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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 3'7]jj  
    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! +SB>>  
     
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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 vGnFX0?h  
    function z = zernfun(n,m,r,theta,nflag) kWacc&*|  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. X`(fJ',  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N MH|F<$42  
    %   and angular frequency M, evaluated at positions (R,THETA) on the [1Aoj|  
    %   unit circle.  N is a vector of positive integers (including 0), and I)kc[/^j$  
    %   M is a vector with the same number of elements as N.  Each element [C/{ru&E  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) !9{hbmF#  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, {r~=mQ  
    %   and THETA is a vector of angles.  R and THETA must have the same WH"'Ju5}  
    %   length.  The output Z is a matrix with one column for every (N,M) {;|pcx\L6~  
    %   pair, and one row for every (R,THETA) pair. {b'  
    % =CW> ;h]  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike n2~WUK  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), dC;&X g`  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral /:^nG+  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +\*b?x  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized }Q*J!OH  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. '"+Gn52#  
    % A.mFa1lH  
    %   The Zernike functions are an orthogonal basis on the unit circle. &8pGq./lr=  
    %   They are used in disciplines such as astronomy, optics, and 6oq5CDoq  
    %   optometry to describe functions on a circular domain. l=t/"M=  
    % cs7^#/3<  
    %   The following table lists the first 15 Zernike functions. -\USDi(  
    % ?lfyC/  
    %       n    m    Zernike function           Normalization Io"3wL)2  
    %       -------------------------------------------------- kBLFK3i  
    %       0    0    1                                 1 +!W:gA  
    %       1    1    r * cos(theta)                    2 y@,PTF  
    %       1   -1    r * sin(theta)                    2 S?6 -I,]h  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) aOw#]pB|  
    %       2    0    (2*r^2 - 1)                    sqrt(3) *~YdL7f)J  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) \#]C !JQ  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) <Y6zJ#BD  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) o>nw~_ H\  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ,(-V<>/*.|  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ]l C2YD}  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 7M _ mR Vh  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :iLRCK3 C  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) "G~!J\  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Trs2M+r)  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) /qJCp![X  
    %       -------------------------------------------------- .p.( \5Fo  
    % 2 S~(P  
    %   Example 1: V ?'p E  
    % {]cr.y]\  
    %       % Display the Zernike function Z(n=5,m=1) =+UtA f<n  
    %       x = -1:0.01:1; ,*dLE   
    %       [X,Y] = meshgrid(x,x); ,Jh#$mil  
    %       [theta,r] = cart2pol(X,Y); .>#O'Z&q9  
    %       idx = r<=1; jl>TZ)4}V  
    %       z = nan(size(X)); BgD3P.;[  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); a] 7g\rg)  
    %       figure Ww60-d}}Q  
    %       pcolor(x,x,z), shading interp 71%$&6  
    %       axis square, colorbar =+K?@;?  
    %       title('Zernike function Z_5^1(r,\theta)') ,`RX~ H=C  
    % cD6^7QF  
    %   Example 2: j{r@>g;3  
    % #;~HoOK*#  
    %       % Display the first 10 Zernike functions ^"D^D`$@  
    %       x = -1:0.01:1; Hi=</ Wy;  
    %       [X,Y] = meshgrid(x,x); 7M4J{}9  
    %       [theta,r] = cart2pol(X,Y); e ><0crb  
    %       idx = r<=1; AX$r,KmE  
    %       z = nan(size(X)); L%(NXSfu7  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Z'j[N4%BK  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; S<NK!89  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ,mHUo4h1O  
    %       y = zernfun(n,m,r(idx),theta(idx)); .{c7 I!8  
    %       figure('Units','normalized') FG[rH]   
    %       for k = 1:10 i0$*):b  
    %           z(idx) = y(:,k); KpYezdPF)  
    %           subplot(4,7,Nplot(k)) - z+,j(@  
    %           pcolor(x,x,z), shading interp ,dTmI{@O  
    %           set(gca,'XTick',[],'YTick',[]) yc~<h/}#  
    %           axis square B~ i  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) $l"%o9ICG  
    %       end xSd&xwP  
    % k9OGnCW\  
    %   See also ZERNPOL, ZERNFUN2. RZV6;=/  
    d1\nMm}v  
    %   Paul Fricker 11/13/2006 G 3,v'D5  
    ssx#|InY  
    K$Vu[!l`  
    % Check and prepare the inputs: GW'v\O  
    % ----------------------------- VqV[ @[P  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) O+|C<;K  
        error('zernfun:NMvectors','N and M must be vectors.') J_Tz\bZ3)  
    end Q17dcgd  
    t4#gW$+^?H  
    if length(n)~=length(m) eGq7+  
        error('zernfun:NMlength','N and M must be the same length.') yD7}  
    end YwET.(oo  
    ~qeFSU(  
    n = n(:); 5Y^"&h[/  
    m = m(:); F/BR#J1  
    if any(mod(n-m,2)) O# ZZ PJ"  
        error('zernfun:NMmultiplesof2', ... X>=`l)ZR  
              'All N and M must differ by multiples of 2 (including 0).') lTqlQ<`V  
    end .gDq+~r8O  
    v.Q#<@B^:  
    if any(m>n) RYEZ'<  
        error('zernfun:MlessthanN', ... 9/{zS3h3  
              'Each M must be less than or equal to its corresponding N.') >":xnX#  
    end a24 AmoWx  
    uStAZ ~b\  
    if any( r>1 | r<0 ) _ C?Wk:Y@  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ) yMrE T m  
    end Y /_CPY  
    F!EiF&[\J  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) c**&,aL  
        error('zernfun:RTHvector','R and THETA must be vectors.') q/U-6A[0  
    end \(P?=] -  
    B??07j  
    r = r(:); &;d N:F;  
    theta = theta(:); :}v-+eIQ  
    length_r = length(r); lUs$I{2_  
    if length_r~=length(theta) ulIEx~qP  
        error('zernfun:RTHlength', ... h9ScN(|0y  
              'The number of R- and THETA-values must be equal.') ZK^cG'^2|  
    end Yu3S3aRE  
    W]ca~%r  
    % Check normalization: Tl2t\z+ps  
    % -------------------- %|(c?`2|  
    if nargin==5 && ischar(nflag) ~SQ xFAto  
        isnorm = strcmpi(nflag,'norm'); +n;nvf}(  
        if ~isnorm lJu^Bcrv  
            error('zernfun:normalization','Unrecognized normalization flag.') 7amVnR1f  
        end ? x #K:a?  
    else dz9U.:C  
        isnorm = false; TsaQR2J@  
    end M/Yr0"%Q<.  
    Xh;.T=/E|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% El<*)  
    % Compute the Zernike Polynomials ^)gyKl:E'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% E:pk'G0bZ  
    `sCaGCp  
    % Determine the required powers of r: 4Lt9Dx1  
    % ----------------------------------- nL:&G'd  
    m_abs = abs(m); ZiJF.(JS  
    rpowers = []; Kt_oo[ey{  
    for j = 1:length(n) mgjJNzclL  
        rpowers = [rpowers m_abs(j):2:n(j)]; `sYFQ+D#O  
    end sh$-}1 ;  
    rpowers = unique(rpowers); `3rwqcxA  
    w'H'o!*/  
    % Pre-compute the values of r raised to the required powers, SO0\d0?u  
    % and compile them in a matrix: luf5-XT  
    % ----------------------------- 46A sD  
    if rpowers(1)==0 R#d~a;j  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); C:J;'[,S  
        rpowern = cat(2,rpowern{:}); `uMEK>b  
        rpowern = [ones(length_r,1) rpowern]; X=$Jp.  
    else .c"nDCFVR  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); :]-oo*xP  
        rpowern = cat(2,rpowern{:}); K.)!qkW-%S  
    end b0$)G-E/Y  
    Q*smH-Sw  
    % Compute the values of the polynomials: 2^WJ1: A  
    % -------------------------------------- k5S;G"i J  
    y = zeros(length_r,length(n)); FXof9fa_B  
    for j = 1:length(n) j?.F-ar  
        s = 0:(n(j)-m_abs(j))/2; tUv>1) [  
        pows = n(j):-2:m_abs(j); K|7"YNohfG  
        for k = length(s):-1:1 4qOzjEQ  
            p = (1-2*mod(s(k),2))* ... >j5\J_( ;D  
                       prod(2:(n(j)-s(k)))/              ... R{#< NE  
                       prod(2:s(k))/                     ... t/i I!}  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... AFz:%m  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); \Z]+j@9  
            idx = (pows(k)==rpowers); a$My6Qa#  
            y(:,j) = y(:,j) + p*rpowern(:,idx); K|P0nJT  
        end <,]:jgX  
         $xbC^ k  
        if isnorm 7=l~fKu  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); p27Dc wov  
        end Hy.u6Jt*/  
    end } e[ E  
    % END: Compute the Zernike Polynomials 0WUBj:@g  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _ .vG)  
    ,"%C.9a  
    % Compute the Zernike functions: ^{+ry<rS>  
    % ------------------------------ pp"X0  
    idx_pos = m>0; 4era5=  
    idx_neg = m<0; 5p0~AN)  
    Q]k< Y  
    z = y; N"S`9B1eD(  
    if any(idx_pos) %~LY'cfPse  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); j_8 YFz5  
    end 5PeS/%uT@  
    if any(idx_neg) 66v,/#K  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); #t+?eye~  
    end MpCPY"WLL  
    zwfft  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) 5GsmBf$RUb  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 7)rQf{q7  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Fy=GU<&AI  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive N1t4o~  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, W5|{A])N  
    %   and THETA is a vector of angles.  R and THETA must have the same P3oYk_oW  
    %   length.  The output Z is a matrix with one column for every P-value, S:xXD^n#H  
    %   and one row for every (R,THETA) pair. T^A(v(^D  
    % aHhLz>H'  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike y1V}c ,  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) TFSdb\g  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) /`PYk]mJh  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 `G\ qGllX  
    %   for all p. }+,Q&]>~  
    % i$Y#7^l%k  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 M Kyj<@[  
    %   Zernike functions (order N<=7).  In some disciplines it is |IAx!Z-P  
    %   traditional to label the first 36 functions using a single mode ,ri&zbB  
    %   number P instead of separate numbers for the order N and azimuthal ?^&ih:"  
    %   frequency M. ^ D0"m>3r  
    % gwj?.7N*k  
    %   Example: </I%VHP,[f  
    % UylIxd  
    %       % Display the first 16 Zernike functions m$8siF{<q  
    %       x = -1:0.01:1; s< tG  
    %       [X,Y] = meshgrid(x,x); )]>t(  
    %       [theta,r] = cart2pol(X,Y); v^9eTeFO  
    %       idx = r<=1; _/>ktYo:  
    %       p = 0:15;  ][ $UN  
    %       z = nan(size(X)); [v1$L p  
    %       y = zernfun2(p,r(idx),theta(idx)); @nH3nn  
    %       figure('Units','normalized') q;K]NP-_p  
    %       for k = 1:length(p) X9*n[ev  
    %           z(idx) = y(:,k); KXWcg#zFY  
    %           subplot(4,4,k) gwaSgV$z  
    %           pcolor(x,x,z), shading interp 4H 6t" X  
    %           set(gca,'XTick',[],'YTick',[]) @]Q4K%1^"  
    %           axis square S^s-md>  
    %           title(['Z_{' num2str(p(k)) '}']) !}=eXDn;A_  
    %       end <"Y>|X  
    % dsIbr"m  
    %   See also ZERNPOL, ZERNFUN. MTYV~S4/  
    ` nX, x-UM  
    %   Paul Fricker 11/13/2006 iwnGWGcuS  
    AbNr]w&pXC  
    FK BRJ5O  
    % Check and prepare the inputs: -Mo4`bN  
    % ----------------------------- Uw4iWcC  
    if min(size(p))~=1 c!@|y E,  
        error('zernfun2:Pvector','Input P must be vector.') {aE[h[=r  
    end EW$drY@  
    fRNj *bIV  
    if any(p)>35 imOIO[<;  
        error('zernfun2:P36', ... ;adZ*'6u  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... $HwF:L)*  
               '(P = 0 to 35).']) d.}65{F,x  
    end .{gDw  
    jTwSyW  
    % Get the order and frequency corresonding to the function number: \J:+Wl.9A  
    % ---------------------------------------------------------------- Rk9n,"xpv  
    p = p(:); Bo:epus}\  
    n = ceil((-3+sqrt(9+8*p))/2); j+!u=E  
    m = 2*p - n.*(n+2); ?g1eW q&  
    \BBs;z[/  
    % Pass the inputs to the function ZERNFUN: Y6wr}U  
    % ---------------------------------------- Y*xgY*K  
    switch nargin .BxI~d^  
        case 3 gLMb,buqC  
            z = zernfun(n,m,r,theta); Lginps[la  
        case 4 14&|(M  
            z = zernfun(n,m,r,theta,nflag); /J}G{Y |n  
        otherwise &zYQ H@  
            error('zernfun2:nargin','Incorrect number of inputs.') J5a8U&A  
    end `n,RC2yo  
    ]Mq-67  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) [Zdrm:=]L  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. OGEe8Z9Jt  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of `C_qqf  
    %   order N and frequency M, evaluated at R.  N is a vector of Na`> pH  
    %   positive integers (including 0), and M is a vector with the ~F@p}u8TV  
    %   same number of elements as N.  Each element k of M must be a L0VZ>!*o  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) q%d,E1  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is cZ%tJ(&\7X  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix !0p K8k&MG  
    %   with one column for every (N,M) pair, and one row for every 7 cV G?Wr  
    %   element in R. %,$xmoj9O]  
    % V+D<626o  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- o(}%b8 K  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is t=eI*M+>h  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to nh7_ jEX  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 DhxS@/  
    %   for all [n,m]. xi"ff .  
    % _{):w~zi  
    %   The radial Zernike polynomials are the radial portion of the 7Z9'Y?[m  
    %   Zernike functions, which are an orthogonal basis on the unit d&G]k!|\  
    %   circle.  The series representation of the radial Zernike z\FBN=54z  
    %   polynomials is _KloX{a  
    % Qu<6X@+5  
    %          (n-m)/2 tvn o3"  
    %            __ W*iTg%a\k  
    %    m      \       s                                          n-2s %J'/cmR&  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r qu#xc0?  
    %    n      s=0 >r X$E<B\  
    % F6J]T6 Y  
    %   The following table shows the first 12 polynomials. +<$nZ=,hsy  
    % )AEtW[~D  
    %       n    m    Zernike polynomial    Normalization g/l:q&Q<  
    %       --------------------------------------------- K%`]HW@I{  
    %       0    0    1                        sqrt(2) ;jx[  +  
    %       1    1    r                           2 |) cJ  
    %       2    0    2*r^2 - 1                sqrt(6) yQ^,>eh  
    %       2    2    r^2                      sqrt(6) |3FGMg%  
    %       3    1    3*r^3 - 2*r              sqrt(8) Qm7];,  
    %       3    3    r^3                      sqrt(8) tKyGD|g S  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) CN` ~DD{  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 9: g]DIL  
    %       4    4    r^4                      sqrt(10) A ?tna6W:  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) g :B4zlKG  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) gP|-A`y  
    %       5    5    r^5                      sqrt(12) s% rmfIp"  
    %       --------------------------------------------- Nk7=[y#z  
    % z80(+ `   
    %   Example: C}uzzG6s  
    % y(iq  
    %       % Display three example Zernike radial polynomials ,j{tGj_  
    %       r = 0:0.01:1; \7h>9}wGf  
    %       n = [3 2 5]; ]5@n`;&#.  
    %       m = [1 2 1]; $;(@0UDE  
    %       z = zernpol(n,m,r); H;<>uE Lie  
    %       figure :B=Gb8?  
    %       plot(r,z) g/68& M  
    %       grid on &:ZR% f  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') <7)sS<I  
    % *@^@7`W  
    %   See also ZERNFUN, ZERNFUN2. K0oF=|  
    %${$P+a`D  
    % A note on the algorithm. AB3OG*C9  
    % ------------------------ o,?G(  
    % The radial Zernike polynomials are computed using the series <L*`WO]\l  
    % representation shown in the Help section above. For many special B1FJAKI);  
    % functions, direct evaluation using the series representation can p<\!{5:   
    % produce poor numerical results (floating point errors), because 7*M-?  
    % the summation often involves computing small differences between 0=U|7%dOL  
    % large successive terms in the series. (In such cases, the functions &RbP N^  
    % are often evaluated using alternative methods such as recurrence KkTE -$-  
    % relations: see the Legendre functions, for example). For the Zernike u^MRKLn  
    % polynomials, however, this problem does not arise, because the qe(gKKA%q  
    % polynomials are evaluated over the finite domain r = (0,1), and \K"7U  
    % because the coefficients for a given polynomial are generally all Vh;|qF 9  
    % of similar magnitude. iF +@aA  
    % y]PuY \+  
    % ZERNPOL has been written using a vectorized implementation: multiple \p.yR.  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] "l-#v| 54  
    % values can be passed as inputs) for a vector of points R.  To achieve Y+),c14#  
    % this vectorization most efficiently, the algorithm in ZERNPOL $aU.M3  
    % involves pre-determining all the powers p of R that are required to DOGGQ$0  
    % compute the outputs, and then compiling the {R^p} into a single xDl; tFI  
    % matrix.  This avoids any redundant computation of the R^p, and dR_6j}  
    % minimizes the sizes of certain intermediate variables. 4 X/UyBk  
    % Nr]Fh  
    %   Paul Fricker 11/13/2006 d^M*%az  
    |cnps$fk~  
    ^>ir&$  
    % Check and prepare the inputs: __7}4mA  
    % ----------------------------- f@Jrbg  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) sG_/E-%5'  
        error('zernpol:NMvectors','N and M must be vectors.') EFx>Hu/ [G  
    end QnP3U  
    4'`P+p"A  
    if length(n)~=length(m) U$OI]Dd9  
        error('zernpol:NMlength','N and M must be the same length.') J;^PM:6  
    end P%Vq#5  
    z k}AGw  
    n = n(:); uY>M3h#qx  
    m = m(:); w1-P6cf  
    length_n = length(n); N>*+Wg$Ne  
    u_+iH$zA  
    if any(mod(n-m,2)) &)+H''JY  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') _4)z:?G5  
    end %1jcY0zEQ  
    >w7KOVbN3  
    if any(m<0) ZQfPDH=  
        error('zernpol:Mpositive','All M must be positive.') -L]-u6kC[  
    end \5!7zPc  
    o<3$|`S&  
    if any(m>n) ILAn2W  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') #z%D d{E  
    end N%Ta. `r  
    >l AtfN='  
    if any( r>1 | r<0 ) "|1iz2L  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ej}S{/<*n  
    end $F# 5/gDVQ  
    g;p} -=  
    if ~any(size(r)==1) >L!c} Ku  
        error('zernpol:Rvector','R must be a vector.') :EQ{7Op`  
    end -j]k^  
    MA:5'n  
    r = r(:); P$k*!j_W  
    length_r = length(r); D@68_sn  
    ,I5SAd|dX  
    if nargin==4 lTq"j?#E]m  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 300w\9fn&  
        if ~isnorm b=/'c Q  
            error('zernpol:normalization','Unrecognized normalization flag.') aif;h! ?y  
        end qT(6TP  
    else h,m 90Hd+  
        isnorm = false; 37jxl+  
    end 0]  
    Z#H<+S(  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% JL1A3G  
    % Compute the Zernike Polynomials ?hkOL$v<9}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]'(D*4  
    \|{/.R  
    % Determine the required powers of r: m?<E >-bI  
    % ----------------------------------- <OGG(dI  
    rpowers = []; Wj(#!\ 7F  
    for j = 1:length(n) qJdlZW<  
        rpowers = [rpowers m(j):2:n(j)]; Is7BJ f  
    end I6f/+;E  
    rpowers = unique(rpowers); .nrllVG%`  
    %k1Pyv;]  
    % Pre-compute the values of r raised to the required powers, '{jr9Vh  
    % and compile them in a matrix: pCh v;  
    % ----------------------------- [TFJb+N&  
    if rpowers(1)==0 l^Rb%?4Z  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 0Z8"f_GK  
        rpowern = cat(2,rpowern{:}); pzz* >Y  
        rpowern = [ones(length_r,1) rpowern]; _/I">/ivlM  
    else =zyA~}M2  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); nlNk  
        rpowern = cat(2,rpowern{:}); .N qXdari  
    end vNv!fkl  
    Y"MHs0O5>  
    % Compute the values of the polynomials: be,Rj,-  
    % -------------------------------------- A<X?1$  
    z = zeros(length_r,length_n); \uJRjw+  
    for j = 1:length_n T[bCY 6  
        s = 0:(n(j)-m(j))/2; 3O/#^~\'hW  
        pows = n(j):-2:m(j); 06S R74  
        for k = length(s):-1:1 ;ItH2Lw<&  
            p = (1-2*mod(s(k),2))* ... *i]?J  
                       prod(2:(n(j)-s(k)))/          ... x)~i`$  
                       prod(2:s(k))/                 ... ;KlYiu  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... aaR& -M@  
                       prod(2:((n(j)+m(j))/2-s(k))); h)HEexyRg  
            idx = (pows(k)==rpowers); -[=eVS.2%  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 4KM-$h,4O  
        end Db,"Gl  
         L"m^LyU  
        if isnorm A I.(}W4]  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); "=djo+y  
        end sE pI)9  
    end }4A] x`3  
    ef7{D P  
    % 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)  -CvmZ:n  
    c8uaZvfW  
    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。 B/a gW  
    x3+ -wv  
    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)