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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 ,mX|TI<*  
    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! I[`2MKh  
     
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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 B#?2,  
    function z = zernfun(n,m,r,theta,nflag) JZ% F  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 6}T%m?/}  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N k2uiu  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 9xQ 8`7  
    %   unit circle.  N is a vector of positive integers (including 0), and T{<@MK%],d  
    %   M is a vector with the same number of elements as N.  Each element &&}5>kg>d  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) [/Z'OV"tU  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, qZJ*J+  
    %   and THETA is a vector of angles.  R and THETA must have the same !"J#,e|  
    %   length.  The output Z is a matrix with one column for every (N,M) dn\F!  
    %   pair, and one row for every (R,THETA) pair. NoO+xLHw8  
    % 8>{W:?I  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /plUzy2Yu  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), F!&pENQ  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral , imvA5  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, L{LU@.;1  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ~J-|,ZMd  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. /HuYduGdP  
    % }#G"!/ZA0:  
    %   The Zernike functions are an orthogonal basis on the unit circle. &U~r}=  
    %   They are used in disciplines such as astronomy, optics, and uT}TSwgp  
    %   optometry to describe functions on a circular domain. T#n1@FgC  
    % vif8 {S  
    %   The following table lists the first 15 Zernike functions. kr(<Y|  
    % B^_Chj*m  
    %       n    m    Zernike function           Normalization F>QT|  
    %       -------------------------------------------------- N+M&d3H`  
    %       0    0    1                                 1 ]rg+n c3  
    %       1    1    r * cos(theta)                    2 [b.'3a++  
    %       1   -1    r * sin(theta)                    2 >I& jurU#  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) K@P`_yxN  
    %       2    0    (2*r^2 - 1)                    sqrt(3) d%lHa??/ h  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) sk ?'^6Xh  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Yv>BOK  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ^ Y7/Ow  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Ok>(>K<r  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) e:J'&r& 1  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 6 r.H8  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) V 7l{hEo3?  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 6"i{P  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) lP*  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) FGwnESCC  
    %       -------------------------------------------------- #<wpSs  
    % 9c6GYWIFt&  
    %   Example 1: A6N~UV*_  
    % '}Wu3X  
    %       % Display the Zernike function Z(n=5,m=1) |[ Ie.&)  
    %       x = -1:0.01:1; *NW QmC~  
    %       [X,Y] = meshgrid(x,x); ^.#X<8hr  
    %       [theta,r] = cart2pol(X,Y); @?Gw|bP  
    %       idx = r<=1; /S]:dDY9K  
    %       z = nan(size(X)); V5O=iMP  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); nU&NopD+*G  
    %       figure {jhmp\PN  
    %       pcolor(x,x,z), shading interp u`Z0{d  
    %       axis square, colorbar {^cF(7p  
    %       title('Zernike function Z_5^1(r,\theta)') q#99iiG1  
    % -XVEV  
    %   Example 2: wb6L? t  
    % @VC .>  
    %       % Display the first 10 Zernike functions *:\:5*SY  
    %       x = -1:0.01:1; A<SOT>m]  
    %       [X,Y] = meshgrid(x,x); a|QE *s.  
    %       [theta,r] = cart2pol(X,Y); 5wH54g j}  
    %       idx = r<=1; EmX>T>~#D  
    %       z = nan(size(X)); dP$8JI{  
    %       n = [0  1  1  2  2  2  3  3  3  3]; zb;(?!Bd#  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; y9C;T(oi;  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; QqiJun_m  
    %       y = zernfun(n,m,r(idx),theta(idx)); =.36y9Mfo  
    %       figure('Units','normalized') K`QOU-M@}  
    %       for k = 1:10 lt{lpH  
    %           z(idx) = y(:,k); Y=vVxVI\  
    %           subplot(4,7,Nplot(k))  R"U/RS  
    %           pcolor(x,x,z), shading interp XM6".eF)M  
    %           set(gca,'XTick',[],'YTick',[]) vi]r  
    %           axis square *jM_wwG  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) =db'#m{$  
    %       end C8IkpAD  
    % M{?zvq?d  
    %   See also ZERNPOL, ZERNFUN2. ,3Wb4so  
    b7B+eN ?z  
    %   Paul Fricker 11/13/2006 E X%6''ys  
    .dx 4,|6  
     0xJ7M.  
    % Check and prepare the inputs: 4q>7OB:e  
    % ----------------------------- {=UFk-$=  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) fdlvn*H  
        error('zernfun:NMvectors','N and M must be vectors.') 6'xomRpYN  
    end 5D,.^a1 A  
    #D+7TWDwNt  
    if length(n)~=length(m) -S"5{N73  
        error('zernfun:NMlength','N and M must be the same length.') @#RuSc  
    end 0b/i r2  
    I eG=J4:*  
    n = n(:); P$Z}  
    m = m(:); {5^K Xj$B  
    if any(mod(n-m,2)) nX0HT )}  
        error('zernfun:NMmultiplesof2', ... !FTNmyM~F  
              'All N and M must differ by multiples of 2 (including 0).') *GQDfs`m  
    end .VT;H1#  
    *YWk1Cwjo  
    if any(m>n) I @ 2uF-  
        error('zernfun:MlessthanN', ... ~C&*.ZR  
              'Each M must be less than or equal to its corresponding N.') aaDP9FW9e  
    end 4/S=5r}  
    Sw~(uH_l  
    if any( r>1 | r<0 ) j'K38@M:MN  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') M)&Io6>  
    end J/2j;,8D  
    U@G"`RYl  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) HcRa`Sfc]/  
        error('zernfun:RTHvector','R and THETA must be vectors.') [J^  
    end @@I7$*  
    vT|`%~Be  
    r = r(:); <5S@ORN  
    theta = theta(:); uG!:Z6%p  
    length_r = length(r); C{ EAmv'  
    if length_r~=length(theta) K]c4"JJ  
        error('zernfun:RTHlength', ... F^QQ0h]2  
              'The number of R- and THETA-values must be equal.') vw2`:]Q+  
    end ' +j<n[JLC  
    -$(Jk<  
    % Check normalization: j~;;l!({i  
    % -------------------- OS-sk!  
    if nargin==5 && ischar(nflag) MtS3p>4  
        isnorm = strcmpi(nflag,'norm'); ~ 3^='o  
        if ~isnorm T*?s@$)m4  
            error('zernfun:normalization','Unrecognized normalization flag.') kH'p\9=  
        end .N,&Uv-  
    else tF*szf|$-  
        isnorm = false; 3 iRA$C-p  
    end As~(7?]r  
    +Y}V3(w9X  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;}qhc l+  
    % Compute the Zernike Polynomials +k.%PO0np  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nd ink$  
    )KE [!ofD  
    % Determine the required powers of r: 5;\gJf  
    % ----------------------------------- 5 c5oSy+  
    m_abs = abs(m); 9T7e\<8"vC  
    rpowers = []; \\,f{?w  
    for j = 1:length(n) %@Oma  
        rpowers = [rpowers m_abs(j):2:n(j)]; \P;rES'  
    end ('O}&F1  
    rpowers = unique(rpowers); Yw'NX5#)g  
    20b<68h$:  
    % Pre-compute the values of r raised to the required powers, &gtG~mp<L  
    % and compile them in a matrix: L[g0&b%%-  
    % ----------------------------- LJFG0 W  
    if rpowers(1)==0 n(1')?"mA  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); (@r `$5D.b  
        rpowern = cat(2,rpowern{:}); #*9-d/K  
        rpowern = [ones(length_r,1) rpowern]; .B72C[' c  
    else `Out(Hn  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3*ixlO:qGk  
        rpowern = cat(2,rpowern{:}); POAw M  
    end U!(@q!>G  
    vAb^]d   
    % Compute the values of the polynomials: J-xS:Ha'l  
    % -------------------------------------- ehNzDr\s  
    y = zeros(length_r,length(n));  Es5f*P0  
    for j = 1:length(n) 7y^%7U \  
        s = 0:(n(j)-m_abs(j))/2; GOT1@.Y  
        pows = n(j):-2:m_abs(j); >&,[H:Z  
        for k = length(s):-1:1 :s={[KBP  
            p = (1-2*mod(s(k),2))* ... q[3x2sR  
                       prod(2:(n(j)-s(k)))/              ... -d+aV1n  
                       prod(2:s(k))/                     ... 5%zXAQD=<  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... C NsNZJ  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); @I`C#~  
            idx = (pows(k)==rpowers); urBc=3Rz  
            y(:,j) = y(:,j) + p*rpowern(:,idx); vb Y3;+M>  
        end 2I/xJ+  
         %" D%:   
        if isnorm 6$U]9D  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); t5B7I59  
        end <TGn=>u  
    end i;/xK=L  
    % END: Compute the Zernike Polynomials n DS}^Ba  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S! Rc|6y%  
    7c|bc6?  
    % Compute the Zernike functions: cD*}..-/4  
    % ------------------------------ k%s_0 @  
    idx_pos = m>0; =m89z}Ot  
    idx_neg = m<0; #Z+i~t{e(  
    r;BT,jiX  
    z = y; ~{hxR)x9  
    if any(idx_pos) E>b2+;Jv  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Zxr!:t7  
    end Vd^g9  
    if any(idx_neg) uvDzKMw~R  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); v ^[39*8  
    end Kt@M)#  
    $rIoHxh. y  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) {3;AwhN0H  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. fjvN$NgVs  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated +$4(zP s@  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Xq'cA9v=$J  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, !It`+0S b  
    %   and THETA is a vector of angles.  R and THETA must have the same B9p?8.[  
    %   length.  The output Z is a matrix with one column for every P-value, *I}`dC[  
    %   and one row for every (R,THETA) pair. w=b)({`M  
    % 8!.ojdyn  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike X%yO5c\l2  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) BA\/YW @  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) HhO".GA  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 J>fQNW!{  
    %   for all p. ?X@fKAj  
    % n>@oBG)!  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 }Zl&]e  
    %   Zernike functions (order N<=7).  In some disciplines it is dJ$"l|$$  
    %   traditional to label the first 36 functions using a single mode )`^p%k  
    %   number P instead of separate numbers for the order N and azimuthal [MuEoWrq(}  
    %   frequency M. OL4z%mDZi  
    % s4&^D<  
    %   Example: @lJzr3}WZ  
    % 8r3A~  
    %       % Display the first 16 Zernike functions UK9@oCIB  
    %       x = -1:0.01:1; 06jqQ-_`h  
    %       [X,Y] = meshgrid(x,x); Uj&W<'I  
    %       [theta,r] = cart2pol(X,Y); +`?Y?L^ J  
    %       idx = r<=1; KNH1#30 K  
    %       p = 0:15; (sVi\R  
    %       z = nan(size(X)); SG6sw]x  
    %       y = zernfun2(p,r(idx),theta(idx)); ^vG8#A}]  
    %       figure('Units','normalized') 9UvXC)R1  
    %       for k = 1:length(p) Mq';S^  
    %           z(idx) = y(:,k); wAnb Di{W  
    %           subplot(4,4,k) =8U&[F  
    %           pcolor(x,x,z), shading interp H'Yh2a`!o  
    %           set(gca,'XTick',[],'YTick',[])  }fp-5  
    %           axis square ,3nN[)dk  
    %           title(['Z_{' num2str(p(k)) '}']) 2<M= L1\  
    %       end 9"g6C<  
    % ?%H):r  
    %   See also ZERNPOL, ZERNFUN. iNMx"F0r  
    Tw +  
    %   Paul Fricker 11/13/2006 as J)4ema  
    +F dB '  
    odIZo|dv  
    % Check and prepare the inputs: GR\5WypoJ  
    % ----------------------------- S_~z-`;h!  
    if min(size(p))~=1 DBLO|&2!z[  
        error('zernfun2:Pvector','Input P must be vector.') .*elggM  
    end ?yh}/T\qp  
    vTv]U5%:>%  
    if any(p)>35 [s<^&WM/  
        error('zernfun2:P36', ... #-h\.#s  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Rb\6;i8R  
               '(P = 0 to 35).']) {d?$m*YR3`  
    end Qt|c1@J  
    np~~mdmRK  
    % Get the order and frequency corresonding to the function number: ;E'"Ks[GH  
    % ---------------------------------------------------------------- o/ui)U_   
    p = p(:); {[+2n]f_G  
    n = ceil((-3+sqrt(9+8*p))/2); ,^S@EDq  
    m = 2*p - n.*(n+2); '= l[;Q^Q  
    /T. KbLx~q  
    % Pass the inputs to the function ZERNFUN: ;'-olW~  
    % ---------------------------------------- Q,80Hor#J  
    switch nargin j2 !3rI  
        case 3 1T:Y0  
            z = zernfun(n,m,r,theta); iDf,e Kk$'  
        case 4 wY"Q o7  
            z = zernfun(n,m,r,theta,nflag); w9|w2UK  
        otherwise H{t_xL)k.  
            error('zernfun2:nargin','Incorrect number of inputs.') @BNEiOAZ#  
    end KM`eIw>8  
     f3UXCp  
    % 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) ++DG5`  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. pFJB'=c  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of w}CmfR  
    %   order N and frequency M, evaluated at R.  N is a vector of 1 `KN]Nt  
    %   positive integers (including 0), and M is a vector with the #Z5}2soA  
    %   same number of elements as N.  Each element k of M must be a M'!U<Y -  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) CA +uKM^"6  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Reu*Pe  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix *OyHHq|>q  
    %   with one column for every (N,M) pair, and one row for every }cN@[3v  
    %   element in R. ~.!c~fke  
    % (#;`"Yu  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- :f$xQr4Qz  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is TXZv2P9  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Yf[Qtmh]I  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 N;+[`l  
    %   for all [n,m]. v 36%Pj`  
    % mRZ :ie  
    %   The radial Zernike polynomials are the radial portion of the ]Ta N{"  
    %   Zernike functions, which are an orthogonal basis on the unit BT@r!>Nl  
    %   circle.  The series representation of the radial Zernike R-P-i0 ~  
    %   polynomials is @UdfAyL  
    % `g,8-  
    %          (n-m)/2 `ImE% r!  
    %            __ 1J' 3g  
    %    m      \       s                                          n-2s 5#QXR+ T  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r FW.$5*f='  
    %    n      s=0 `N5|Ho*C  
    % Sv;_HZ  
    %   The following table shows the first 12 polynomials. l (3bW1{n  
    % ./$cMaDJ  
    %       n    m    Zernike polynomial    Normalization q=lAb\i  
    %       --------------------------------------------- 4GB7A]^E  
    %       0    0    1                        sqrt(2) TW^/sx  
    %       1    1    r                           2 pZU9^Z?~6  
    %       2    0    2*r^2 - 1                sqrt(6) Dn>%%K@0  
    %       2    2    r^2                      sqrt(6) C^ )*Dsp  
    %       3    1    3*r^3 - 2*r              sqrt(8) ko^\ HSXl  
    %       3    3    r^3                      sqrt(8) R CkaJ3  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) w4LScvBg  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) %2V-~.Ro6  
    %       4    4    r^4                      sqrt(10) 8KH\`5<  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Oq3A#6~  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) `~X!Ll  
    %       5    5    r^5                      sqrt(12) ZR\VCVH\^  
    %       --------------------------------------------- gqWupL  
    % `|Or{ih  
    %   Example: vp(;W,ba:|  
    % [NFNzwUB  
    %       % Display three example Zernike radial polynomials 6K-5g/hL  
    %       r = 0:0.01:1; +S))3 5N[  
    %       n = [3 2 5]; ( 9]_ HW[  
    %       m = [1 2 1]; MfmACd^3$  
    %       z = zernpol(n,m,r); P+)DsZ0ig  
    %       figure ".dZn6"mI  
    %       plot(r,z) N D<HXO  
    %       grid on ` }3qhar  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') P4Th_B7  
    % C.kxQ<  
    %   See also ZERNFUN, ZERNFUN2. 2<hpK!R  
    {hJXj,  
    % A note on the algorithm. V_Wwrhua  
    % ------------------------ 0 u?{ \  
    % The radial Zernike polynomials are computed using the series ,hVvve,j}  
    % representation shown in the Help section above. For many special .I@CS>j  
    % functions, direct evaluation using the series representation can 3~#h|?  
    % produce poor numerical results (floating point errors), because Z/ Tm)Xd  
    % the summation often involves computing small differences between 7n90f2"m  
    % large successive terms in the series. (In such cases, the functions {-A^g!jT&  
    % are often evaluated using alternative methods such as recurrence n$[f94d=  
    % relations: see the Legendre functions, for example). For the Zernike ^V|Oxp'7_  
    % polynomials, however, this problem does not arise, because the %0Y=WYUH>  
    % polynomials are evaluated over the finite domain r = (0,1), and D3c2^r $Z  
    % because the coefficients for a given polynomial are generally all 2r%lA\,h$  
    % of similar magnitude. <94_@3  
    % %?e(hnM  
    % ZERNPOL has been written using a vectorized implementation: multiple ,|88r=}  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Iyo@r%I  
    % values can be passed as inputs) for a vector of points R.  To achieve H'qG/@u-l  
    % this vectorization most efficiently, the algorithm in ZERNPOL ?:Y#Tbi3  
    % involves pre-determining all the powers p of R that are required to Yhp]x   
    % compute the outputs, and then compiling the {R^p} into a single vzn{h)D  
    % matrix.  This avoids any redundant computation of the R^p, and rDdzxrKg{  
    % minimizes the sizes of certain intermediate variables. %Qmk2  
    % sK=0Np=`  
    %   Paul Fricker 11/13/2006 _uc\ D R  
    r 6eb}z!i  
    Z@gnsPN^r  
    % Check and prepare the inputs: m4:^}O-#  
    % ----------------------------- 1&:@  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ?m c%.Bt  
        error('zernpol:NMvectors','N and M must be vectors.') gDIBnH  
    end CB~Q%QLG  
    5b/ojr7  
    if length(n)~=length(m) hAj1{pA,  
        error('zernpol:NMlength','N and M must be the same length.') c)&>$S8*  
    end 4'p=p#o  
    R4Rb73o  
    n = n(:); :SV>+EDY   
    m = m(:); ouHu8)q'r  
    length_n = length(n); _j>;ipTb+  
    8^B;1`#  
    if any(mod(n-m,2)) MCh#="L2  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') .qob_dRA  
    end -|Kzo_" v5  
    _ IeU+tS  
    if any(m<0) ]4 (?BJ  
        error('zernpol:Mpositive','All M must be positive.') !jqWwi  
    end V\K<$?oUb  
    Gs2p5nL<  
    if any(m>n) dd|W@Xp -  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') S-+M;@'Rl  
    end TzBzEiANn  
    -=698h*  
    if any( r>1 | r<0 ) bAr` E  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') w{ `|N$  
    end S=3^Q;V/1  
    ):EBgg4-N  
    if ~any(size(r)==1) }1-I[q6  
        error('zernpol:Rvector','R must be a vector.') [0[M'![8M  
    end -RJE6~>'\  
    m=qOg>k  
    r = r(:); KjB/.4lLq  
    length_r = length(r); 4e9q`~ sO  
    %]p6Kn/>  
    if nargin==4 `B8tmW#  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ;3C:%!CdA]  
        if ~isnorm N~ANjn/wL  
            error('zernpol:normalization','Unrecognized normalization flag.') @6o]chJo  
        end DG;y6#|p  
    else fRTo.u  
        isnorm = false; bl/,*Wx:4.  
    end /NF#+bx  
    dV8iwI  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;1DdjETr  
    % Compute the Zernike Polynomials uHNpfKnZ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jw6Tj;c  
    (P6vOo  
    % Determine the required powers of r: v[<Bjs\q5  
    % ----------------------------------- 0=v{RQ;W4  
    rpowers = []; z2/!m[U  
    for j = 1:length(n) 8n4V cu  
        rpowers = [rpowers m(j):2:n(j)]; t^E hE  
    end `#IcxweA  
    rpowers = unique(rpowers); oQ+61!5>  
    Gt/4F-Gn  
    % Pre-compute the values of r raised to the required powers, `0#H]=$2h  
    % and compile them in a matrix: N<z`yV  
    % ----------------------------- DlE_W+F  
    if rpowers(1)==0 -T/W:-M(  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 6-,m}Ce\  
        rpowern = cat(2,rpowern{:}); IPA*-I57  
        rpowern = [ones(length_r,1) rpowern]; !D.0 (J  
    else TA}UY7v  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); >Cd9fJ&0gP  
        rpowern = cat(2,rpowern{:}); hv)7H)|l~]  
    end Qu{c B^Ga*  
    Uedvc5><t  
    % Compute the values of the polynomials: oUW<4l  
    % -------------------------------------- 7y*ZXT]f  
    z = zeros(length_r,length_n);  [~Hg}-c  
    for j = 1:length_n gp|1?L 54  
        s = 0:(n(j)-m(j))/2; B94 &elu  
        pows = n(j):-2:m(j); SlT*C6f  
        for k = length(s):-1:1 1(`M~vFDK  
            p = (1-2*mod(s(k),2))* ... k ~6- cx  
                       prod(2:(n(j)-s(k)))/          ... Ri?\m!o  
                       prod(2:s(k))/                 ... 1"K*._K  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... :X .,  
                       prod(2:((n(j)+m(j))/2-s(k))); OKwOugi0  
            idx = (pows(k)==rpowers); T%CxvZ  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 7S2C/f  
        end |9NIGg'n  
         ,J9}.}Hd  
        if isnorm -4L!k'uR  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); =L 0fZf  
        end , &' Y  
    end 5A_4\YpDR  
    9F_6}.O  
    % 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)  KZW'O b>[  
    Q;Q%SI`yT  
    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。 jwq\stjD  
    8Kk3_ y  
    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)