切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 10475阅读
    • 9回复

    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

    上一主题 下一主题
    离线niuhelen
     
    发帖
    19
    光币
    28
    光券
    0
    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 fBHkLRFH  
    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! 73'U#@g6  
     
    分享到
    离线phility
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    19
    光币
    28
    光券
    0
    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 V\iIvBpWg  
    function z = zernfun(n,m,r,theta,nflag) ~)! V8  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. xWC\954  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N WU+Jo@]y  
    %   and angular frequency M, evaluated at positions (R,THETA) on the >K_$[qP3  
    %   unit circle.  N is a vector of positive integers (including 0), and XPc9z}/(e  
    %   M is a vector with the same number of elements as N.  Each element J[<D/WIH  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) O4b-A3:  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, F8|5_214'  
    %   and THETA is a vector of angles.  R and THETA must have the same vOvxQS}dBp  
    %   length.  The output Z is a matrix with one column for every (N,M) P+*rWJ8gQ  
    %   pair, and one row for every (R,THETA) pair. ]X >QLD0W  
    % k$UzBxR  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Xa?6#  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), "6I-]:K-  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral !T#8N7J>  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 9sfB+]}h  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized '-nuH;r  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. giPhW>  
    % 4'}_qAT  
    %   The Zernike functions are an orthogonal basis on the unit circle. UtW"U0A  
    %   They are used in disciplines such as astronomy, optics, and Z3X&<Y5  
    %   optometry to describe functions on a circular domain. u>3&.t@hU1  
    % NE=#5?6%g7  
    %   The following table lists the first 15 Zernike functions. fwnYzd3  
    % M0;t%*1  
    %       n    m    Zernike function           Normalization Y1U"HqNl*  
    %       -------------------------------------------------- V<~_OF  
    %       0    0    1                                 1 HdY3DdC%q  
    %       1    1    r * cos(theta)                    2 bG?WB,1  
    %       1   -1    r * sin(theta)                    2 OIXAjU*N  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ~kSnXJv  
    %       2    0    (2*r^2 - 1)                    sqrt(3) QigoRB!z#9  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) '1kj:Np  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) XoH[MJC  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 0w'y#U)&8  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) {d?4;Kd  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 6'No4[F 4n  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) U!;aM*67  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7=QC+XSO  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) RIVL 0Ig  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f@F^W YQm  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) }]39 iK`w  
    %       -------------------------------------------------- Vlp*'2VO  
    % [o[v"e\w  
    %   Example 1: 7n\j"0z  
    % 0ez i?Um  
    %       % Display the Zernike function Z(n=5,m=1) ?,i#B'Z^  
    %       x = -1:0.01:1; '^-4{Y^2E  
    %       [X,Y] = meshgrid(x,x); x^='pEt{  
    %       [theta,r] = cart2pol(X,Y); ~*cY&  9  
    %       idx = r<=1; yqVaA 'w5  
    %       z = nan(size(X)); Zjp5\+hHV  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); P/gb+V=g!  
    %       figure @]ptY*   
    %       pcolor(x,x,z), shading interp d4/`:?w  
    %       axis square, colorbar gGiV1jN _  
    %       title('Zernike function Z_5^1(r,\theta)') v_@#hf3  
    % YP\4XI  
    %   Example 2: xXY)KI N[  
    % xo)?XFM2  
    %       % Display the first 10 Zernike functions S] K6qY  
    %       x = -1:0.01:1; GdfK xSO  
    %       [X,Y] = meshgrid(x,x); YnO1Lf@  
    %       [theta,r] = cart2pol(X,Y); &6|^~(P?  
    %       idx = r<=1; h@>rjeY@  
    %       z = nan(size(X)); 9i2vWSga  
    %       n = [0  1  1  2  2  2  3  3  3  3]; a9@l8{)RX  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; sNk>0 X[  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Y(I*%=:$  
    %       y = zernfun(n,m,r(idx),theta(idx)); H:{(CY?t  
    %       figure('Units','normalized') :DX/r  
    %       for k = 1:10 vu.S>2Wv  
    %           z(idx) = y(:,k); ]N(zom_0d  
    %           subplot(4,7,Nplot(k)) ">D(+ xr!)  
    %           pcolor(x,x,z), shading interp %dk$K!5D0  
    %           set(gca,'XTick',[],'YTick',[]) *l?% o{  
    %           axis square <>*''^  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) gH{\y5%rO  
    %       end WfjUJw5x"s  
    % sm&rR=b  
    %   See also ZERNPOL, ZERNFUN2. CO%O<_C  
    "w|k\1D  
    %   Paul Fricker 11/13/2006 BE2\?q-  
    a+a%}76N  
    z`r4edk3  
    % Check and prepare the inputs: GLKN<2|2@y  
    % ----------------------------- (27F   
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) aXK%m  
        error('zernfun:NMvectors','N and M must be vectors.') ,tR'0&=  
    end O*n%2Mam  
    Y`O}]*{>8R  
    if length(n)~=length(m) A_q3p\b  
        error('zernfun:NMlength','N and M must be the same length.') %k;FxUKi  
    end v!'@NW_  
    OB  i!fLa  
    n = n(:); z#E,96R  
    m = m(:); )RCqsFjK  
    if any(mod(n-m,2)) h9n<ped`A;  
        error('zernfun:NMmultiplesof2', ... \=G Xe.}4d  
              'All N and M must differ by multiples of 2 (including 0).') )J6b:W  
    end eg~^wi  
    ]zMBZs  
    if any(m>n) JK8@J9(#  
        error('zernfun:MlessthanN', ... MVL }[J  
              'Each M must be less than or equal to its corresponding N.') xo_k"'f+  
    end "vRqtEBO@  
    a3 _0F@I  
    if any( r>1 | r<0 ) BiLreZ~"  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') .idl@%  
    end 4a\+o]  
    O>F.Wf5g  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Cg\)BHv~  
        error('zernfun:RTHvector','R and THETA must be vectors.') xY'YbHFz  
    end  iIEIGQx  
    Joo)GIB  
    r = r(:); vAhO!5]>\  
    theta = theta(:); oJu4vGy0  
    length_r = length(r); %C][E^9  
    if length_r~=length(theta) x w83K  
        error('zernfun:RTHlength', ... wkpVX*DfRE  
              'The number of R- and THETA-values must be equal.') U)%u`C0  
    end ~u`! Gi  
    !<PTsk F  
    % Check normalization: qmyZbo|8&  
    % -------------------- &E '>+6  
    if nargin==5 && ischar(nflag) `IRT w"  
        isnorm = strcmpi(nflag,'norm'); 9*Twx&  
        if ~isnorm 6)<oO(  
            error('zernfun:normalization','Unrecognized normalization flag.') o%>nu  
        end >)Z2bCe  
    else WIl S^?5I<  
        isnorm = false; ]G&\L~P  
    end 44{:UhJkx  
    vlyNQ7"%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% cCKda3v!O  
    % Compute the Zernike Polynomials <4HuV.K  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G8-d%O p  
    daJ-H  
    % Determine the required powers of r: m/B9)JzY  
    % ----------------------------------- ';!UJWYl  
    m_abs = abs(m); J 2~B<=V  
    rpowers = []; I}0 -  
    for j = 1:length(n) p 8Hv7*  
        rpowers = [rpowers m_abs(j):2:n(j)]; AG%es0D[H  
    end |-Klh  
    rpowers = unique(rpowers); )4~XZt1r  
    s/^= WV  
    % Pre-compute the values of r raised to the required powers, *<5lx[:4/x  
    % and compile them in a matrix: d }CMX$1  
    % ----------------------------- XxQ2g&USk  
    if rpowers(1)==0 'N/%SRk  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); n ,%^R  
        rpowern = cat(2,rpowern{:}); 8(K~QvE~  
        rpowern = [ones(length_r,1) rpowern]; >Nqkz?67  
    else =n?@My?;  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); #!j&L6  
        rpowern = cat(2,rpowern{:}); 5d;K.O  
    end [beuDZA  
    g+{MvSj$  
    % Compute the values of the polynomials: r 24]2A  
    % -------------------------------------- ;b2>y>?[  
    y = zeros(length_r,length(n)); UM^hF%  
    for j = 1:length(n) l%w|f`B:  
        s = 0:(n(j)-m_abs(j))/2; ~%q e,  
        pows = n(j):-2:m_abs(j); u-cC}DP  
        for k = length(s):-1:1 kQcQi}e  
            p = (1-2*mod(s(k),2))* ... 2a}_|#*  
                       prod(2:(n(j)-s(k)))/              ... .SFwjriZ  
                       prod(2:s(k))/                     ... 8u23@?  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... J sdEA  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); imuHSxcaV  
            idx = (pows(k)==rpowers); !LESRh?  
            y(:,j) = y(:,j) + p*rpowern(:,idx); :udZfA\sW  
        end _+7f+eB  
         @}}1xP4Sr  
        if isnorm y!Eh /KD  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 9$t@Gmn  
        end }Q*ec/^{f  
    end !2,.C+,  
    % END: Compute the Zernike Polynomials <m\TZQBD  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &$ 9bC 't6  
    a#@ opUn-  
    % Compute the Zernike functions: * tqeq y-X  
    % ------------------------------ {GY$J<5=  
    idx_pos = m>0; P|4a}SWU  
    idx_neg = m<0; Cq'r 'cBZ  
    _z< q9:  
    z = y; A-5%_M3\G  
    if any(idx_pos) HxAa,+k  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ijT^gsLL  
    end }\*|b@)]  
    if any(idx_neg) 8A=(,)`}9  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); f5eX%FR  
    end oWT0WS  
    !h;VdCCi#  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) @A%`\Ea%  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. :>u{BG;=79  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 5VS<I\o}  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive a7R7Ks|q  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, # jyAq$I0  
    %   and THETA is a vector of angles.  R and THETA must have the same g>{=R|uO5  
    %   length.  The output Z is a matrix with one column for every P-value, Yy5F'RY  
    %   and one row for every (R,THETA) pair. o@-cT`HP  
    % HvU)GJ u b  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike *HUqW}_r  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) j@f(cRAf#  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) N~_gT Jr~P  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 >3/<goXk7  
    %   for all p. :/08}!_:  
    % S45jY=)z  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 m;|I}{r  
    %   Zernike functions (order N<=7).  In some disciplines it is dcsd//E  
    %   traditional to label the first 36 functions using a single mode 01b0;|  
    %   number P instead of separate numbers for the order N and azimuthal 5Dd;?T>  
    %   frequency M. MH-,+-Eq  
    % s5 BV8 M  
    %   Example: CEiG jo^  
    % ">7 bnOJ  
    %       % Display the first 16 Zernike functions b?l\Q Mvi  
    %       x = -1:0.01:1; =6hf'lP  
    %       [X,Y] = meshgrid(x,x); Yi3DoaS;"  
    %       [theta,r] = cart2pol(X,Y); 5;+Bl@zGu  
    %       idx = r<=1; -#@;-2w  
    %       p = 0:15; f sMF46  
    %       z = nan(size(X)); `O F\f  
    %       y = zernfun2(p,r(idx),theta(idx)); YR>xh2< 9  
    %       figure('Units','normalized') u=5^xpI<D  
    %       for k = 1:length(p) tBt\&{=|D  
    %           z(idx) = y(:,k); wS*UXF&f  
    %           subplot(4,4,k) Mh\c+1MFs  
    %           pcolor(x,x,z), shading interp G9]GK+@&F  
    %           set(gca,'XTick',[],'YTick',[]) !q?}[E2  
    %           axis square 3z#16*  
    %           title(['Z_{' num2str(p(k)) '}']) >8c9-dTmf  
    %       end ay2.C BF  
    % wcO_;1_ H  
    %   See also ZERNPOL, ZERNFUN. ;)*Drk*t,  
    `%VrT`  
    %   Paul Fricker 11/13/2006 NB[b[1 Ch  
    >A6lX)  
    %NuS!v>  
    % Check and prepare the inputs: d] {^  
    % ----------------------------- 3:r;(IaX  
    if min(size(p))~=1 %~@}wHMB  
        error('zernfun2:Pvector','Input P must be vector.') 3Dy.mtP  
    end `R\0g\  
    5_PD ?lg  
    if any(p)>35 z`W$/tw"  
        error('zernfun2:P36', ... z;LntQZp-  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... !.!Ervi!N  
               '(P = 0 to 35).']) awUIYAgJ3  
    end N?aU<-Tn  
    3>Yec6Hs  
    % Get the order and frequency corresonding to the function number: Q'Q^K  
    % ---------------------------------------------------------------- k&^fIz  
    p = p(:); Q%6*S!~  
    n = ceil((-3+sqrt(9+8*p))/2); %NKf@If)  
    m = 2*p - n.*(n+2); N:0mjHG  
    m]85F^R0  
    % Pass the inputs to the function ZERNFUN: $WDa} ~j~^  
    % ---------------------------------------- z}Q54,9m  
    switch nargin \ /o`CV{O  
        case 3 Gk<h_1WWK  
            z = zernfun(n,m,r,theta); h4]yIM `8d  
        case 4 w%kxY5q  
            z = zernfun(n,m,r,theta,nflag); <)&;9C  
        otherwise 0HE@L_$;2  
            error('zernfun2:nargin','Incorrect number of inputs.') E[ ,Ur`>:  
    end Rh%x5RFFc  
    -*3wNGh {  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) 8|gwH2 st~  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. lO[[iMHl<  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ka655O/)&  
    %   order N and frequency M, evaluated at R.  N is a vector of :\>@yCD  
    %   positive integers (including 0), and M is a vector with the W EZ)7H  
    %   same number of elements as N.  Each element k of M must be a Fq:BRgCE  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) @xR=bWY  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is I;9>$?t[  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix  RCKb5p9  
    %   with one column for every (N,M) pair, and one row for every Bf.@B0\  
    %   element in R. uN2Ck  
    % 46sV\In>?  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- S U04q+  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is B\`4TU}kE  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to /g@!#Dt  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 XI@;;>D1=U  
    %   for all [n,m]. pxjb^GZ0  
    % _O{3bIay3!  
    %   The radial Zernike polynomials are the radial portion of the !c/G'se  
    %   Zernike functions, which are an orthogonal basis on the unit X&b)E0]pR  
    %   circle.  The series representation of the radial Zernike 7vZznN8e  
    %   polynomials is U*b1yxt  
    % yal T6  
    %          (n-m)/2 _46 y  
    %            __ edD19A  
    %    m      \       s                                          n-2s p@0Va  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r xj\! Sn2  
    %    n      s=0 jt=%oa  
    % eT0Yp  
    %   The following table shows the first 12 polynomials. ?U$H`[VF}  
    % 0e-M 24,C  
    %       n    m    Zernike polynomial    Normalization u[k0z!p_ c  
    %       --------------------------------------------- K?`Fpg (  
    %       0    0    1                        sqrt(2)  [,JUC<  
    %       1    1    r                           2 >|IUjv2L  
    %       2    0    2*r^2 - 1                sqrt(6) Pv@Lx+ k  
    %       2    2    r^2                      sqrt(6) j#6@ cO'`  
    %       3    1    3*r^3 - 2*r              sqrt(8) <A"[Wk  
    %       3    3    r^3                      sqrt(8) RDGefxv  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) cgY + xd@  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) AUBZ7*VO  
    %       4    4    r^4                      sqrt(10) EbXWCD  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) H}vq2|MN  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) GI']&{  
    %       5    5    r^5                      sqrt(12) f-$%Ck$%,  
    %       --------------------------------------------- kzozjh%`9h  
    % Q6%dM'fR  
    %   Example: NpqK+GO  
    % {-a8^IK,  
    %       % Display three example Zernike radial polynomials 3M~*4  
    %       r = 0:0.01:1; _=$:<wIE[  
    %       n = [3 2 5]; ?y"= jn  
    %       m = [1 2 1]; H.-VfROi2  
    %       z = zernpol(n,m,r); GE?M. '!{{  
    %       figure `?P)RS30  
    %       plot(r,z) [ H|ifi  
    %       grid on U}hQVpP#  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 3;v%78[&P  
    % @ (4$<><  
    %   See also ZERNFUN, ZERNFUN2. 'Sk-L 5  
    @?bO@  
    % A note on the algorithm. `YL)[t? V  
    % ------------------------ #u]'3en  
    % The radial Zernike polynomials are computed using the series zw ,( kv  
    % representation shown in the Help section above. For many special ,.6)y1!  
    % functions, direct evaluation using the series representation can | 6/ # H*  
    % produce poor numerical results (floating point errors), because !N"Y  
    % the summation often involves computing small differences between ?OD43y1rzd  
    % large successive terms in the series. (In such cases, the functions >/J!:Htk+K  
    % are often evaluated using alternative methods such as recurrence #"&<^  
    % relations: see the Legendre functions, for example). For the Zernike 5,vw%F-m  
    % polynomials, however, this problem does not arise, because the mqrV:3}  
    % polynomials are evaluated over the finite domain r = (0,1), and k z{_H`5.  
    % because the coefficients for a given polynomial are generally all D^dos`L0b  
    % of similar magnitude. R-[t 4BHn  
    % `%@| sK2  
    % ZERNPOL has been written using a vectorized implementation: multiple crvq]J5  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] n.L/Xp@gc  
    % values can be passed as inputs) for a vector of points R.  To achieve ]'q"Kw/10  
    % this vectorization most efficiently, the algorithm in ZERNPOL n=_jmR1  
    % involves pre-determining all the powers p of R that are required to yUY* l@v]  
    % compute the outputs, and then compiling the {R^p} into a single MqKf'6z  
    % matrix.  This avoids any redundant computation of the R^p, and vXI2u;=y  
    % minimizes the sizes of certain intermediate variables. 6v1F. u  
    % @a~GHG[x  
    %   Paul Fricker 11/13/2006 P[q 'Y^\  
    ))9w)A@  
    _-6IB>  
    % Check and prepare the inputs: )y#~eYn  
    % ----------------------------- A,fPl R  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) |K(j XZ)  
        error('zernpol:NMvectors','N and M must be vectors.') CO5>Q o  
    end qi51'@  
    dsrKHi  
    if length(n)~=length(m) =CqZ$  
        error('zernpol:NMlength','N and M must be the same length.') F4X0DRC,G  
    end oj$^87KX  
    09_5niaz[  
    n = n(:); 6C@W6DR3N  
    m = m(:); ?yNg5z  
    length_n = length(n); $C.;GUEQ  
    qvHRP@  
    if any(mod(n-m,2)) '.$va<  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') T*3>LY+bb  
    end n-)Xs;`2  
    ] -}Zd\Rs  
    if any(m<0) J{w[vcf  
        error('zernpol:Mpositive','All M must be positive.') \g;o9}@3~  
    end ud`!X#e~  
    c|hT\1XR,  
    if any(m>n) <$+Cd=71\  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') N3U.62  
    end ;;{!wA+"D  
    =jEh#  
    if any( r>1 | r<0 ) *f ;">(`o*  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') F?y4 L9|e  
    end iVdY\+N!<  
    ^hyY,X  
    if ~any(size(r)==1) 0Z,a3)jcc  
        error('zernpol:Rvector','R must be a vector.') ~9Jlb-*I5  
    end }<7S% ?TY  
    dd> qy  
    r = r(:); BXj]]S2  
    length_r = length(r); uW{;@ 7N  
    %Bf;F;xuB  
    if nargin==4 Xe. az  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); [+8in\T i  
        if ~isnorm Z8xKg  
            error('zernpol:normalization','Unrecognized normalization flag.') :GBM`f@  
        end 8~@?cy1j!  
    else !kG2$/lR  
        isnorm = false; <RaUs2Q3.  
    end ?nc:B]=pTY  
    nMT"Rp  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s k>E(Myo  
    % Compute the Zernike Polynomials @4FG & >kQ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^V;h>X|  
    =_)yV0  
    % Determine the required powers of r: Y Z.? k4>  
    % ----------------------------------- '2=$pw  
    rpowers = []; x(r~<a[  
    for j = 1:length(n) @ )< 3Z  
        rpowers = [rpowers m(j):2:n(j)]; </B<=tc  
    end >u=Dc.lX  
    rpowers = unique(rpowers); kS)azV  
    KP*cb6vA  
    % Pre-compute the values of r raised to the required powers, 41oXOB  
    % and compile them in a matrix: ;GF+0~5>  
    % ----------------------------- F15Yn  
    if rpowers(1)==0 zxhE9 [`*e  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); gAxf5 A_x)  
        rpowern = cat(2,rpowern{:}); 8Ts_;uId  
        rpowern = [ones(length_r,1) rpowern]; s-lNpOi  
    else *^=zQ~  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Z6\H4,k&  
        rpowern = cat(2,rpowern{:}); D.:6X'hp  
    end ^Yg}>?0  
    vOV$Hle  
    % Compute the values of the polynomials: !QXPn}q^0  
    % -------------------------------------- )wdTs>W7  
    z = zeros(length_r,length_n); W9M~2< L  
    for j = 1:length_n 8{)j"rghah  
        s = 0:(n(j)-m(j))/2; )z Hib;O  
        pows = n(j):-2:m(j); zg+6< .Sf  
        for k = length(s):-1:1 )z=L^ot  
            p = (1-2*mod(s(k),2))* ... 0E^6"nt7N  
                       prod(2:(n(j)-s(k)))/          ... mR3-+dB/  
                       prod(2:s(k))/                 ... 1n-+IR"  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... !U[/P6 +0  
                       prod(2:((n(j)+m(j))/2-s(k))); ^X\SwgD2w  
            idx = (pows(k)==rpowers); Q xm:5P  
            z(:,j) = z(:,j) + p*rpowern(:,idx); (Ee5Af,4  
        end 7%)KB4(\_  
         =6H  
        if isnorm i8Xz'Sw07  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); Ef2i#BoZ  
        end T6^ H%;G  
    end /O {iL:`  
    2Sb68hJIE  
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    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
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
    发帖
    850
    光币
    833
    光券
    0
    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
    发帖
    4352
    光币
    5476
    光券
    1
    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)  J~lKN <w  
    7'RU\0QG  
    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。 cdTG ]n  
    #K\;)z(?  
    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)