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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 `8$:F4%P  
    就是我用ansys计算出了镜面的面型的数据,怎样可以得到zernike多项式系数,然后用zemax各阶得到像差!谢谢啦! c+)36/; X  
     
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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 1A)wbH)  
    function z = zernfun(n,m,r,theta,nflag) ^G7n#  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. |2+F I<v4  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N dH2j*G Ij  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Z7KB?1{G  
    %   unit circle.  N is a vector of positive integers (including 0), and ~,`\D7Z3  
    %   M is a vector with the same number of elements as N.  Each element 2S7H_qo$  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 7Dx .;  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, .LGkr@P  
    %   and THETA is a vector of angles.  R and THETA must have the same >gS5[`xRE  
    %   length.  The output Z is a matrix with one column for every (N,M) +i q+  
    %   pair, and one row for every (R,THETA) pair. 4/mj"PBKL  
    % q)z1</B-  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 9^C!,A{u4  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ~YT>:Np  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral &a2V-|G',  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +gD)Yd  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized -V<=`e  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. _6QLnr&@j  
    % RL]lt0O{  
    %   The Zernike functions are an orthogonal basis on the unit circle. ?Ss RN jeL  
    %   They are used in disciplines such as astronomy, optics, and oN1wrf}Sh  
    %   optometry to describe functions on a circular domain. {ZBb. $}RC  
    % zvQ^f@lq2  
    %   The following table lists the first 15 Zernike functions. d@q t%r3;  
    % 61eKGcjs:  
    %       n    m    Zernike function           Normalization |]2eGrGj4  
    %       -------------------------------------------------- fi-&[llg  
    %       0    0    1                                 1 d= T9mj.@  
    %       1    1    r * cos(theta)                    2 )lngef /D_  
    %       1   -1    r * sin(theta)                    2 >/OXC+=^4  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) [#3Cg%V  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Q+%m+ /Zq  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) Q,M/R6i-  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ~M9 n<kmE  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 5)SZd)  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) .o,51dn+ s  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) )1tnZ=&  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) WY. \<$7  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) hz~CW-47  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) q NQ3(1xW  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) DHujpZXQ  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) BoiIr[ (  
    %       -------------------------------------------------- X m:gD6;9  
    % 'm p{O  
    %   Example 1: dW=D]  
    % 5KPPZmO  
    %       % Display the Zernike function Z(n=5,m=1) da86Jj=k  
    %       x = -1:0.01:1; 2O)Kn q  
    %       [X,Y] = meshgrid(x,x); O'sr[  
    %       [theta,r] = cart2pol(X,Y); Uub%s`O  
    %       idx = r<=1; %[ bO\,  
    %       z = nan(size(X)); bEXHB  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); p~zTRnm  
    %       figure 0 f#a_  
    %       pcolor(x,x,z), shading interp HEfA c  
    %       axis square, colorbar `\u),$  
    %       title('Zernike function Z_5^1(r,\theta)') z1KC$~{O  
    % H/la'f#o%  
    %   Example 2: a!J ow?(  
    % Kd[`mkmS  
    %       % Display the first 10 Zernike functions 02 c.;ka3  
    %       x = -1:0.01:1; &+r ;>  
    %       [X,Y] = meshgrid(x,x); Px?At5  
    %       [theta,r] = cart2pol(X,Y); AYQh=$)(  
    %       idx = r<=1; \S@=zII_  
    %       z = nan(size(X)); `::(jW.KO  
    %       n = [0  1  1  2  2  2  3  3  3  3]; KL\=:iWA  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; t:j07 ,1~  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; cS;=_%~  
    %       y = zernfun(n,m,r(idx),theta(idx)); <IkD=X  
    %       figure('Units','normalized') D30Z9_^%:  
    %       for k = 1:10 u9~V2>r\  
    %           z(idx) = y(:,k); wT AEJ{p  
    %           subplot(4,7,Nplot(k)) r L|BkN  
    %           pcolor(x,x,z), shading interp k49n9EX  
    %           set(gca,'XTick',[],'YTick',[]) ZYt"=\_  
    %           axis square .+~kJ0~Y  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @_:?N(%(  
    %       end D2*Q1n  
    % IhKas4  
    %   See also ZERNPOL, ZERNFUN2. Fu$Gl$qV?%  
    K.L+; nQ  
    %   Paul Fricker 11/13/2006 L5|;VH  
    27i<6PAC[A  
    M #Ru I%  
    % Check and prepare the inputs: ia.B@u1/  
    % ----------------------------- +#"Ic:  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) yT9RNo/w  
        error('zernfun:NMvectors','N and M must be vectors.') ?|rw=%  
    end -+2xdLa63  
    BCDf9]X  
    if length(n)~=length(m) 0J,d9a [1  
        error('zernfun:NMlength','N and M must be the same length.') !Fs) "?  
    end 0JL6EL>_  
    hQLx"R$  
    n = n(:); #Lt+6sa]2@  
    m = m(:); sEi.f(WA  
    if any(mod(n-m,2)) X1QZEl  
        error('zernfun:NMmultiplesof2', ... cx%9UK*c  
              'All N and M must differ by multiples of 2 (including 0).') k yA(m;r  
    end _[Wrd?Z  
    3T^dgWXEG  
    if any(m>n) >!.lr9(l  
        error('zernfun:MlessthanN', ... !x / Z"  
              'Each M must be less than or equal to its corresponding N.') +GtGyp  
    end gG>^h1_o~  
    N28?JQha  
    if any( r>1 | r<0 ) _@?Jx/`;bk  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') yFtf~8s3  
    end n& &U9sf?  
    nk.E q[08  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) &=O1Qg=K  
        error('zernfun:RTHvector','R and THETA must be vectors.') ]*Ki7h |B  
    end gxtbu$  
    7n]%`Yb  
    r = r(:); l'8wPmy%N  
    theta = theta(:); JT_B@TO\  
    length_r = length(r); ~TIZumGB  
    if length_r~=length(theta) 'U Cx^-  
        error('zernfun:RTHlength', ... 9 9BK/>R  
              'The number of R- and THETA-values must be equal.') l+qtA~V&2  
    end Pu*UZcXY  
    VQ}3r)ch  
    % Check normalization: md LJ,w?{  
    % -------------------- f=Y9a$.:M  
    if nargin==5 && ischar(nflag) y{P9k8v!z  
        isnorm = strcmpi(nflag,'norm'); HBGA lZ  
        if ~isnorm UHHKI)(  
            error('zernfun:normalization','Unrecognized normalization flag.') 70(?X/5#  
        end =xP{f<`   
    else % E_{L  
        isnorm = false; 4'| :SyOm  
    end 6;V 1PK>9  
    IcA~f@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% HKw4}FC*  
    % Compute the Zernike Polynomials BVeNK=7m%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xGk4KcxKs  
    h(up1(x  
    % Determine the required powers of r: DMW:%h{  
    % ----------------------------------- GQWTQIl]  
    m_abs = abs(m); a}hM}U!  
    rpowers = []; b;ZAz  
    for j = 1:length(n) =_3qUcOP  
        rpowers = [rpowers m_abs(j):2:n(j)]; ~[6|VpGc:  
    end cNv c pv  
    rpowers = unique(rpowers);  p$v +L  
    H.K`#W&  
    % Pre-compute the values of r raised to the required powers, oPZ4}>uV  
    % and compile them in a matrix: ?!-im*~w  
    % ----------------------------- -2d&Aq4m)  
    if rpowers(1)==0 ZK*aVYnu  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); >Sah\u`  
        rpowern = cat(2,rpowern{:}); !7?wd^C'f  
        rpowern = [ones(length_r,1) rpowern]; N Q=YTRU  
    else G"w Q(6J@  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); `^{P,N>X  
        rpowern = cat(2,rpowern{:}); ZeV)/g,w  
    end 6>J #M  
    4f,x@:Jw  
    % Compute the values of the polynomials: L,L7WObA  
    % -------------------------------------- F tjm@:X  
    y = zeros(length_r,length(n)); GrC")Z|3u  
    for j = 1:length(n) net9K X4\  
        s = 0:(n(j)-m_abs(j))/2; rfpxE>_|G  
        pows = n(j):-2:m_abs(j); `$-  Ib^  
        for k = length(s):-1:1 b*fflJ  
            p = (1-2*mod(s(k),2))* ... iq -o$6Pg  
                       prod(2:(n(j)-s(k)))/              ... k =_@1b-  
                       prod(2:s(k))/                     ... ,iUx'U  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... U7?ez  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); xM\ApN~W  
            idx = (pows(k)==rpowers); 3}Qh`+Yj]  
            y(:,j) = y(:,j) + p*rpowern(:,idx); # w6CL  
        end pT tX[CE  
         ~yN,FpD  
        if isnorm \f#ao<vQm  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Jmx }r,j  
        end W9"I++~f  
    end ") D!OW]  
    % END: Compute the Zernike Polynomials 6Tnzg`0I  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O6]~5&8U.  
    [DwB7l)O(  
    % Compute the Zernike functions: V ;jz0B  
    % ------------------------------ g!ww;_  
    idx_pos = m>0; -&}E:zoe  
    idx_neg = m<0; ZbUf|#GTB  
    KHtY +93  
    z = y; K-3 _4As  
    if any(idx_pos) RSC-+c6 1  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); .<dmdqk]  
    end ~JpUO~i/  
    if any(idx_neg) KG$2u:n  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 3I+pe;  
    end .>n|#XK  
    6* 7&X#gG  
    % 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) 4KCxhJq  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ?}[keSEh>  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ?F/3]lsggT  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive |k+^D:  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, jPfoI-  
    %   and THETA is a vector of angles.  R and THETA must have the same @zbXG_J  
    %   length.  The output Z is a matrix with one column for every P-value, GSp1,E2J  
    %   and one row for every (R,THETA) pair. PW}.`  
    % P*>V6SK>b  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 7 <xxOY>y  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) U{EW +>  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Y{KJk'xN5W  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 U,'n}]=4A3  
    %   for all p. Y~Rwsx  
    % w8qI7/  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 cu-WY8n  
    %   Zernike functions (order N<=7).  In some disciplines it is V~;YV]1Y  
    %   traditional to label the first 36 functions using a single mode v,;?+Ck  
    %   number P instead of separate numbers for the order N and azimuthal 5 S 1m&s5k  
    %   frequency M. t(Uoi~#[  
    % qb Q> z+c  
    %   Example: )-(NL!?`  
    % DjIs"5Iei  
    %       % Display the first 16 Zernike functions (rJvE*  
    %       x = -1:0.01:1; _bX)fnUu  
    %       [X,Y] = meshgrid(x,x); Q*I/mUP&f  
    %       [theta,r] = cart2pol(X,Y); xk/(| f{L  
    %       idx = r<=1; h>wU';5#f  
    %       p = 0:15; $IHa]9 {  
    %       z = nan(size(X)); [# :k3aFz  
    %       y = zernfun2(p,r(idx),theta(idx)); `d8TA#|`  
    %       figure('Units','normalized') 6XP>p$-  
    %       for k = 1:length(p) zy|hf<V  
    %           z(idx) = y(:,k); J ?&9ofj&  
    %           subplot(4,4,k) 4:.M*Dz  
    %           pcolor(x,x,z), shading interp wQ5__"D  
    %           set(gca,'XTick',[],'YTick',[]) $)U RY~;i  
    %           axis square 5GKz@as8  
    %           title(['Z_{' num2str(p(k)) '}']) 6 Qmtb2  
    %       end Yr+&|;DB  
    % 6R25Xfm_|  
    %   See also ZERNPOL, ZERNFUN. FB@G.f  
    { }z7N~  
    %   Paul Fricker 11/13/2006 "k.<"pf  
    PF.HYtZqK  
    O'k"6sBb  
    % Check and prepare the inputs: KnuqU2< {  
    % ----------------------------- (f)QEho7  
    if min(size(p))~=1 B-RaAiE@  
        error('zernfun2:Pvector','Input P must be vector.') iY="M_kQ_  
    end 8:f( PN  
    u% FA.  
    if any(p)>35 zIu1oF4[  
        error('zernfun2:P36', ... Q8_d]V=X:  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... V{][{5SR  
               '(P = 0 to 35).']) /O9z-!Jz  
    end 8 =d9*lm  
    ~Y{K ^:wN^  
    % Get the order and frequency corresonding to the function number: uB\A8zC  
    % ---------------------------------------------------------------- =_5-z|<  
    p = p(:); -{dw Ll_  
    n = ceil((-3+sqrt(9+8*p))/2); $3So`8Bm[$  
    m = 2*p - n.*(n+2); [8ih-k  
    >1YJETysO  
    % Pass the inputs to the function ZERNFUN: hGsY u)  
    % ---------------------------------------- %NvY~,  
    switch nargin kp[Jl0K5  
        case 3 +7=3[K  
            z = zernfun(n,m,r,theta); .A E(D7d6  
        case 4 C[75 !F   
            z = zernfun(n,m,r,theta,nflag); [ACa<U/  
        otherwise nkxVc  
            error('zernfun2:nargin','Incorrect number of inputs.') y@F{pr+dA  
    end ;X8eZQ  
    Au9Rr3n  
    % 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) C-&\qAo?<:  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. fTMn  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of W;Ct[Y 8m  
    %   order N and frequency M, evaluated at R.  N is a vector of F8nR.|  
    %   positive integers (including 0), and M is a vector with the ) }(Po_  
    %   same number of elements as N.  Each element k of M must be a `ml  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) BKi@c\Wb  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 1GE[*$vuq  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix f]Xh7m(Gh  
    %   with one column for every (N,M) pair, and one row for every \Cx2$<8  
    %   element in R. FG/1!8F  
    % Kqm2TMO]>V  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- PY[nnoF"|  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is Ejmpg_kux  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to /@ m]@  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ''tCtG" Xi  
    %   for all [n,m]. {{qu:(_g  
    % Z ):q1:y  
    %   The radial Zernike polynomials are the radial portion of the 8MU+i%hd  
    %   Zernike functions, which are an orthogonal basis on the unit #ozui-u>  
    %   circle.  The series representation of the radial Zernike vhvFBx0  
    %   polynomials is DZ"'GQSg  
    % #A\@)wJ  
    %          (n-m)/2 $Y,y~4I  
    %            __ evjj~xkte  
    %    m      \       s                                          n-2s f?W_/daP  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ]Q$Sei5  
    %    n      s=0 pLSh +*F  
    % k'QI`@l&l  
    %   The following table shows the first 12 polynomials. TophV}@B`  
    % iSbPOC7  
    %       n    m    Zernike polynomial    Normalization Ogv9_ X8  
    %       --------------------------------------------- *O_>3Hgl  
    %       0    0    1                        sqrt(2) 2)hfYLi  
    %       1    1    r                           2 Q<0X80w>  
    %       2    0    2*r^2 - 1                sqrt(6) OY Sq)!:  
    %       2    2    r^2                      sqrt(6) 0RgE~x!hI  
    %       3    1    3*r^3 - 2*r              sqrt(8) jh(T?t$&  
    %       3    3    r^3                      sqrt(8) K @RGvP  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 0nbY~j$A=  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) L>LIN 1A  
    %       4    4    r^4                      sqrt(10) Fs"i fn0  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) &+ "<ia(  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) bMn)lrsX  
    %       5    5    r^5                      sqrt(12) #lR-?Uh  
    %       --------------------------------------------- _iu~vU)r  
    % ZLP/&`>8  
    %   Example: F/x2}'  
    % DL`8qJ'mJs  
    %       % Display three example Zernike radial polynomials p]0`rf!|  
    %       r = 0:0.01:1; S /"G=^~  
    %       n = [3 2 5]; i-PK59VZ8f  
    %       m = [1 2 1]; djH&)&q!  
    %       z = zernpol(n,m,r); v*[UG^+)  
    %       figure i\<S ;  
    %       plot(r,z) d a<>a  
    %       grid on `WIZY33V  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 3\2%i 6W6  
    % zOd* >  
    %   See also ZERNFUN, ZERNFUN2. tn#cVB3  
    yDfH`]i)U  
    % A note on the algorithm. h4jo<yp\  
    % ------------------------ |.VSw  
    % The radial Zernike polynomials are computed using the series FQJiLb._Z  
    % representation shown in the Help section above. For many special Fei5'  
    % functions, direct evaluation using the series representation can "4\k1H"_  
    % produce poor numerical results (floating point errors), because EB| iW2'  
    % the summation often involves computing small differences between r4pX4 7H  
    % large successive terms in the series. (In such cases, the functions tL3R<'  
    % are often evaluated using alternative methods such as recurrence oUwo!n}  
    % relations: see the Legendre functions, for example). For the Zernike 3m= _a  
    % polynomials, however, this problem does not arise, because the +j{(NwsX  
    % polynomials are evaluated over the finite domain r = (0,1), and b"uO BB  
    % because the coefficients for a given polynomial are generally all <pfl>Uf  
    % of similar magnitude. u$ yXuFj/  
    % PChew3  
    % ZERNPOL has been written using a vectorized implementation: multiple .u>[m.  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] rCwE$5 b  
    % values can be passed as inputs) for a vector of points R.  To achieve hs[x\:})/  
    % this vectorization most efficiently, the algorithm in ZERNPOL d\uN  
    % involves pre-determining all the powers p of R that are required to  4d\^  
    % compute the outputs, and then compiling the {R^p} into a single ?TeozhUY  
    % matrix.  This avoids any redundant computation of the R^p, and 5mQ@&E~#W  
    % minimizes the sizes of certain intermediate variables. pJx7S sW  
    % +l^tT&s;f  
    %   Paul Fricker 11/13/2006 ffG<hclk  
    +@=V}IO  
    u8T@W}FX  
    % Check and prepare the inputs: P&sWn?q Ol  
    % ----------------------------- ~4khIz  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) XjF@kQeM=  
        error('zernpol:NMvectors','N and M must be vectors.') *#'j0;2F  
    end PQDLbSe)\  
    N]w_9p~=1  
    if length(n)~=length(m) h*%FZ}}`q  
        error('zernpol:NMlength','N and M must be the same length.') 3("C'(W  
    end g35!a<JW  
    uG^CyM>R`  
    n = n(:); gYhY1Mym  
    m = m(:); GuO}CQs^W  
    length_n = length(n); r5DR F4,7  
    LP6 p  
    if any(mod(n-m,2)) H0;Iv#S!  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') EW|$qLg  
    end qS#G7~ur>y  
    _;B wP  
    if any(m<0) -T,?'J0 2  
        error('zernpol:Mpositive','All M must be positive.') 9a=Ll]=\  
    end nd]SI;<  
    R3~,&ab  
    if any(m>n) C< 9x\JY%  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 8@;]@c)m  
    end g%&E~V/g$  
    se\fbe^0  
    if any( r>1 | r<0 ) N[p o)}hp  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') D`T;j[SsS#  
    end SI=$s>1  
    !l9{R8m>eJ  
    if ~any(size(r)==1) ^+SE_-+]  
        error('zernpol:Rvector','R must be a vector.') Z^_qXerjP  
    end !&{rnK  
    ZeP=}0TGjn  
    r = r(:); s0/O/G?  
    length_r = length(r); uZC=]Ieh  
    v>_@D@pr  
    if nargin==4 H)Btm  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); UGSZg|&6#*  
        if ~isnorm &"^F;z/  
            error('zernpol:normalization','Unrecognized normalization flag.') hZx&j{  
        end 8M99cx*K  
    else L}`/v]E"eU  
        isnorm = false; t(u2%R4<d  
    end B?! L~J@p  
    U?UU] >Q  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Wef%f] u  
    % Compute the Zernike Polynomials J!2Z9<q5  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $fmTa02q>  
    e$Ksn_wEq  
    % Determine the required powers of r: 4j#y?^s  
    % ----------------------------------- vqSpF6F q  
    rpowers = []; BpZ~6WtBq  
    for j = 1:length(n) ?{ N,&d  
        rpowers = [rpowers m(j):2:n(j)]; ./#YUIC  
    end l~i?  
    rpowers = unique(rpowers); pey=zR!  
    *'>_XX  
    % Pre-compute the values of r raised to the required powers, 7?*+,Fo#  
    % and compile them in a matrix: eFG/!b<17  
    % ----------------------------- 2? qC8eC  
    if rpowers(1)==0 YzESV Th  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); /65YHXg,  
        rpowern = cat(2,rpowern{:}); <tD,Uu{P  
        rpowern = [ones(length_r,1) rpowern]; gXxi; g  
    else Y4rxnXGw  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); BU:;;iV8  
        rpowern = cat(2,rpowern{:}); /eV)5`V  
    end !*-|!Vz  
    qYK^S4L  
    % Compute the values of the polynomials: 5#GMp  
    % -------------------------------------- #`VAw ) eV  
    z = zeros(length_r,length_n); ]SQ+r*a  
    for j = 1:length_n K!6T8^JH  
        s = 0:(n(j)-m(j))/2; B[N]=V  
        pows = n(j):-2:m(j); M~A# _%2U  
        for k = length(s):-1:1 q`9.@u@a  
            p = (1-2*mod(s(k),2))* ... _4by3?<c  
                       prod(2:(n(j)-s(k)))/          ... B!wN%> U  
                       prod(2:s(k))/                 ... tu\XuDk y  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... B4y_{V  
                       prod(2:((n(j)+m(j))/2-s(k))); 2FMmANH0ev  
            idx = (pows(k)==rpowers); 3"h*L8No  
            z(:,j) = z(:,j) + p*rpowern(:,idx); &<t%u[3  
        end t]h_w7!U  
         I~&*^q6 |  
        if isnorm K(+=V)'Dz  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); A (2 0+  
        end W Z'<iI  
    end ]64Pk9z=  
    L1SX2F8  
    % 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)  -J=6)  
    %7 -(c  
    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。 pq]z%\$u  
    NA$)qX_  
    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)