切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11074阅读
    • 9回复

    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

    上一主题 下一主题
    离线niuhelen
     
    发帖
    19
    光币
    28
    光券
    0
    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 FSQ&J|O  
    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! o6y,M!p@  
     
    分享到
    离线phility
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    19
    光币
    28
    光券
    0
    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 i$ L]X[  
    function z = zernfun(n,m,r,theta,nflag)  y"\,%.  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 16QbB;  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N jAK{<7v4U  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ,%h!%nz!  
    %   unit circle.  N is a vector of positive integers (including 0), and B#aH\$_U  
    %   M is a vector with the same number of elements as N.  Each element -(w~LT$ "  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) F4rKFMr  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, sJb)HQ,7x  
    %   and THETA is a vector of angles.  R and THETA must have the same A$~xG(  
    %   length.  The output Z is a matrix with one column for every (N,M) <s8? Z1  
    %   pair, and one row for every (R,THETA) pair. 5$oewjLO  
    % "PyWo  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ~g1, !Wl  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^$IZLM?E~  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral V-_/(xt*  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, p08kZ  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ns#~}2"d  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ]iDJ*!I  
    % VI24+h'J  
    %   The Zernike functions are an orthogonal basis on the unit circle. NLQE"\#a  
    %   They are used in disciplines such as astronomy, optics, and poD \C;o"  
    %   optometry to describe functions on a circular domain. 7%9)C[6NSs  
    % Ka]@[R6e  
    %   The following table lists the first 15 Zernike functions. 8lOI\-  
    % f<89$/w  
    %       n    m    Zernike function           Normalization w(/DTQc~d  
    %       -------------------------------------------------- mP pvZ  
    %       0    0    1                                 1 e40udLH~x  
    %       1    1    r * cos(theta)                    2 .Z=Ce!  
    %       1   -1    r * sin(theta)                    2 PW%1xHLfk  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) / Mo d=/e  
    %       2    0    (2*r^2 - 1)                    sqrt(3) BWUt{,?KU  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) cJ(BiL-uF  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) u 1ZJHry  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Xsd $*F@<  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) /vjGjb=3U  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ;1W6"3t-Y  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) gYatsFyL  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (kIz  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) lC#RNjDp/~  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) bD35JG^&i  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Xb}!0k/{  
    %       -------------------------------------------------- &%^K,Q"  
    % nr OqH  
    %   Example 1: DVI7]+=nV  
    % sV u k  
    %       % Display the Zernike function Z(n=5,m=1) 3AURzU  
    %       x = -1:0.01:1; {odA[H  
    %       [X,Y] = meshgrid(x,x); Wo{K}  
    %       [theta,r] = cart2pol(X,Y); HZ }6Q  
    %       idx = r<=1; ]pnYvXf>!  
    %       z = nan(size(X)); 9x=3W?K:,  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); "B#Y-  
    %       figure nbGoJC:U  
    %       pcolor(x,x,z), shading interp .Gh%p`<  
    %       axis square, colorbar Fn!SGX~kx$  
    %       title('Zernike function Z_5^1(r,\theta)') P@gt di(Q  
    % '[nmFCG%m*  
    %   Example 2: -Q$b7*"z(  
    % =}v ;1m  
    %       % Display the first 10 Zernike functions 66Gx.tE  
    %       x = -1:0.01:1; ^agj4$  
    %       [X,Y] = meshgrid(x,x);  \~>e_;  
    %       [theta,r] = cart2pol(X,Y); OV[`|<C '  
    %       idx = r<=1; ?E<c[*F05  
    %       z = nan(size(X)); R:/ha(+  
    %       n = [0  1  1  2  2  2  3  3  3  3]; p<KIF>rf|  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 3B{[%#vO  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; nUc;/  
    %       y = zernfun(n,m,r(idx),theta(idx)); KCUU#t|8V\  
    %       figure('Units','normalized') BwxnDeG)  
    %       for k = 1:10 3OP.12^  
    %           z(idx) = y(:,k); \jyjQ,v)  
    %           subplot(4,7,Nplot(k)) QU/fT_ORw  
    %           pcolor(x,x,z), shading interp O8lFx_N7Q  
    %           set(gca,'XTick',[],'YTick',[]) , T\-;7  
    %           axis square Uoj i@  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])  sTkkM9  
    %       end IU#x[P!  
    % p@ygne 4  
    %   See also ZERNPOL, ZERNFUN2. :l,OalO  
    %d;<2b0  
    %   Paul Fricker 11/13/2006 \a))  
    y-9+a7j  
    Ei5wel6!  
    % Check and prepare the inputs: ;`(R7X *3  
    % ----------------------------- 3c#s|qW  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ;YyXT"6/p  
        error('zernfun:NMvectors','N and M must be vectors.') >72JV; W]  
    end !aNh!  
    qX#MV>1  
    if length(n)~=length(m) CO^Jz  
        error('zernfun:NMlength','N and M must be the same length.') =Z,5$6%)  
    end ".U^if F  
    [}2Z/   
    n = n(:); '6-$Xq0^E  
    m = m(:); p &(OZJT  
    if any(mod(n-m,2)) kV&9`c+  
        error('zernfun:NMmultiplesof2', ... md bp8,O  
              'All N and M must differ by multiples of 2 (including 0).') 1n=_y o  
    end 60}! LmL  
    / T c=  
    if any(m>n) ^3]UZ@  
        error('zernfun:MlessthanN', ... ?j O 5 9n  
              'Each M must be less than or equal to its corresponding N.') lr@#^  
    end -YGbfd<wq  
    _. V?A*  
    if any( r>1 | r<0 ) FjFMR 63  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') OJO!FH)  
    end {~Tg7<\L  
    |L6&Gf]#5  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 1zxq^BI  
        error('zernfun:RTHvector','R and THETA must be vectors.') Shr,#wwM`B  
    end EbY,N:LK  
    }8K4-[\  
    r = r(:); [bz T& o  
    theta = theta(:); MCTsi:V>+  
    length_r = length(r); w.{&=WTr  
    if length_r~=length(theta) 1bnBji  
        error('zernfun:RTHlength', ... t= #&fSR  
              'The number of R- and THETA-values must be equal.') *fMpZ+;[m  
    end uQ1@b-e`5  
    }_'IE1bA  
    % Check normalization: 4u;9J*r4  
    % -------------------- }T2xXbU  
    if nargin==5 && ischar(nflag) A7_4 .VH  
        isnorm = strcmpi(nflag,'norm'); =8Jfgq9E  
        if ~isnorm "r4AY  
            error('zernfun:normalization','Unrecognized normalization flag.') b}^S.;vNj  
        end BaI $S>/Q  
    else }|OaL*|u  
        isnorm = false; uA t V".  
    end /1=4"|q>h'  
    C*=Xk/0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f"G-  
    % Compute the Zernike Polynomials |~`as(@Ih  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 96cJ8I8  
    B(a-k?  
    % Determine the required powers of r: 5@IB39  
    % ----------------------------------- 2"P 99$"  
    m_abs = abs(m); E907fX[R~  
    rpowers = []; =uk0@hy9b  
    for j = 1:length(n) G'2#9<c*  
        rpowers = [rpowers m_abs(j):2:n(j)]; K5ZC:Ks  
    end q\Q{sv_  
    rpowers = unique(rpowers); =!O*/6rz  
    0]KraLu"N  
    % Pre-compute the values of r raised to the required powers, KH)D 08  
    % and compile them in a matrix: <~}7Mxn%x@  
    % ----------------------------- +d+@u)6  
    if rpowers(1)==0 ;{ i'#rn{  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); t"hYcnC  
        rpowern = cat(2,rpowern{:}); H1<>NWm!v7  
        rpowern = [ones(length_r,1) rpowern]; q"O.Cbk  
    else !TZhQiorC  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )vmA^nU>  
        rpowern = cat(2,rpowern{:}); _Jwq`]Z  
    end EXP%Mk/  
    s]mo$ _na  
    % Compute the values of the polynomials: E$W{8?:{  
    % -------------------------------------- }I 3gU  
    y = zeros(length_r,length(n)); Cm$.<CV  
    for j = 1:length(n) h\plQ[T  
        s = 0:(n(j)-m_abs(j))/2; JnHo9K2.  
        pows = n(j):-2:m_abs(j); >fH=DOz$&  
        for k = length(s):-1:1 a+hd(JX0~  
            p = (1-2*mod(s(k),2))* ... -.g|l\  
                       prod(2:(n(j)-s(k)))/              ... |mdi]TL  
                       prod(2:s(k))/                     ... g{W;I_P^9  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 3qY K_M^[  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); KD/V aN  
            idx = (pows(k)==rpowers); [V4{c@  
            y(:,j) = y(:,j) + p*rpowern(:,idx); : ^ 8  
        end USFD y  
         K3\#E/Ox  
        if isnorm u:aW 8  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ,J`'Y+7W  
        end p>_;^&>&  
    end P&Pj>!T5  
    % END: Compute the Zernike Polynomials Vd=yr'?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c/_ +o;Bc  
    YeF1C/'hy  
    % Compute the Zernike functions: LH:i| I  
    % ------------------------------ @ym/27cRE  
    idx_pos = m>0; p{PE@KO:  
    idx_neg = m<0; Q}S_%I}u:  
    }K8/-d6  
    z = y; Y?ez9o:/#  
    if any(idx_pos) ID.n1i3  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Q5%#^ZdsTd  
    end %*#n d  
    if any(idx_neg) E~LT b) !  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); %lg=YGLQB  
    end _-5,zP R  
    z&V+#Ws/  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) ]0L&v7[  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. * !^<m0  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Jjl`_X$CB  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive m[k_>e\ u  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, f@co<iA  
    %   and THETA is a vector of angles.  R and THETA must have the same |a{Q0:  
    %   length.  The output Z is a matrix with one column for every P-value, h=_mNG>R)  
    %   and one row for every (R,THETA) pair. sl*5Y#,|1  
    % 0EJ(.8hwm  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike >[6{LAe~hp  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) !;_H$r0  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 5[n(7;+gw  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 "`[!Lz  
    %   for all p. DS|KkTy3  
    % {-BRt)L[  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 16SOIT  
    %   Zernike functions (order N<=7).  In some disciplines it is HHjt/gc}`  
    %   traditional to label the first 36 functions using a single mode rWh6RYd<T  
    %   number P instead of separate numbers for the order N and azimuthal lg pW@g  
    %   frequency M. qaim6a  
    % !gnj]k&/c  
    %   Example: wM_c48|d  
    % jp' K%P  
    %       % Display the first 16 Zernike functions kM`l  
    %       x = -1:0.01:1; |y<),j6  
    %       [X,Y] = meshgrid(x,x); "7>>I D  
    %       [theta,r] = cart2pol(X,Y); wv^rS^~  
    %       idx = r<=1; Q%aU42?_1  
    %       p = 0:15; |w}xl'>q  
    %       z = nan(size(X)); ~ d^<_R  
    %       y = zernfun2(p,r(idx),theta(idx)); ?X?&~3iD%  
    %       figure('Units','normalized') E2L(wt}^  
    %       for k = 1:length(p) *)]"27^  
    %           z(idx) = y(:,k); d3^OEwe  
    %           subplot(4,4,k) DP/J (>eG  
    %           pcolor(x,x,z), shading interp 6mH/ m&  
    %           set(gca,'XTick',[],'YTick',[]) i* R,QN)  
    %           axis square OdI\B   
    %           title(['Z_{' num2str(p(k)) '}']) )rLMIk  
    %       end BK,sc'b  
    % .k4W_9  
    %   See also ZERNPOL, ZERNFUN. |lH;Fq{\  
    drwgjLC+  
    %   Paul Fricker 11/13/2006 >\ST-7[^L  
    k#bu#YZk  
    FZiW|G  
    % Check and prepare the inputs: ENh!N4vbO  
    % ----------------------------- |_u8mV  
    if min(size(p))~=1 M\b")Tu{0  
        error('zernfun2:Pvector','Input P must be vector.') l!E7A Kk8  
    end =OJ;0 /$6  
    ZV}X'qGaq  
    if any(p)>35 iAMtejw  
        error('zernfun2:P36', ... @+U,Nzd  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 4\.V   
               '(P = 0 to 35).']) b=a!j=-D  
    end +5v}q.:+  
    ;evCW$G=  
    % Get the order and frequency corresonding to the function number: wxLXh6|6%_  
    % ---------------------------------------------------------------- iV.p5FD  
    p = p(:); v;y0jD#b  
    n = ceil((-3+sqrt(9+8*p))/2); 64Gi8|P  
    m = 2*p - n.*(n+2); (\puf+  
    ydWtvFuS  
    % Pass the inputs to the function ZERNFUN: MQ][mMM;w  
    % ---------------------------------------- z}}]jR \y?  
    switch nargin LU!1s@  
        case 3 ZeasYSo4P  
            z = zernfun(n,m,r,theta); X_; *`,<T  
        case 4 HW=xvA+  
            z = zernfun(n,m,r,theta,nflag); kR.wOJ7'  
        otherwise ]0c Pml  
            error('zernfun2:nargin','Incorrect number of inputs.') #:3r4J%+~  
    end QL"gWr`R  
    oL/o*^  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) o0;7b>Tv  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. z<C[nR$N  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of >.dHt\  
    %   order N and frequency M, evaluated at R.  N is a vector of ='/Z;3jt]x  
    %   positive integers (including 0), and M is a vector with the ;cSGlE |  
    %   same number of elements as N.  Each element k of M must be a M1=_^f=&.  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) Md(JIlh3  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is *M!kA65'  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix <A~GW 'HB  
    %   with one column for every (N,M) pair, and one row for every J6|JWp  
    %   element in R. I{n;4?  
    % cIw eBDl  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- %<U{K;  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is WEa>)@  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 4UCwT1  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 >_Uj?F:  
    %   for all [n,m]. ko[TDh$T5  
    % RebTg1vGu  
    %   The radial Zernike polynomials are the radial portion of the #4y,a_)  
    %   Zernike functions, which are an orthogonal basis on the unit )bW5yG!  
    %   circle.  The series representation of the radial Zernike sMi{"`37  
    %   polynomials is vj3isI4lU  
    % \RE c8nsLy  
    %          (n-m)/2 J/S{FxNe]  
    %            __ qc0 B<,x7  
    %    m      \       s                                          n-2s A^pW]r=Xtk  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Z\!rH "8  
    %    n      s=0 :HZ;Po   
    % "|4jP za  
    %   The following table shows the first 12 polynomials. {,f[r*{Y  
    % zUQe0Gc.b^  
    %       n    m    Zernike polynomial    Normalization v' 0!=r  
    %       --------------------------------------------- <|JU(B  
    %       0    0    1                        sqrt(2) ~L){O*Z  
    %       1    1    r                           2 PPgW ^gj  
    %       2    0    2*r^2 - 1                sqrt(6) ,Oi^ySn  
    %       2    2    r^2                      sqrt(6) 4$i}Xk#3  
    %       3    1    3*r^3 - 2*r              sqrt(8) 5Bd(>'ig_  
    %       3    3    r^3                      sqrt(8) 7)PJ:4IqS  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 5DSuUEvWcL  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Q [:<S/w  
    %       4    4    r^4                      sqrt(10) /| f[us-w  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) H XP;0B%4  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) .Cfp'u%\;  
    %       5    5    r^5                      sqrt(12) T&4fBMBp,%  
    %       --------------------------------------------- IozNjII$:.  
    % Cgo XZX  
    %   Example: w -dI<s  
    % ;Xh5oB\)W  
    %       % Display three example Zernike radial polynomials BK*UR+,  
    %       r = 0:0.01:1; pE(sV{PD  
    %       n = [3 2 5]; j]4,6` b\  
    %       m = [1 2 1]; {r{>?)O  
    %       z = zernpol(n,m,r); Loc8eToZ  
    %       figure )]}$   
    %       plot(r,z) y^YVo^3  
    %       grid on p|s2G~0<  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') I}ndRDz[  
    % Cg*kN"8q  
    %   See also ZERNFUN, ZERNFUN2. }6@%((9E 2  
    +Wd L  
    % A note on the algorithm. x Zg7Jg  
    % ------------------------ K /ZHJkJ7  
    % The radial Zernike polynomials are computed using the series wKLN:aRF2  
    % representation shown in the Help section above. For many special /=- h:0{M  
    % functions, direct evaluation using the series representation can "Y(%oJS]D  
    % produce poor numerical results (floating point errors), because ?M]u$Te/.  
    % the summation often involves computing small differences between ex:3ua$N  
    % large successive terms in the series. (In such cases, the functions |QMT A5  
    % are often evaluated using alternative methods such as recurrence G "c/a8  
    % relations: see the Legendre functions, for example). For the Zernike ME,duY/>Q  
    % polynomials, however, this problem does not arise, because the klo^K9!  
    % polynomials are evaluated over the finite domain r = (0,1), and W7T" d4  
    % because the coefficients for a given polynomial are generally all (Y>U6  
    % of similar magnitude. 0*{@E%9  
    % MKe *f%  
    % ZERNPOL has been written using a vectorized implementation: multiple "|\94  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ^[*AK_o_DQ  
    % values can be passed as inputs) for a vector of points R.  To achieve Upu%.[7  
    % this vectorization most efficiently, the algorithm in ZERNPOL )Jmw|B  
    % involves pre-determining all the powers p of R that are required to DSTx#*  
    % compute the outputs, and then compiling the {R^p} into a single 6XZN>#  
    % matrix.  This avoids any redundant computation of the R^p, and 0x6@{0  
    % minimizes the sizes of certain intermediate variables. l2r>|CGQ[  
    % iAg}pwU  
    %   Paul Fricker 11/13/2006 sbpu qOL  
    U <|B7t4M  
    br10ptEx  
    % Check and prepare the inputs: hN!.@L  
    % ----------------------------- lej^gxj/2  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) `c>A >c|  
        error('zernpol:NMvectors','N and M must be vectors.') DU(X,hDBF  
    end S [ i$e  
    ;Xz(B4N~o  
    if length(n)~=length(m) "_ Zh5 g  
        error('zernpol:NMlength','N and M must be the same length.') E=3#TBd  
    end Jr|K>  
    5N Fq7&rJ6  
    n = n(:); *:k~g].Iz  
    m = m(:); j`Tm\!q  
    length_n = length(n); tYF$#Nor#k  
    qNhH%tYQ  
    if any(mod(n-m,2)) N" 8*FiZ|  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') L,tZh0  
    end q=J9L Q  
    wrhBH;3  
    if any(m<0) T<! \B]  
        error('zernpol:Mpositive','All M must be positive.') c,+iU R<  
    end nqBG]y aI  
    oA5<[&~<  
    if any(m>n) Jx)~kK  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') @263)`9G  
    end {H/8#y4qp&  
    *1%e%G  
    if any( r>1 | r<0 ) X^u4%O['  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') wV7@D[8  
    end <.y;&a o  
    MYKs??]Y1  
    if ~any(size(r)==1) 0-6:AHix  
        error('zernpol:Rvector','R must be a vector.') v#{G8'+%  
    end &h98.A*&  
    6tDg3`w>  
    r = r(:); 5)h+(u C3  
    length_r = length(r); -Y5YCY!`  
    g| _HcaW  
    if nargin==4 nNkyOaK*4  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); i7Y s_8A"9  
        if ~isnorm f$~ _FX  
            error('zernpol:normalization','Unrecognized normalization flag.') 7Y$4MMNQ  
        end nL "g23  
    else *C[4 (DmB  
        isnorm = false; S.zg&   
    end ,+swH;=7#r  
    EOMuqP)  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% oAIY=z  
    % Compute the Zernike Polynomials 5 ]A$P\7~1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% IBn'iE[>  
    JNU"5sB  
    % Determine the required powers of r: (MF+/fi  
    % ----------------------------------- PEOM1oY)w  
    rpowers = []; 5i}CzA96  
    for j = 1:length(n) xMO[3 D&D  
        rpowers = [rpowers m(j):2:n(j)]; SaX,^_GY  
    end ~*,Ddwr0a  
    rpowers = unique(rpowers); ]{q- Y<{"  
    -N /8Ho  
    % Pre-compute the values of r raised to the required powers, /h.:br?M#P  
    % and compile them in a matrix: =%:n0S0C"  
    % ----------------------------- bUY:XmA  
    if rpowers(1)==0 #U\&i`  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 0xvMR&.H  
        rpowern = cat(2,rpowern{:}); "u3fs2  
        rpowern = [ones(length_r,1) rpowern]; =|j*VF2y"  
    else %8rr*l5  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,6y-.m7>  
        rpowern = cat(2,rpowern{:}); Mo:!jS~a(Z  
    end @'k,\$/  
    : 9djMsd  
    % Compute the values of the polynomials: k'WS"<-  
    % -------------------------------------- +j)-L \  
    z = zeros(length_r,length_n); T2/v}  
    for j = 1:length_n S\yu%=h  
        s = 0:(n(j)-m(j))/2; >uP{9kDm  
        pows = n(j):-2:m(j); t ?Njw7  
        for k = length(s):-1:1 &"~,V6,q  
            p = (1-2*mod(s(k),2))* ... dd?x5|/#  
                       prod(2:(n(j)-s(k)))/          ... k=ior  
                       prod(2:s(k))/                 ... IxK 3,@d  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... e$p1Th*|]4  
                       prod(2:((n(j)+m(j))/2-s(k))); cAsSN.HFS  
            idx = (pows(k)==rpowers); 9}'l=b:Jms  
            z(:,j) = z(:,j) + p*rpowern(:,idx); }5fI*v  
        end [7SI<xkv  
         @vQ;>4i.  
        if isnorm n{qa]3  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); U&d-?PI  
        end 3Aqe;Wf9%+  
    end fmZzBZ_  
    5sA>O2Rt>  
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
    发帖
    856
    光币
    846
    光券
    0
    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
    发帖
    4352
    光币
    5478
    光券
    1
    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  )^@V*$D  
    VqT[ca\  
    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。 [&}<! :9'  
    Dq1XZ%8  
    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)