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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 jJ2rfdfj  
    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! h% -=8l,  
     
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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 OTYkJEC8\N  
    function z = zernfun(n,m,r,theta,nflag) _E9[4%f  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. /K2[`+-  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N "y8W5R5kL4  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ,tXI*R  
    %   unit circle.  N is a vector of positive integers (including 0), and %Ja0:e  
    %   M is a vector with the same number of elements as N.  Each element 2?qT,pN  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 0=+feB1T  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 8A0a/ 7Lj  
    %   and THETA is a vector of angles.  R and THETA must have the same 2(uh7#Q  
    %   length.  The output Z is a matrix with one column for every (N,M) sC"w{_D@*4  
    %   pair, and one row for every (R,THETA) pair. 0`pCgF  
    % A#`$#CO  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike WXo bh  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), +L=Xc^  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral \hBzQ%0  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, a?ete9Q+  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ]fDb|s48  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. uNEl]Q]<e]  
    % aY4v'[  
    %   The Zernike functions are an orthogonal basis on the unit circle. ;0| :.q  
    %   They are used in disciplines such as astronomy, optics, and j0LZ )V  
    %   optometry to describe functions on a circular domain. ;eo}/-a_Xw  
    % {^Q,G x(  
    %   The following table lists the first 15 Zernike functions. O:'qwJ# ~  
    % N=U`BhL_  
    %       n    m    Zernike function           Normalization ~p'|A}9[/  
    %       -------------------------------------------------- AP`1hz4].-  
    %       0    0    1                                 1 g3Q;]8Y&  
    %       1    1    r * cos(theta)                    2 s3sD7 @  
    %       1   -1    r * sin(theta)                    2 {ZdF6~+H(!  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6)  +mft  
    %       2    0    (2*r^2 - 1)                    sqrt(3) k{{ Y2B?C  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) e1b?TF@lz  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 0i5S=L`j  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) u)zv`m  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) `'3&tAy  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) xVYa-I[Z  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) !ni 1 qM  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) GwA\>qXw  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) #I MaN%  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) $v_&j E  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) iD cYyNE  
    %       -------------------------------------------------- c om4@NK  
    % l['p^-I  
    %   Example 1: Q(Yn8t  
    % O46v  
    %       % Display the Zernike function Z(n=5,m=1) ;,uATd|  
    %       x = -1:0.01:1; { 2Ew^Li  
    %       [X,Y] = meshgrid(x,x); -Ju;i<  
    %       [theta,r] = cart2pol(X,Y); +BO kHXk1  
    %       idx = r<=1; `t9k!y!GV  
    %       z = nan(size(X)); hwvitD!0  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); S~H>MtX(<  
    %       figure y8C8~-&OK  
    %       pcolor(x,x,z), shading interp 86cnEj=   
    %       axis square, colorbar QrFKjmD<  
    %       title('Zernike function Z_5^1(r,\theta)') R'vNJDFY  
    % R-<8j`[0  
    %   Example 2: ? [5>!  
    % "1XTgCu\  
    %       % Display the first 10 Zernike functions ~xDu2 -5  
    %       x = -1:0.01:1; gH,Pz  
    %       [X,Y] = meshgrid(x,x); 0Ntvd7"`}  
    %       [theta,r] = cart2pol(X,Y); _O Jfd  
    %       idx = r<=1; m<k6oev$  
    %       z = nan(size(X)); $;$vcV9*  
    %       n = [0  1  1  2  2  2  3  3  3  3]; _ iDVd2X"H  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 9 !UNO  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; yJ ]Va $M  
    %       y = zernfun(n,m,r(idx),theta(idx)); >z/.8!#Q  
    %       figure('Units','normalized') ]t&^o**  
    %       for k = 1:10 ;ThFB  
    %           z(idx) = y(:,k); ? F!c"+C  
    %           subplot(4,7,Nplot(k)) N(yd<M w  
    %           pcolor(x,x,z), shading interp V?0IMc  
    %           set(gca,'XTick',[],'YTick',[])  ~H   
    %           axis square `:EhYj.   
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) oclU)f.,  
    %       end Fv)E:PnKC  
    % -F*vN'  
    %   See also ZERNPOL, ZERNFUN2. 01&E.A  
    <s\ZqL$ f  
    %   Paul Fricker 11/13/2006 z%T|L[(6  
    $`%Om WW{  
    3O Ks?i3A  
    % Check and prepare the inputs: zG/? wP"  
    % ----------------------------- G3]#Du  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) h 6?Z  
        error('zernfun:NMvectors','N and M must be vectors.') _emW#*V  
    end QY<5o;m`  
    .L;e:cvx  
    if length(n)~=length(m) nN-S5?X#  
        error('zernfun:NMlength','N and M must be the same length.') d+5~^\lV  
    end /NiD#s0t  
    RP+)sCh  
    n = n(:); YAeF*vP  
    m = m(:); E,K>V:P*  
    if any(mod(n-m,2)) Y6)o7t  
        error('zernfun:NMmultiplesof2', ... i'>5vU0?3  
              'All N and M must differ by multiples of 2 (including 0).') 4$ihnb`DQN  
    end e3p:lu  
    ,d*hhe  
    if any(m>n) 3Z me?o*bY  
        error('zernfun:MlessthanN', ... *TI?tD  
              'Each M must be less than or equal to its corresponding N.') |</)6r  
    end dT?3Q;>B?  
    PXJ7Ek*/  
    if any( r>1 | r<0 ) pWv1XTs@t:  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') %.$7-+:7A  
    end 5U+4vV/*  
    yf8kBT:&S  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) SA=>9L,2  
        error('zernfun:RTHvector','R and THETA must be vectors.') 8 Zp^/43  
    end ~Fwbi  
    esx/{j;<u  
    r = r(:); 3/ }  
    theta = theta(:); K r|.I2?"  
    length_r = length(r); ,5ZQPICF  
    if length_r~=length(theta) q-_!&kDK"  
        error('zernfun:RTHlength', ... NV9JMB{q  
              'The number of R- and THETA-values must be equal.') +DR$>a  
    end \M._x"  
    b l+g7g;  
    % Check normalization: y35~bz^2  
    % -------------------- 7[u>#8  
    if nargin==5 && ischar(nflag) ^i!6z2/  
        isnorm = strcmpi(nflag,'norm'); u-4@[*^T$  
        if ~isnorm !3mt<i]a"  
            error('zernfun:normalization','Unrecognized normalization flag.') Myiv#rQ)  
        end A%$~  
    else S >CKm:7  
        isnorm = false; w( XZSE  
    end +0UBP7kn  
    G\;6n  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x(eX.>o\  
    % Compute the Zernike Polynomials c-Yd> 4+ 1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Rq[d\BN0.d  
    ~eoM 2XlW  
    % Determine the required powers of r: h ! R=t  
    % ----------------------------------- 7X/t2Vih@  
    m_abs = abs(m); p e+h8  
    rpowers = []; fbOqxF"?we  
    for j = 1:length(n) lG94^|U  
        rpowers = [rpowers m_abs(j):2:n(j)]; emnT;kJ>  
    end }b&S3?ONt  
    rpowers = unique(rpowers); Q!U}  
    (uDd_@a9t  
    % Pre-compute the values of r raised to the required powers, q^EY?;Y  
    % and compile them in a matrix: !%('8-x%  
    % ----------------------------- 6:Z8d%Z  
    if rpowers(1)==0 PzNPwd  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ~tW~%]bs2Q  
        rpowern = cat(2,rpowern{:}); %>i:C-l8  
        rpowern = [ones(length_r,1) rpowern]; [p`5$\e  
    else !@8i(!xb  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); :Z6j5V;s  
        rpowern = cat(2,rpowern{:}); VLkAsM5}%  
    end zN|k*}j1J  
    5Q"w{ n  
    % Compute the values of the polynomials: |.UY' B  
    % -------------------------------------- !+^'Ej)z  
    y = zeros(length_r,length(n)); /+SLq`'u)  
    for j = 1:length(n) ~S\L(B(  
        s = 0:(n(j)-m_abs(j))/2; =huV(THU  
        pows = n(j):-2:m_abs(j); +W*~=*h|  
        for k = length(s):-1:1 `;;l {8  
            p = (1-2*mod(s(k),2))* ... Hn(1_I%zF  
                       prod(2:(n(j)-s(k)))/              ... 'Uf?-t*LT@  
                       prod(2:s(k))/                     ... k<^M >` $  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... R54[U  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); vb6EO[e% I  
            idx = (pows(k)==rpowers); ~!r;?38V`  
            y(:,j) = y(:,j) + p*rpowern(:,idx); #T^2=7 w  
        end t n5  
         crP2jF!  
        if isnorm &R_7]f+%)  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); m3lz#Pm'0  
        end jBw)8~tYm  
    end $Xu3s~:S  
    % END: Compute the Zernike Polynomials - Fbp!*. u  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [c 8=b,EI  
    &S*~EM.l8  
    % Compute the Zernike functions: Wx GD*%  
    % ------------------------------ hb5K"9Y  
    idx_pos = m>0; $El-pMq  
    idx_neg = m<0; :V)jm`)#+  
    ([u|j  
    z = y; "[|b,fxR  
    if any(idx_pos) x [FLV8`b|  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 'Be'!9K*d  
    end n_e'n|T  
    if any(idx_neg) UUJQc ~=  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); L9 D`hefz  
    end kk3^m1  
    sV  
    % 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) AC) M2;  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Dsg>~J'  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated _95296  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive g\JJkXjD#  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, `a& kD|Yh  
    %   and THETA is a vector of angles.  R and THETA must have the same zN4OrG 0  
    %   length.  The output Z is a matrix with one column for every P-value, Zh<;r;2  
    %   and one row for every (R,THETA) pair. @^W`Yg)C  
    % #@S%?`4,  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 86r5!@WN  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) afEa@et'  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 9Eyx Ob  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 6Xlzdt  
    %   for all p. 9t"Rw ns  
    % V8?}I)#(7  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 NI >%v  
    %   Zernike functions (order N<=7).  In some disciplines it is RdTM5ANT  
    %   traditional to label the first 36 functions using a single mode a:h<M^n049  
    %   number P instead of separate numbers for the order N and azimuthal S(Yd.Sp  
    %   frequency M. >gk_klLh  
    % :gh[BeqQ)  
    %   Example: e3?=1ZB  
    % ETe4I`d{  
    %       % Display the first 16 Zernike functions y>^^.  
    %       x = -1:0.01:1; Ey46JO"  
    %       [X,Y] = meshgrid(x,x); d8j1L/e  
    %       [theta,r] = cart2pol(X,Y); xP9(J 0y  
    %       idx = r<=1; "F<CGSo  
    %       p = 0:15; ~Iu!B Y  
    %       z = nan(size(X)); z$32rt8{`v  
    %       y = zernfun2(p,r(idx),theta(idx)); gE-y`2SU  
    %       figure('Units','normalized') WSkGVQu  
    %       for k = 1:length(p) _u`YjzK  
    %           z(idx) = y(:,k); j2Zp#E!  
    %           subplot(4,4,k) H",B[ YK  
    %           pcolor(x,x,z), shading interp n_8[bkbi  
    %           set(gca,'XTick',[],'YTick',[]) /0h *(nL  
    %           axis square kNEEu! G  
    %           title(['Z_{' num2str(p(k)) '}']) b'H'QY   
    %       end d2 ^}ooE  
    % C_.9qo]DT7  
    %   See also ZERNPOL, ZERNFUN. g,Z A\R~  
    ie}O ZM  
    %   Paul Fricker 11/13/2006 gV_/t+jI  
    9(CvGzco <  
    y7R#PkQ~  
    % Check and prepare the inputs: ^NW[)Dq1<  
    % ----------------------------- p5Q]/DhG  
    if min(size(p))~=1 Qw5nfg3T  
        error('zernfun2:Pvector','Input P must be vector.') 3dShznlf_*  
    end (L_-!=e  
    NWBYpGZx  
    if any(p)>35 ^[L(kHOGzk  
        error('zernfun2:P36', ... =8kmFXo  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Kz4S6N c  
               '(P = 0 to 35).']) :QCL9QZ'  
    end fCl}eXg6w  
    )*|/5wW1  
    % Get the order and frequency corresonding to the function number: Q=\ Oa(I  
    % ---------------------------------------------------------------- c}x1-d8  
    p = p(:); G7-BeA8  
    n = ceil((-3+sqrt(9+8*p))/2); z`y9<+  
    m = 2*p - n.*(n+2); CUA @CZ6{  
    !XA3G`}p6s  
    % Pass the inputs to the function ZERNFUN: PJnC  
    % ---------------------------------------- ?0tg}0|  
    switch nargin }kbSbRH43  
        case 3 D7ex{SVA)  
            z = zernfun(n,m,r,theta); R;& >PFmq  
        case 4 H#- 3  
            z = zernfun(n,m,r,theta,nflag); KWwtL"3  
        otherwise Hh<H~s [  
            error('zernfun2:nargin','Incorrect number of inputs.') r );R/)&  
    end t<%0eu|  
    uGKjZi  
    % 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) Urj*V0^  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. =O/Bte.  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of x "W~m.y$h  
    %   order N and frequency M, evaluated at R.  N is a vector of ]]xKc5CT  
    %   positive integers (including 0), and M is a vector with the 8$Q`wRt(%  
    %   same number of elements as N.  Each element k of M must be a HN47/]"*  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) O-- p)\   
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 61\u{@o$  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 1I Yip\:lS  
    %   with one column for every (N,M) pair, and one row for every St=nf\P&F  
    %   element in R. gfk)`>E  
    % `i{d"H0E  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- (5a73%>@  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is Q&m85'r5X  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Re%[t9 F&  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 vr!J3H f  
    %   for all [n,m]. [f6uwp  
    % <+8'H:wz  
    %   The radial Zernike polynomials are the radial portion of the ,'NasL8?We  
    %   Zernike functions, which are an orthogonal basis on the unit  >DL  
    %   circle.  The series representation of the radial Zernike :I"CQ C[Z  
    %   polynomials is |@ mz@  
    % npP C;KD  
    %          (n-m)/2 ];r! M0  
    %            __ Z:b?^u4.  
    %    m      \       s                                          n-2s OhF55,[  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 3CUQQ_  
    %    n      s=0 Z[vx0[av&  
    % M,Gy.ivz  
    %   The following table shows the first 12 polynomials. gv!8' DKn  
    % !}*N';  
    %       n    m    Zernike polynomial    Normalization 6fwNlC/9  
    %       --------------------------------------------- yUoR6w  
    %       0    0    1                        sqrt(2) 0'q4=!l  
    %       1    1    r                           2 ,5'o>Y  
    %       2    0    2*r^2 - 1                sqrt(6) Y#U.9>h  
    %       2    2    r^2                      sqrt(6) Q G) s  
    %       3    1    3*r^3 - 2*r              sqrt(8) N#w5}It  
    %       3    3    r^3                      sqrt(8) G  hM  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) jKSj);  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) d[9,J?'OQ  
    %       4    4    r^4                      sqrt(10) MVatV[G  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) QE<Z@/V*a  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) mY|c7}>V;  
    %       5    5    r^5                      sqrt(12) I2}W/}  
    %       --------------------------------------------- N,t9X7G&  
    % KbJ6U75|f  
    %   Example: rcnH^P  
    % PZ[-a-p40  
    %       % Display three example Zernike radial polynomials ZvY"yl?e  
    %       r = 0:0.01:1; U#<d",I  
    %       n = [3 2 5]; fif;n[<  
    %       m = [1 2 1]; +]l?JKV  
    %       z = zernpol(n,m,r); YOxgpQ:i  
    %       figure [o^$WL?c  
    %       plot(r,z) SH*'<  
    %       grid on *i#2>=)  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') /08FV|tX)  
    % HOW<IZ^  
    %   See also ZERNFUN, ZERNFUN2. \ %-<O  
    {R$`YWk  
    % A note on the algorithm. -:}vf?  
    % ------------------------ a%go[_w  
    % The radial Zernike polynomials are computed using the series j/v>,MM  
    % representation shown in the Help section above. For many special ;W0J  
    % functions, direct evaluation using the series representation can bGh&@&dHr  
    % produce poor numerical results (floating point errors), because g pciv  
    % the summation often involves computing small differences between 2 BY|Cp4R  
    % large successive terms in the series. (In such cases, the functions s/1r{;q  
    % are often evaluated using alternative methods such as recurrence 3}L3n*Ft#.  
    % relations: see the Legendre functions, for example). For the Zernike ];.5 *a%*  
    % polynomials, however, this problem does not arise, because the 3mgvWR  
    % polynomials are evaluated over the finite domain r = (0,1), and &BVUK"}P  
    % because the coefficients for a given polynomial are generally all K^{j$  
    % of similar magnitude. U$:^^Zt`B  
    % %Z;RY5  
    % ZERNPOL has been written using a vectorized implementation: multiple 1N/4W6  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] C&O8fNB_  
    % values can be passed as inputs) for a vector of points R.  To achieve %Tp9G Gt  
    % this vectorization most efficiently, the algorithm in ZERNPOL v]JET9hY  
    % involves pre-determining all the powers p of R that are required to >^8O:.  
    % compute the outputs, and then compiling the {R^p} into a single Rsx6vF8]5  
    % matrix.  This avoids any redundant computation of the R^p, and mF gqM:  
    % minimizes the sizes of certain intermediate variables. $.,PteYK  
    % )\U:e:Zae  
    %   Paul Fricker 11/13/2006 =B&|\2`{)  
    YB*)&@yx  
    +m_ .?V6  
    % Check and prepare the inputs: Vmz#u1gGT6  
    % ----------------------------- \mF-L,yu  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) `(W"wC   
        error('zernpol:NMvectors','N and M must be vectors.') ?!ap @)9  
    end M]8>5Zx.  
    mtUiO p  
    if length(n)~=length(m) XRaGV~  
        error('zernpol:NMlength','N and M must be the same length.') c<13r=+  
    end $$AZ)#t[  
    Fd8nR9A  
    n = n(:); Ehy(;n)\  
    m = m(:); <n_? $ TJ  
    length_n = length(n); h!B{7J  
    `!8\ |/  
    if any(mod(n-m,2)) hC-uz _/3  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 9^^\Z5  
    end 1dD%a91  
    +5fB?0D;  
    if any(m<0) 1D%P;eUDp  
        error('zernpol:Mpositive','All M must be positive.') /G5KNSi  
    end Z%#e* O0  
    FC 8<D  
    if any(m>n) E2Sj IR}  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') tFcQ.1  
    end :b9#e g  
    <v ub Q4  
    if any( r>1 | r<0 ) [,bJKz)a  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') azZ|T{S  
    end _9oKW;7f7  
    k r$)nf  
    if ~any(size(r)==1) J rK{MhO  
        error('zernpol:Rvector','R must be a vector.') ,):aU  
    end !19T=p/:$  
    Gn7\4,C  
    r = r(:); W3l[a^1d  
    length_r = length(r); 9#H0|zL  
    H:b"Vd"x9  
    if nargin==4 xpZ@DK;  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ^uo,LTq+  
        if ~isnorm %2RXrH2&H  
            error('zernpol:normalization','Unrecognized normalization flag.') Qfeu3AT  
        end %X\J%Fj  
    else [voc_o7AI  
        isnorm = false; -0uGzd+m*  
    end Zn1((J7  
    ^PdD-tY<  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wz+mFf  
    % Compute the Zernike Polynomials tzl,r"k3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (9bU\4F\  
    F_u ?.6e]  
    % Determine the required powers of r: bSM|"  
    % ----------------------------------- W)`>'X`  
    rpowers = []; |yNyk7~  
    for j = 1:length(n) ;8VZsh  
        rpowers = [rpowers m(j):2:n(j)]; 7<kr|-  
    end !}A`6z  
    rpowers = unique(rpowers); gy1kb,MO  
    0#c-qy  
    % Pre-compute the values of r raised to the required powers, &2@"zD  
    % and compile them in a matrix: y;0Zk~R$  
    % ----------------------------- 'bv(T2d~~  
    if rpowers(1)==0 zCq6k7u  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 'Q'-7z-6  
        rpowern = cat(2,rpowern{:}); W9Us I  
        rpowern = [ones(length_r,1) rpowern]; vU/sQt8  
    else wV-N\5!r%H  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 2AqcabI9  
        rpowern = cat(2,rpowern{:}); 3b?-83a  
    end mGz'%?zj  
    AB!({EIi  
    % Compute the values of the polynomials: 7F~Jz*,B*W  
    % -------------------------------------- NVVAh5R  
    z = zeros(length_r,length_n); yokZ>+jb  
    for j = 1:length_n a"+/fC`  
        s = 0:(n(j)-m(j))/2; | rpMwkR  
        pows = n(j):-2:m(j); P^ -x  
        for k = length(s):-1:1 :U1V 2f'l3  
            p = (1-2*mod(s(k),2))* ... R]kH$0`  
                       prod(2:(n(j)-s(k)))/          ... ?`*`A9@  
                       prod(2:s(k))/                 ... 4pDZ +}p  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... U:/_T>f%  
                       prod(2:((n(j)+m(j))/2-s(k))); \  }-v  
            idx = (pows(k)==rpowers); 4yu=e;C wy  
            z(:,j) = z(:,j) + p*rpowern(:,idx); |bRi bB  
        end { F0"U=  
         d76C ]R5L  
        if isnorm "| oW6@  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); BZQJ@lk5  
        end B]D51R\}VE  
    end a(U/70j  
    fQU_A  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  Ybx4 Up@  
    e$tKKcj0T  
    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。 Qo{^jDe,c*  
    f=J#mmH w$  
    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)