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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 3 1KMn  
    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! 2<>n8K  
     
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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 tf|/_Y2  
    function z = zernfun(n,m,r,theta,nflag) j/3827jw=  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. (S0MqX*  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N .x$+R%5U  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 4pV.R5:  
    %   unit circle.  N is a vector of positive integers (including 0), and ~/Aw[>_;  
    %   M is a vector with the same number of elements as N.  Each element  ;4 R1  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) IGEf*!  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 6xr$  
    %   and THETA is a vector of angles.  R and THETA must have the same Un^QNd>  
    %   length.  The output Z is a matrix with one column for every (N,M) ?;,s=2  
    %   pair, and one row for every (R,THETA) pair. h|yv*1/|  
    % [|d:QFx  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike C/"fS#<  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Ge@./SGT  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral eJilSFp1  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ldrKk'S,B  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Im{50%Y  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. oaHg6PT!  
    % jU)r~QhN  
    %   The Zernike functions are an orthogonal basis on the unit circle. TU$/3fp*  
    %   They are used in disciplines such as astronomy, optics, and &zlwV"W  
    %   optometry to describe functions on a circular domain. tq}sXt  
    % ;TF(opW:  
    %   The following table lists the first 15 Zernike functions. 24Z7;'  
    % ylLQKdcL  
    %       n    m    Zernike function           Normalization 9bl&\Ykt.  
    %       -------------------------------------------------- '{\VO U  
    %       0    0    1                                 1 #R"9(Q&  
    %       1    1    r * cos(theta)                    2 %CfJ.;BDNE  
    %       1   -1    r * sin(theta)                    2 ,G e7 9(  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Tc,Bv7:  
    %       2    0    (2*r^2 - 1)                    sqrt(3) cE/7B'cR  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) UAnq|NJO  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Zn1+} Z@I  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Z8(1QU,~2  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 8;P8CKe  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) S9 <J \`FG  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) IQMk:  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,]i ^/fT  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) JHwkLAuz  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) $@FD01h.t3  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 2JYp.CJv  
    %       -------------------------------------------------- %Xh/16X${  
    % [^A.$,  
    %   Example 1: {0q;:7Bt  
    % ElZ'/l*\  
    %       % Display the Zernike function Z(n=5,m=1) F}DdErd!f  
    %       x = -1:0.01:1; vpFN{UfD  
    %       [X,Y] = meshgrid(x,x); Id *Gs>4U  
    %       [theta,r] = cart2pol(X,Y); lInq=  
    %       idx = r<=1; Ra'0 ^4t  
    %       z = nan(size(X)); A)2vjM9}K  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); AEX]_1TG  
    %       figure iH#~eg  
    %       pcolor(x,x,z), shading interp A,W-=TC  
    %       axis square, colorbar yX,2`&c  
    %       title('Zernike function Z_5^1(r,\theta)') QN9$n%Z  
    % mk~i (Ee  
    %   Example 2: `FH Hh  
    % MxuwEV|^  
    %       % Display the first 10 Zernike functions }e6Ta_Z~  
    %       x = -1:0.01:1; C (vi ns  
    %       [X,Y] = meshgrid(x,x); -9~kp'_a  
    %       [theta,r] = cart2pol(X,Y); 9<k<HmkD  
    %       idx = r<=1; [3nhf<O  
    %       z = nan(size(X)); _J 6|ju\  
    %       n = [0  1  1  2  2  2  3  3  3  3]; o*:VG\#Z6  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; p.n]y=o.)  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; r) T^ Td1  
    %       y = zernfun(n,m,r(idx),theta(idx)); ZD6rD (l9  
    %       figure('Units','normalized') i6-q%%]6  
    %       for k = 1:10 GfUIF]X  
    %           z(idx) = y(:,k); :4}?%3&;  
    %           subplot(4,7,Nplot(k)) a_^3:}i~D  
    %           pcolor(x,x,z), shading interp }9R45h}{<  
    %           set(gca,'XTick',[],'YTick',[]) F@kOj*5,[  
    %           axis square #^gn,^QQ  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])  .LEQ r)  
    %       end ,ZJI]Q=!  
    % CM>/b3nOW  
    %   See also ZERNPOL, ZERNFUN2. V5i_\A  
    i/Q*AG>b  
    %   Paul Fricker 11/13/2006 /R8>f  
    I--WS[  
    yUq,9.6Ig  
    % Check and prepare the inputs: GI WgfE?  
    % ----------------------------- Q nDymVF  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) I}p uN!  
        error('zernfun:NMvectors','N and M must be vectors.') N:)`+}  
    end I.fV_ H^  
    n4 KiC!*i0  
    if length(n)~=length(m) Bg-C:Ok 2'  
        error('zernfun:NMlength','N and M must be the same length.') - DlKFN  
    end k)'hNk"x  
    $G"PZ7  
    n = n(:); K)]7e?:Wu  
    m = m(:); Y:FV+ SI  
    if any(mod(n-m,2)) X8ev uN  
        error('zernfun:NMmultiplesof2', ... U_ V0  
              'All N and M must differ by multiples of 2 (including 0).') N;F1Z-9  
    end 6]\F_Z41  
    kN`[Q$B  
    if any(m>n) C(3yJzg>y  
        error('zernfun:MlessthanN', ... r%xp^j}  
              'Each M must be less than or equal to its corresponding N.') uwj/]#`  
    end \_!FOUPz(  
    G`R Ed-Z[  
    if any( r>1 | r<0 ) a)(j68c  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') M`FsKK`  
    end F] +t/  
    9HLn_|yU  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) YdV5\!  
        error('zernfun:RTHvector','R and THETA must be vectors.') R# 8D}5[&  
    end ,M>W)TSH  
    C N"V w  
    r = r(:); hAOXOj1  
    theta = theta(:); Gc~A,_(  
    length_r = length(r); $.QnM  
    if length_r~=length(theta) fm;1Iu#  
        error('zernfun:RTHlength', ... :GIY"l'  
              'The number of R- and THETA-values must be equal.') V{HZ/p_Y  
    end *-ZD-B*?  
    itm;,Sbg  
    % Check normalization: q+~z# jFX  
    % -------------------- GLwL'C'591  
    if nargin==5 && ischar(nflag) =P'=P0G  
        isnorm = strcmpi(nflag,'norm'); {uM0J$P:  
        if ~isnorm 6O"Vy  
            error('zernfun:normalization','Unrecognized normalization flag.') ;G0~f9  
        end ~`#.ZMO  
    else MCurKT<pQ  
        isnorm = false; 56G5JSB=\  
    end R=i$*6}a  
    MQQiQ 2  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Sj+ gf~~  
    % Compute the Zernike Polynomials !H~!i.m'-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <z#r3J  
    /_*:  
    % Determine the required powers of r: ;p BXAl  
    % ----------------------------------- [_|i W%<`  
    m_abs = abs(m); %saTyF,  
    rpowers = []; N?kXATB  
    for j = 1:length(n) \tyL`& )  
        rpowers = [rpowers m_abs(j):2:n(j)]; %p/Qz|W  
    end ~NpnRIt  
    rpowers = unique(rpowers); E-*udQ  
    3 V8SKBS  
    % Pre-compute the values of r raised to the required powers, \z:p"eua z  
    % and compile them in a matrix: `*KS` z?  
    % ----------------------------- >/6v` 8F  
    if rpowers(1)==0 7vNS@[8  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); y3 LWh}~E  
        rpowern = cat(2,rpowern{:}); +O j28vR  
        rpowern = [ones(length_r,1) rpowern]; HjGT{o  
    else \Y>^L{  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); :7W5R  
        rpowern = cat(2,rpowern{:}); ] X%bU*4  
    end qf2{Te1  
    Oq*a4_R'YV  
    % Compute the values of the polynomials: Vn];vN  
    % -------------------------------------- Cla Yy58v  
    y = zeros(length_r,length(n)); E4}MvV=  
    for j = 1:length(n) &|9mM=^  
        s = 0:(n(j)-m_abs(j))/2; QdUl-(  
        pows = n(j):-2:m_abs(j); *:BN LM  
        for k = length(s):-1:1 )lB-D;3[_  
            p = (1-2*mod(s(k),2))* ... @a%,0Wn  
                       prod(2:(n(j)-s(k)))/              ... %04>R'mN  
                       prod(2:s(k))/                     ... I #1_  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... TCmWn$LeE  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); nqgfAQsE)  
            idx = (pows(k)==rpowers); U!3nn#!yE  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ?B@hCd)  
        end MMhd-B1O&  
         #kLM=a/_NO  
        if isnorm 8'^eH1d'  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); (C6Y*Zm\  
        end u>k;P UH4  
    end \Q^\z   
    % END: Compute the Zernike Polynomials 5Tn4iyg;B  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5:ir il  
    O- |RPW}  
    % Compute the Zernike functions: Q>TaaGc  
    % ------------------------------ #n2GW^x  
    idx_pos = m>0; fQOaTsyA  
    idx_neg = m<0; o }Tv^>L  
    HFo}r~  
    z = y; FuEHO6nx  
    if any(idx_pos) s15f <sp  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); -7=pb#y  
    end =%2 E|/  
    if any(idx_neg) \sp7[}Sw  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ;pAkdX&b  
    end g](m& O  
    dE ^(KBF  
    % 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) 8d Ftp3(  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ?hfos Bn&[  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ZQ~?  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Or_9KX2  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, . `hlw'20  
    %   and THETA is a vector of angles.  R and THETA must have the same h[XGFz  
    %   length.  The output Z is a matrix with one column for every P-value, *}mtVa_|  
    %   and one row for every (R,THETA) pair. tH W"eag  
    % ]}'WNy6c&x  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 3%cNePlr  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) vo0[Z,aH5  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) v- {kPc=:#  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 gO$!_!@LM  
    %   for all p. !w C4ei`  
    % Y61E|:fV!  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Crww\#E;  
    %   Zernike functions (order N<=7).  In some disciplines it is mtTJm4  
    %   traditional to label the first 36 functions using a single mode c)E'',-J_2  
    %   number P instead of separate numbers for the order N and azimuthal 0K#dWc}"a  
    %   frequency M. ZnmBb_eX  
    % 08.dV<P  
    %   Example: %mxG;w$  
    % Jwa2Y0  
    %       % Display the first 16 Zernike functions >6rPDzW`Dx  
    %       x = -1:0.01:1; l@Ml8+  
    %       [X,Y] = meshgrid(x,x); ;dPLi4=o  
    %       [theta,r] = cart2pol(X,Y); Kt`0vwkjvI  
    %       idx = r<=1; JH?[hb  
    %       p = 0:15; L[O+9Yh  
    %       z = nan(size(X)); aEL6-['(  
    %       y = zernfun2(p,r(idx),theta(idx)); ?!A{n3\<  
    %       figure('Units','normalized') P@*whjPmo  
    %       for k = 1:length(p) vWj|[| <rX  
    %           z(idx) = y(:,k); IHB{US1G  
    %           subplot(4,4,k) 5gEUE{S  
    %           pcolor(x,x,z), shading interp OSq"q-Q  
    %           set(gca,'XTick',[],'YTick',[]) 2QBq  
    %           axis square 3UH=wmG0w  
    %           title(['Z_{' num2str(p(k)) '}']) 4~1_%wb  
    %       end E%40u.0  
    % O #0:6QX  
    %   See also ZERNPOL, ZERNFUN. nQ/El&{  
    .m+KXlP  
    %   Paul Fricker 11/13/2006 Ag?@fuk$J  
    f+F /`P%  
    R%5\1!Fl=G  
    % Check and prepare the inputs: UUA7m$F1  
    % ----------------------------- |yqx ]  
    if min(size(p))~=1 -5E%f|U  
        error('zernfun2:Pvector','Input P must be vector.') YZmD:P  
    end 5[;p<GqGN  
    rL_AqSGAK1  
    if any(p)>35 ]Oe#S"-Oo  
        error('zernfun2:P36', ... Z!hDTT  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... o[8Y%3  
               '(P = 0 to 35).']) Kk#8r+ ,  
    end B:SzCC.B  
    o&X!75^G>  
    % Get the order and frequency corresonding to the function number: Y&+<'FA  
    % ---------------------------------------------------------------- O":x$>'t  
    p = p(:); 3+(Fq5I  
    n = ceil((-3+sqrt(9+8*p))/2); #t{?WkO[  
    m = 2*p - n.*(n+2); ``zg |h  
    .YLg^JfZ  
    % Pass the inputs to the function ZERNFUN: 0HF",:yl  
    % ---------------------------------------- *<BasP  
    switch nargin -3bl !9h^  
        case 3 YSeXCJ:Iy  
            z = zernfun(n,m,r,theta); cMtkdIO  
        case 4 6rPe\'n=B  
            z = zernfun(n,m,r,theta,nflag); c\-I+lMBi  
        otherwise  "X}!j>-  
            error('zernfun2:nargin','Incorrect number of inputs.') <5q:mG88  
    end l-M~e]  
    .F> c Z,  
    % 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) ^[d)Hk}L  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. whxE[Xnv  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Ugrcy7  
    %   order N and frequency M, evaluated at R.  N is a vector of ")cdY) 14"  
    %   positive integers (including 0), and M is a vector with the b9`MUkGGd  
    %   same number of elements as N.  Each element k of M must be a y{5ZC~Z<!  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) .4.zy]I  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is #;2Ju'e#z  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix v#=-  
    %   with one column for every (N,M) pair, and one row for every &!m;s_gi  
    %   element in R. TRX; m|   
    % piY=(y&3  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- WG(tt.  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is A1Rt  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ezy0m}@   
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 [u/g =^+u  
    %   for all [n,m]. &LHQ) ?  
    % 8?P@<Do%  
    %   The radial Zernike polynomials are the radial portion of the >qCUs3}C{*  
    %   Zernike functions, which are an orthogonal basis on the unit S}ZM;M  
    %   circle.  The series representation of the radial Zernike e9"<.:&  
    %   polynomials is \jZvP`.2  
    % D.U)R7(  
    %          (n-m)/2 uppA`>  
    %            __ C$ nT&06o  
    %    m      \       s                                          n-2s R:Z{,R+  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r wD}[XE?S  
    %    n      s=0 VO[s:e9L  
    % uu]<R@!J  
    %   The following table shows the first 12 polynomials. !<@k\~9^D  
    % C6:<.`iD87  
    %       n    m    Zernike polynomial    Normalization SJj0*ry:  
    %       --------------------------------------------- ImyB4welo  
    %       0    0    1                        sqrt(2) OB l-6W  
    %       1    1    r                           2 >*{\N^:z  
    %       2    0    2*r^2 - 1                sqrt(6) Y7]N.G3,]  
    %       2    2    r^2                      sqrt(6) j`ggg]"&$  
    %       3    1    3*r^3 - 2*r              sqrt(8) W UDQb5k  
    %       3    3    r^3                      sqrt(8) %/-Z1Nv*#  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) r9z/hm}E  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) IHMZE42  
    %       4    4    r^4                      sqrt(10) doVBVTk^  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) FC/m,D50oI  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 4E&URl0Bh  
    %       5    5    r^5                      sqrt(12) >mi%L3Pk  
    %       --------------------------------------------- xG\&QE  
    % Bp>%'L  
    %   Example: "JKrbgN@;L  
    % ld$LG6[PA  
    %       % Display three example Zernike radial polynomials OGrp {s  
    %       r = 0:0.01:1; ={YW*1Xw  
    %       n = [3 2 5]; 0;} 9XZ  
    %       m = [1 2 1]; x]XhWScr '  
    %       z = zernpol(n,m,r); thl{IU  
    %       figure 2< w/GX.  
    %       plot(r,z) >}43MxU?  
    %       grid on K{t7_i#tv  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') qun#z$  
    % /`?i&\C3r  
    %   See also ZERNFUN, ZERNFUN2. ?_(0cVi  
    z?Hvh  
    % A note on the algorithm. )CYSU(YTD  
    % ------------------------ 8s6[?=nM  
    % The radial Zernike polynomials are computed using the series WojZ[j>  
    % representation shown in the Help section above. For many special K q: +{'  
    % functions, direct evaluation using the series representation can |9{l8`9}_  
    % produce poor numerical results (floating point errors), because Xu3o,k  
    % the summation often involves computing small differences between vZq7U]RW  
    % large successive terms in the series. (In such cases, the functions M)!8 `]  
    % are often evaluated using alternative methods such as recurrence =YE"6iU  
    % relations: see the Legendre functions, for example). For the Zernike cRDjpc]  
    % polynomials, however, this problem does not arise, because the p&_Kb\} U  
    % polynomials are evaluated over the finite domain r = (0,1), and S%R:GZEf_  
    % because the coefficients for a given polynomial are generally all VSc;}LH  
    % of similar magnitude. "=MRzSke3  
    % .3Jggp  
    % ZERNPOL has been written using a vectorized implementation: multiple Z; r}G m  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] xoA\^AA  
    % values can be passed as inputs) for a vector of points R.  To achieve yOxJx7uD  
    % this vectorization most efficiently, the algorithm in ZERNPOL O\q|b#q}/  
    % involves pre-determining all the powers p of R that are required to V+W,# 5  
    % compute the outputs, and then compiling the {R^p} into a single X0* y8"  
    % matrix.  This avoids any redundant computation of the R^p, and e(@YBQ/Z  
    % minimizes the sizes of certain intermediate variables. XuVbi=pN.2  
    % bT@3fuL4  
    %   Paul Fricker 11/13/2006 EXK~Zf|&Z  
    Ha)eeE$  
    aqK<}jy  
    % Check and prepare the inputs: 4xjk^N9  
    % ----------------------------- 24}r;=U  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) J'sVT{@GS  
        error('zernpol:NMvectors','N and M must be vectors.') .\7R/cP}{A  
    end ]/XNfb  
    vClD)Ar  
    if length(n)~=length(m) =q.2S; ?  
        error('zernpol:NMlength','N and M must be the same length.') X8~ cWW  
    end  I@08F  
    _S7GkpoK  
    n = n(:); s_y Y,Z:  
    m = m(:); T_lexX[\  
    length_n = length(n); {*bXO8vi((  
    KA# 4iu{  
    if any(mod(n-m,2)) ^sY ]N77  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') m,K0BL  
    end *6cP-Vzd  
    40<ifz[7  
    if any(m<0) 'QFf 7A  
        error('zernpol:Mpositive','All M must be positive.') S^HuQe!#  
    end oC#@9>+@+"  
    '-p<E"#4Z  
    if any(m>n) r]iec{ ^  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') i:0~%X  
    end U-6b><  
    ]7}2"?J4v  
    if any( r>1 | r<0 ) 1tHTjEG4^3  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 7rw}q~CE5  
    end 6Daz1Pxd+  
    KGS=(z  
    if ~any(size(r)==1) %,g6:Zc@  
        error('zernpol:Rvector','R must be a vector.') ?*zRM?*  
    end ZY-W~p1:G  
    i9[=x(-@  
    r = r(:); |_{-hNiz0  
    length_r = length(r); g!(j.xe  
    |tC!`.^\  
    if nargin==4 BaIH7JLZ8  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); f' Dl*d  
        if ~isnorm Ouc=4'$-  
            error('zernpol:normalization','Unrecognized normalization flag.') 5K|1Y#X  
        end nyD(G=Q5  
    else #8z2>&:|  
        isnorm = false; a938l^@;s8  
    end $rD&rsx6  
    @x3x/g U  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ";AM3  
    % Compute the Zernike Polynomials n KC$ KC  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% D|} y{~  
    Z.Z+cFi  
    % Determine the required powers of r: h1} x2  
    % ----------------------------------- hVo]fD|W  
    rpowers = [];  T},Nqt<  
    for j = 1:length(n) {.v-  
        rpowers = [rpowers m(j):2:n(j)]; 73OFFKbsk  
    end C vfm ,BL  
    rpowers = unique(rpowers); z@iu$DZ  
    y[BUWas(  
    % Pre-compute the values of r raised to the required powers, @2c Gx/1#  
    % and compile them in a matrix: ;0(|06=  
    % ----------------------------- (Vnv"= (  
    if rpowers(1)==0 N '2Nv  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); V\r!H>  
        rpowern = cat(2,rpowern{:}); 7'\<\oT  
        rpowern = [ones(length_r,1) rpowern]; yyb8l l?@a  
    else _"%mLH=!8  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 4Z5ZV!  
        rpowern = cat(2,rpowern{:}); #?-2f{  
    end |pqc(B u  
    *}DCxv  
    % Compute the values of the polynomials: //S/pCqED  
    % -------------------------------------- cL}} ^  
    z = zeros(length_r,length_n); 8%q:lI  
    for j = 1:length_n i;>Yx#  
        s = 0:(n(j)-m(j))/2; 6Ty;m>j  
        pows = n(j):-2:m(j); H5j6$y|I|N  
        for k = length(s):-1:1 qKag'0e  
            p = (1-2*mod(s(k),2))* ... D&KRJQ/  
                       prod(2:(n(j)-s(k)))/          ... {Hg.ctam  
                       prod(2:s(k))/                 ... yU]NgG=z:-  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... f-&4x_5  
                       prod(2:((n(j)+m(j))/2-s(k))); \7o&'zEw  
            idx = (pows(k)==rpowers); Gv?3T Am8  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ".Z|zt6C  
        end 31v0V:j  
         LPO:K a  
        if isnorm }xXUCU<  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ^ur?da9z'  
        end o|FjNL  
    end ,xmmS\  
    ErmlM#u  
    % 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)  =".sCV9"N  
    AqZ()p*z  
    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。 F'^y?UP[  
    ^D]y<@01  
    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)