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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 t|Ipxk.)  
    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! }&mFpc  
     
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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 q-.e9eoc\  
    function z = zernfun(n,m,r,theta,nflag) UEq;}4Bo  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. x Qh?  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N G@) I  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 4pF U`g=  
    %   unit circle.  N is a vector of positive integers (including 0), and @HfWAFT  
    %   M is a vector with the same number of elements as N.  Each element I~R<}volu  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) LaZF=<w(  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, -%=StWdb   
    %   and THETA is a vector of angles.  R and THETA must have the same fxDY:l  
    %   length.  The output Z is a matrix with one column for every (N,M) t#y   
    %   pair, and one row for every (R,THETA) pair.  afEp4(X~  
    % xrT_ro8  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike +fhyw{  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), L-d8bA  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral wYf=(w \c  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, >5Zp x8W  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized K)qbd~<\  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. a{h(BI^~  
    % `~(C\+gUp  
    %   The Zernike functions are an orthogonal basis on the unit circle. yvxC/Jo4  
    %   They are used in disciplines such as astronomy, optics, and We]X+>BlO  
    %   optometry to describe functions on a circular domain. !dLz ?0  
    % 5Ag>,>kJ6  
    %   The following table lists the first 15 Zernike functions. );h\0w>3  
    % 1V`]sfRK  
    %       n    m    Zernike function           Normalization <LW|m7  
    %       -------------------------------------------------- 4(4JQ(5  
    %       0    0    1                                 1  &1Fcwj  
    %       1    1    r * cos(theta)                    2 N,ik&NIWy  
    %       1   -1    r * sin(theta)                    2 2LYd # !i  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) uz4mHyS6  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ?E2k]y6<  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) LM'` U-/e$  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) }bznx[4?I  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ; _i0@@J  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) s/[i>`g/9  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) V8&/O)}o  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) wZa;cg.-q  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) z s"AYxr  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) +>qBK}`  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) T *t$   
    %       4    4    r^4 * sin(4*theta)             sqrt(10) |->y'V  
    %       -------------------------------------------------- ~+7yi4(i  
    % ~v>w%]  
    %   Example 1: Xy*X4JJh^  
    % ,.FTw,<  
    %       % Display the Zernike function Z(n=5,m=1) %Y Rg1UKY  
    %       x = -1:0.01:1; k7{fkl9|#  
    %       [X,Y] = meshgrid(x,x); >q &ouVE  
    %       [theta,r] = cart2pol(X,Y); K=5_jE^e  
    %       idx = r<=1; J-PzIFWd  
    %       z = nan(size(X)); HHnabSn}{q  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 0acY@_  
    %       figure @?]-5~3;  
    %       pcolor(x,x,z), shading interp e3>Re![_.  
    %       axis square, colorbar GPx S.&  
    %       title('Zernike function Z_5^1(r,\theta)') /1li^</|p`  
    % L7Oytdc<  
    %   Example 2: IPxfjBC+J  
    % eBAB7r/7  
    %       % Display the first 10 Zernike functions 3`9*Hoy0c  
    %       x = -1:0.01:1; .`'SL''c  
    %       [X,Y] = meshgrid(x,x); M<$l&%<`G  
    %       [theta,r] = cart2pol(X,Y); ,t+ATaOF  
    %       idx = r<=1; 3X!~*_i C  
    %       z = nan(size(X)); F[=m|MZb  
    %       n = [0  1  1  2  2  2  3  3  3  3]; @&ZTEznbyt  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 3+|6])Hi1  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; jATU b-  
    %       y = zernfun(n,m,r(idx),theta(idx)); tiE+x|Ju"  
    %       figure('Units','normalized') 'c$9[|x  
    %       for k = 1:10 1UM]$$:i  
    %           z(idx) = y(:,k); J/<`#XZB   
    %           subplot(4,7,Nplot(k)) iz^wBQ  
    %           pcolor(x,x,z), shading interp 78QFaN$  
    %           set(gca,'XTick',[],'YTick',[]) Wq9s[)F"Z  
    %           axis square >Ed^dsb&  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Z],"<[E  
    %       end *Yr-:s9J9  
    % @E>^\!nH  
    %   See also ZERNPOL, ZERNFUN2. _@OYC<  
    /MU<)[*Ro  
    %   Paul Fricker 11/13/2006 CXQ ?P  
    t!u*6 W|@  
    4a @iR2e  
    % Check and prepare the inputs: sMS`-,37u  
    % ----------------------------- &"kx (B  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) {f&ga  
        error('zernfun:NMvectors','N and M must be vectors.') ^r& {V"l]  
    end )[K3p{4  
    (KQt%]  
    if length(n)~=length(m) }1W$9\%  
        error('zernfun:NMlength','N and M must be the same length.') rOD KM-7+  
    end v4zd x)  
    =0)^![y]v  
    n = n(:); u=l(W(9=  
    m = m(:); y^A $bTQq  
    if any(mod(n-m,2)) k`AJ$\=  
        error('zernfun:NMmultiplesof2', ... OWjZ)f/  
              'All N and M must differ by multiples of 2 (including 0).') p_AV3   
    end +-nQ, fOV  
    >eTlew<5  
    if any(m>n) !qpu /  
        error('zernfun:MlessthanN', ... ^"l$p,P+  
              'Each M must be less than or equal to its corresponding N.') @iRVY|t/  
    end |d3agfS[n  
    ~?&ijhZ  
    if any( r>1 | r<0 ) f5a](&  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') b tu:@s8ci  
    end X2uX+}h*tA  
    3PA'Uk"5Z  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 7asq]Y}<  
        error('zernfun:RTHvector','R and THETA must be vectors.') R,\ r{@yrz  
    end $a A.d^  
    itF+6wv~  
    r = r(:); VL{#.;QQa  
    theta = theta(:); HIq1/)  
    length_r = length(r); W =zG  
    if length_r~=length(theta) cBI )?  
        error('zernfun:RTHlength', ... UYQ$c }Z5  
              'The number of R- and THETA-values must be equal.') 8[C6LG  
    end v/czW\z  
    Ds87#/Yfv  
    % Check normalization: ~{+{pcO}  
    % -------------------- ja;5:=8A5  
    if nargin==5 && ischar(nflag) 2f!oA~|2  
        isnorm = strcmpi(nflag,'norm'); RNdnlD#P  
        if ~isnorm Wn^^Q5U#  
            error('zernfun:normalization','Unrecognized normalization flag.') ]K7  64}  
        end |&Pl4P  
    else A,{D9-%  
        isnorm = false; B0i}Y-Z  
    end >y9o&D  
    lAk1ncx  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'u[o`31.  
    % Compute the Zernike Polynomials fqb$_>3Ol  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8q3TeMYV  
    .dCP8|  
    % Determine the required powers of r: $t$f1?  
    % ----------------------------------- C):d9OI?  
    m_abs = abs(m); U_- K6:tr  
    rpowers = []; pYVy(]1I(3  
    for j = 1:length(n) H040-Q;S'  
        rpowers = [rpowers m_abs(j):2:n(j)]; ? ~Zrd  
    end ?Q)Z..7  
    rpowers = unique(rpowers); udGGDH  
    M:M>@|)  
    % Pre-compute the values of r raised to the required powers, WdqK/s<jM  
    % and compile them in a matrix: C[nr>   
    % ----------------------------- 0xUj#)  
    if rpowers(1)==0 l :Nxl  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); :WIf$P?X  
        rpowern = cat(2,rpowern{:}); va(9{AXI  
        rpowern = [ones(length_r,1) rpowern]; \hW73a!  
    else Ro]IE|Fv  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ?ev G=S4>  
        rpowern = cat(2,rpowern{:}); IKDjatn  
    end |u;BAb  
    wmE,k1G  
    % Compute the values of the polynomials: htYrv5q=M  
    % -------------------------------------- FRt/{(jro  
    y = zeros(length_r,length(n)); ^3|$wB=  
    for j = 1:length(n) 4sBoD=e  
        s = 0:(n(j)-m_abs(j))/2; Kw0V4UF  
        pows = n(j):-2:m_abs(j); DD 5EHJR  
        for k = length(s):-1:1 ]8>UII,US  
            p = (1-2*mod(s(k),2))* ... MD4 j~q\ g  
                       prod(2:(n(j)-s(k)))/              ... DG*o w^  
                       prod(2:s(k))/                     ... ~_db<!a  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... = )l:^+q  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 8+a<#? ;  
            idx = (pows(k)==rpowers); k*3_) S -  
            y(:,j) = y(:,j) + p*rpowern(:,idx); c9TAV,/fF*  
        end pZ~> l=-  
         T{4fa^c2J  
        if isnorm (vsk^3R[6  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); $v<hW A]>  
        end 11<@++,i  
    end PnIvk]"Ab  
    % END: Compute the Zernike Polynomials wu!_BCIy  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^xw [d}0 S  
    q\t>D _lU  
    % Compute the Zernike functions: 8^/Ek<Q b|  
    % ------------------------------ &iiK ZZ`_o  
    idx_pos = m>0; <<On*#80w  
    idx_neg = m<0; 0/P-> n~  
    bC4* w O  
    z = y; f93rY<  
    if any(idx_pos) ,cy/fW  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); AzO3(1:  
    end ]7S7CVDk4  
    if any(idx_neg) $ l sRg:J  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Rc:cVK  
    end BdB`  
    aRO_,n9  
    % 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) Z!RRe]"y  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. GkGC4*n  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated )' x/q  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive K!- &Zv  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, :S99}pgY  
    %   and THETA is a vector of angles.  R and THETA must have the same A.$VM#  
    %   length.  The output Z is a matrix with one column for every P-value, z)W#&JFF  
    %   and one row for every (R,THETA) pair. g?A5'o&Yu  
    % x)#<.DX  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike $-fjrQ  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) +NLQYuN  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 3<)@ll  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 K+ 7yUF8XP  
    %   for all p. g=oeS%>E  
    % wwK~H  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ndKvJH4  
    %   Zernike functions (order N<=7).  In some disciplines it is Ic{'H2~4,  
    %   traditional to label the first 36 functions using a single mode q]iKz%|Z/  
    %   number P instead of separate numbers for the order N and azimuthal @wB'3q}(  
    %   frequency M. m.HX2(&\3  
    % .sJys SA\  
    %   Example: *3F /Ft5  
    % fV A=<:  
    %       % Display the first 16 Zernike functions W p7@  
    %       x = -1:0.01:1; > G4HZE  
    %       [X,Y] = meshgrid(x,x); CFkW@\]  
    %       [theta,r] = cart2pol(X,Y); #.MIW*==  
    %       idx = r<=1; VeD+U~ d  
    %       p = 0:15; nv_m!JG7  
    %       z = nan(size(X)); zO).<xIq+  
    %       y = zernfun2(p,r(idx),theta(idx)); FU]8.)`G  
    %       figure('Units','normalized') 6cQeL$,SQ  
    %       for k = 1:length(p) GLaZN4`  
    %           z(idx) = y(:,k); w8ZHk?:  
    %           subplot(4,4,k) \'It,PN  
    %           pcolor(x,x,z), shading interp Y @XkqvX  
    %           set(gca,'XTick',[],'YTick',[]) 'XP>} m  
    %           axis square 75\RG+kQ  
    %           title(['Z_{' num2str(p(k)) '}']) <@U.   
    %       end {m_A1D/_  
    % \'s$ZN$k  
    %   See also ZERNPOL, ZERNFUN. "UhK]i*@l  
    9&O#+FU  
    %   Paul Fricker 11/13/2006 +K$5tT6b  
    ;<bj{#mMv  
    vB{; N  
    % Check and prepare the inputs: Nh1e1m?  
    % ----------------------------- NRHr6!f>  
    if min(size(p))~=1 ~{1/*&P  
        error('zernfun2:Pvector','Input P must be vector.') teq^xTUF[  
    end 8m/FKO (r  
    v2M"b?Q  
    if any(p)>35 |n|U;|'^  
        error('zernfun2:P36', ... 3r~>~ueZ  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 1EC-e|M.  
               '(P = 0 to 35).']) Qm35{^p+  
    end "4Lg8qm  
    9atjK4+o  
    % Get the order and frequency corresonding to the function number: B^8ZoF  
    % ---------------------------------------------------------------- gZ`32fB%  
    p = p(:); Eu`2w%qz  
    n = ceil((-3+sqrt(9+8*p))/2); c W81  
    m = 2*p - n.*(n+2); * 1 |YLy  
    ":UWowJO  
    % Pass the inputs to the function ZERNFUN: P3wU#qU  
    % ---------------------------------------- LPq*ZZK  
    switch nargin Cbgj@4H  
        case 3 pr62:  
            z = zernfun(n,m,r,theta); (TT3(|v  
        case 4 5`4}A%@&  
            z = zernfun(n,m,r,theta,nflag); 6)=](VmNL`  
        otherwise $ 7U Dz  
            error('zernfun2:nargin','Incorrect number of inputs.') Y=P9:unG  
    end Ph(]?MG\_  
    T7>4 8eH  
    % 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) HBt|}uZ?6i  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. bWGyLo,  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of :wQC_;  
    %   order N and frequency M, evaluated at R.  N is a vector of +IwdMJ8&8  
    %   positive integers (including 0), and M is a vector with the P_,v5Qx"-  
    %   same number of elements as N.  Each element k of M must be a R<0Fy=z  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ry$tK"v/  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 4^ c!_K&&  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 2-B6IPeI  
    %   with one column for every (N,M) pair, and one row for every +\+Uz!YS  
    %   element in R. $cRcap  
    % [NQmL=l  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- +:Lk^Ny  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is sFbfFUd  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 8B}'\e4i  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 DBQOxryP>o  
    %   for all [n,m]. J)-T:.i|0  
    % p G)9=X!9  
    %   The radial Zernike polynomials are the radial portion of the l'|E,N>X  
    %   Zernike functions, which are an orthogonal basis on the unit C}n'>],p  
    %   circle.  The series representation of the radial Zernike LiiK3!^i  
    %   polynomials is uQeqnGp  
    % }BA9Ka#%  
    %          (n-m)/2 * eA{[  
    %            __ IO}+[%ptc*  
    %    m      \       s                                          n-2s gsnP!2cR  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r /m97CC#+  
    %    n      s=0 ZrFr`L5F;  
    % &foD&  
    %   The following table shows the first 12 polynomials. 3Z_t%J5QZ$  
    % # .~ga7Q  
    %       n    m    Zernike polynomial    Normalization _GE=kw;:  
    %       --------------------------------------------- ?lF mXZy`  
    %       0    0    1                        sqrt(2) pNP_f:A|  
    %       1    1    r                           2 $kD7y5  
    %       2    0    2*r^2 - 1                sqrt(6) 7@FDBjq  
    %       2    2    r^2                      sqrt(6) S  <2}8D  
    %       3    1    3*r^3 - 2*r              sqrt(8) %_>Tcm=  
    %       3    3    r^3                      sqrt(8) Ynvj;  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) wHA/b.jH  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) h8em\<;  
    %       4    4    r^4                      sqrt(10) 1Wv{xML"  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) _~juv&  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) b2G2c L-(  
    %       5    5    r^5                      sqrt(12) Ud$Q0m&  
    %       --------------------------------------------- Cy`26[E$S  
    % *U M! (  
    %   Example: |pBMrN+is  
    % &j3` )N  
    %       % Display three example Zernike radial polynomials nlaG<L#  
    %       r = 0:0.01:1; I=U+GY:  
    %       n = [3 2 5]; 8B j4 _!g  
    %       m = [1 2 1]; kzMa+(fu  
    %       z = zernpol(n,m,r); 4 ^4d9?c  
    %       figure h iAxh Y  
    %       plot(r,z) hXNH"0VCV  
    %       grid on ~ W@X-  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') Gv;;!sZ  
    % = PV/`I_h  
    %   See also ZERNFUN, ZERNFUN2. h(_P9E[g  
    "t=UX -3  
    % A note on the algorithm. n|6?J_{<b>  
    % ------------------------ #hpIyy%n  
    % The radial Zernike polynomials are computed using the series L1rwIOgq^  
    % representation shown in the Help section above. For many special <3lUV7!  
    % functions, direct evaluation using the series representation can %06vgjOa (  
    % produce poor numerical results (floating point errors), because Vz'HM$  
    % the summation often involves computing small differences between &2Q*1YXj  
    % large successive terms in the series. (In such cases, the functions  U7E  
    % are often evaluated using alternative methods such as recurrence *_ PPrx5  
    % relations: see the Legendre functions, for example). For the Zernike 3&$Nd  
    % polynomials, however, this problem does not arise, because the wEE2a56L-  
    % polynomials are evaluated over the finite domain r = (0,1), and #XcU{5Qm5  
    % because the coefficients for a given polynomial are generally all eI0F!Yon  
    % of similar magnitude. ]Dh1~k.Kp  
    % lu]o34  
    % ZERNPOL has been written using a vectorized implementation: multiple '[Xl>Z[  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] BMdSf(l  
    % values can be passed as inputs) for a vector of points R.  To achieve xkM] J)C  
    % this vectorization most efficiently, the algorithm in ZERNPOL (|dPeix|  
    % involves pre-determining all the powers p of R that are required to 9_GokU P_  
    % compute the outputs, and then compiling the {R^p} into a single Q{[@`bZB  
    % matrix.  This avoids any redundant computation of the R^p, and %MbyKz:X  
    % minimizes the sizes of certain intermediate variables. a&C.=  
    % ;Xyte  
    %   Paul Fricker 11/13/2006 , |l@j%  
     #I;D  
    +1/b^Ac  
    % Check and prepare the inputs: &Xv1[nByU  
    % ----------------------------- c yP,[?N  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 8bt53ta  
        error('zernpol:NMvectors','N and M must be vectors.') }"n7~|  
    end v77fQ0w3  
    x/xb1"  
    if length(n)~=length(m) R]Ek}1~?  
        error('zernpol:NMlength','N and M must be the same length.') -TTs.O8P|<  
    end HxZ.OZbR  
    MxTmWsaW  
    n = n(:); 0cFn{q'u  
    m = m(:); ] IS;\~  
    length_n = length(n); Ig9d#c  
    #]y5z i  
    if any(mod(n-m,2)) p,;mYms  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') [Tp%"f1  
    end x,\!DLq:p  
    #uKWuGz]  
    if any(m<0) (ii( yz|  
        error('zernpol:Mpositive','All M must be positive.') 4-V)_U#8  
    end `|EH[W&y  
    s"coQ!e1.  
    if any(m>n) 3;l"=#5  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') I+",b4  
    end 88l,&2q  
    B.*"Xfr8  
    if any( r>1 | r<0 ) 'E-FO_N  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') \ I:.<2i  
    end 'I v_mig  
    +/y]h 0aa  
    if ~any(size(r)==1) DsGI/c  
        error('zernpol:Rvector','R must be a vector.') Y)Tl<  
    end [;^,CD|P  
    ^N-'xy  
    r = r(:); PS@*qTin  
    length_r = length(r); qfr Ni1\9-  
    X;VQEDMPU  
    if nargin==4 k':s =IXW  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); cDeZMsV  
        if ~isnorm [zh"x#AyI  
            error('zernpol:normalization','Unrecognized normalization flag.') R=M!e<'  
        end [PWL<t::c  
    else 8TPN#"  
        isnorm = false; 3m=2x5 {L  
    end 7ZsA5%s=,  
    [/$N!2'5  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,{KCY[}|  
    % Compute the Zernike Polynomials $r79n-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N4wA#\-  
    1bSD,;$sQ  
    % Determine the required powers of r: 9M'DC^x*T  
    % ----------------------------------- ;#/0b{XFj  
    rpowers = []; EB,4PEe:  
    for j = 1:length(n) Q9slfQ  
        rpowers = [rpowers m(j):2:n(j)]; *aRX \ TnN  
    end re`t ]gzb  
    rpowers = unique(rpowers); CW`!}yu%  
    Z0* %Rq  
    % Pre-compute the values of r raised to the required powers, N wtg%;  
    % and compile them in a matrix: o`6|ba  
    % ----------------------------- cj g.lzY H  
    if rpowers(1)==0 Vz"u>BP3~  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); /;oqf4MF  
        rpowern = cat(2,rpowern{:}); z<&m*0WYA  
        rpowern = [ones(length_r,1) rpowern]; T)SbHp Y  
    else h{_*oBa  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); CDdkoajBa  
        rpowern = cat(2,rpowern{:}); Eju~}:Lo  
    end j*3}1L4P  
    v}[dnG  
    % Compute the values of the polynomials: 6+` tn  
    % -------------------------------------- n:4uA`Vg  
    z = zeros(length_r,length_n); a$JLc a  
    for j = 1:length_n i9m*g*"2  
        s = 0:(n(j)-m(j))/2; b{5K2k&,  
        pows = n(j):-2:m(j); o#D.9K(  
        for k = length(s):-1:1 yPgmg@G@/  
            p = (1-2*mod(s(k),2))* ... XG 0v  
                       prod(2:(n(j)-s(k)))/          ... }}T,W.#%u  
                       prod(2:s(k))/                 ... C@gXT]Q 0}  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... !')y&7a~  
                       prod(2:((n(j)+m(j))/2-s(k))); '\~^TFi  
            idx = (pows(k)==rpowers); YnTB&GPxl  
            z(:,j) = z(:,j) + p*rpowern(:,idx); #YK5WTn5  
        end ~?U*6P)o  
         I1"MPx{  
        if isnorm UVEz;<5@\  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); CxF-Z7 '  
        end &`!^Zq vG  
    end $nPAm6mH  
    `G$1n#&  
    % 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)  7*H:Ob)9k  
    &XLD S=j  
    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。 $ 0Yh!L?\  
    Es5p}uh.[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)