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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 r{~K8!=oU]  
    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! m<ZwbD  
     
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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 ff"Cl p  
    function z = zernfun(n,m,r,theta,nflag) ;2RCgX!'%  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. tZ1iaYbvV  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 9s)YPlDz  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Dbr(Wg  
    %   unit circle.  N is a vector of positive integers (including 0), and lkp!S3,  
    %   M is a vector with the same number of elements as N.  Each element kl[bDb1p  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ?Gr<9e2Eo  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, m^_)aS  
    %   and THETA is a vector of angles.  R and THETA must have the same )|/t}|DIx  
    %   length.  The output Z is a matrix with one column for every (N,M) ))63?_  
    %   pair, and one row for every (R,THETA) pair. =Fea vyx  
    % 5}e-~-  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike GpF,=:  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), C78d29  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral e*vSGT$KgL  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ihH!"HH+  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized GMOv$Tn-_L  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m].  ]\P  
    % [AR$Sw60  
    %   The Zernike functions are an orthogonal basis on the unit circle. ?A Y596  
    %   They are used in disciplines such as astronomy, optics, and V&KH{j/P  
    %   optometry to describe functions on a circular domain. ,=>O/!s  
    % =fBJQK2sk  
    %   The following table lists the first 15 Zernike functions. C%#C|X193  
    % ]8YHA}P  
    %       n    m    Zernike function           Normalization >T~{_|N  
    %       -------------------------------------------------- ~C=`yj  
    %       0    0    1                                 1 c#9 zw[y-L  
    %       1    1    r * cos(theta)                    2 r3ZY` zf  
    %       1   -1    r * sin(theta)                    2 Q}]:lmqH  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) r3Z-mJ$:  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Ltcr]T(Ic  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) @tjC{?5Y  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) CNcH)2Mk  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) SVXey?A;CJ  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ZH:#~Zyj  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 6@o_MtI  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) $yaE!.Kc  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) snj4MA@I]  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) y9\s[}c_  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) U$VTk  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) L6$,<}l  
    %       -------------------------------------------------- oB9Fas!N  
    % 2T?t[;-  
    %   Example 1: Q;r 0#"  
    % */\dH<  
    %       % Display the Zernike function Z(n=5,m=1) v-G(bw3  
    %       x = -1:0.01:1; 9FV#@uA}D  
    %       [X,Y] = meshgrid(x,x); w/G5I )G  
    %       [theta,r] = cart2pol(X,Y); pS%,wjb&P  
    %       idx = r<=1; 4KybN  
    %       z = nan(size(X)); O";r\Z  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); "cJ5Fd:*  
    %       figure shn`>=0.&  
    %       pcolor(x,x,z), shading interp .M:,pw"S]  
    %       axis square, colorbar W,Dr2$V  
    %       title('Zernike function Z_5^1(r,\theta)') aKCCFHq t!  
    % w #(XiH*  
    %   Example 2: E pM 4 +  
    % WHAEB1c#Q  
    %       % Display the first 10 Zernike functions ^@X =v`C  
    %       x = -1:0.01:1; nk-6W4  
    %       [X,Y] = meshgrid(x,x); 9M01}  
    %       [theta,r] = cart2pol(X,Y); NqqLRgMOR'  
    %       idx = r<=1; V=(4 c  
    %       z = nan(size(X)); Bfw>2  
    %       n = [0  1  1  2  2  2  3  3  3  3]; oF ,8j1  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; D"1ciO8^I]  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; %_tL}m{?  
    %       y = zernfun(n,m,r(idx),theta(idx)); >y]YF3?  
    %       figure('Units','normalized') )J#@L*  
    %       for k = 1:10 RFA5vCG  
    %           z(idx) = y(:,k); *QLl jGe  
    %           subplot(4,7,Nplot(k)) \UB<'~z6!  
    %           pcolor(x,x,z), shading interp L**!$k"{5  
    %           set(gca,'XTick',[],'YTick',[]) Fd'Ang6"  
    %           axis square &5d>jEaB}  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) U?|s/U  
    %       end >Ckb9A  
    % )_bXKYUX*0  
    %   See also ZERNPOL, ZERNFUN2. TS3 00F  
    <j,7Z>Rk\x  
    %   Paul Fricker 11/13/2006 %8{' XJ!  
    $g|g}>Sc  
    /h2`?~k+  
    % Check and prepare the inputs: kt;X|`V{5z  
    % ----------------------------- )SDGj;j+  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) )XO2DY1/&  
        error('zernfun:NMvectors','N and M must be vectors.') $h_@`j  
    end g>f(5  
    VCc4nn#  
    if length(n)~=length(m) Mu:*(P/  
        error('zernfun:NMlength','N and M must be the same length.') };zF&  
    end 7?hC t  
    e[e2X<&0RT  
    n = n(:); @&M $`b ^  
    m = m(:); g]d"d  
    if any(mod(n-m,2)) L YH9P-5H  
        error('zernfun:NMmultiplesof2', ... * rs_k/2(  
              'All N and M must differ by multiples of 2 (including 0).') 'Y"q=@Ei9  
    end `C!Pe84(  
    o-)E_X  
    if any(m>n) Z.R^@@RqJ  
        error('zernfun:MlessthanN', ... "sHD8TUX  
              'Each M must be less than or equal to its corresponding N.') {h@R\bU  
    end 6(ja5)sn*  
    D %5 0  
    if any( r>1 | r<0 ) |`O7> (h  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') sHEISNj/^  
    end c8}1-MKs_R  
     d;CD~s  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) #vS>^OyP  
        error('zernfun:RTHvector','R and THETA must be vectors.') fwl RwH(  
    end zSq+#O1#  
    9'4cqR  
    r = r(:); aX  ?ON  
    theta = theta(:); ul% q6=f)  
    length_r = length(r); X$st{@}ZB  
    if length_r~=length(theta) wL%>  
        error('zernfun:RTHlength', ... m*I5 \  
              'The number of R- and THETA-values must be equal.') }QC: !e,yG  
    end 1P[!B[;c  
    4`*jF'N[  
    % Check normalization: * |,V$  
    % -------------------- wPG3Ap8L  
    if nargin==5 && ischar(nflag) q+m&V#FT%  
        isnorm = strcmpi(nflag,'norm'); 8"S0E(,mu  
        if ~isnorm 7tt&/k?Q  
            error('zernfun:normalization','Unrecognized normalization flag.') *?i~AXJm  
        end 9h9Y:i*Gh5  
    else xwz2N5  
        isnorm = false; lFRgyEPH  
    end hy6px  
    -EL"Sv?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% thq(tK7  
    % Compute the Zernike Polynomials :z^c<KFX  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r\em-%:  
    s=KA(4p  
    % Determine the required powers of r: F!Nx^M1  
    % ----------------------------------- &Wn!W  
    m_abs = abs(m); U :IQWlC  
    rpowers = []; +i K.+B  
    for j = 1:length(n) Z?^AX&F  
        rpowers = [rpowers m_abs(j):2:n(j)]; UDxfS4yI  
    end e+&/ Tq'2  
    rpowers = unique(rpowers); r?[Zf2&  
    XfY]qQP  
    % Pre-compute the values of r raised to the required powers, ]i{-@Ven  
    % and compile them in a matrix: $osDw1C  
    % ----------------------------- t4 aa5@r  
    if rpowers(1)==0 , {]>U'-  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ="XxS|Mq3  
        rpowern = cat(2,rpowern{:}); ==Y^~ab;K  
        rpowern = [ones(length_r,1) rpowern]; rVZk G,Q  
    else &}*[-z  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); PY) 74sa  
        rpowern = cat(2,rpowern{:}); )C <sj   
    end EpPKo  
    [dUW3}APV  
    % Compute the values of the polynomials: kkh#VGh"  
    % -------------------------------------- FVHEb\Z  
    y = zeros(length_r,length(n)); )2:d8J\  
    for j = 1:length(n) sx|=*j,_  
        s = 0:(n(j)-m_abs(j))/2; ,.DU)Wi?}  
        pows = n(j):-2:m_abs(j); t*n!kXa  
        for k = length(s):-1:1 Wny{qj)=  
            p = (1-2*mod(s(k),2))* ... V<(cW'zA/  
                       prod(2:(n(j)-s(k)))/              ... Z(CzU{7c  
                       prod(2:s(k))/                     ... :5p`H  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... bY]aADv\  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); KZ&8aulP  
            idx = (pows(k)==rpowers); ^F_c'  
            y(:,j) = y(:,j) + p*rpowern(:,idx); d+z8^$z"  
        end * y u|]T  
         X(N!y"z  
        if isnorm OBu$T&  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); i]YH"t8GY  
        end @_0XK)pW  
    end UDGVq S!,E  
    % END: Compute the Zernike Polynomials 4fp}`U  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0(HUy`]>  
    Sh=z  
    % Compute the Zernike functions: j#.Aiy:,  
    % ------------------------------ 3-z57f,}6~  
    idx_pos = m>0; /2 WGo-  
    idx_neg = m<0; UG 9uNgzQ/  
    l2z@t3{  
    z = y; }zj_Pp  
    if any(idx_pos) Un@dWf6'  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 5_0Eh!sx  
    end THkg,*;:  
    if any(idx_neg) qy/xJ>:  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); kp LDK81I  
    end +<&_1% 5+  
    XeJn,=  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) e?,n>  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. BF b<"!Y  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Nu/wjx$b  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive d)1 d0ES  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ,` 6O{Z~  
    %   and THETA is a vector of angles.  R and THETA must have the same +DU^"q=  
    %   length.  The output Z is a matrix with one column for every P-value, o#fr5>h-w  
    %   and one row for every (R,THETA) pair. s'b 4Me  
    % gF# HNv  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike %68'+qz  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) C.].HQ  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) |Bz1u|uc  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ,)@njC?J  
    %   for all p. -n FKP&P  
    % m Q4(<,F  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 >J*x` a3Q  
    %   Zernike functions (order N<=7).  In some disciplines it is d<K2 \:P{}  
    %   traditional to label the first 36 functions using a single mode ~@=(#tO.  
    %   number P instead of separate numbers for the order N and azimuthal > ~:Md  
    %   frequency M. & %A&&XT9  
    % Xdh2  
    %   Example: 2*Zk^h=  
    % p>_Qns7W  
    %       % Display the first 16 Zernike functions & OYo  
    %       x = -1:0.01:1; K*I!:1;3N  
    %       [X,Y] = meshgrid(x,x); mX 3p   
    %       [theta,r] = cart2pol(X,Y); Znr@-=xZO*  
    %       idx = r<=1; +`,;tz=?  
    %       p = 0:15; :^7>kJ5?  
    %       z = nan(size(X)); hh8Grl;  
    %       y = zernfun2(p,r(idx),theta(idx)); xP4}LL9)  
    %       figure('Units','normalized') P q0 %oz  
    %       for k = 1:length(p) |;:g7eb  
    %           z(idx) = y(:,k); d|?Xo\+  
    %           subplot(4,4,k) I+=+ ,iXhB  
    %           pcolor(x,x,z), shading interp Ps!umV  
    %           set(gca,'XTick',[],'YTick',[]) $vrkxn  
    %           axis square 5skxixG  
    %           title(['Z_{' num2str(p(k)) '}']) 0vETg'r  
    %       end 3xg9D.A  
    % is }>+&_  
    %   See also ZERNPOL, ZERNFUN. ijsoY\V50  
    @A4$k dJ2  
    %   Paul Fricker 11/13/2006 .7]P-]uOZ  
    3xT9/8*  
    rk6K0TQ8  
    % Check and prepare the inputs: <d @9[]  
    % ----------------------------- /~M H]Gh  
    if min(size(p))~=1 N=AHS  
        error('zernfun2:Pvector','Input P must be vector.') 2n)?)w]!M  
    end KL3Z(  
    h PL]B_<  
    if any(p)>35 C];P yQS  
        error('zernfun2:P36', ... >`r3@|UY  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Z\?2"4H  
               '(P = 0 to 35).']) AtewC Yo  
    end u\V^g   
    lD[37U!  
    % Get the order and frequency corresonding to the function number: P#O2MiG  
    % ---------------------------------------------------------------- H4s~=iB  
    p = p(:); 3 P9ux  
    n = ceil((-3+sqrt(9+8*p))/2); DmpT<SI+!  
    m = 2*p - n.*(n+2); #=t/wAE y:  
    Q_U.J0  
    % Pass the inputs to the function ZERNFUN: y{S8?$dU$:  
    % ---------------------------------------- "$XX4w M  
    switch nargin RWc<CQcL"  
        case 3 -QroT`gy  
            z = zernfun(n,m,r,theta); H T|DT  
        case 4 I]^>>>p$  
            z = zernfun(n,m,r,theta,nflag); gs5(~YiT6  
        otherwise =A.$~9P  
            error('zernfun2:nargin','Incorrect number of inputs.') 8LbwEKl  
    end 8 q@Z  
    !bP%\)5  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) GC{M"q|_  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. F!]Sr'UA  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of {drc}BL_  
    %   order N and frequency M, evaluated at R.  N is a vector of Ho>Np&  
    %   positive integers (including 0), and M is a vector with the (k?H T'3)  
    %   same number of elements as N.  Each element k of M must be a );$99t  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) "j@\a)a  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Ax;=Zh<DAv  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix :OG I|[  
    %   with one column for every (N,M) pair, and one row for every c-sjYJXKM*  
    %   element in R. U[@y 8yN6M  
    % Y()" 2CCV  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 1^!SuAA@  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is -QrC>3xZR  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to p49]{2GXb  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 QO2cTk m  
    %   for all [n,m]. [={mCGU  
    % Z!)~?<gcq:  
    %   The radial Zernike polynomials are the radial portion of the 5~L]zE  
    %   Zernike functions, which are an orthogonal basis on the unit s/E|Z1pg3  
    %   circle.  The series representation of the radial Zernike uu9M}]mDl  
    %   polynomials is <kak9 6A  
    % ;-+q*@sa]  
    %          (n-m)/2 X\AH^I6S  
    %            __ ,#K/+T  
    %    m      \       s                                          n-2s A&Y5z[p  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Zh=a rlk  
    %    n      s=0 ^[M{s(b  
    % 7?$?Yu  
    %   The following table shows the first 12 polynomials. \$ytmtf5  
    % F5h/>  
    %       n    m    Zernike polynomial    Normalization i[v4[C=WB!  
    %       --------------------------------------------- [nTI\17iA  
    %       0    0    1                        sqrt(2) =p+y$  
    %       1    1    r                           2 p+VU:%.t  
    %       2    0    2*r^2 - 1                sqrt(6) Mo,&h?VOM?  
    %       2    2    r^2                      sqrt(6) ZJYn[\]  
    %       3    1    3*r^3 - 2*r              sqrt(8) =V^-@ji)b  
    %       3    3    r^3                      sqrt(8) 9)Fx;GxL  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) CMa6':~  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 2 !s&|lI  
    %       4    4    r^4                      sqrt(10) |$RNY``J  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) M/zO|-j&  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Zf'*pp T&q  
    %       5    5    r^5                      sqrt(12) IH]9%d)  
    %       --------------------------------------------- *'%V}R[>  
    % %FO{:@CH  
    %   Example: (l{vlFWd  
    % TNX9Z)=>g  
    %       % Display three example Zernike radial polynomials b)LT[>f  
    %       r = 0:0.01:1; !"rPSGK*  
    %       n = [3 2 5]; #B `?}a=  
    %       m = [1 2 1]; =!q% 1mP  
    %       z = zernpol(n,m,r); w!.@64-  
    %       figure al2t\Iq90  
    %       plot(r,z) bSB%hFp=Cp  
    %       grid on }qNc `8h  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') }E'0vf /  
    % fab'\|Y   
    %   See also ZERNFUN, ZERNFUN2. =BJe)!b  
    O$Rz/&  
    % A note on the algorithm. 3/G^V'Yu  
    % ------------------------ ~zVxprEf_  
    % The radial Zernike polynomials are computed using the series 7=!9kk0  
    % representation shown in the Help section above. For many special $#Pxf  
    % functions, direct evaluation using the series representation can x9_mlZ  
    % produce poor numerical results (floating point errors), because AK:cDKBO  
    % the summation often involves computing small differences between U7r8FLl  
    % large successive terms in the series. (In such cases, the functions hXW` n*Zw  
    % are often evaluated using alternative methods such as recurrence /:{%X(8  
    % relations: see the Legendre functions, for example). For the Zernike swKkY`g  
    % polynomials, however, this problem does not arise, because the *rxr:y#Ve  
    % polynomials are evaluated over the finite domain r = (0,1), and gFDnt  
    % because the coefficients for a given polynomial are generally all N]<(cG&p  
    % of similar magnitude. S@qp_!  
    % :^;c(>u{  
    % ZERNPOL has been written using a vectorized implementation: multiple O9=/\Kc  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] h^M_yz-f  
    % values can be passed as inputs) for a vector of points R.  To achieve !jCgTo y  
    % this vectorization most efficiently, the algorithm in ZERNPOL |VfEp  
    % involves pre-determining all the powers p of R that are required to hW^,' m  
    % compute the outputs, and then compiling the {R^p} into a single K/Q"Z*  
    % matrix.  This avoids any redundant computation of the R^p, and (O.%Xbx3  
    % minimizes the sizes of certain intermediate variables. Cux(v8=n  
    % P3M$&::D-  
    %   Paul Fricker 11/13/2006 !Ok(mgV$/  
    |3H+b,M5  
    1+l8%G=hB  
    % Check and prepare the inputs: Dk1& <} I  
    % ----------------------------- u v%Q5O4  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) $( hT{C,K  
        error('zernpol:NMvectors','N and M must be vectors.') 0-2|(9 Kc  
    end *Gsj pNr-  
    Q&9 yrx.  
    if length(n)~=length(m) $a(-r-_Fi]  
        error('zernpol:NMlength','N and M must be the same length.') BZR{}Aj4pa  
    end .~z'm$s1o  
    E$8JrL  
    n = n(:); l_B735  
    m = m(:); fi+}hGj(r  
    length_n = length(n); j\>LJai"  
    Xn7G2Yp  
    if any(mod(n-m,2)) 7& M-^Ev  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') <izQ]\kL  
    end i"{O~[  
    h`[$ Bp  
    if any(m<0)   f XD+  
        error('zernpol:Mpositive','All M must be positive.') %#6@PQ[R.  
    end =c8}^3L~7  
     q+P@2FL  
    if any(m>n) f/Gx}x=  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.')  Rr) 5 [  
    end ]#!uke Q  
    #Z&/w.D2  
    if any( r>1 | r<0 ) [&x9<f6  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ?h<4trYcv  
    end 6].[z+  
    }nmlN  
    if ~any(size(r)==1) yR}. Xq/  
        error('zernpol:Rvector','R must be a vector.') [4)Oi-_Y>  
    end t,/ G  
    "*g+qll!5d  
    r = r(:); i'~-\F!  
    length_r = length(r); $|4@Zx4vf  
    86&M Zdv6  
    if nargin==4 !_I1=yi  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); (vXr2Z<l  
        if ~isnorm EF/d7  
            error('zernpol:normalization','Unrecognized normalization flag.') hlVye&;b8  
        end *$p*'vR  
    else J^!;$Hkd  
        isnorm = false; 5_yQI D%Sq  
    end cNll??j  
    =8FvkNr  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .i0K-B  
    % Compute the Zernike Polynomials {Jr1K,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "ra$x2|=}  
     01kRe  
    % Determine the required powers of r: #2Rz=QI  
    % ----------------------------------- UQVL)-Z  
    rpowers = []; Ee>VA_ss  
    for j = 1:length(n) H MOIUd  
        rpowers = [rpowers m(j):2:n(j)]; O\}C`CiC  
    end +Y;P*U}Qg[  
    rpowers = unique(rpowers); _3IT3mb2n  
    1{uxpYAP=  
    % Pre-compute the values of r raised to the required powers, 4.A^5J'W  
    % and compile them in a matrix: # :+Nr  
    % ----------------------------- 5epI'D  
    if rpowers(1)==0 3:nhZN/95T  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?0qVyK_1  
        rpowern = cat(2,rpowern{:}); @N'n>8Wn  
        rpowern = [ones(length_r,1) rpowern]; _[:6.oNjIe  
    else *,u3Wm|7  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); $QbJT`,mr  
        rpowern = cat(2,rpowern{:});  5"%.8P  
    end @2(u=E:^  
    uWS]l[Ga  
    % Compute the values of the polynomials: sG g458  
    % -------------------------------------- )k<cd.MX  
    z = zeros(length_r,length_n); ?G{0{ c2  
    for j = 1:length_n jCd]ENl+_  
        s = 0:(n(j)-m(j))/2; zCs34=3 D[  
        pows = n(j):-2:m(j); )@]%:m!ER  
        for k = length(s):-1:1 iSfRJ:_&6  
            p = (1-2*mod(s(k),2))* ... &1 \/B  
                       prod(2:(n(j)-s(k)))/          ... O]:9va  
                       prod(2:s(k))/                 ... dJuyJl$*  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Jv~R/qaaD  
                       prod(2:((n(j)+m(j))/2-s(k))); _|~Dj)z  
            idx = (pows(k)==rpowers); i?L=8+9f  
            z(:,j) = z(:,j) + p*rpowern(:,idx); =%_=!%  
        end w4 yrAj 2  
         T!/o^0w  
        if isnorm  lKbWQ>  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); VuLb9Kn  
        end QO@86{u#Y  
    end rfV'EjiM}  
    hs6pp/h>  
    % 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)  -$"$r ~ad  
    W8S sv  
    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。 ,=\.L_'  
    <y8oYe_!  
    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)