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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 Oc; G(l(  
    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! S Z$Kz n  
     
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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 s}% M4  
    function z = zernfun(n,m,r,theta,nflag) fx>4  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 'y3!fN =h  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N MFAH%Z$  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ';=O 0)u  
    %   unit circle.  N is a vector of positive integers (including 0), and <<R*2b  
    %   M is a vector with the same number of elements as N.  Each element V% 6I\G2/:  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) KNIn:K^/  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, QW(Mz Hg  
    %   and THETA is a vector of angles.  R and THETA must have the same 8q}q{8  
    %   length.  The output Z is a matrix with one column for every (N,M) W]5w \  
    %   pair, and one row for every (R,THETA) pair.  O+Y6N  
    % h{HHLR  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ` v@m-j6  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), p sMvq@>  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral (c &mCJN  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, tHwMX1 IG  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized "mvt>X  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. (rm?jDm   
    % JB[~;nLlC  
    %   The Zernike functions are an orthogonal basis on the unit circle. EGF '"L  
    %   They are used in disciplines such as astronomy, optics, and \Et3|Iv  
    %   optometry to describe functions on a circular domain.  o!ebs0  
    % X LOh7(  
    %   The following table lists the first 15 Zernike functions. 6.nCV 0xA  
    % o]M5b;1  
    %       n    m    Zernike function           Normalization <s<n  
    %       -------------------------------------------------- PKg@[<g43  
    %       0    0    1                                 1 ]a*d#  
    %       1    1    r * cos(theta)                    2 wHMX=N1/  
    %       1   -1    r * sin(theta)                    2 .Od !0(0  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) MC.) 2B7  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 4-H+vNG{%  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) LR.<&m%~.  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) CSq4x5!_7>  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) )g#T9tx2D  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) *@=/qkaJaI  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) !0LWa"  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) dufu|BL|}  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) zFff`]^`  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) c>:wd@w  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 3>`mI8 $t  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) .Una+Z  
    %       -------------------------------------------------- RF53Jyt  
    % 9BBmw(M}  
    %   Example 1: ( !fKNia@S  
    % peuZ&yK+"  
    %       % Display the Zernike function Z(n=5,m=1) EPM-df!=  
    %       x = -1:0.01:1; Y}|X|!0x  
    %       [X,Y] = meshgrid(x,x); ca*DZG/  
    %       [theta,r] = cart2pol(X,Y); tKx~1-  
    %       idx = r<=1; MSqVlj  
    %       z = nan(size(X)); 4`]^@"{  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); D_^ nI:  
    %       figure  J^5So  
    %       pcolor(x,x,z), shading interp *>'V1b4}  
    %       axis square, colorbar ?u=Fj_N_  
    %       title('Zernike function Z_5^1(r,\theta)') d#rf5<i  
    % aPfO$b:  
    %   Example 2: 6J6BF%  
    % 1 A !bE  
    %       % Display the first 10 Zernike functions udUyh%n  
    %       x = -1:0.01:1; JZ*/,|1}EC  
    %       [X,Y] = meshgrid(x,x); =llvuUd\n  
    %       [theta,r] = cart2pol(X,Y); u jq=F  
    %       idx = r<=1; FvXZ<(A{  
    %       z = nan(size(X)); KNpl:g3{<Q  
    %       n = [0  1  1  2  2  2  3  3  3  3]; "ZoRZ'i  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; >#~& -3  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; A) %/[GD2  
    %       y = zernfun(n,m,r(idx),theta(idx)); xU>WEm2  
    %       figure('Units','normalized') ,nLy4T&"  
    %       for k = 1:10 0g y/:T  
    %           z(idx) = y(:,k); u#;7<.D  
    %           subplot(4,7,Nplot(k)) xH(lm2kvT  
    %           pcolor(x,x,z), shading interp }`QUHIF  
    %           set(gca,'XTick',[],'YTick',[]) ag#S6E^%S  
    %           axis square )Y6 +  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) G"U9E5O  
    %       end w/S%YW3*  
    % A8fOQ  
    %   See also ZERNPOL, ZERNFUN2. so)[59M7  
    H*&f:mfq  
    %   Paul Fricker 11/13/2006 (*nT(Adk  
    6YLj^w] %  
    QP^Cx=  
    % Check and prepare the inputs: 3kIN~/<R+7  
    % ----------------------------- gG:Vt}N  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ZaDyg"Tw+  
        error('zernfun:NMvectors','N and M must be vectors.') C]eSizS.  
    end v/0QOp  
    ~u!|qM  
    if length(n)~=length(m) N^ds RYC  
        error('zernfun:NMlength','N and M must be the same length.') W*4-.*U8a  
    end #ASz;$P  
    7>|J8*/Nd  
    n = n(:); )}]g] g  
    m = m(:); gA5/,wDO  
    if any(mod(n-m,2)) {M$1N5Eh  
        error('zernfun:NMmultiplesof2', ... _ Yx]_Y9I  
              'All N and M must differ by multiples of 2 (including 0).') i]y<|W)Q3  
    end +p_CN*10H  
    |vwVghC  
    if any(m>n) 5C*Pd Wpl  
        error('zernfun:MlessthanN', ... [vK ^Um  
              'Each M must be less than or equal to its corresponding N.') YTpSHpf@  
    end b#Z{{eLny  
    *@r/5pM2}  
    if any( r>1 | r<0 ) zh`<WN&H  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') }DE g-j,F  
    end Xe'x[(l  
    f ue(UMF~  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) C'@i/+  
        error('zernfun:RTHvector','R and THETA must be vectors.') <#y[gTJ<'>  
    end 3cyHfpx-W  
    -.@r#d/  
    r = r(:); e&F8m%t  
    theta = theta(:); Y3cMC)  
    length_r = length(r); o&zJ=k[4  
    if length_r~=length(theta) N1S{suic  
        error('zernfun:RTHlength', ... Nw/  ku  
              'The number of R- and THETA-values must be equal.') qIE9$7*X  
    end +z\^t_"f  
    Nk 8B_{  
    % Check normalization: 3{^9]7UC  
    % -------------------- Mj?`j_X  
    if nargin==5 && ischar(nflag) g i-$Z FzB  
        isnorm = strcmpi(nflag,'norm'); ]G= L=D^cK  
        if ~isnorm omu|yCK  
            error('zernfun:normalization','Unrecognized normalization flag.') V-2(?auZd  
        end 6NuD4Ga  
    else F>6|3bOR  
        isnorm = false; cJ n=  
    end Y>T-af49  
    o.g V4%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L>NL:68yN  
    % Compute the Zernike Polynomials ~&_z2|UXp  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wn, KY$/  
    l^-];|Y  
    % Determine the required powers of r: D~iz+{Q4  
    % ----------------------------------- 9@:2wR |  
    m_abs = abs(m); 7~% ?#  
    rpowers = []; (ejvF):|  
    for j = 1:length(n) xY8$I6  
        rpowers = [rpowers m_abs(j):2:n(j)]; r:'.nhe  
    end ,vawzq[oSy  
    rpowers = unique(rpowers); :$|HNeDO  
    z`}qkbvi  
    % Pre-compute the values of r raised to the required powers, o]_dJB  
    % and compile them in a matrix: t%FwXaO#  
    % ----------------------------- TR`U-= jH,  
    if rpowers(1)==0 1~`fVg  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); EhvX)s  
        rpowern = cat(2,rpowern{:}); KYhwOGN  
        rpowern = [ones(length_r,1) rpowern]; E\EsWb  
    else OU.6bmWy|  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ^j7Vt2-  
        rpowern = cat(2,rpowern{:}); }W8;=$jr  
    end nYSiS}?S .  
    cn3\kT*  
    % Compute the values of the polynomials: 3m)0z{n  
    % -------------------------------------- gp?uHKsM  
    y = zeros(length_r,length(n)); 6OIte -c  
    for j = 1:length(n) EU;9 *W<  
        s = 0:(n(j)-m_abs(j))/2; yu|8_<bq  
        pows = n(j):-2:m_abs(j); :#ik. D  
        for k = length(s):-1:1 D%SlAzZ3  
            p = (1-2*mod(s(k),2))* ... ]Sz:|%JP1  
                       prod(2:(n(j)-s(k)))/              ... Yn Mvl  
                       prod(2:s(k))/                     ... "| g>'wM*  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... E GS)b  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); (OL4Ex']  
            idx = (pows(k)==rpowers); 6l1jMm|= X  
            y(:,j) = y(:,j) + p*rpowern(:,idx); |F[+k e  
        end djG*YM\B  
         {9pZ)tB  
        if isnorm 5d^sA;c  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 69NeQ$](  
        end Vwf$JdK%&l  
    end  A,<E\  
    % END: Compute the Zernike Polynomials WDD%Q8ejV&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [yQt^!;  
    p38-l'{#  
    % Compute the Zernike functions: "m8^zg hL  
    % ------------------------------ 6l x>>J!H  
    idx_pos = m>0; &`r-.&Y  
    idx_neg = m<0; "|q& ea rc  
    o#Dk& cH  
    z = y;  8q!]y6  
    if any(idx_pos) lgy <?LI\  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); `HSKQ52  
    end %)1?TU  
    if any(idx_neg) I;(L%TT `  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); BwpqNQN  
    end b. '-?Nn  
    ?e4YGOe.  
    % 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) Nh|uO?&C6  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 9W5lSX#^;  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated <'Eme  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive T?0eVvM  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, "HIRTE;&  
    %   and THETA is a vector of angles.  R and THETA must have the same %wy.TN  
    %   length.  The output Z is a matrix with one column for every P-value, Nai2W<,  
    %   and one row for every (R,THETA) pair. qe#tj/aZ  
    % ,&.!?0+  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike %sP*=5?vA  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 6d}lw6L  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi)  <kqo^  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 IEi^kJflU  
    %   for all p. /,yd+wcW#  
    % S%>]q s  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 7n<{tM  
    %   Zernike functions (order N<=7).  In some disciplines it is p]TAELy  
    %   traditional to label the first 36 functions using a single mode 7JH6A'&  
    %   number P instead of separate numbers for the order N and azimuthal q]-r@yF  
    %   frequency M. Yj49t_$b  
    % M6j y\<a  
    %   Example: C&%_a~  
    % ^ZcGY+/~  
    %       % Display the first 16 Zernike functions g! |kp?  
    %       x = -1:0.01:1; 0{D'n@veP  
    %       [X,Y] = meshgrid(x,x); %tGO?JMkd  
    %       [theta,r] = cart2pol(X,Y); $U WZDD  
    %       idx = r<=1; oG\Vxg*  
    %       p = 0:15; _G@GpkSe>  
    %       z = nan(size(X)); -Q*gW2KmV  
    %       y = zernfun2(p,r(idx),theta(idx)); *g2x%aZWbG  
    %       figure('Units','normalized') XRi8Gpg  
    %       for k = 1:length(p) 4D4j7  
    %           z(idx) = y(:,k); u6JM]kR  
    %           subplot(4,4,k) U[MA)41  
    %           pcolor(x,x,z), shading interp &h/X ku&0  
    %           set(gca,'XTick',[],'YTick',[]) m-, x<bM?  
    %           axis square g2_"zDiw2  
    %           title(['Z_{' num2str(p(k)) '}']) f]CXu3w(J  
    %       end k9!{IScq  
    % ~c `l@:  
    %   See also ZERNPOL, ZERNFUN. } q8ASYNc  
    UaeXY+O  
    %   Paul Fricker 11/13/2006 I efn$  
    e9 B064  
    6i/(5 nQ  
    % Check and prepare the inputs: 5\ nAeP  
    % ----------------------------- |CyE5i0  
    if min(size(p))~=1 ~4'$yWG  
        error('zernfun2:Pvector','Input P must be vector.') rey!{3U  
    end evmeqQG=  
    > ~O.@|  
    if any(p)>35 1yhDrpm  
        error('zernfun2:P36', ... bk[!8- b/a  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ;4\;mmLVk  
               '(P = 0 to 35).']) ww1[rCh\+  
    end (sZ"iGn%  
    3Y$GsN4ln  
    % Get the order and frequency corresonding to the function number: O=7CMbS3  
    % ---------------------------------------------------------------- J|73.&B  
    p = p(:); K-Ef%a2#`  
    n = ceil((-3+sqrt(9+8*p))/2); tCt#%7J;a  
    m = 2*p - n.*(n+2); &oMh]Z*:  
    5{,<j\#L  
    % Pass the inputs to the function ZERNFUN: (tW`=]z-<  
    % ---------------------------------------- ~P-mC@C  
    switch nargin ox.F%)eQ  
        case 3 8}:nGK|kx  
            z = zernfun(n,m,r,theta); %xLh Z\  
        case 4 ~k5W@`"W  
            z = zernfun(n,m,r,theta,nflag); @6-jgw>W2  
        otherwise )^hbsMhO  
            error('zernfun2:nargin','Incorrect number of inputs.') }jPSUdo  
    end N;%6:I./  
    [:*)XeRK  
    % 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) />>\IR  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. _@/8gPT*i  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of DtnEi4h,  
    %   order N and frequency M, evaluated at R.  N is a vector of xgtR6E^k  
    %   positive integers (including 0), and M is a vector with the /Z4et'Lo  
    %   same number of elements as N.  Each element k of M must be a 3Zh)]^  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ;dhQN }7  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is L}NSR  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Etm?'  
    %   with one column for every (N,M) pair, and one row for every zbPqYhJzA  
    %   element in R. 1h5 Akq  
    % m#p'iU*va,  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly-  9gZ$   
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is d'sZxU  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to +"VP-s0  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 RG`1en  
    %   for all [n,m]. xkA K!uVy  
    % $ME)#(  
    %   The radial Zernike polynomials are the radial portion of the Z?z.?a r  
    %   Zernike functions, which are an orthogonal basis on the unit vvOV2n .WD  
    %   circle.  The series representation of the radial Zernike #e5\j\#.  
    %   polynomials is Z/J y'$x  
    % 5kXYeP3:  
    %          (n-m)/2 rrv%~giU  
    %            __ <9 ;!3xG  
    %    m      \       s                                          n-2s HpnWo DM  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Rx}Gz$   
    %    n      s=0 w%sT{(Vd`C  
    % 40 0#v|b  
    %   The following table shows the first 12 polynomials. NaCy@  
    % `P ,d$H "  
    %       n    m    Zernike polynomial    Normalization xN%K^Tree  
    %       --------------------------------------------- CJI~_3+K  
    %       0    0    1                        sqrt(2) xkR0  
    %       1    1    r                           2 v*yuE5{  
    %       2    0    2*r^2 - 1                sqrt(6) 4E?Oky#}-  
    %       2    2    r^2                      sqrt(6) wlmRe`R  
    %       3    1    3*r^3 - 2*r              sqrt(8) pb=h/8R  
    %       3    3    r^3                      sqrt(8) POR\e|hRT]  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) X[TR3[1}  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) FC"8#*x  
    %       4    4    r^4                      sqrt(10) ?0xgRe<  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 29q _BR *:  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) { [>Kob1  
    %       5    5    r^5                      sqrt(12) ;jTN | i'  
    %       --------------------------------------------- 4xJQ!>6  
    % WMP,\=6k0  
    %   Example: <rSF*  
    % RCLeA=/N@0  
    %       % Display three example Zernike radial polynomials Xb,3Dvf  
    %       r = 0:0.01:1; pY$Q  
    %       n = [3 2 5]; g$o&Udgs  
    %       m = [1 2 1]; A. w:h;7  
    %       z = zernpol(n,m,r); CVR3 A'  
    %       figure '[O;zJN;  
    %       plot(r,z) *_g$MI  
    %       grid on dnuu&Rv  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') W`*r>`krVJ  
    % s&J]zb`  
    %   See also ZERNFUN, ZERNFUN2. & "B=/-(  
    9Lfv^V0  
    % A note on the algorithm. e(8Ba X _  
    % ------------------------ Ld-_,-n  
    % The radial Zernike polynomials are computed using the series ?P c'C  
    % representation shown in the Help section above. For many special Q)z8PQl O  
    % functions, direct evaluation using the series representation can ]"1DGg \A  
    % produce poor numerical results (floating point errors), because eKqk= (  
    % the summation often involves computing small differences between $, fX:x  
    % large successive terms in the series. (In such cases, the functions cPc</[x[W  
    % are often evaluated using alternative methods such as recurrence w:l V"]1  
    % relations: see the Legendre functions, for example). For the Zernike $ o#V#  
    % polynomials, however, this problem does not arise, because the y$R_.KbO  
    % polynomials are evaluated over the finite domain r = (0,1), and vgN&K@hJ  
    % because the coefficients for a given polynomial are generally all $7A8/#  
    % of similar magnitude. -RK- Fu<e  
    % ef E.&]  
    % ZERNPOL has been written using a vectorized implementation: multiple b*Q&CL  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] mU9kVx1+  
    % values can be passed as inputs) for a vector of points R.  To achieve %GIr&V4|  
    % this vectorization most efficiently, the algorithm in ZERNPOL lCHO;7YHX  
    % involves pre-determining all the powers p of R that are required to ps DetP  
    % compute the outputs, and then compiling the {R^p} into a single '(jG[ry&T  
    % matrix.  This avoids any redundant computation of the R^p, and R .2wqkY  
    % minimizes the sizes of certain intermediate variables. {P#|zp4C{  
    % ',5 ky{  
    %   Paul Fricker 11/13/2006 ^D-/`d  
    *bpD`s @  
    2jCfT>`3  
    % Check and prepare the inputs: QoH6  
    % ----------------------------- d4z/5Oa  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) VBcPu  
        error('zernpol:NMvectors','N and M must be vectors.') N_q|\S>t/  
    end tcog'nAz  
    #@nezu2  
    if length(n)~=length(m) K@w{"7}  
        error('zernpol:NMlength','N and M must be the same length.') k~FRD?[u  
    end ?p8_AL'RS  
    "i W"NFO  
    n = n(:); ,U dVNA  
    m = m(:); WQO) =n  
    length_n = length(n); K8Y=S12Ti  
    2P{Gxz<#  
    if any(mod(n-m,2)) "|KP'<8%  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') i,9)\1R  
    end P\)iZiGc  
    ijx0gh`~  
    if any(m<0) 6<(.4a?  
        error('zernpol:Mpositive','All M must be positive.') :tv,]05t  
    end FH+s s!  
    zjoq6  
    if any(m>n) \=0Vi6!Mc  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') Rr|VD@%  
    end kt$jm)UI~l  
    rguCp}r  
    if any( r>1 | r<0 ) 'F#KM1s  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') $l&(%\pp  
    end 2x0<&Xy#P  
    _b;{_g  
    if ~any(size(r)==1) / FEVmH?  
        error('zernpol:Rvector','R must be a vector.') aPbE;" f  
    end KRDmY+  
    "C0Q(dr/n  
    r = r(:); }Y36C.@H  
    length_r = length(r); 5iyd Z  
    _~iw[*#u  
    if nargin==4 $[=%R`~w  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ]}2ZttQ?  
        if ~isnorm )~ h}  
            error('zernpol:normalization','Unrecognized normalization flag.') [-oc>; `=l  
        end t" Z6[XG  
    else ZoZ| M a  
        isnorm = false; Pdt vU-(  
    end `~CQU  
    q;>7*Y&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /{--+ C  
    % Compute the Zernike Polynomials !^Y(^RS@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =h73s0 ]  
    tS8u  
    % Determine the required powers of r: B%+T2=&$7  
    % ----------------------------------- V-L"gnd&2  
    rpowers = []; ?R.j^ S^  
    for j = 1:length(n) )+t0:GwP`:  
        rpowers = [rpowers m(j):2:n(j)]; 2u*KM`fa`  
    end 'qX|jtdM  
    rpowers = unique(rpowers); 7,9=uk>0\  
    Q9G;V]./  
    % Pre-compute the values of r raised to the required powers, CooQ>f  
    % and compile them in a matrix: SPmq4  
    % ----------------------------- |T)6yDL  
    if rpowers(1)==0 2Gaa(rJ5o  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); h6`6tk  
        rpowern = cat(2,rpowern{:}); @xYlS5{  
        rpowern = [ones(length_r,1) rpowern]; >y:,9;  
    else \<TXS)w]  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); !LN?PKJ  
        rpowern = cat(2,rpowern{:}); FNY8tv*/x  
    end _#8RSr8'y  
    6h,(wo3Y  
    % Compute the values of the polynomials: ~d*(=G  
    % -------------------------------------- !s?nJ(p  
    z = zeros(length_r,length_n); K':;%~I  
    for j = 1:length_n cI?8RF(;  
        s = 0:(n(j)-m(j))/2; !AfHk|  
        pows = n(j):-2:m(j); Z+. '>  
        for k = length(s):-1:1 {vyv7L  
            p = (1-2*mod(s(k),2))* ... ia E^a^*  
                       prod(2:(n(j)-s(k)))/          ... q}#6e]t  
                       prod(2:s(k))/                 ... Jk=I^%~  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ...  e5*hE  
                       prod(2:((n(j)+m(j))/2-s(k))); OCaq3_#tZ  
            idx = (pows(k)==rpowers); A='N=^Pm  
            z(:,j) = z(:,j) + p*rpowern(:,idx); FOy|F-j  
        end F Yzi~L  
         ,a]?S^:y]  
        if isnorm rb+j*5Es  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); v4c[(&  
        end L4z ~B!uvF  
    end 3L}!RB  
    w\i\Wp,FP  
    % 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)  1jPJw3"3h  
    '+vA\(K  
    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。 AUNQA  
    rEv*)W  
    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)