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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  MK"  
    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! vfUfrk@D~  
     
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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 4Wla&yy  
    function z = zernfun(n,m,r,theta,nflag) mvTyx7 h=  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 60,-\h  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N }-{b$6]  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ";_K x={  
    %   unit circle.  N is a vector of positive integers (including 0), and 5B>Q 6  
    %   M is a vector with the same number of elements as N.  Each element oB0 8  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) !jAWNK6  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, UOu6LD/|h  
    %   and THETA is a vector of angles.  R and THETA must have the same &*aer5?`  
    %   length.  The output Z is a matrix with one column for every (N,M) D#d8^U  
    %   pair, and one row for every (R,THETA) pair. 0ck&kpL:9  
    % L8:]`M Q0  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike h7EUIlh"  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), p fL2v,]g  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ~Un64M?  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, R2N^'  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 8Da(tS  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. xHv|ca.E  
    % i$[,-4 v  
    %   The Zernike functions are an orthogonal basis on the unit circle. 3q#"i&  
    %   They are used in disciplines such as astronomy, optics, and 8B*E+f0  
    %   optometry to describe functions on a circular domain. emv;m/&8  
    % m|[\F#+C  
    %   The following table lists the first 15 Zernike functions. QJ a4R  
    % p*pn@z  
    %       n    m    Zernike function           Normalization 0 OAqA?Z  
    %       -------------------------------------------------- |"CJ  
    %       0    0    1                                 1 $/[Gys3"  
    %       1    1    r * cos(theta)                    2 _\,rX\  
    %       1   -1    r * sin(theta)                    2 (B>)2:T1  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) k;;nE o~6  
    %       2    0    (2*r^2 - 1)                    sqrt(3) iN<(O7B;  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) e86Aqehle  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) S)"##-~`T  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) K08 iPIkQ  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) _kn]#^ucCe  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) #0P!xZ'|{  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) GFd Z`i  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 3TU'*w &  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) |x d@M-ln  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) v]WH8GI  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) nU} ~I)@V  
    %       -------------------------------------------------- %<aImR]  
    % ?_VRfeztw  
    %   Example 1: kF+ZW%6N  
    % j6n2dMRvSE  
    %       % Display the Zernike function Z(n=5,m=1) Az U|p  
    %       x = -1:0.01:1; PSW #^o  
    %       [X,Y] = meshgrid(x,x); QjQ4Z'.r>  
    %       [theta,r] = cart2pol(X,Y); LIr(mB"Y0  
    %       idx = r<=1; u=vh Z%A]  
    %       z = nan(size(X)); U:qF/%w  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); d4d\0[  
    %       figure TkA9tFi  
    %       pcolor(x,x,z), shading interp UUl*f!& o  
    %       axis square, colorbar {V[Ha~b%*  
    %       title('Zernike function Z_5^1(r,\theta)') jo_o` j  
    % ER{yuw  
    %   Example 2: 7k 3p'FeS  
    % [/?c@N,  
    %       % Display the first 10 Zernike functions Ip>^O/}$1  
    %       x = -1:0.01:1; GSQfg  
    %       [X,Y] = meshgrid(x,x); c2/FHI0J;  
    %       [theta,r] = cart2pol(X,Y); 5+`=t07^et  
    %       idx = r<=1; gk"mr_03  
    %       z = nan(size(X)); = Q@6c   
    %       n = [0  1  1  2  2  2  3  3  3  3]; ?LM:RADCm  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 5QR}IxQ  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ?hKm&B;d  
    %       y = zernfun(n,m,r(idx),theta(idx)); +q7qK*  
    %       figure('Units','normalized') iNt 4>  
    %       for k = 1:10 ;JYoW{2  
    %           z(idx) = y(:,k); pNuqT*  
    %           subplot(4,7,Nplot(k)) Wt(Kd5k0'2  
    %           pcolor(x,x,z), shading interp . /Y&\<  
    %           set(gca,'XTick',[],'YTick',[]) ^ b@!dS  
    %           axis square /n(9&'H<  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Bgf=\7;5  
    %       end VW{,:Ya  
    % {-Yee[d<?  
    %   See also ZERNPOL, ZERNFUN2. 7 xUE,)?  
    l7ZB3'  
    %   Paul Fricker 11/13/2006 N9pwWg&<+  
    fO #?k<p  
    1XCmM Z  
    % Check and prepare the inputs: O"qR}W  
    % ----------------------------- HQl~Dh0DJ  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) rxs8De  
        error('zernfun:NMvectors','N and M must be vectors.') A hR0zg  
    end ikr7DBLt  
    =9(tsB gTX  
    if length(n)~=length(m) :xM}gPj"  
        error('zernfun:NMlength','N and M must be the same length.') Gp,'kw"I  
    end =C#*!N73  
    ":V%(c  
    n = n(:); X3AwM%,!  
    m = m(:); Jns/v6  
    if any(mod(n-m,2)) Y3<b~!f  
        error('zernfun:NMmultiplesof2', ... \p3v#0R{  
              'All N and M must differ by multiples of 2 (including 0).') Mo_$b8i  
    end hl**zF  
    Uh}+"h5  
    if any(m>n) w [L&*  
        error('zernfun:MlessthanN', ... 2qlIy  
              'Each M must be less than or equal to its corresponding N.') ,aWCiu}  
    end ^( DL+r,  
    5~QT g  
    if any( r>1 | r<0 ) SetX#e?q~  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') D&-vq,c  
    end Tv1]v.  
    $C$ub&D ~"  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) R1Yqz $#  
        error('zernfun:RTHvector','R and THETA must be vectors.') 1U'ZVJ5bpK  
    end UG #X/%p  
    j$mz3Yk  
    r = r(:); zC#%6@P\  
    theta = theta(:); m2Q$+p@  
    length_r = length(r); L?Cjo4xS  
    if length_r~=length(theta) aDh|48}X  
        error('zernfun:RTHlength', ... )T/J  
              'The number of R- and THETA-values must be equal.') >4M<W4  
    end m@[3~ 6A  
    f7 wm w2  
    % Check normalization: x)$2nonM  
    % -------------------- ki#bPgT  
    if nargin==5 && ischar(nflag) LZa% x  
        isnorm = strcmpi(nflag,'norm'); ?M~  k$  
        if ~isnorm =9<$eLE0  
            error('zernfun:normalization','Unrecognized normalization flag.') Z0W0uP;J  
        end #2N_/J(U  
    else "[.ne)/MC  
        isnorm = false; r>O|L%xpv  
    end 9DPb|+O-  
    djGs~H>;U_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~ aA;<#  
    % Compute the Zernike Polynomials 7@3sUA_Go  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f"P$f8$  
    #N9d$[R*  
    % Determine the required powers of r: U6c@Et,  
    % ----------------------------------- `2e_ L  
    m_abs = abs(m); yquAr$L!  
    rpowers = []; 0 u2Ny&6w  
    for j = 1:length(n) }*Zo6{B-  
        rpowers = [rpowers m_abs(j):2:n(j)]; 5*1#jiq  
    end q5?{ 1  
    rpowers = unique(rpowers); =x#&\ui  
    IM]h*YV'  
    % Pre-compute the values of r raised to the required powers, Bq{ ]Eh0%  
    % and compile them in a matrix: ~ k<SbFp  
    % ----------------------------- 73)Ll"(  
    if rpowers(1)==0 .pW o>`"  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); p&O8qAaO  
        rpowern = cat(2,rpowern{:}); {$|/|*  
        rpowern = [ones(length_r,1) rpowern]; O4!9{  
    else $P;UoqG<&  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); +4HlRGH  
        rpowern = cat(2,rpowern{:}); ^vW$XRnt  
    end B j=@&;  
    j/' g$  
    % Compute the values of the polynomials: KC]tY9 FK  
    % -------------------------------------- P9s_2KOF  
    y = zeros(length_r,length(n)); B%mtp;) P  
    for j = 1:length(n) ;AJ< LC  
        s = 0:(n(j)-m_abs(j))/2; om>VQ3  
        pows = n(j):-2:m_abs(j); gCL{Cw  
        for k = length(s):-1:1 vnZ4(  
            p = (1-2*mod(s(k),2))* ... s-%J 5_d f  
                       prod(2:(n(j)-s(k)))/              ... 7*MU2gb  
                       prod(2:s(k))/                     ... P=Puaz5&{  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... k:mlt:  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); pl?kS8#U?  
            idx = (pows(k)==rpowers); + ~~ Z0.[  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Z'e\_C  
        end F+3!uWUK  
         *l {4lu  
        if isnorm (V)9s\Le_  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); )WmZP3$^TX  
        end .aJ%am/:%  
    end B*2{M  
    % END: Compute the Zernike Polynomials nd;O(s;  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |eF.ZC)QWh  
    <"A#Eok|4  
    % Compute the Zernike functions: L&QtHSzy  
    % ------------------------------ &1~Re.* B  
    idx_pos = m>0; v4D!7 t&v"  
    idx_neg = m<0; AoIc9E lEX  
    0JyqCb l  
    z = y; pagC(F  
    if any(idx_pos) [WYJrk.  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); m|mG;8}pI  
    end <ZV7|'^  
    if any(idx_neg) f}%sO  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); DP0Z*8Ia  
    end ]o `4Z"  
    .01TTK*  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) 2AmR(vVa"  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. }WoX9M; 1  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated /1U,+g^O>  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive m[{nm95QZ  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, =\*S'Ded  
    %   and THETA is a vector of angles.  R and THETA must have the same 7~:>WMv9  
    %   length.  The output Z is a matrix with one column for every P-value, =GLYDV  
    %   and one row for every (R,THETA) pair. []!tT-Gzy  
    % - f+CyhR"*  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike @i;LZa  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) p {w}  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ud 5x$`  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 5QNBB|X@  
    %   for all p. j^:b-:F  
    % )_WH#-}  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Sv~PXi^`H  
    %   Zernike functions (order N<=7).  In some disciplines it is ">03~:oA  
    %   traditional to label the first 36 functions using a single mode 9[B<rz  
    %   number P instead of separate numbers for the order N and azimuthal TZ)(ZKX*R  
    %   frequency M. jD$;q7fB  
    % V>DXV-%&C  
    %   Example: PsacXZNs\N  
    % "bL P3  
    %       % Display the first 16 Zernike functions @?(nwj~ s`  
    %       x = -1:0.01:1; MA l{66  
    %       [X,Y] = meshgrid(x,x); ,!xz*o+#@  
    %       [theta,r] = cart2pol(X,Y);  eYPt  
    %       idx = r<=1; K#%O3RRs  
    %       p = 0:15; jqV)V>M.  
    %       z = nan(size(X)); yN%3w0v  
    %       y = zernfun2(p,r(idx),theta(idx)); ytuWT,u  
    %       figure('Units','normalized') 2&3eAJC  
    %       for k = 1:length(p) WlF+unB!9  
    %           z(idx) = y(:,k); Djg 1Qh  
    %           subplot(4,4,k) R;5QD`  
    %           pcolor(x,x,z), shading interp Ih9ORp7  
    %           set(gca,'XTick',[],'YTick',[]) x0N-[//YV  
    %           axis square :..E:HdYO  
    %           title(['Z_{' num2str(p(k)) '}']) NXC~#oG  
    %       end &VtWSq-)  
    % Iqe=#hUFe!  
    %   See also ZERNPOL, ZERNFUN. Fj4l %=  
    3(Hj7d7'}  
    %   Paul Fricker 11/13/2006 f z/?=  
    k}+MvGq  
    %Hh &u .  
    % Check and prepare the inputs: Rz])wBv e  
    % ----------------------------- =X%R*~!#Of  
    if min(size(p))~=1 ,& \&::R  
        error('zernfun2:Pvector','Input P must be vector.') q_%w l5\F  
    end W? 6  
    vh:UXE lm  
    if any(p)>35 oK(W)[u  
        error('zernfun2:P36', ... .wt>.mUH  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... w2M IY_N?  
               '(P = 0 to 35).']) ps{&WT3a  
    end ?$`1%Y9  
    8O;rp(N.n  
    % Get the order and frequency corresonding to the function number: lL(}dbT~N  
    % ---------------------------------------------------------------- ,i$(yx?  
    p = p(:); !pF KC)  
    n = ceil((-3+sqrt(9+8*p))/2); =_H*fhXS  
    m = 2*p - n.*(n+2); i&SBW0)  
    M7cI$=G  
    % Pass the inputs to the function ZERNFUN: eW }jS/g`  
    % ---------------------------------------- $O8EiC!f6  
    switch nargin S3JygN*  
        case 3 +2_6C;_DX  
            z = zernfun(n,m,r,theta); D=pI'5&  
        case 4 iA{chQBr  
            z = zernfun(n,m,r,theta,nflag); <|>:UGAR  
        otherwise f@x( ,p  
            error('zernfun2:nargin','Incorrect number of inputs.') M%Kx{*aw&  
    end G3^n_]Jb  
    .ON$vn7  
    % 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) . AQ3zpy5B  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. `'k's]Y  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of #.t$A9'  
    %   order N and frequency M, evaluated at R.  N is a vector of G4`sRaT.  
    %   positive integers (including 0), and M is a vector with the iRr& 'k  
    %   same number of elements as N.  Each element k of M must be a PTV`=vtj  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 6;=wuoJi  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is :92a34  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix [8J}da}  
    %   with one column for every (N,M) pair, and one row for every h/y}  
    %   element in R. gu%'M:Xe  
    % @BQB NGR1  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- p-03V"^&  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 9,a,A6xry  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to M&\?)yG  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 j!8+|eA kk  
    %   for all [n,m]. s$y#Ufz  
    % !{ )AV/\D  
    %   The radial Zernike polynomials are the radial portion of the )cmLo0`$  
    %   Zernike functions, which are an orthogonal basis on the unit YV!V9   
    %   circle.  The series representation of the radial Zernike kx#L<   
    %   polynomials is Xs,PT  
    % r#w_=h)  
    %          (n-m)/2 Xq)%w#l5?  
    %            __ -v+^x`HR  
    %    m      \       s                                          n-2s #3[b|cL  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Kxaz^$5Y$  
    %    n      s=0 "9T`3cM0  
    % D\&y(=fzf  
    %   The following table shows the first 12 polynomials. N S}`(N  
    % ~acK$.#  
    %       n    m    Zernike polynomial    Normalization ^3s&90  
    %       --------------------------------------------- M[N.H9  
    %       0    0    1                        sqrt(2) eu|q {p  
    %       1    1    r                           2 iBW6<2@oZF  
    %       2    0    2*r^2 - 1                sqrt(6) =sVt8FWGY  
    %       2    2    r^2                      sqrt(6) "@? kxRn!  
    %       3    1    3*r^3 - 2*r              sqrt(8) ,%G2>PBt  
    %       3    3    r^3                      sqrt(8) |(ju!&  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) b1^Yxe#L  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) *K^O oS  
    %       4    4    r^4                      sqrt(10) 9F1stT0G%  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) M{RZ-)IC  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) O!+5As  
    %       5    5    r^5                      sqrt(12) exKmK!FT  
    %       --------------------------------------------- FAl6  
    % 1>{-wL4rc  
    %   Example: O7*i;$!R  
    % V xs`w  
    %       % Display three example Zernike radial polynomials z(68^-V=:  
    %       r = 0:0.01:1; xyWdzc] (p  
    %       n = [3 2 5]; ^TuEp$Z=  
    %       m = [1 2 1]; E }j8p_p  
    %       z = zernpol(n,m,r); F@K;A%us)  
    %       figure sBI%lrO  
    %       plot(r,z) 5kNs@FP  
    %       grid on RYaof W  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ;7*@Gf}R  
    % 0! %}  
    %   See also ZERNFUN, ZERNFUN2. s hvcc  
    <&Xq`i/(  
    % A note on the algorithm. uL AXN  
    % ------------------------ ]}y'3aW  
    % The radial Zernike polynomials are computed using the series f+~!s 2uw  
    % representation shown in the Help section above. For many special !jnIXvT1qy  
    % functions, direct evaluation using the series representation can 0J</`/gH  
    % produce poor numerical results (floating point errors), because *lO+^\HXD  
    % the summation often involves computing small differences between WfO$q^'?DP  
    % large successive terms in the series. (In such cases, the functions 8{ t&8Ql n  
    % are often evaluated using alternative methods such as recurrence 74Wg@! P  
    % relations: see the Legendre functions, for example). For the Zernike [i#Gqx>'w  
    % polynomials, however, this problem does not arise, because the YcZ4y@6"  
    % polynomials are evaluated over the finite domain r = (0,1), and 1\{F.v  
    % because the coefficients for a given polynomial are generally all RyD$4jk+T"  
    % of similar magnitude. P?7b,a95O  
    % PaJwM%s)L  
    % ZERNPOL has been written using a vectorized implementation: multiple H;&t"Ql.  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 0+i,,^x.  
    % values can be passed as inputs) for a vector of points R.  To achieve !S{<Xc'wv  
    % this vectorization most efficiently, the algorithm in ZERNPOL  2oASz|  
    % involves pre-determining all the powers p of R that are required to 1zW6Pb  
    % compute the outputs, and then compiling the {R^p} into a single S,%HW87  
    % matrix.  This avoids any redundant computation of the R^p, and XePBA J  
    % minimizes the sizes of certain intermediate variables. nP31jm+A  
    % ^u,x~nPXg  
    %   Paul Fricker 11/13/2006 Or>[_3  
    !YHu  
    Ij_`=w<  
    % Check and prepare the inputs: e84TL U?~  
    % ----------------------------- 0yNlf-O  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) (B#|3o  
        error('zernpol:NMvectors','N and M must be vectors.') T,>e\  
    end sAlgp2-  
    RoRVu,1  
    if length(n)~=length(m) TD7ONa-,  
        error('zernpol:NMlength','N and M must be the same length.') &r%3)Z8Et  
    end c*Nbz,:  
    `_`,XkpzCJ  
    n = n(:); ;Vt u8f  
    m = m(:); Um<vsR  
    length_n = length(n); mNKa~E  
    >m!.l{*j>N  
    if any(mod(n-m,2)) FU3B;Fn^Z(  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') M czWg  
    end )I4tl/  
    06$9Uz9  
    if any(m<0) oMbCljUC  
        error('zernpol:Mpositive','All M must be positive.') Ls{fCi/2F  
    end 6 -}gqkR  
    H_FhHX.2(  
    if any(m>n) 8>9+w/DL  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') {9MYEN}FO  
    end r N7"%dx  
    V^i3:'  
    if any( r>1 | r<0 ) p%-9T>og  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') (Q+3aEUE  
    end (tvh9 o  
    r "R\  
    if ~any(size(r)==1) X'm2uOEj  
        error('zernpol:Rvector','R must be a vector.') e+[J9;g  
    end G w[&P%  
    i_"I"5pBF  
    r = r(:); nC^'2z  
    length_r = length(r); xo$ZPnf(zv  
    ?H&p zY~H  
    if nargin==4 TfPx   
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); |=POV]K  
        if ~isnorm KJiwM(o  
            error('zernpol:normalization','Unrecognized normalization flag.') H=jnCGk  
        end J"y@n ~*0  
    else jVz1`\Nje  
        isnorm = false; %#,BvQz~  
    end qJ@?[|2R  
    _,^sI%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% H &JKja}`  
    % Compute the Zernike Polynomials ? &O$ayG77  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sAN#j {  
    e9d~Xi16KY  
    % Determine the required powers of r: ,#G@ri:B  
    % ----------------------------------- CS/-:>s%  
    rpowers = []; TI332,eL  
    for j = 1:length(n) NmQ]qv  
        rpowers = [rpowers m(j):2:n(j)]; W5p}oN  
    end kBzzi^cl  
    rpowers = unique(rpowers); i ,'~Ds  
    1 .M?Hp9i  
    % Pre-compute the values of r raised to the required powers, v09f#t$;5  
    % and compile them in a matrix: UTPl7po5D  
    % ----------------------------- fHigLL0B  
    if rpowers(1)==0 &a`-NRU#  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); v>XE]c_  
        rpowern = cat(2,rpowern{:}); Ssj'1[%  
        rpowern = [ones(length_r,1) rpowern]; Cv|:.y  
    else vzw\f   
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); sR6 (8  
        rpowern = cat(2,rpowern{:}); +3C S3fTq  
    end L6a8%%`  
    Y%faf.$/9  
    % Compute the values of the polynomials: 1pV"< ,t  
    % -------------------------------------- n'ro5D  
    z = zeros(length_r,length_n); g=pDC+  
    for j = 1:length_n z,9qAts?mh  
        s = 0:(n(j)-m(j))/2; 8^{BuUA  
        pows = n(j):-2:m(j); (:\hor%  
        for k = length(s):-1:1 a5'QL(IX  
            p = (1-2*mod(s(k),2))* ... ty78)XI  
                       prod(2:(n(j)-s(k)))/          ... d^w_rL  
                       prod(2:s(k))/                 ... MiC&av  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 6"DvdJ0MB  
                       prod(2:((n(j)+m(j))/2-s(k))); d|TIrlA  
            idx = (pows(k)==rpowers); G > ,rf ]N  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 3EyN"Lvp{o  
        end E8xXr>j>#  
         "CaVT7L  
        if isnorm |0&S>%=  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 4Mprc~ 7vr  
        end bfJDF(=h  
    end vmoqsdZ/  
    K)qmJ-Gub  
    % 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)  OY?uqP}c  
    RPLr7Lb  
    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。 $'e.bh  
    W[YcYa_tQ  
    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)