }MoB  
    end A7~)h}~   
    kZSe#'R's  
    if length(n)~=length(m) #d(6q$IE  
        error('zernpol:NMlength','N and M must be the same length.') aN%t>*?Xa  
    end 8t0i j  
    JnV$)EYi  
    n = n(:); $?ke "  
    m = m(:); :8yrtbf$  
    length_n = length(n); f5mk\^  
    -D38>#Y  
    if any(mod(n-m,2)) {l\v J#r:  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') -V_e=Y<J/  
    end 7o0e j#  
    "c1vW<;  
    if any(m<0) WNlWigwYl  
        error('zernpol:Mpositive','All M must be positive.') T-f+<Cxf  
    end AUzJ:([V  
    '00DUUa  
    if any(m>n) kZPj{^c:  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') Eu1s  
    end nb(#;3DQ  
    \muyL?  
    if any( r>1 | r<0 ) j$N`JiKM  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') %6kD^K-  
    end pd>EUdbrp&  
    h#;fBQ]   
    if ~any(size(r)==1) n3~xiQ'  
        error('zernpol:Rvector','R must be a vector.') ~A>3k2 N/e  
    end ~wh8)rm  
    ~cU,3g  
    r = r(:); Gd:fWz(  
    length_r = length(r); /`:5#O  
    F RS@-P  
    if nargin==4 sN^R Z0!>  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); #HM0s~^w&  
        if ~isnorm 9~Q.[ A  
            error('zernpol:normalization','Unrecognized normalization flag.') }SUe 4r&4}  
        end EDL<J1%  
    else ,i,f1XJ|  
        isnorm = false; yd`.Rb&V  
    end evu@uq  
    Tet,mzVuu  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j~Rh_\>Q  
    % Compute the Zernike Polynomials J|,| *t  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CNf eHMT  
    G)'cd D1  
    % Determine the required powers of r: {Qlvj.Xw  
    % ----------------------------------- HO & #Lv  
    rpowers = []; vseuk@>  
    for j = 1:length(n) A%%WPBk{O  
        rpowers = [rpowers m(j):2:n(j)];   7&l  
    end _oe2 pL&  
    rpowers = unique(rpowers); !oM 1  
    *gVRMSrx4  
    % Pre-compute the values of r raised to the required powers, 3 T& m  
    % and compile them in a matrix: DQKhR sC  
    % ----------------------------- )CihqsA2  
    if rpowers(1)==0 a"#5JcR3  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); tw\/1wa.  
        rpowern = cat(2,rpowern{:}); "d%":F(  
        rpowern = [ones(length_r,1) rpowern]; o`hF1*yp  
    else %UgyGQeo  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); CW, Kw  
        rpowern = cat(2,rpowern{:}); M0"xDvQ  
    end  $p}7CP  
    S(9fGh  
    % Compute the values of the polynomials: /v=MGX@r  
    % -------------------------------------- V4ayewVX  
    z = zeros(length_r,length_n); } Tp!Ub\Cc  
    for j = 1:length_n gQ,PG  
        s = 0:(n(j)-m(j))/2; viY _Y.Yjy  
        pows = n(j):-2:m(j); mA3C)V  
        for k = length(s):-1:1 LT# *nr  
            p = (1-2*mod(s(k),2))* ... ^EM##Ss_  
                       prod(2:(n(j)-s(k)))/          ... DkQy.  
                       prod(2:s(k))/                 ... @/B&R^aVZ  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ks 3<zW(  
                       prod(2:((n(j)+m(j))/2-s(k)));  8(5}Jo+  
            idx = (pows(k)==rpowers); lE$X9yIt  
            z(:,j) = z(:,j) + p*rpowern(:,idx); F7cv`i?2."  
        end cFD(Ap  
         N\<M4 fn  
        if isnorm Rf2;O<  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); vYrqZie<  
        end /_aFQ>.4n  
    end kCLz@9>FQ  
    n\wO[l)  
    % 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)  >k)}R|tJ  
    !C]0l  
    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。 +F= j1*'&  
    u/-u l  
    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)