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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 {3aua:q  
    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! & >fQp(f  
     
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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 ;#< 0<  
    function z = zernfun(n,m,r,theta,nflag) ?(_08O  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. SNk=b6`9  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Z6MO^_m2  
    %   and angular frequency M, evaluated at positions (R,THETA) on the J\=*#*rJ1  
    %   unit circle.  N is a vector of positive integers (including 0), and 5'u<iSmBo  
    %   M is a vector with the same number of elements as N.  Each element ="l/klYV  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) )MT}+ai  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, {Ou1KDy#)  
    %   and THETA is a vector of angles.  R and THETA must have the same &s!@29DXR  
    %   length.  The output Z is a matrix with one column for every (N,M) +G>\-tjSD  
    %   pair, and one row for every (R,THETA) pair. "qy,*{~  
    % S~G ]~gt  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike t\O16O7S  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi),  &q*Aj17  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral QIFgQ0{  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, rEz^  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized k$:|-_(w  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. p0eX{xm  
    % FW DNpr  
    %   The Zernike functions are an orthogonal basis on the unit circle. {R{=+2K!|k  
    %   They are used in disciplines such as astronomy, optics, and a(ZcmYzXU  
    %   optometry to describe functions on a circular domain. +:/%3}`  
    % 2y1Sne=<Kb  
    %   The following table lists the first 15 Zernike functions. DzRFMYBR  
    % pEz_qy[#  
    %       n    m    Zernike function           Normalization %E;'ln4h&,  
    %       -------------------------------------------------- cPQiUU~W@  
    %       0    0    1                                 1 \o3gKoL%  
    %       1    1    r * cos(theta)                    2 +&H4m=D-#a  
    %       1   -1    r * sin(theta)                    2 '$+ogBS  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 1X1dG#:  
    %       2    0    (2*r^2 - 1)                    sqrt(3) hOK8(U0  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 4s oJ.j8  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) E=O\0!F|b  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) [()koU#w.  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) uCB=u[]y4  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) &5!8F(7  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) j_j]"ew)  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >y+B  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) tfWS)y7  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) dlnX_+((KC  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) b| (: [nB  
    %       -------------------------------------------------- 8H`[*|{'  
    % llDkJ)\  
    %   Example 1: `XDl_E+>l  
    % uhq8   
    %       % Display the Zernike function Z(n=5,m=1) w&.a QGR#  
    %       x = -1:0.01:1; 7a}k  
    %       [X,Y] = meshgrid(x,x); F((4U"   
    %       [theta,r] = cart2pol(X,Y); x.4m|f0;  
    %       idx = r<=1; y8xE 6i  
    %       z = nan(size(X)); cm+P]8o%{  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); \z)%$#I  
    %       figure K:WDl;8 (d  
    %       pcolor(x,x,z), shading interp `@yp+8  
    %       axis square, colorbar ue>D 7\8  
    %       title('Zernike function Z_5^1(r,\theta)') :rP=t ,  
    % \GU<43J2uo  
    %   Example 2: f%8C!W]Dm  
    % $<OD31T  
    %       % Display the first 10 Zernike functions o{[qZc_%  
    %       x = -1:0.01:1; l%=;  
    %       [X,Y] = meshgrid(x,x); >@Kx>cg+  
    %       [theta,r] = cart2pol(X,Y); &xExyz~`  
    %       idx = r<=1; tT._VK]o&R  
    %       z = nan(size(X)); /zox$p$?h  
    %       n = [0  1  1  2  2  2  3  3  3  3]; vw@S>G lGg  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; qcRs$-J  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; :~SyL!  
    %       y = zernfun(n,m,r(idx),theta(idx)); c[s4EUG  
    %       figure('Units','normalized') [_:nHZb  
    %       for k = 1:10 3iU=c&P  
    %           z(idx) = y(:,k); U%/+B]6jP  
    %           subplot(4,7,Nplot(k)) &9>vl*  
    %           pcolor(x,x,z), shading interp CNx8] _2  
    %           set(gca,'XTick',[],'YTick',[]) &,)&%Sg[  
    %           axis square &6k3*dq  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) fTX;.M/%   
    %       end 6E}qL8'5x  
    % o,wUc"CE  
    %   See also ZERNPOL, ZERNFUN2. q0 \6F^;M  
    ,iwp,=h=  
    %   Paul Fricker 11/13/2006 /<BI46B\  
    OB}Ib]  
    EEL,^3KR  
    % Check and prepare the inputs: (Awm9|.{+  
    % ----------------------------- I*^Ta{j[  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) D3K8F@d  
        error('zernfun:NMvectors','N and M must be vectors.') Xlt|nX~#;  
    end i{qgn%#}Y  
    ( uidNq  
    if length(n)~=length(m) 8a"%0d#  
        error('zernfun:NMlength','N and M must be the same length.') S`]k>' l  
    end '4<1 1(U  
    S5EK~#-L[  
    n = n(:); ijU*|8n{>  
    m = m(:); K~ EmD9  
    if any(mod(n-m,2)) 2b8L\$1q  
        error('zernfun:NMmultiplesof2', ... SZCze"`[  
              'All N and M must differ by multiples of 2 (including 0).') rQ snhv  
    end @=f\<"$vt  
    j*m%*_kO  
    if any(m>n) -`6+UkOV[x  
        error('zernfun:MlessthanN', ... (&x['IR  
              'Each M must be less than or equal to its corresponding N.') 6;5Ss?ep  
    end "5$B>S(Q  
    Ny)X+2Ae  
    if any( r>1 | r<0 ) Z;)%%V%o  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 1[-tD 0{H  
    end ZqO^f*F>h  
    zT-_5uZQ  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) KJZ4AWH`  
        error('zernfun:RTHvector','R and THETA must be vectors.') 7"D.L-H  
    end BTrn0  
    )dd@\n$6  
    r = r(:); %ULr8)R;  
    theta = theta(:); mpJ#:}n  
    length_r = length(r); d m%8K6|  
    if length_r~=length(theta) ^pk7"l4Xm  
        error('zernfun:RTHlength', ... Aq7osU1B  
              'The number of R- and THETA-values must be equal.') ufT`"i  
    end X!g#T9kG  
    Jxm.cC5z.  
    % Check normalization: @U}1EC{A  
    % -------------------- Pk)1WK7E  
    if nargin==5 && ischar(nflag) GWip-wI  
        isnorm = strcmpi(nflag,'norm'); u\JNr}bL  
        if ~isnorm 8}| (0mC  
            error('zernfun:normalization','Unrecognized normalization flag.') W `}Rf\g  
        end =_u4=4  
    else JqiP>4Uwm^  
        isnorm = false; VyGJ=[ ]  
    end *-p}z@8  
    8)I^ t81  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 45>?o  
    % Compute the Zernike Polynomials !%0 * z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L*JjG sTH  
    lHX72s|V  
    % Determine the required powers of r: kMd.h[X~  
    % ----------------------------------- H7:] ]j1  
    m_abs = abs(m); 4HA<P6L  
    rpowers = []; B^9j@3Ux  
    for j = 1:length(n) ?6Y?a2 |  
        rpowers = [rpowers m_abs(j):2:n(j)]; rw #$lP  
    end | Xy6PN8  
    rpowers = unique(rpowers); 5XB H$&Td  
    V "h +L7T  
    % Pre-compute the values of r raised to the required powers, J/*`7Pd  
    % and compile them in a matrix: CeC6hGR5  
    % ----------------------------- E?0%Z&1h  
    if rpowers(1)==0 0"bcdG<}  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?5 7Sk+  
        rpowern = cat(2,rpowern{:}); ,nm*q#R,0  
        rpowern = [ones(length_r,1) rpowern]; ~Jz6O U*z  
    else "#\ ;H$+  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ySDH "|0  
        rpowern = cat(2,rpowern{:}); _aT5jR=  
    end :6\qpex  
    9qG6Pb  
    % Compute the values of the polynomials: *!7 O~yQ  
    % -------------------------------------- ~R92cH>L  
    y = zeros(length_r,length(n)); dlTt _.  
    for j = 1:length(n) \P`hq^;  
        s = 0:(n(j)-m_abs(j))/2; .0]<k,JZZ  
        pows = n(j):-2:m_abs(j); k+pr \d~  
        for k = length(s):-1:1 W:L AP R  
            p = (1-2*mod(s(k),2))* ... Q$@I"V&G.  
                       prod(2:(n(j)-s(k)))/              ... yO~Ig `w  
                       prod(2:s(k))/                     ... hQDXlFHT  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... jtc]>]6i  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); @6T/Tdz  
            idx = (pows(k)==rpowers); !d0kV,F:  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ;MdlwQ$`  
        end FQ5U$x. [P  
         Z>5b;8  
        if isnorm E09 :E  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); :&9s,l   
        end }S<2A7)el  
    end 7E~;xn;  
    % END: Compute the Zernike Polynomials N5b!.B x-w  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ._{H~R|  
    VS8Rx.?  
    % Compute the Zernike functions: &FN.:_E  
    % ------------------------------ -C?ZB}`   
    idx_pos = m>0; ?+}_1x`  
    idx_neg = m<0; Y glmX"fLf  
    2!=f hN  
    z = y; E#N|w q  
    if any(idx_pos) l]l'4@1   
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); QE`bSI  
    end .jWC$SVR  
    if any(idx_neg) n]o<S+z  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); L>4"(  
    end 68WO~*  
    vuY~_  
    % 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) ,uhb~N<  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Zw S F^  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated EDl!w:  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive sLT3Y}IO  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, B !L{  
    %   and THETA is a vector of angles.  R and THETA must have the same !Pfr,a  
    %   length.  The output Z is a matrix with one column for every P-value, L2i_X@/  
    %   and one row for every (R,THETA) pair. SP_75BJ  
    % a![{M<Y~  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike j[J-f@F \Y  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) #r~# I}U  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) q\4Xs$APq  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1  B Qxs~  
    %   for all p. Zaf:fsj>  
    % .2Elr(&*h  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ? _9  
    %   Zernike functions (order N<=7).  In some disciplines it is u(F_oZ~  
    %   traditional to label the first 36 functions using a single mode bUdLs.:  
    %   number P instead of separate numbers for the order N and azimuthal fW1CFRHH  
    %   frequency M. %axh`xK#  
    % U@)eTHv}6  
    %   Example: ,~@X{7U  
    % WUXx;9>  
    %       % Display the first 16 Zernike functions '"/=f\)u  
    %       x = -1:0.01:1; [ =9T*Sp  
    %       [X,Y] = meshgrid(x,x); sW'AjI  
    %       [theta,r] = cart2pol(X,Y); bSi%2Onj  
    %       idx = r<=1; WH@,kH@  
    %       p = 0:15; Ma']?Rb`  
    %       z = nan(size(X)); g63(E,;;J  
    %       y = zernfun2(p,r(idx),theta(idx)); s.QwSbw-g  
    %       figure('Units','normalized') =M [bnq*\  
    %       for k = 1:length(p) -[9JJ/7y  
    %           z(idx) = y(:,k); Q}K"24`=  
    %           subplot(4,4,k) ^Hnb }L  
    %           pcolor(x,x,z), shading interp P90yI  
    %           set(gca,'XTick',[],'YTick',[]) )|R)Q6UJ  
    %           axis square li'YDtMKCY  
    %           title(['Z_{' num2str(p(k)) '}']) J~ zUp(>K  
    %       end '/n1IM$7  
    % Sc1 8dC0  
    %   See also ZERNPOL, ZERNFUN. { VfXsI  
    H.|#c^I  
    %   Paul Fricker 11/13/2006 f<fXsSv(  
    D4lG[qb  
    /h H  
    % Check and prepare the inputs: p6]1w]*R  
    % ----------------------------- [" )o.(  
    if min(size(p))~=1 ~IfJwBn-i  
        error('zernfun2:Pvector','Input P must be vector.') j<99FW"@e  
    end "ESwA  
    (ybI\UI  
    if any(p)>35 n,V[eW#m'L  
        error('zernfun2:P36', ... j@U]'5EVB  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... qYjce]c  
               '(P = 0 to 35).']) "fCu=@i  
    end ll<Xz((o  
    *yt=_Q  
    % Get the order and frequency corresonding to the function number: k}kQI~S9  
    % ---------------------------------------------------------------- iohop(LZ  
    p = p(:); 7u S~MW  
    n = ceil((-3+sqrt(9+8*p))/2); jrlVvzZ  
    m = 2*p - n.*(n+2); :I j{s  
    Jr ,;>   
    % Pass the inputs to the function ZERNFUN: n.`($yR_  
    % ---------------------------------------- {W =%U|f  
    switch nargin dGYn4i2k?  
        case 3 1R{!]uh  
            z = zernfun(n,m,r,theta); q77;ZPfs8  
        case 4 hl7bzKO*w  
            z = zernfun(n,m,r,theta,nflag); pMx*F@&nU  
        otherwise j9x<Y]  
            error('zernfun2:nargin','Incorrect number of inputs.') &I+5  
    end .779pT!,M  
    L%*!`TN  
    % 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) G\/zkrxmv  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. F 5bj=mI  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ~rE|%o  
    %   order N and frequency M, evaluated at R.  N is a vector of }l(&}#dY  
    %   positive integers (including 0), and M is a vector with the M)J5;^["  
    %   same number of elements as N.  Each element k of M must be a DbBcQ%  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) &Cq`Y !y  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 8\@m - E!{  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix T6y\|  
    %   with one column for every (N,M) pair, and one row for every !=*g@mgF  
    %   element in R. o8V5w!+#  
    % xBThq?N?  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 0rQMLx  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is |B?m,U$A!  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to  <$A  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 p,5i)nEFj  
    %   for all [n,m]. 59LZv-l  
    % vjbASFF0=  
    %   The radial Zernike polynomials are the radial portion of the ,8S/t+H  
    %   Zernike functions, which are an orthogonal basis on the unit O@T9x$  
    %   circle.  The series representation of the radial Zernike |k )=0mCz  
    %   polynomials is YFLZ%(  
    % SB;&GHq"n  
    %          (n-m)/2 pz!Zs."f)  
    %            __ Avge eJi  
    %    m      \       s                                          n-2s |PvPAPy)uu  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r g+8OekzB5  
    %    n      s=0 : Xda1S  
    % j nkR}wAA  
    %   The following table shows the first 12 polynomials. I13y6= d  
    % %^)fmu  
    %       n    m    Zernike polynomial    Normalization !j8FIY'[  
    %       --------------------------------------------- @+&LYy72  
    %       0    0    1                        sqrt(2) .Yamc#A-  
    %       1    1    r                           2 /H[=5  
    %       2    0    2*r^2 - 1                sqrt(6) G*?8MTP8![  
    %       2    2    r^2                      sqrt(6) mxvp3t \  
    %       3    1    3*r^3 - 2*r              sqrt(8) 8 `v-<J  
    %       3    3    r^3                      sqrt(8) h/QXPdV  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ^rB8? kt  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 6iry6wcHm  
    %       4    4    r^4                      sqrt(10) F#3Q_G^/  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) aG-vtld  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 3<e=g)F  
    %       5    5    r^5                      sqrt(12) lB8-Z ow  
    %       --------------------------------------------- %e8@*~h@  
    % vO H4#  
    %   Example: y B81f  
    % 0.Q Ujw  
    %       % Display three example Zernike radial polynomials RF?`vRZOe  
    %       r = 0:0.01:1; v8wq,CYV  
    %       n = [3 2 5]; G~]Uk*M q  
    %       m = [1 2 1]; `_6C {<O  
    %       z = zernpol(n,m,r); [@_Jj3`4  
    %       figure ]"pVj6O  
    %       plot(r,z) v1#otrf  
    %       grid on WSP I|#Xr%  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') zF@/K`  
    % _f7 9wx\B  
    %   See also ZERNFUN, ZERNFUN2. "-E\[@/  
    =?5]()'*n  
    % A note on the algorithm. h! ,v/7=  
    % ------------------------ <q)#  
    % The radial Zernike polynomials are computed using the series p . %]Q*8  
    % representation shown in the Help section above. For many special 3RUy, s  
    % functions, direct evaluation using the series representation can b3P+H r  
    % produce poor numerical results (floating point errors), because Q*GN`07@?d  
    % the summation often involves computing small differences between 2/U.| *mH  
    % large successive terms in the series. (In such cases, the functions ; t)3F  
    % are often evaluated using alternative methods such as recurrence 3h]g}&k  
    % relations: see the Legendre functions, for example). For the Zernike k<z )WNBf  
    % polynomials, however, this problem does not arise, because the M.JA.I@XC  
    % polynomials are evaluated over the finite domain r = (0,1), and +l42Awl>K  
    % because the coefficients for a given polynomial are generally all M+oHtX$  
    % of similar magnitude. E[OJ+ ;c  
    % )|cc X  
    % ZERNPOL has been written using a vectorized implementation: multiple GWGSd\z  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] "BAK !N$9  
    % values can be passed as inputs) for a vector of points R.  To achieve Oz.HH  
    % this vectorization most efficiently, the algorithm in ZERNPOL eB2a-,  
    % involves pre-determining all the powers p of R that are required to (xycJ`N  
    % compute the outputs, and then compiling the {R^p} into a single //B&k`u  
    % matrix.  This avoids any redundant computation of the R^p, and 6]i-E>p3R  
    % minimizes the sizes of certain intermediate variables. k``_EiV4t  
    % }ZYd4h|g\z  
    %   Paul Fricker 11/13/2006 @ 8(q$  
    L]7=?vN=8  
    @?ebuj5{e  
    % Check and prepare the inputs: zE*li`@  
    % ----------------------------- }f%}v  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) C-xr"]#]  
        error('zernpol:NMvectors','N and M must be vectors.') *9 {PEx  
    end O}gV`q;  
    xl{=Y< ;  
    if length(n)~=length(m) ^+ml5m  
        error('zernpol:NMlength','N and M must be the same length.') #-rH1h3*q  
    end <(#(hDwy  
    qyb?49I  
    n = n(:); 'JtBZFq  
    m = m(:); . P viA  
    length_n = length(n); _=r6=.  
    e v}S+!|U  
    if any(mod(n-m,2)) D'>_I.  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') _1X!EH"  
    end Xc ++b|k  
    ?Rb9|`6  
    if any(m<0) P.se'z)E  
        error('zernpol:Mpositive','All M must be positive.') hw uiu*  
    end xH4m|  
    QP==?g3  
    if any(m>n) s3N'02G  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 8bGd} (  
    end 1}+3dB_s  
    \0gis#  
    if any( r>1 | r<0 ) Ng&%o  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 2YL?,uLS  
    end cdH>n)  
    g @Z))M+  
    if ~any(size(r)==1) bG"~"ipn%  
        error('zernpol:Rvector','R must be a vector.') _oL?*ks  
    end j a[Et/r  
    yZ7&b&2nLn  
    r = r(:); iO$8:mxm0?  
    length_r = length(r); PN%zIkbo  
    OG~gFZr)6  
    if nargin==4 5&g@3j]  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); YRk(u7:0  
        if ~isnorm -/B+T>[nTb  
            error('zernpol:normalization','Unrecognized normalization flag.') f^ZRT@`O  
        end wSL}`CgU  
    else C.:<-xo  
        isnorm = false; 2ACCh4(/P  
    end eu|YCYj)g  
    !.$I["/=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m,28u3@r  
    % Compute the Zernike Polynomials 1#g2A0U,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h f)?1z4  
    yF:1( 4  
    % Determine the required powers of r: T~?Ff|qFC  
    % ----------------------------------- P{`C^W$J^  
    rpowers = []; E`JI>7  
    for j = 1:length(n) g'f@H-KCD  
        rpowers = [rpowers m(j):2:n(j)]; @u+]aI!`-  
    end <{p4V|:  
    rpowers = unique(rpowers); pQ">UL*  
    ]#<4vl\  
    % Pre-compute the values of r raised to the required powers, PQt")[  
    % and compile them in a matrix: f5"k55}  
    % ----------------------------- GKqm&/M*=  
    if rpowers(1)==0 KkyVSoD\  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); tFn)aa~L  
        rpowern = cat(2,rpowern{:}); (#c*M?g3  
        rpowern = [ones(length_r,1) rpowern]; &E F!OBR  
    else R{4^t97wH{  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); pX<`+t[  
        rpowern = cat(2,rpowern{:}); 2|y"!JqE1  
    end _ye |Y  
    MKCsv+   
    % Compute the values of the polynomials: mIvx1_[  
    % -------------------------------------- K3&qq[8.e  
    z = zeros(length_r,length_n); c]<5zyl"j1  
    for j = 1:length_n wu6;.xTLl  
        s = 0:(n(j)-m(j))/2; s) t@ol  
        pows = n(j):-2:m(j); wm@@$  
        for k = length(s):-1:1 MY)O^I X$  
            p = (1-2*mod(s(k),2))* ... C& f= ywi0  
                       prod(2:(n(j)-s(k)))/          ... qwcD`HV,  
                       prod(2:s(k))/                 ... @{e}4s?7od  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... qZh/IW  
                       prod(2:((n(j)+m(j))/2-s(k))); 1\m[$Gs:  
            idx = (pows(k)==rpowers); {z|)Njhg  
            z(:,j) = z(:,j) + p*rpowern(:,idx); a!SiX  
        end <=&`ZH   
         I,DS@SK  
        if isnorm uMv,zO5  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); :4w ?#  
        end ?R 'r4P,  
    end #z%fx   
    MJ)RvNF  
    % 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)  oB7_O-3z  
    yG{TH0tq  
    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。  RRJ%:5&  
    ,P0) 6>  
    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)