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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 (95|DCL  
    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! Bqf(6\)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 f_c\uN@f  
    function z = zernfun(n,m,r,theta,nflag) T?8BAxC?K  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ]#o;`5'  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N KuR]X``2  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 9Yt|Wj  
    %   unit circle.  N is a vector of positive integers (including 0), and kV'zA F v  
    %   M is a vector with the same number of elements as N.  Each element  /YJo"\7  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) !>48`o ^  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, <cTX;&0=  
    %   and THETA is a vector of angles.  R and THETA must have the same $kUB%\`  
    %   length.  The output Z is a matrix with one column for every (N,M) q{w|`vIb  
    %   pair, and one row for every (R,THETA) pair. !tq]kKJ3:  
    % <B6md i'R  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike LUQ.=:mBR  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 8"h;+;  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral V(ELrjB0  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Cy-p1s  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized SeHrj&5U  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. L^qCE-[  
    % 13?:a[~=Y  
    %   The Zernike functions are an orthogonal basis on the unit circle. qiz(k:\o  
    %   They are used in disciplines such as astronomy, optics, and mV}bQ^*?Z  
    %   optometry to describe functions on a circular domain. y[7M(K  
    % GCl *x:  
    %   The following table lists the first 15 Zernike functions. wDvu2iC=  
    % bF _]j/  
    %       n    m    Zernike function           Normalization { j_-iF  
    %       -------------------------------------------------- 8F[ ];LF>  
    %       0    0    1                                 1 aE0R{yupZ  
    %       1    1    r * cos(theta)                    2 \GEz.Vb  
    %       1   -1    r * sin(theta)                    2 'Xik2PaO  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) [{Wo:c9Qq1  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Ta[2uv>  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 0moAmfc  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) jf)cDj2  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) EjfQF C  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) uO4 LD}A  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 2TGND-(j  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 2/3yW.C  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7rD 8  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) A;8kC}  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5WI bnV@  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) /Xi21W/  
    %       -------------------------------------------------- / =9Y(v  
    % p&I>xu8fl  
    %   Example 1: q{h,}[U=  
    % 3$"V,_TBZ  
    %       % Display the Zernike function Z(n=5,m=1) :2j`NyLI.  
    %       x = -1:0.01:1; 6aB]&WO1@  
    %       [X,Y] = meshgrid(x,x); //NV_^$y  
    %       [theta,r] = cart2pol(X,Y); (rFkXK4^J  
    %       idx = r<=1; d'(n/9K  
    %       z = nan(size(X)); /T6bc^nOW  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); H!Gw@u]E  
    %       figure pj_W^,*/  
    %       pcolor(x,x,z), shading interp vyS>3(NZ  
    %       axis square, colorbar #~p;s>  
    %       title('Zernike function Z_5^1(r,\theta)') +mjwX?yF  
    % $'l<2h>4  
    %   Example 2: B-g-T>8  
    % @ 95p[  
    %       % Display the first 10 Zernike functions @7}XBg[pI  
    %       x = -1:0.01:1; ou0TKE9 _  
    %       [X,Y] = meshgrid(x,x); TDw~sxtv&  
    %       [theta,r] = cart2pol(X,Y); >V8!OaY5n  
    %       idx = r<=1; A$p&<#  
    %       z = nan(size(X)); }Bv1fbD4U  
    %       n = [0  1  1  2  2  2  3  3  3  3]; OGcdv{ ,P  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; -`8@  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ft7M9<#v  
    %       y = zernfun(n,m,r(idx),theta(idx)); g5U,   
    %       figure('Units','normalized') 8^EWD3N`  
    %       for k = 1:10 y9mV6.r  
    %           z(idx) = y(:,k); AyQ5jkIE^{  
    %           subplot(4,7,Nplot(k)) u$tst_y-  
    %           pcolor(x,x,z), shading interp OybmyGHY  
    %           set(gca,'XTick',[],'YTick',[]) P, ZQ*Ju  
    %           axis square uPl7u 1c  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Q"s6HZ"YI  
    %       end Ak3^en  
    % G\tN(%.f  
    %   See also ZERNPOL, ZERNFUN2. iJdJP)!tz6  
    .WSn Y71  
    %   Paul Fricker 11/13/2006 W/A@qo"  
    < e3] pM  
    <mP_K^9c  
    % Check and prepare the inputs: _3W .:  
    % ----------------------------- r;b`@ .  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +s_a{iMVP  
        error('zernfun:NMvectors','N and M must be vectors.') I!Dx)>E&  
    end G8]{pbX  
    XR8`,qH>  
    if length(n)~=length(m) De 3;}]wC  
        error('zernfun:NMlength','N and M must be the same length.') Q~"Lyy8  
    end X*#\JF4$i  
    5M>p%/  
    n = n(:); zEQQ4)mA  
    m = m(:); auIW>0?}  
    if any(mod(n-m,2)) _"F=4`lJ  
        error('zernfun:NMmultiplesof2', ... ~i?Jg/qcxN  
              'All N and M must differ by multiples of 2 (including 0).') t{UWb~"  
    end A' ![*O  
    [qxpu{  
    if any(m>n) Q,9KLi3  
        error('zernfun:MlessthanN', ... Uf_mwEE  
              'Each M must be less than or equal to its corresponding N.') qm#?DSLap  
    end pqvl,G5  
    sAO/yG  
    if any( r>1 | r<0 ) U(+QrC:  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') us5Zi#}  
    end & :W6O)uY  
    Te!eM{_$T  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) StR)O))I  
        error('zernfun:RTHvector','R and THETA must be vectors.') S&=@Hj-  
    end 08@4u L  
    ej7N5~!,s  
    r = r(:); i`6utOq  
    theta = theta(:); r_ m|?U %  
    length_r = length(r); h+d k2|a  
    if length_r~=length(theta) ,]qc#KDq-1  
        error('zernfun:RTHlength', ...  ZJ)>gV  
              'The number of R- and THETA-values must be equal.') #mioT",bm=  
    end ;=%cA#}_0  
    i< imE#  
    % Check normalization: * XDe:A  
    % -------------------- `{yD\qDyX  
    if nargin==5 && ischar(nflag) @w%kOX  
        isnorm = strcmpi(nflag,'norm'); }#g &l*P  
        if ~isnorm kVeY} 8  
            error('zernfun:normalization','Unrecognized normalization flag.') ?TDmW8G}J  
        end Ozulp(8*  
    else Ir` l*:j$  
        isnorm = false;  OvC@E]/+  
    end _MQh<,Z8  
    .GYdC '  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )abH//Pps.  
    % Compute the Zernike Polynomials b!QRD'31'j  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N>s3tGh  
    p&xj7qwp@F  
    % Determine the required powers of r: :hB6-CZkqN  
    % ----------------------------------- qbD_  
    m_abs = abs(m); ,o `tRh<  
    rpowers = []; *!NW!,R  
    for j = 1:length(n) J| 46i  
        rpowers = [rpowers m_abs(j):2:n(j)]; D!)h92CIDm  
    end (t"|XSF  
    rpowers = unique(rpowers); _+~jZ]o N  
    Z0~,cO8~  
    % Pre-compute the values of r raised to the required powers, 8SiWAOQAL  
    % and compile them in a matrix: 2*-qEUl1  
    % ----------------------------- ;8]Hw a1!  
    if rpowers(1)==0 >FFp"%%  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);  W"~"R  
        rpowern = cat(2,rpowern{:});  Cb|R  
        rpowern = [ones(length_r,1) rpowern]; hR>`I0|p&  
    else -&y&b-  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); <qoPBm])  
        rpowern = cat(2,rpowern{:}); 1JGww]JZo  
    end Me+)2S 9  
    EL *l5!Iu  
    % Compute the values of the polynomials: *z'Rl'j9[  
    % -------------------------------------- #\}xyPS  
    y = zeros(length_r,length(n)); +b dnTV6  
    for j = 1:length(n) M7gqoJM'Q  
        s = 0:(n(j)-m_abs(j))/2; ]#rmk!VT?  
        pows = n(j):-2:m_abs(j); O4W 2X@  
        for k = length(s):-1:1 57N<OQWf  
            p = (1-2*mod(s(k),2))* ... *; 6LX  
                       prod(2:(n(j)-s(k)))/              ... fb!>@@9Z  
                       prod(2:s(k))/                     ... w (,x{Bg\  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... UXS+GAWU  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); cPl$N5/5  
            idx = (pows(k)==rpowers); wD<W'K   
            y(:,j) = y(:,j) + p*rpowern(:,idx); oFu( J  
        end $O9#4A;  
         i:^ 8zW  
        if isnorm J s,.$t  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); a3_pF~Qx  
        end pmDFmES  
    end }Do$oyAV$G  
    % END: Compute the Zernike Polynomials E-#}.}i5  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,Ct1)%   
    Znh<r[p<  
    % Compute the Zernike functions: PkdL] !:  
    % ------------------------------ ,NU`aG-  
    idx_pos = m>0; u,Cf4H*xS  
    idx_neg = m<0; Z1+1>|-iW  
    L q;=UE  
    z = y; Czd)AVK  
    if any(idx_pos) {X&H  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); I6 Q{ Axy  
    end 5q.)K f+  
    if any(idx_neg) Ivc/g,  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); k!wEPi]  
    end $)M 5@KT  
    RZ:= ';  
    % EOF zernfun
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) IX 2 dic'  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 5'[b:YC  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated  F]KAnEf  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive nHF%PH#|o  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, &X OFc.u  
    %   and THETA is a vector of angles.  R and THETA must have the same /~;om\7r  
    %   length.  The output Z is a matrix with one column for every P-value, QNgfvy  
    %   and one row for every (R,THETA) pair. tx||<8  
    % mvt%3zCB!  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Ex}hk!  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) _T.k/a  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 5Z*6,P0  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 c^EU &q{4  
    %   for all p. phqmr5s^H  
    % ^;a~_9 m-  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 @j=Q$k.GF  
    %   Zernike functions (order N<=7).  In some disciplines it is FkY <I]F  
    %   traditional to label the first 36 functions using a single mode S;I}:F#5  
    %   number P instead of separate numbers for the order N and azimuthal 58o'Q  
    %   frequency M.  MON]rj7  
    % 0hM!#BU5K  
    %   Example: C0%yGLh&  
    % w j<fi  
    %       % Display the first 16 Zernike functions fy|$A@f  
    %       x = -1:0.01:1; gano>W0  
    %       [X,Y] = meshgrid(x,x); 4|Ay;}X \  
    %       [theta,r] = cart2pol(X,Y); [e;c)XS[  
    %       idx = r<=1; TnQ"c)ta  
    %       p = 0:15; T43Jgk,  
    %       z = nan(size(X)); Av_JcH  
    %       y = zernfun2(p,r(idx),theta(idx)); $ucA.9pJ  
    %       figure('Units','normalized') @PAT|6  
    %       for k = 1:length(p) _%:$sAj  
    %           z(idx) = y(:,k); ^n&_JQIXb  
    %           subplot(4,4,k) 5v,_ Hgh  
    %           pcolor(x,x,z), shading interp sA!$}W  
    %           set(gca,'XTick',[],'YTick',[]) mL]a_S{H  
    %           axis square K:(E"d;  
    %           title(['Z_{' num2str(p(k)) '}']) OV,t|  
    %       end )4e?-?bK!  
    % <S68UN(Ke  
    %   See also ZERNPOL, ZERNFUN.  jWqjGX`  
    kqQT^6S   
    %   Paul Fricker 11/13/2006 6,a:s:$>}R  
    +Fa!<txn  
    (]N- HN]v  
    % Check and prepare the inputs: M2lvD&  
    % ----------------------------- jiqE^j3;  
    if min(size(p))~=1 ]-"G:r  
        error('zernfun2:Pvector','Input P must be vector.') h1 pEC  
    end K$/&C:,Q  
    liU8OXBl  
    if any(p)>35 Bht!+  
        error('zernfun2:P36', ... p(Ux]_s%  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 7m:ZG  
               '(P = 0 to 35).']) dTZ$92<  
    end IRyZ0$r:e\  
    cPy/}A  
    % Get the order and frequency corresonding to the function number: Mqv[7.|  
    % ---------------------------------------------------------------- I>JBGR`j  
    p = p(:); }\0ei(%H  
    n = ceil((-3+sqrt(9+8*p))/2); *WaqNMD[%  
    m = 2*p - n.*(n+2); qsWy <yL+  
    LY;Fjb yU  
    % Pass the inputs to the function ZERNFUN: ->L>`<7(  
    % ---------------------------------------- e2qSU[  
    switch nargin +h08uo5c  
        case 3 a'Yi^;2+\  
            z = zernfun(n,m,r,theta); L@s_)?x0  
        case 4 S`8Iu[Ma  
            z = zernfun(n,m,r,theta,nflag); "fu:hHq  
        otherwise * o{7 a$V  
            error('zernfun2:nargin','Incorrect number of inputs.') T/]f5/  
    end N@0cn q:"  
    ZeLed[J^xJ  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) S5V:HRj{?  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 'o~gT ;T#  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of LcZ|A;it  
    %   order N and frequency M, evaluated at R.  N is a vector of 7=%Oev&0g-  
    %   positive integers (including 0), and M is a vector with the k/(]1QnW  
    %   same number of elements as N.  Each element k of M must be a YjH~8==  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) m*jTvn  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is !Au#j^5K-o  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix XO+rg&Pu  
    %   with one column for every (N,M) pair, and one row for every +Qf}&D_  
    %   element in R. T*nP-b  
    % jD${ZIv  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- \<ysJgqUG  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is l0C`teO  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 4(p`xdr}K  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 2vWn(6`  
    %   for all [n,m]. c]zFZJ6M  
    % 3~VV2O  
    %   The radial Zernike polynomials are the radial portion of the C~R ?iZ.&U  
    %   Zernike functions, which are an orthogonal basis on the unit J#t-." f6^  
    %   circle.  The series representation of the radial Zernike MqJ5|C.q  
    %   polynomials is =3 Vug2*wd  
    % ^GyZycch  
    %          (n-m)/2 Qe{w)e0}`  
    %            __ BlQ X$s]  
    %    m      \       s                                          n-2s 5VJe6i9;  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r $ dF3@(p  
    %    n      s=0 yOt#6Vw  
    % rlD!%gG2x  
    %   The following table shows the first 12 polynomials. &a;?o~%*]i  
    % IzJq:G.  
    %       n    m    Zernike polynomial    Normalization Z`u$#<ukX  
    %       --------------------------------------------- [iEz?1.,  
    %       0    0    1                        sqrt(2) bMxK@$G~  
    %       1    1    r                           2 ~`C _B]3|  
    %       2    0    2*r^2 - 1                sqrt(6) SI=u-'%  
    %       2    2    r^2                      sqrt(6) X fqhD&g  
    %       3    1    3*r^3 - 2*r              sqrt(8) -sfv"?  
    %       3    3    r^3                      sqrt(8) A4cOnG,  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) DL?nvH  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) P6R_W  
    %       4    4    r^4                      sqrt(10) h='F,r5#2  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 4A6Yl6\Y  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) c~?Zmdn:  
    %       5    5    r^5                      sqrt(12) `/ayg:WSU  
    %       --------------------------------------------- 5OppK(Oi*C  
    % Rh?bBAn8  
    %   Example: tw>2<zmSi%  
    % EgU#r@7I  
    %       % Display three example Zernike radial polynomials u;gO+)wqv  
    %       r = 0:0.01:1; i*Ldec^  
    %       n = [3 2 5]; (XQuRL<X  
    %       m = [1 2 1]; uTxa5j  
    %       z = zernpol(n,m,r); m$kQbPlatN  
    %       figure Ph1XI&us9  
    %       plot(r,z) L]|mWyzT  
    %       grid on QCF'/G  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') n+Kv^Y`qxO  
    % B+j]C$8}  
    %   See also ZERNFUN, ZERNFUN2. J+J,W5t^  
    -(8I?{"4i  
    % A note on the algorithm. `(sb  
    % ------------------------ [/UchU]DT  
    % The radial Zernike polynomials are computed using the series jb -kg</A  
    % representation shown in the Help section above. For many special ||_hET  
    % functions, direct evaluation using the series representation can :q]9F4im  
    % produce poor numerical results (floating point errors), because /v8Q17O?e  
    % the summation often involves computing small differences between =O![>Fu5  
    % large successive terms in the series. (In such cases, the functions |zYOCDFf  
    % are often evaluated using alternative methods such as recurrence ^,acU\}VqP  
    % relations: see the Legendre functions, for example). For the Zernike La;G S  
    % polynomials, however, this problem does not arise, because the )rD!4"8/A  
    % polynomials are evaluated over the finite domain r = (0,1), and #_@cI(P  
    % because the coefficients for a given polynomial are generally all @#)` -]g  
    % of similar magnitude. KLg1(W(  
    % rP3HR 5  
    % ZERNPOL has been written using a vectorized implementation: multiple .$%p0Yx+  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] K>@yk9)vi  
    % values can be passed as inputs) for a vector of points R.  To achieve \ ;npdFy  
    % this vectorization most efficiently, the algorithm in ZERNPOL xzm]v9k&  
    % involves pre-determining all the powers p of R that are required to Nr4:Gih  
    % compute the outputs, and then compiling the {R^p} into a single ff\~`n~WZ  
    % matrix.  This avoids any redundant computation of the R^p, and )U?_&LY)[M  
    % minimizes the sizes of certain intermediate variables. l@4_D;b3o"  
    % EiL#Dwx  
    %   Paul Fricker 11/13/2006 xzOa9w/  
    3/kT'r  
    QPJ \Iu@D$  
    % Check and prepare the inputs: /SD}`GxH  
    % ----------------------------- Az`Aa0h]7  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) dAh.I3  
        error('zernpol:NMvectors','N and M must be vectors.') Gt9$hB7  
    end HTJ2D@h  
    E-bswUVaEE  
    if length(n)~=length(m) }02`ve*   
        error('zernpol:NMlength','N and M must be the same length.') _M&TT]a  
    end _FY&XL=  
    }TmOoi(X@  
    n = n(:);  Y'iX   
    m = m(:); 2bp@m;g$  
    length_n = length(n); t4WB^dHYp  
    :v&[ !  
    if any(mod(n-m,2)) 2$JGhgDI  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') /4:bx#;A  
    end Z!1D4`w  
    |*&l?S  
    if any(m<0) ]gk1q{Ql<  
        error('zernpol:Mpositive','All M must be positive.') ~VGnE:  
    end /`*{57/3  
    B$4*U"tk  
    if any(m>n) ,fkvvM{mq  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') > ;,S||  
    end uN|A}/hr]  
    l!mbpFt  
    if any( r>1 | r<0 ) Mt[yY|Ec|  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 3Vb4zZsl  
    end "yn~axk7  
    k ut=( ;  
    if ~any(size(r)==1) -aoYoJ '  
        error('zernpol:Rvector','R must be a vector.') WE<?y_0y&  
    end ;[v!#+yml  
    +)Z]<O  
    r = r(:); D*XrK0#Z`  
    length_r = length(r); SepjF  
    oaY_6  
    if nargin==4 NB-dlv1  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); {ZJO5*  
        if ~isnorm @T'^V0!-q:  
            error('zernpol:normalization','Unrecognized normalization flag.') Hq3|>OqC2Q  
        end (o^tmH*  
    else @,vmX z  
        isnorm = false; tcD7OC:"6  
    end (m~>W"x/  
    88g3<&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^|Ap_!t$;  
    % Compute the Zernike Polynomials XHr*Rs.[=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h3ygL"k  
    o-,."|6  
    % Determine the required powers of r: vCzZjGBY  
    % ----------------------------------- p~pD`'%  
    rpowers = []; [);oj<  
    for j = 1:length(n) D31X {dJ  
        rpowers = [rpowers m(j):2:n(j)]; uZqL'l+/y  
    end 7#V7D6j1  
    rpowers = unique(rpowers); :ym?]EL4o  
     *}?[tR5  
    % Pre-compute the values of r raised to the required powers, \Z-th,t  
    % and compile them in a matrix: L 4By5)  
    % ----------------------------- dbp\tWaW  
    if rpowers(1)==0 -'rdN i  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); `MtzA^Xr  
        rpowern = cat(2,rpowern{:}); B}e/MlX3M  
        rpowern = [ones(length_r,1) rpowern]; w:&" "'E  
    else L*@`i ]jl  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); w^yb`\$  
        rpowern = cat(2,rpowern{:}); X Xque-  
    end -IPo/?}  
    wi(Y=?=  
    % Compute the values of the polynomials: ( }-*irSsj  
    % -------------------------------------- !Sc"V.o @!  
    z = zeros(length_r,length_n); =|3BkmO  
    for j = 1:length_n GO"`{|o  
        s = 0:(n(j)-m(j))/2; r`H}f#.KR  
        pows = n(j):-2:m(j); "<,lqIqA;  
        for k = length(s):-1:1 I!x.bp~V!  
            p = (1-2*mod(s(k),2))* ... ,OE&e* 1  
                       prod(2:(n(j)-s(k)))/          ... /6x&%G:m#  
                       prod(2:s(k))/                 ... !09)WtsEfx  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... C@(@n!o:!  
                       prod(2:((n(j)+m(j))/2-s(k))); {)YbksrJ{  
            idx = (pows(k)==rpowers); !_QI<=X  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ;pnF%co9  
        end Z,&O8Jelf  
         %(CC  
        if isnorm ZmA}i`  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ,Qj G|P  
        end ``MO5${  
    end f:o.[4p2  
    y>o:5':;'  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  yR5XcPoKI  
    FOD'&Yb&  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 z<Z0/a2'1  
    "IS; o o$g  
    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)