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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 J@ x%TA  
    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! 15Vb`Vf`N  
     
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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 ODK$G [-  
    function z = zernfun(n,m,r,theta,nflag) OKfJ  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. (#* 7LdZ  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N kVs'>H@FY  
    %   and angular frequency M, evaluated at positions (R,THETA) on the >{i/LC^S  
    %   unit circle.  N is a vector of positive integers (including 0), and b:.aZ7+4  
    %   M is a vector with the same number of elements as N.  Each element A87JPX#R?  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) n(.y_NEgV!  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, I0 a,mO;m  
    %   and THETA is a vector of angles.  R and THETA must have the same bs!N~,6h  
    %   length.  The output Z is a matrix with one column for every (N,M) 0es[!  
    %   pair, and one row for every (R,THETA) pair. u2 a U0k:  
    % *6~ODiB  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike FjIS:9^)t5  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Uw^`_\si  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral c 6sGjZdR  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, #|fa/kb~  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized |R:gu\gG  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 0!F"s>(H  
    % |ofegO}W7  
    %   The Zernike functions are an orthogonal basis on the unit circle. v4!zB9d  
    %   They are used in disciplines such as astronomy, optics, and Ed9ynJ~)X  
    %   optometry to describe functions on a circular domain. b:/;  
    % 0 Vv 6B2<  
    %   The following table lists the first 15 Zernike functions. J& }/Xw)  
    % kH1hsDe|&y  
    %       n    m    Zernike function           Normalization mD-qJ6AM  
    %       -------------------------------------------------- 6V\YYrUz  
    %       0    0    1                                 1 R0y={\*B5k  
    %       1    1    r * cos(theta)                    2 `m?%{ \  
    %       1   -1    r * sin(theta)                    2 IbC(/i#%`  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Ed,`1+  
    %       2    0    (2*r^2 - 1)                    sqrt(3) :G9+-z{Y&  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) SCE5|3j  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) L+Yn}"gIs  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) !s#25}9zX5  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) tWQ_.,ld  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 8RWfv}:X  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) WS8m^~S@\  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) VO3&!uOd  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) }\}pSqW  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) wXp A1,i  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) <qN0Q7  
    %       -------------------------------------------------- Xn-GSW3{  
    % <y=VDb/  
    %   Example 1: zu'Uau  
    % |WH'aGG  
    %       % Display the Zernike function Z(n=5,m=1) 'gk.J  
    %       x = -1:0.01:1; PHl{pE*  
    %       [X,Y] = meshgrid(x,x); c4ptY5R),  
    %       [theta,r] = cart2pol(X,Y); .MkHB0 2N  
    %       idx = r<=1; ^pZ1uN!b  
    %       z = nan(size(X)); !/+ZKx("9  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); n"8vlNeW  
    %       figure 1o)@{x/pd  
    %       pcolor(x,x,z), shading interp Ov"]&e(I[  
    %       axis square, colorbar \#.,@g  
    %       title('Zernike function Z_5^1(r,\theta)') LnIln[g:  
    % 8A}w}h  
    %   Example 2: }]_/:KUt  
    % Wr Ht  
    %       % Display the first 10 Zernike functions zvV<0 Z  
    %       x = -1:0.01:1; QQUeY2}  
    %       [X,Y] = meshgrid(x,x); /^^t>L  
    %       [theta,r] = cart2pol(X,Y); ,dn9tY3  
    %       idx = r<=1; n4Nb,)M  
    %       z = nan(size(X)); n/#zx:d?  
    %       n = [0  1  1  2  2  2  3  3  3  3]; t!RR5!  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 0 3fCn"  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; O6Bs!0,  
    %       y = zernfun(n,m,r(idx),theta(idx)); ~Q"3#4l  
    %       figure('Units','normalized') E8gXa-hv  
    %       for k = 1:10 nmZz`P9g  
    %           z(idx) = y(:,k); yQE|FbiA  
    %           subplot(4,7,Nplot(k)) j78WPG  
    %           pcolor(x,x,z), shading interp hc OT+L>  
    %           set(gca,'XTick',[],'YTick',[]) &<6E*qM  
    %           axis square `s5<PCq  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) d4KT wn5g  
    %       end K}"xZy Tm1  
    % RUqN,C,m5I  
    %   See also ZERNPOL, ZERNFUN2. ,?k[<C  
    D ]Q,~Y&'  
    %   Paul Fricker 11/13/2006 VZo[\sWf  
    )QYg[<e6  
    -V0_%Smc  
    % Check and prepare the inputs: 4-;"w;  
    % ----------------------------- Fw5|_@&k  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) |S.G#za  
        error('zernfun:NMvectors','N and M must be vectors.') O 4zD >O  
    end |U{9Yy6p  
    li'h&!|]  
    if length(n)~=length(m) G2 A#&86J{  
        error('zernfun:NMlength','N and M must be the same length.') WLl_;BgN  
    end TI4#A E  
    ~!UC:&UKo  
    n = n(:); `G*7y7  
    m = m(:); (5- w>(  
    if any(mod(n-m,2)) !>QS746S@  
        error('zernfun:NMmultiplesof2', ... -n&g**\w  
              'All N and M must differ by multiples of 2 (including 0).') ]D?//  
    end  [U9b_`  
    x|4m*>Ke  
    if any(m>n) yUV0{A-q{0  
        error('zernfun:MlessthanN', ... Z(DCR/U=(>  
              'Each M must be less than or equal to its corresponding N.') ~C[p}MED  
    end vD<6BQR  
    &*2\1;1tB  
    if any( r>1 | r<0 ) D.d(D:  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') (:9yeP1  
    end V]I@&*O~ r  
    s~e<Pr?yu  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) |^fubQs;2  
        error('zernfun:RTHvector','R and THETA must be vectors.') S(NH# ^  
    end +8qtFog$\g  
    ;pe1tp  
    r = r(:); Z] ?Tx2|7  
    theta = theta(:); O/g|E47  
    length_r = length(r); PWeCk2xH  
    if length_r~=length(theta) x/~qyX8vo  
        error('zernfun:RTHlength', ... g4b-~1[S  
              'The number of R- and THETA-values must be equal.') 5ncjv@Aa  
    end 0XouHU  
    vHR-mQUs  
    % Check normalization: ;:<z hO  
    % -------------------- -7MR2)U  
    if nargin==5 && ischar(nflag) :"m~tU3&  
        isnorm = strcmpi(nflag,'norm'); \8j5b+  
        if ~isnorm <Z{pjJ/  
            error('zernfun:normalization','Unrecognized normalization flag.') &L7u//  
        end wq yw#)S  
    else + *u'vt?  
        isnorm = false; _N8Tu~lqV  
    end F`!B!uY  
    A:|dY^,:?*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% w2*.3I,~)B  
    % Compute the Zernike Polynomials Oi#4|*b{W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U'(Exr[  
    n(X{|?  
    % Determine the required powers of r: 1.S7MSpTV  
    % ----------------------------------- 0$=Uhi  
    m_abs = abs(m); EQQ/E!N8l  
    rpowers = []; /<[S> ;!kr  
    for j = 1:length(n) (!b_o A8V  
        rpowers = [rpowers m_abs(j):2:n(j)]; TUE*mDRmP  
    end mjgwU8'![  
    rpowers = unique(rpowers); 0e./yPTT  
    i4<&zj})  
    % Pre-compute the values of r raised to the required powers, fZQL!j4  
    % and compile them in a matrix: 'iQ  
    % ----------------------------- 1XfH,6\8i  
    if rpowers(1)==0 \9;SOAv  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); :r4]8X-  
        rpowern = cat(2,rpowern{:}); %>,B1nt  
        rpowern = [ones(length_r,1) rpowern]; #Z;6f{yWf  
    else 8H2zM IB  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); I+JWDYk  
        rpowern = cat(2,rpowern{:}); ku2g FO  
    end q%H`/~AYM  
    Kmy'z  
    % Compute the values of the polynomials: \.0cA4)[$  
    % -------------------------------------- m(2(Caz{  
    y = zeros(length_r,length(n)); NO$n-<ag  
    for j = 1:length(n) GCrIa Z  
        s = 0:(n(j)-m_abs(j))/2; )q.Z}_,)@  
        pows = n(j):-2:m_abs(j); 'K|Jg.2  
        for k = length(s):-1:1 [^N8v;O  
            p = (1-2*mod(s(k),2))* ... NxOiT#YH  
                       prod(2:(n(j)-s(k)))/              ... 8]SJ=c"}Xf  
                       prod(2:s(k))/                     ... [cJQ"G '  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Mn)>G36(  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ,/m@<NyK  
            idx = (pows(k)==rpowers); !WTZ =|  
            y(:,j) = y(:,j) + p*rpowern(:,idx); .`I;qF  
        end fj 14'T  
         j@w+>h  
        if isnorm =1!,A  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Vgh;w-a  
        end Drn{ucIs  
    end 8 `\^wG$W  
    % END: Compute the Zernike Polynomials 25bbuhss  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "o| f  
    "hE/f~\  
    % Compute the Zernike functions: @k< e]@r  
    % ------------------------------ 4blw9x N  
    idx_pos = m>0; JpI(Vcd  
    idx_neg = m<0; 33R1<dRk  
    tQ:g#EqL9B  
    z = y; A~2U9f+\  
    if any(idx_pos) O?p8Gjf  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); X jJV  
    end q+j.)e  
    if any(idx_neg) >rbHpLm1`  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); E$u9Jbe  
    end ,^Cl?\9"  
    Mz?xvP?z  
    % 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) .Lwp`{F/  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. .o27uB.  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 7o+JQ&fF;  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive f *Xum[  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, @yGK $<R  
    %   and THETA is a vector of angles.  R and THETA must have the same Lip(r3  
    %   length.  The output Z is a matrix with one column for every P-value, {Df97n%h;  
    %   and one row for every (R,THETA) pair. 8fG$><@  
    % ]+U:8*  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 3`Ug]<m  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) F!>92H~3G  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) .C 6wsmQ  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 I.4o9Z[?  
    %   for all p. iY|zv|;]=  
    % LTn@OhC  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36  (0wQ [(  
    %   Zernike functions (order N<=7).  In some disciplines it is A^m]DSFOO  
    %   traditional to label the first 36 functions using a single mode @;6I94Bp  
    %   number P instead of separate numbers for the order N and azimuthal x4_xl .  
    %   frequency M. z: ;ZPSn  
    % YK=o[nPmK  
    %   Example: B-R& v8F  
    % Zb \E!>V  
    %       % Display the first 16 Zernike functions sI/]pgt2  
    %       x = -1:0.01:1; _v[yY3=3  
    %       [X,Y] = meshgrid(x,x); fGwRv% $^  
    %       [theta,r] = cart2pol(X,Y); /LH# 3  
    %       idx = r<=1; s(0S)l<  
    %       p = 0:15; a>05Yxw  
    %       z = nan(size(X)); =do*(  
    %       y = zernfun2(p,r(idx),theta(idx)); :jKiHeBQu?  
    %       figure('Units','normalized') 7Gos-_s  
    %       for k = 1:length(p) ;Dw6pmZ  
    %           z(idx) = y(:,k); T z`O+fx &  
    %           subplot(4,4,k) A"Prgf eT  
    %           pcolor(x,x,z), shading interp u|.c?fW'3  
    %           set(gca,'XTick',[],'YTick',[]) o+w G6 9  
    %           axis square O<*l"fw3  
    %           title(['Z_{' num2str(p(k)) '}']) <FkoWN  
    %       end qe/|u3I<lF  
    % u|G&CV#r  
    %   See also ZERNPOL, ZERNFUN. nfldj33*  
    >~%EB?8  
    %   Paul Fricker 11/13/2006  9Kpzj43  
    wU"0@^k]<  
    7])cu>/  
    % Check and prepare the inputs: fQ[& ^S$  
    % ----------------------------- Vgj&h dbd  
    if min(size(p))~=1 b|rMmx8vA  
        error('zernfun2:Pvector','Input P must be vector.') MF41q%9p  
    end 'XbrO|%  
    TJ5g? #Wul  
    if any(p)>35 ^xNs^wC.  
        error('zernfun2:P36', ... "3?N*,U_  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ].!^BYNht  
               '(P = 0 to 35).']) ?*nFz0cs^  
    end m|CB')  
    z5> {(iY;,  
    % Get the order and frequency corresonding to the function number: + Cf  
    % ---------------------------------------------------------------- YF4?3K0F:k  
    p = p(:); NCXr$ES{  
    n = ceil((-3+sqrt(9+8*p))/2); &A1~x!`  
    m = 2*p - n.*(n+2); cu@i;Hb@  
    +H4H$H  
    % Pass the inputs to the function ZERNFUN: _Yms]QEZ  
    % ---------------------------------------- `pTCK9  
    switch nargin AeZ__X  
        case 3 Y30T>5  
            z = zernfun(n,m,r,theta); kp$w)%2JW  
        case 4 k$NNpv&;d  
            z = zernfun(n,m,r,theta,nflag); b@> MA  
        otherwise ^p}S5,  
            error('zernfun2:nargin','Incorrect number of inputs.') K<g<xW*X  
    end p$:ERI  
    c*@#0B  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) <aPbKDF~V  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. ~Q3y3,x  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Y+kfMAv  
    %   order N and frequency M, evaluated at R.  N is a vector of W[R^5{k`  
    %   positive integers (including 0), and M is a vector with the L T2UY*  
    %   same number of elements as N.  Each element k of M must be a '  ~F  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) h8)m2KrZ!.  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is _[:>!ekx  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix )[]*Y]vSx  
    %   with one column for every (N,M) pair, and one row for every :p|wo"=@Ge  
    %   element in R. w{$X :Z  
    % {~y,.[Ga  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- Y48MCL  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is YR? ujN  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to {: H&2iF  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 >2)`/B9f4  
    %   for all [n,m]. iu$:_W_  
    % qtI42u{  
    %   The radial Zernike polynomials are the radial portion of the Pqtk1=U  
    %   Zernike functions, which are an orthogonal basis on the unit p3q >a<  
    %   circle.  The series representation of the radial Zernike cOz/zD f5  
    %   polynomials is A7c*qBt  
    % vhz[H  
    %          (n-m)/2 ]aDU*tk  
    %            __ /9 ^F_2'_  
    %    m      \       s                                          n-2s %vZTD +i  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r G oHdhne3  
    %    n      s=0 WW&ag r  
    % KccIYn~  
    %   The following table shows the first 12 polynomials. 7/>#yR  
    % G< _<j}=  
    %       n    m    Zernike polynomial    Normalization j YVR"D;  
    %       --------------------------------------------- [zw0'-h.  
    %       0    0    1                        sqrt(2) 0hB9D{`,{  
    %       1    1    r                           2 \YZ7  
    %       2    0    2*r^2 - 1                sqrt(6) 1<(('H  
    %       2    2    r^2                      sqrt(6) qZwqnH  
    %       3    1    3*r^3 - 2*r              sqrt(8) Gtm|aR{OS  
    %       3    3    r^3                      sqrt(8) g7-*WN<  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) '&+5L.  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 'lIj89h<E  
    %       4    4    r^4                      sqrt(10) E/:mO~1< c  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Q8GI;`Rb  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) H(rK39Q  
    %       5    5    r^5                      sqrt(12) g[%^OT#  
    %       --------------------------------------------- @GyxOc@6  
    % V\6V&_  
    %   Example: >&Ios<67g  
    % gZW(z  
    %       % Display three example Zernike radial polynomials =g3o@WD/G  
    %       r = 0:0.01:1; pj9*$.{  
    %       n = [3 2 5]; ,3P@5Ef  
    %       m = [1 2 1]; >2BWie?T  
    %       z = zernpol(n,m,r); l6~wm1vO  
    %       figure 6>)oG6  
    %       plot(r,z) fP>~ @^  
    %       grid on A1p87o>  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 98ot{+/LK  
    % $s S;#r0  
    %   See also ZERNFUN, ZERNFUN2. Ucqn 3&  
    *I<L1g%9d  
    % A note on the algorithm. 69iY)Ob/  
    % ------------------------ l+XTn;cS  
    % The radial Zernike polynomials are computed using the series =#so[Pd  
    % representation shown in the Help section above. For many special VNT*@^O_=  
    % functions, direct evaluation using the series representation can 7]F@ g}8  
    % produce poor numerical results (floating point errors), because xN +Oca  
    % the summation often involves computing small differences between 3IyNnm=u  
    % large successive terms in the series. (In such cases, the functions O]cuJp  
    % are often evaluated using alternative methods such as recurrence !3;KC"o  
    % relations: see the Legendre functions, for example). For the Zernike ggL^*MV  
    % polynomials, however, this problem does not arise, because the o$rA;^2X  
    % polynomials are evaluated over the finite domain r = (0,1), and +L hV4@zC  
    % because the coefficients for a given polynomial are generally all KSgYf;  
    % of similar magnitude. VOkSR6  
    % $_Kcm"oj  
    % ZERNPOL has been written using a vectorized implementation: multiple x"83[0ib  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] )[np{eF.k  
    % values can be passed as inputs) for a vector of points R.  To achieve N?j#=b+D  
    % this vectorization most efficiently, the algorithm in ZERNPOL B }t529Z  
    % involves pre-determining all the powers p of R that are required to 5i1E 5@~  
    % compute the outputs, and then compiling the {R^p} into a single Q^} Ib[  
    % matrix.  This avoids any redundant computation of the R^p, and AO~f=GW  
    % minimizes the sizes of certain intermediate variables. k esuM3  
    % 76eF6N+%}t  
    %   Paul Fricker 11/13/2006 ^hRx{A  
    FnWN]9  
    @aC9O 9|~  
    % Check and prepare the inputs: !e?2 x@J  
    % ----------------------------- ,vcd>"PK  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) &\m=|S  
        error('zernpol:NMvectors','N and M must be vectors.') YwU[kr-i  
    end TMw6 EM  
    ,TlYQ/j%h  
    if length(n)~=length(m) ,JqCxb9  
        error('zernpol:NMlength','N and M must be the same length.') #D ]P3  
    end yB5JvD ?  
    :v B9z  
    n = n(:); Y_= ]w1  
    m = m(:); )F'r-I%Hi  
    length_n = length(n); >!3r7LgK  
    Y{I,ipU.  
    if any(mod(n-m,2)) e5$S2o~JF  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') bR'UhPs-8;  
    end A/sM ?!p>_  
    21sXCmYR,t  
    if any(m<0) +[2ep"5H  
        error('zernpol:Mpositive','All M must be positive.') IBYSI0  
    end r) g:-[Ox9  
    ni?5h5-  
    if any(m>n) sH51 .JG  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') `q m$2  
    end Y6RbRcJw  
    [79iC$8B|  
    if any( r>1 | r<0 ) ,B1~6y\b  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') &cZl2ynPi  
    end T!X`"rI  
    2?nEHIUT  
    if ~any(size(r)==1) })umg8s  
        error('zernpol:Rvector','R must be a vector.') S0w:R:q}L  
    end `5 Iaz  
    Q" G;L  
    r = r(:); c&'5r OY~  
    length_r = length(r); j1O_Az|3  
    x4XCR,-  
    if nargin==4 #CRd@k ?  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ^4Tf6Fw#  
        if ~isnorm PVaqKCj:6W  
            error('zernpol:normalization','Unrecognized normalization flag.') DCKH^J   
        end )1gOO{T]h?  
    else Kh7C7[&  
        isnorm = false; %[x PyqX  
    end & ^;3S*p  
    W!V-m  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% cRv#aV  
    % Compute the Zernike Polynomials ?izl#?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R!rj:f!>  
    c@Xb6z_>  
    % Determine the required powers of r: n;LjKE  
    % ----------------------------------- >e!Y63`  
    rpowers = []; j8W<iy  
    for j = 1:length(n) nL+y"O  
        rpowers = [rpowers m(j):2:n(j)]; 6h7TM?lt  
    end (bAw>  
    rpowers = unique(rpowers); t"?)x&dS  
    sBa&]9>m  
    % Pre-compute the values of r raised to the required powers, elz0t<V  
    % and compile them in a matrix: \)i,`bz  
    % ----------------------------- }H:wgy`  
    if rpowers(1)==0 ) uTFId  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Y=D\  
        rpowern = cat(2,rpowern{:}); hv*XuT/  
        rpowern = [ones(length_r,1) rpowern]; YySo%\d  
    else _&N}.y)+t  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); :!wl/X ~  
        rpowern = cat(2,rpowern{:}); Ey)ey-'\  
    end ~\+Bb8+hpJ  
    3F32 /_`  
    % Compute the values of the polynomials: :,V&P_  
    % -------------------------------------- PR7B Cxm  
    z = zeros(length_r,length_n); Muyi2F)j  
    for j = 1:length_n KNjU!Z/4  
        s = 0:(n(j)-m(j))/2; ~l}\K10L*  
        pows = n(j):-2:m(j); > D%  
        for k = length(s):-1:1 L *cP8v4  
            p = (1-2*mod(s(k),2))* ... 06z+xxCo  
                       prod(2:(n(j)-s(k)))/          ... hdwF;  
                       prod(2:s(k))/                 ... uH)?`I\zrd  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... z9E*1B+  
                       prod(2:((n(j)+m(j))/2-s(k))); tLcw?aB  
            idx = (pows(k)==rpowers); NAOCQDk{  
            z(:,j) = z(:,j) + p*rpowern(:,idx); -vv_6Z L[  
        end CA5T3J@vAQ  
         P!I Lji!  
        if isnorm $b)t`r+  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); .DM-&P  
        end S6Y:Z0  
    end 5+UNLvsZ  
    e,MgR\F}  
    % 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)  {:#nrD"  
    &"I csxG  
    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。 2%6 >)|  
    >KvK'Mus/  
    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)