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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 #4lHaFq  
    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! `U!(cDY  
     
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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 3HXh6( e  
    function z = zernfun(n,m,r,theta,nflag) YHJ'  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. B6-AIPb  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N FyQOa)5  
    %   and angular frequency M, evaluated at positions (R,THETA) on the G &m>Ov$#&  
    %   unit circle.  N is a vector of positive integers (including 0), and \]Kq(k[p  
    %   M is a vector with the same number of elements as N.  Each element Z=0iPy,m>  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) "MW55OWYU  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, //VG1@vaVX  
    %   and THETA is a vector of angles.  R and THETA must have the same (69kvA&|q  
    %   length.  The output Z is a matrix with one column for every (N,M) M_yZR^;^-  
    %   pair, and one row for every (R,THETA) pair. :p,c%"8  
    % wHq('+{=&  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike hU |LFjc  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), GcPB'`!M  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ~_ (!}V  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, \?aOExG I  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized v\J!yz  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ~4l6unCI  
    % goG] WGVr  
    %   The Zernike functions are an orthogonal basis on the unit circle. r7zf+a]  
    %   They are used in disciplines such as astronomy, optics, and 9t,aT!f  
    %   optometry to describe functions on a circular domain. Vx0MG{vG1  
    % F I80vV7  
    %   The following table lists the first 15 Zernike functions. @oUf}rMiDa  
    % Fj '\v#h  
    %       n    m    Zernike function           Normalization Vjv6\;tt8  
    %       -------------------------------------------------- IO?~b XP  
    %       0    0    1                                 1 "-G.V#zI  
    %       1    1    r * cos(theta)                    2 ch%Q'DR_I)  
    %       1   -1    r * sin(theta)                    2 8f5%xY$  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) C6Um6 X9/i  
    %       2    0    (2*r^2 - 1)                    sqrt(3) rjq -ZrC%  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) O%r S;o  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) +:j4G^V  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) :FEd:0TS  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) MZgmv  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ={e#lC  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) FZ;Y vdX6  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1 8l~4"|fk  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) pP=_@ 3 D  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) U`},)$  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) C`=`Ce~|d  
    %       -------------------------------------------------- (cbB %  
    % O% j,:t'"  
    %   Example 1: rElG7[+)p  
    % P7M0Ce~iW  
    %       % Display the Zernike function Z(n=5,m=1) 7!]k#|u  
    %       x = -1:0.01:1; hfVzzVX:  
    %       [X,Y] = meshgrid(x,x); 6EW"8RG`  
    %       [theta,r] = cart2pol(X,Y); p;)klH@X  
    %       idx = r<=1; 9}7oKlyk  
    %       z = nan(size(X)); 6"#Tvj~-8  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); B)LXxdkOn  
    %       figure *GY,h$Ul  
    %       pcolor(x,x,z), shading interp y"{UN M|R  
    %       axis square, colorbar dW] Ej"W  
    %       title('Zernike function Z_5^1(r,\theta)') GLoL4el  
    % |2+c DR  
    %   Example 2: ^+YGSg7  
    % >xk:pL*o`  
    %       % Display the first 10 Zernike functions `qQQQ.K7)z  
    %       x = -1:0.01:1; 2g`uC}  
    %       [X,Y] = meshgrid(x,x); Fp* &os  
    %       [theta,r] = cart2pol(X,Y); la6e`  
    %       idx = r<=1; WoN]eO  
    %       z = nan(size(X)); eFeCS{LV+  
    %       n = [0  1  1  2  2  2  3  3  3  3]; V3. vE,  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; G!fE'B  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; [K^q: 3R  
    %       y = zernfun(n,m,r(idx),theta(idx)); Nc^b8& 2J  
    %       figure('Units','normalized') ]MBJ"1F  
    %       for k = 1:10 G/^5P5y%@  
    %           z(idx) = y(:,k); <{P^W;N7  
    %           subplot(4,7,Nplot(k)) et7T)(k0  
    %           pcolor(x,x,z), shading interp t2U]CI%  
    %           set(gca,'XTick',[],'YTick',[]) D(2kb  
    %           axis square NC#kI3{  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ^U|CNB%.  
    %       end m78MWz]Yo  
    % knj,[7uh  
    %   See also ZERNPOL, ZERNFUN2. c"_H%x<[  
    aF_ZV bS  
    %   Paul Fricker 11/13/2006 KfN`ZZ<  
    R&d_ WB4w  
    s`7 _J9  
    % Check and prepare the inputs: pu m9x)y1  
    % ----------------------------- 7Ohu$5\  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) &Cn9 k3E\R  
        error('zernfun:NMvectors','N and M must be vectors.') 2+hfbFu,1  
    end Hr64M0V3B  
    }][|]/s?42  
    if length(n)~=length(m) Toa#>Z*+Rb  
        error('zernfun:NMlength','N and M must be the same length.') DdA}A>47  
    end 0zk T8'v  
    -^NAHE$bW  
    n = n(:); q2"'W|I  
    m = m(:); "Ezr-4  
    if any(mod(n-m,2)) "=0 lcb C  
        error('zernfun:NMmultiplesof2', ... 9 h{:!  
              'All N and M must differ by multiples of 2 (including 0).') +xu/RY_  
    end E;+OD&|  
    #+5mpDh  
    if any(m>n) ]idD&5gd  
        error('zernfun:MlessthanN', ...  z]R!l%`  
              'Each M must be less than or equal to its corresponding N.') 6d?2{_},  
    end bm]dz;ljh  
    hSf#;=9'  
    if any( r>1 | r<0 ) @=| b$E  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') %A Du[M.  
    end fgz'C?  
    2$/gg"g+  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) h,RUL  
        error('zernfun:RTHvector','R and THETA must be vectors.') (YWc%f4  
    end X +  
    Gxt<kz  
    r = r(:); x;b+gIz*  
    theta = theta(:); 88L bO(q\d  
    length_r = length(r); u:>3j,Cs  
    if length_r~=length(theta) ^# g;"K0  
        error('zernfun:RTHlength', ... lDM~Z3(/b  
              'The number of R- and THETA-values must be equal.') WoT z'  
    end XQoT},C  
    UK9MWC5g9  
    % Check normalization: # ;KG6IE  
    % -------------------- &+|4(d1  
    if nargin==5 && ischar(nflag) 6}FDLBA  
        isnorm = strcmpi(nflag,'norm'); 2ZIY{lBe  
        if ~isnorm W;9X*I8f8  
            error('zernfun:normalization','Unrecognized normalization flag.') 7)8}8tY^{  
        end X;a{JjN  
    else 4Xho0lO&  
        isnorm = false; #YMp,i  
    end GP k Cgb(  
    vCe<-k  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <("w'd}  
    % Compute the Zernike Polynomials L5P}%1 _  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mZJzBYM)  
    B*?PB]  
    % Determine the required powers of r: 2A;[Ek6{q  
    % ----------------------------------- u z2s-,  
    m_abs = abs(m); 7%x+7  
    rpowers = []; uM6!RR!~  
    for j = 1:length(n)  V# %spW  
        rpowers = [rpowers m_abs(j):2:n(j)]; 'ah0IYe  
    end >u[1v  
    rpowers = unique(rpowers); gd,%H@3  
    93eqFCF.  
    % Pre-compute the values of r raised to the required powers, ])l[tVHm  
    % and compile them in a matrix: 2%yJo7f$[  
    % ----------------------------- 7%FZXsD  
    if rpowers(1)==0 p%y\`Nlgdx  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); t'/;Z:  
        rpowern = cat(2,rpowern{:}); e{+{,g{iu  
        rpowern = [ones(length_r,1) rpowern]; VYQbyD{V w  
    else g>-[-z$E3  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); `ha:Gf  
        rpowern = cat(2,rpowern{:}); ^W05Z!}  
    end ._nKM5.  
    IbaL.t\>  
    % Compute the values of the polynomials: WQC6{^/4[1  
    % -------------------------------------- T1di$8  
    y = zeros(length_r,length(n)); oVsazYJ|?  
    for j = 1:length(n) #E@i@'T  
        s = 0:(n(j)-m_abs(j))/2; (`Mz.VN  
        pows = n(j):-2:m_abs(j); A)\DPLAG  
        for k = length(s):-1:1 Bx!` UdRn  
            p = (1-2*mod(s(k),2))* ... Z69 IHA[  
                       prod(2:(n(j)-s(k)))/              ... m =F@CA~C  
                       prod(2:s(k))/                     ... ?7ZlX?D[  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... N6 8>`  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); vfDb9QP  
            idx = (pows(k)==rpowers); .*7UT~o=CS  
            y(:,j) = y(:,j) + p*rpowern(:,idx); WkIV  
        end ,F Vy:"FR  
         dkp[?f)x  
        if isnorm ay|{!MkQ  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); cTTE] ix]  
        end p>O< "X@  
    end nv{4 U}&P  
    % END: Compute the Zernike Polynomials 5z>\'a1U  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I@M^Wu]wW  
    ~B\:  
    % Compute the Zernike functions: 9iNns;^`q  
    % ------------------------------ OFbg]{ub?  
    idx_pos = m>0; 9v2 ;  
    idx_neg = m<0; r2'rf pQ  
    [wG%@0\  
    z = y; >MrU^t  
    if any(idx_pos) x@}Fn:c!5  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); @v=q,A8_  
    end 2H "iN[2A  
    if any(idx_neg) ~=ys~em e  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ~m U_ `o  
    end elB 8   
    W fNMyI  
    % 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) uJ[Vv4N%9  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. V/e_:xECC  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated AgJ~6tK  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Am  $L  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, +Bfi/>  
    %   and THETA is a vector of angles.  R and THETA must have the same "M &4c:cz  
    %   length.  The output Z is a matrix with one column for every P-value, a6P.Zf7  
    %   and one row for every (R,THETA) pair. fk1f'M)/8  
    % -~fI|A^  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ,[ L$  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) q04Dj-2<  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) -+_&#twU  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 3PffQ,c[~  
    %   for all p. p\ S3A(  
    % )7J>:9h  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 nDy=ZsK  
    %   Zernike functions (order N<=7).  In some disciplines it is 7!;/w;C  
    %   traditional to label the first 36 functions using a single mode -+|[0hpw  
    %   number P instead of separate numbers for the order N and azimuthal Kf~+jYobO  
    %   frequency M. |vzWSm  
    % >oDP(]YGg  
    %   Example: A!yLwkc:5  
    % 'bPo 5V|  
    %       % Display the first 16 Zernike functions k)Wz b  
    %       x = -1:0.01:1; 'O9=*L) X  
    %       [X,Y] = meshgrid(x,x); d 4R+gIA  
    %       [theta,r] = cart2pol(X,Y); G|_aU8b|t  
    %       idx = r<=1; 3~rc=e  
    %       p = 0:15; 1A-EP@# J  
    %       z = nan(size(X)); &y\2:IyA  
    %       y = zernfun2(p,r(idx),theta(idx)); DU8LU*q'  
    %       figure('Units','normalized') "~ stZ.  
    %       for k = 1:length(p) ~7'.{VrU  
    %           z(idx) = y(:,k); H_nJST<v`  
    %           subplot(4,4,k) MDt?7c  
    %           pcolor(x,x,z), shading interp o#/iR]3  
    %           set(gca,'XTick',[],'YTick',[]) sb.SpF>   
    %           axis square d.o FlT  
    %           title(['Z_{' num2str(p(k)) '}'])  ?Nql7F4  
    %       end 3>v0W@C  
    % !H\GHA'DO]  
    %   See also ZERNPOL, ZERNFUN. 38i,\@p`9$  
    ped Yf{T  
    %   Paul Fricker 11/13/2006 QPE.b-S  
    tC-KW~&  
    k|'Mh0G0  
    % Check and prepare the inputs: [)vwg`]   
    % ----------------------------- ~1sl.8tF  
    if min(size(p))~=1 *?Ef}:]  
        error('zernfun2:Pvector','Input P must be vector.') RQNi&zX/  
    end % 6.jh#C  
    rF3]AW(  
    if any(p)>35 1Z8oN3  
        error('zernfun2:P36', ... S'p`ECfVMA  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... -$ z"74  
               '(P = 0 to 35).']) LfXr(2u  
    end T?{9Z  
    o{W]mr3D  
    % Get the order and frequency corresonding to the function number: n ]}2O 4j  
    % ---------------------------------------------------------------- /+O8A}  
    p = p(:); N~_jiVD>  
    n = ceil((-3+sqrt(9+8*p))/2); 1[9j`~[([  
    m = 2*p - n.*(n+2); Nj&%xe>].  
    ld:alEo  
    % Pass the inputs to the function ZERNFUN: z ]N~_9w  
    % ---------------------------------------- KXCmCn  
    switch nargin K/ m)f#  
        case 3 pu*u[n  
            z = zernfun(n,m,r,theta); kA=~ 8N  
        case 4 E?U]w0g  
            z = zernfun(n,m,r,theta,nflag); 0.+eF }'H  
        otherwise fO!O" D5  
            error('zernfun2:nargin','Incorrect number of inputs.') ]GKx[F{)  
    end ~c$ts&Cl  
    [j U  
    % 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) mX.3R+t  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Zbh]SF{3F  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of fB,1s}3Hn  
    %   order N and frequency M, evaluated at R.  N is a vector of yx w27~  
    %   positive integers (including 0), and M is a vector with the $"{3yLg  
    %   same number of elements as N.  Each element k of M must be a B~g05`s  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) #Y>%Dr&  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 'Mx K}9  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix R:BBNzY}f  
    %   with one column for every (N,M) pair, and one row for every KSB_%OI1  
    %   element in R. 'S4EKV]  
    % rspoSPnY1  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- >dvWa-rNUT  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is &DQ4=/Z  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to K#f`_SCW  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 +[8Kl=]L  
    %   for all [n,m]. K[>@'P}y  
    % xD= qU  
    %   The radial Zernike polynomials are the radial portion of the X$|TN+Ub  
    %   Zernike functions, which are an orthogonal basis on the unit 5ZyBP~  
    %   circle.  The series representation of the radial Zernike 26#Jhb E+  
    %   polynomials is 6SBvn%  
    % <_a70"i  
    %          (n-m)/2 H;*a:tbxO+  
    %            __ M?~<w)L}  
    %    m      \       s                                          n-2s hp]ng!I{\u  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r <+-Yh_D  
    %    n      s=0 ,rB9esxic  
    % [[0bhmG)  
    %   The following table shows the first 12 polynomials. k4F"UG-`  
    % U|Z>SE<k  
    %       n    m    Zernike polynomial    Normalization =Kt9,d08x  
    %       --------------------------------------------- /q"d`!h)w  
    %       0    0    1                        sqrt(2) ,D@ ;i  
    %       1    1    r                           2 V)1:LLRW  
    %       2    0    2*r^2 - 1                sqrt(6) ,8=`*  
    %       2    2    r^2                      sqrt(6) Q),3&4pM  
    %       3    1    3*r^3 - 2*r              sqrt(8) cR=94i=t  
    %       3    3    r^3                      sqrt(8) ]oas  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ,v}?{p c  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 0ve`  
    %       4    4    r^4                      sqrt(10) ,P@/=I5  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) >)n4s Mq  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) #mRFUA  
    %       5    5    r^5                      sqrt(12) .qIy7_^  
    %       --------------------------------------------- ~C"k$;(n  
    % c.8((h/  
    %   Example: :(l $^ M  
    % Y1fy2\<'  
    %       % Display three example Zernike radial polynomials b$goF }b'g  
    %       r = 0:0.01:1; j FPU zB"  
    %       n = [3 2 5]; oGJ*Rn)Z  
    %       m = [1 2 1]; T}t E/  
    %       z = zernpol(n,m,r); =CKuiO.j  
    %       figure '6o`^u>  
    %       plot(r,z) 1qLl^DW  
    %       grid on i+)}aA  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') [*9YIjn  
    % !]rETP_  
    %   See also ZERNFUN, ZERNFUN2. :>P4L,Da]  
    U R1JbyT  
    % A note on the algorithm. S$jV|xK B  
    % ------------------------ r:c@17  
    % The radial Zernike polynomials are computed using the series fou_/Nrue  
    % representation shown in the Help section above. For many special <Qcex3  
    % functions, direct evaluation using the series representation can RGl=7^M  
    % produce poor numerical results (floating point errors), because b46[fa   
    % the summation often involves computing small differences between ~_ u*\]-  
    % large successive terms in the series. (In such cases, the functions "?.'{,Q  
    % are often evaluated using alternative methods such as recurrence c Pq Dsl3  
    % relations: see the Legendre functions, for example). For the Zernike \LdmGv@ &  
    % polynomials, however, this problem does not arise, because the cBLR#Yu;O5  
    % polynomials are evaluated over the finite domain r = (0,1), and m"gni #  
    % because the coefficients for a given polynomial are generally all s&dO/}3uR]  
    % of similar magnitude. ^)f{q)to  
    % ~!]&>n;=G  
    % ZERNPOL has been written using a vectorized implementation: multiple _{LN{iqDv  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] G$;] ?g  
    % values can be passed as inputs) for a vector of points R.  To achieve KE/-VjZu  
    % this vectorization most efficiently, the algorithm in ZERNPOL ~A`&/U  
    % involves pre-determining all the powers p of R that are required to 9Fy\t{ks  
    % compute the outputs, and then compiling the {R^p} into a single nT.L}1@  
    % matrix.  This avoids any redundant computation of the R^p, and gppBFS  
    % minimizes the sizes of certain intermediate variables. 1R=)17'O  
    % =tr1*s{  
    %   Paul Fricker 11/13/2006 `z|= ~  
    bZNIxkc[Dh  
    {OB-J\7Y  
    % Check and prepare the inputs: Em e'Gk  
    % ----------------------------- jM5_8nS&d  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 4S,.R  
        error('zernpol:NMvectors','N and M must be vectors.') r]A" Og_U  
    end lLuID  
    uY^v"cw/F  
    if length(n)~=length(m) L`9TB"0R+  
        error('zernpol:NMlength','N and M must be the same length.') -VS9`7k  
    end dB@Wn!Y  
    #yW.o'S+  
    n = n(:); ([:]T$0 #  
    m = m(:); qbS'|--wH  
    length_n = length(n); v5(q) h  
    ;i<$7MR.e  
    if any(mod(n-m,2)) g%`i=s&N%  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ecr886  
    end bTZ>@~$  
    ^"3\iA:  
    if any(m<0) ;~ W8v.EW  
        error('zernpol:Mpositive','All M must be positive.') Ho 3dsh)  
    end 0B=[80K;8  
    \Sg<='/{L;  
    if any(m>n) ;wJ~haC  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ePf+[pV3  
    end exfm q  
    W7H&R,  
    if any( r>1 | r<0 ) V,V*30K5  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') q`XW5VV{K  
    end <&4nOt  
    <0CzB"Ap  
    if ~any(size(r)==1) h}<0/  
        error('zernpol:Rvector','R must be a vector.') 3pvYi<<D'  
    end ]b3/Es+  
    >A-<ZS*N  
    r = r(:); 6gXIt9B.h$  
    length_r = length(r); $tI]rU  
    Y4d3n  
    if nargin==4 >D 97c|?c  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); g3Z:{@m  
        if ~isnorm wZ#Rlv,3Wa  
            error('zernpol:normalization','Unrecognized normalization flag.') #Mh{<gk%ax  
        end Ab/j(xr=  
    else 1%%'6cWWu  
        isnorm = false; O7%2v@j|8  
    end  3P1&;  
    F8H'^3`b`U  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% oBr.S_Qe  
    % Compute the Zernike Polynomials #O"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9Gca6e3  
    iZaeoy  
    % Determine the required powers of r: S=' wJ@?;  
    % ----------------------------------- :- ?Ct  
    rpowers = []; oK2pM18  
    for j = 1:length(n) -T7%dLHY  
        rpowers = [rpowers m(j):2:n(j)]; ;6ky5}z  
    end -D^L}b  
    rpowers = unique(rpowers); =VNSi K>F  
    'Gjq/L/x  
    % Pre-compute the values of r raised to the required powers, 'n0 .#E_  
    % and compile them in a matrix: 1"}cdq.  
    % ----------------------------- Wqra8u#  
    if rpowers(1)==0 9Y/L?km_(  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); in<}fAro6  
        rpowern = cat(2,rpowern{:}); 5!Bktgk.  
        rpowern = [ones(length_r,1) rpowern]; 5o#Yt  
    else _d@=nK)  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Y>B P?l  
        rpowern = cat(2,rpowern{:}); JWROYED  
    end {^5?)/<  
    #]9hTa IR  
    % Compute the values of the polynomials: Gih[i\%Q  
    % -------------------------------------- -I":Z2.fR  
    z = zeros(length_r,length_n); 6 {}JbRNf  
    for j = 1:length_n Y#FO5O%W  
        s = 0:(n(j)-m(j))/2; ubYG  
        pows = n(j):-2:m(j); N L'R\R  
        for k = length(s):-1:1 M"{uX  
            p = (1-2*mod(s(k),2))* ... oE?QnH3R  
                       prod(2:(n(j)-s(k)))/          ... Z)pz,  
                       prod(2:s(k))/                 ... 09S6#;N&  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... e}0:"R%E  
                       prod(2:((n(j)+m(j))/2-s(k))); aE|OTm+@9;  
            idx = (pows(k)==rpowers); vMla'5|l  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Ue*C>F   
        end |Ps% M|8~  
         $Z?\>K0i  
        if isnorm ar.AL'  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); W2Luz;(U  
        end ?m0IehI  
    end 5\Fz!  
    9b;A1gu  
    % 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)  ;z M*bWh9  
    R8rfM?"W  
    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。 f7h*Vu`>  
    mWR4|1(  
    07年就写过这方面的计算程序了。
    让光学不再神秘,让光学变得容易,快速实现客户关于光学的设想与愿望。