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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 "-0;#&!  
    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! 7"K^H]6u30  
     
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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 )&z4_l8`=  
    function z = zernfun(n,m,r,theta,nflag) L#ZLawG  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. "mt p0  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N D$hQyhz'  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ~6sE an3p  
    %   unit circle.  N is a vector of positive integers (including 0), and :~33U)?{T  
    %   M is a vector with the same number of elements as N.  Each element <r;o6>+  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) PkJcd->  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, +6376$dC  
    %   and THETA is a vector of angles.  R and THETA must have the same 50,Y  
    %   length.  The output Z is a matrix with one column for every (N,M) ZpWu,1  
    %   pair, and one row for every (R,THETA) pair. nsl*Dm"*F  
    % 1J'pB;.]s  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike n^Vxi;F  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), :l`i4kx  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ,R}Z=w#  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, |[ocyUsxX  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized }P. K2ku  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 0I^Eo|  
    % *%?d\8d  
    %   The Zernike functions are an orthogonal basis on the unit circle. 9v$qrM`8  
    %   They are used in disciplines such as astronomy, optics, and T3rn+BxF7  
    %   optometry to describe functions on a circular domain. {,Fcd(MU  
    % A6i et~h[  
    %   The following table lists the first 15 Zernike functions. )-q\aX$])  
    % OHhs y|W  
    %       n    m    Zernike function           Normalization n}:t<  
    %       -------------------------------------------------- gn`zy9PU  
    %       0    0    1                                 1 OAVQ`ek  
    %       1    1    r * cos(theta)                    2 :MBS>owR  
    %       1   -1    r * sin(theta)                    2 R'Eq:Rv~;^  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) sX5sL  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 8nsZ+,@+[  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) J | q^+K  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) C#$6O8O  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ^]7,1dH}M  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) (Y)!"_|  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) <tW:LU(!  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) "Y(^F bs  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Xy!&^C` J`  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ]9@X? q  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) %yvA   
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ENyAF%6  
    %       -------------------------------------------------- $l#{_~ "m7  
    % _25d%Ne0  
    %   Example 1: UM`nq;>  
    % ]hKgA~;  
    %       % Display the Zernike function Z(n=5,m=1) >[8#hSk  
    %       x = -1:0.01:1; 2/EK`S  
    %       [X,Y] = meshgrid(x,x); 3`ml; L?D  
    %       [theta,r] = cart2pol(X,Y); [9HYO  
    %       idx = r<=1; =%L@WVbM  
    %       z = nan(size(X)); /sV?JV[t  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 0# l#,Y6#I  
    %       figure EIPnm%{1  
    %       pcolor(x,x,z), shading interp oR#my ^  
    %       axis square, colorbar Oa1'oYIHg  
    %       title('Zernike function Z_5^1(r,\theta)') k{{hZ/om  
    % 2!idy]vy_  
    %   Example 2: hbH#Co~o4#  
    % s,kU*kHn  
    %       % Display the first 10 Zernike functions q- H&5K  
    %       x = -1:0.01:1; 5pmQp}}R  
    %       [X,Y] = meshgrid(x,x); 7O9n!aJ  
    %       [theta,r] = cart2pol(X,Y); dEG ]riO  
    %       idx = r<=1; }>,CUz  
    %       z = nan(size(X)); `1q|F9D  
    %       n = [0  1  1  2  2  2  3  3  3  3]; m\?\6W k  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; *7_@7=W,  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; @sdS 0pC  
    %       y = zernfun(n,m,r(idx),theta(idx)); |e+aZ%g  
    %       figure('Units','normalized') u6pIdt  
    %       for k = 1:10 dxntGH< O  
    %           z(idx) = y(:,k); Y.X4*B  
    %           subplot(4,7,Nplot(k)) /L$NE$D} "  
    %           pcolor(x,x,z), shading interp D Kq-C%  
    %           set(gca,'XTick',[],'YTick',[]) pkW5D  
    %           axis square &\c5!xQ9*  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) a-:pJE.'p  
    %       end +NT:<(;|i5  
    % "5h_8k~sQ  
    %   See also ZERNPOL, ZERNFUN2.  +xq=<jy  
    T1bFxim#b  
    %   Paul Fricker 11/13/2006 I^@.Aw t  
    ~Zu}M>-^c,  
    0H<4+ *`K  
    % Check and prepare the inputs: LC76Qi;|k  
    % ----------------------------- {>A 8g({i  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 9j2\y=<&  
        error('zernfun:NMvectors','N and M must be vectors.') t%:G|n Sz  
    end `;e^2  
    Q<C@KBiVE  
    if length(n)~=length(m) g*28L[Q~  
        error('zernfun:NMlength','N and M must be the same length.') x~nQm]@`h  
    end c<>y!^g  
    h)P]gT0f/  
    n = n(:); C-&#r."L  
    m = m(:); @| P3  
    if any(mod(n-m,2)) 4[Z1r~t\L  
        error('zernfun:NMmultiplesof2', ... xp(mB7;:  
              'All N and M must differ by multiples of 2 (including 0).') %~G0[fG  
    end wCC-Y kA  
    K# /Ch5?  
    if any(m>n) $=lJG(2%  
        error('zernfun:MlessthanN', ... FJW`$5?  
              'Each M must be less than or equal to its corresponding N.') ~%/'0}F  
    end 0T=jR{j!o  
    tgc@7  
    if any( r>1 | r<0 ) Iht@mE  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ]2P/G5C3tU  
    end Xa>}4j.  
    }0vtc[!  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) W;91H'`?H  
        error('zernfun:RTHvector','R and THETA must be vectors.') Bg5;Q)  
    end 8dlInms  
    z(#=tC|  
    r = r(:); ??q!jm-m  
    theta = theta(:); jzQ9zy_  
    length_r = length(r); cK/PQsMP  
    if length_r~=length(theta) o%$<LaQG5  
        error('zernfun:RTHlength', ... F W/)uf3I  
              'The number of R- and THETA-values must be equal.') .\)--+(  
    end ~T;K-9R  
    r,QJG$ Jo  
    % Check normalization: py}.00it  
    % -------------------- dy'X<o^?W  
    if nargin==5 && ischar(nflag) )Gx": D  
        isnorm = strcmpi(nflag,'norm'); .0?ss0~  
        if ~isnorm >c&4_?d&,A  
            error('zernfun:normalization','Unrecognized normalization flag.') J6= w:c  
        end *1R##9\jU7  
    else ]j72P  
        isnorm = false; )H.ubM1  
    end S$Qr@5  
    'M47'{7T  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z3Bo@`&?  
    % Compute the Zernike Polynomials {6-;P#Q0_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h5<T.vV  
    UUZ6N ZQI  
    % Determine the required powers of r: $,p.=j;P  
    % ----------------------------------- aB/{ %%o  
    m_abs = abs(m); $:xF)E  
    rpowers = []; []^PJ  
    for j = 1:length(n) z<FV1niE  
        rpowers = [rpowers m_abs(j):2:n(j)]; sj#{TTW  
    end c1gz #,  
    rpowers = unique(rpowers); h4J{jh.  
    p)K9 ZI  
    % Pre-compute the values of r raised to the required powers, {yGZc3e1j  
    % and compile them in a matrix: ;bUJ+6f:  
    % ----------------------------- tn(f rccy  
    if rpowers(1)==0 BDarJY  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?v0A/68s#  
        rpowern = cat(2,rpowern{:}); wjN`EF5$}&  
        rpowern = [ones(length_r,1) rpowern]; o'9OPoof:.  
    else FSI]k:  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 1\M"`L/  
        rpowern = cat(2,rpowern{:}); ]C9%]`  
    end 5q0BG!A%T  
    IwZZewb-a  
    % Compute the values of the polynomials: aNuZ/9O  
    % -------------------------------------- WO.}DUfG+  
    y = zeros(length_r,length(n));  |JirBz  
    for j = 1:length(n) C5.\;;7^&  
        s = 0:(n(j)-m_abs(j))/2; p,M3#^ q  
        pows = n(j):-2:m_abs(j); p~v2XdR  
        for k = length(s):-1:1 AH"g^ gw~T  
            p = (1-2*mod(s(k),2))* ... PPuXas?i  
                       prod(2:(n(j)-s(k)))/              ... I,?Fqg'sq  
                       prod(2:s(k))/                     ... D2hAlV)i(  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... (cPeee%Q  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); xfbK eS8  
            idx = (pows(k)==rpowers); 3fbD"gL  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 6E)uu; 8  
        end +MOe{:/6  
         H]T2$'U6  
        if isnorm =woqHTR  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); aPcGI  
        end y<IZ|f  
    end /j=DC9_  
    % END: Compute the Zernike Polynomials %XDip]+rb  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% H4,.H,PZ  
    z=- 8iks|  
    % Compute the Zernike functions: 4iL.4Uj{N  
    % ------------------------------ (;Dn%kK  
    idx_pos = m>0; Zu [?'  
    idx_neg = m<0; %WJ\'@O\  
    -.+KCt G$+  
    z = y; A{{q'zb!  
    if any(idx_pos) a!hI${Xn  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); TnMVHO-  
    end ;|;h9"  
    if any(idx_neg) FrAqTz  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); `E4!u=%  
    end iuH8g  
    ~L4*b *W  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) {C,1w  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. /!&b'7y  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated R"\u b"]  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive U#~nN+SIt  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, `|ck5DZT5L  
    %   and THETA is a vector of angles.  R and THETA must have the same QB.*R?A  
    %   length.  The output Z is a matrix with one column for every P-value, #P,[fgNy  
    %   and one row for every (R,THETA) pair. J*6n6  
    % )W}/k$S  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike wr;|\<c  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) JK^B+.  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 0[$Mo3c+'  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 9-Nq[i"  
    %   for all p. >w]k3MC  
    % '#An+;x{  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 tr9_bl&z  
    %   Zernike functions (order N<=7).  In some disciplines it is v[3hnLN%  
    %   traditional to label the first 36 functions using a single mode -XDP-Trk  
    %   number P instead of separate numbers for the order N and azimuthal * F%ol;|Q  
    %   frequency M. 9'/|?I  
    % T0]%(F/8  
    %   Example: ]`i@~Z h\  
    % pb2{J#  
    %       % Display the first 16 Zernike functions }T1Xds8w)t  
    %       x = -1:0.01:1; E't G5,/m  
    %       [X,Y] = meshgrid(x,x); b1 ['uJF  
    %       [theta,r] = cart2pol(X,Y); ^?S@v1~7d  
    %       idx = r<=1; L_zmU_zD  
    %       p = 0:15; Zy+QA>d|  
    %       z = nan(size(X)); i&s=!`  
    %       y = zernfun2(p,r(idx),theta(idx)); 2I(@aB+  
    %       figure('Units','normalized') #3:'lGBIK  
    %       for k = 1:length(p) J^+$L"K  
    %           z(idx) = y(:,k); >-@{vyoOy  
    %           subplot(4,4,k) 3 Q%k (,  
    %           pcolor(x,x,z), shading interp HV.|Eh_7  
    %           set(gca,'XTick',[],'YTick',[]) tx&U"]  
    %           axis square Mpl,}Q!c  
    %           title(['Z_{' num2str(p(k)) '}']) JjTzq2'%  
    %       end J-%PyvK$?  
    % Z :i"|;  
    %   See also ZERNPOL, ZERNFUN. )]>Y*<s }  
    %_J/&{6G  
    %   Paul Fricker 11/13/2006 $j4?'-i=e  
    <"|<)BGeI  
    ZD] '$  
    % Check and prepare the inputs: ]!Aze^7;  
    % ----------------------------- WSB|-Qj}W  
    if min(size(p))~=1 ht cO ~b  
        error('zernfun2:Pvector','Input P must be vector.') 2T2#HP  
    end -mYI[AG)  
    XJ1nhE  
    if any(p)>35 4.kkxQR7r  
        error('zernfun2:P36', ... s8I77._s  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... !=Vh2UbC3  
               '(P = 0 to 35).']) w!h{P38  
    end 3SB7)8Id1  
    Kmf-l*7}  
    % Get the order and frequency corresonding to the function number: _<~Vxz9  
    % ---------------------------------------------------------------- 4}YT@={g}  
    p = p(:); >_% g8T'  
    n = ceil((-3+sqrt(9+8*p))/2); Bma|!p{  
    m = 2*p - n.*(n+2); bDh(;%=  
    H&Lbdu~E  
    % Pass the inputs to the function ZERNFUN: *l8vCa9Y  
    % ---------------------------------------- d_|v=^;  
    switch nargin P>,D$-3  
        case 3 s?I=}  
            z = zernfun(n,m,r,theta); Z5^ UF2`Q  
        case 4 /(L1!BPP9m  
            z = zernfun(n,m,r,theta,nflag); xMGd'l?  
        otherwise 7Qztc?XK  
            error('zernfun2:nargin','Incorrect number of inputs.') v__Go kj-  
    end R =c  
    X OtS+p  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) X)+sHcE~#  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. ] :#IZ0#  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of H;te)km}  
    %   order N and frequency M, evaluated at R.  N is a vector of 13@| {H CB  
    %   positive integers (including 0), and M is a vector with the ;rdLYmmx^  
    %   same number of elements as N.  Each element k of M must be a iiFKt(  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ,Yt&PE  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is HpC4$JMm  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 'bO? =+c  
    %   with one column for every (N,M) pair, and one row for every 71ybZ 0  
    %   element in R. D[y|y 3F  
    % laJ%fBWmbi  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- P z!yIj  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is !{0!G  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to bW3o%srxa  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ^>28>!"1  
    %   for all [n,m]. p=T\3_q  
    % 6Ky"4\e  
    %   The radial Zernike polynomials are the radial portion of the daNIP1Qn  
    %   Zernike functions, which are an orthogonal basis on the unit 2DQC)Pe+z  
    %   circle.  The series representation of the radial Zernike iKKWn*u  
    %   polynomials is Jxp'.oo[  
    %  ]bSt[  
    %          (n-m)/2 $KFWV2P  
    %            __ 5!%/j,?  
    %    m      \       s                                          n-2s aE$p;I  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r $ wB  
    %    n      s=0 V lNzm  
    % Agf!6kh  
    %   The following table shows the first 12 polynomials. U#4W"1~iX  
    % b)@x@3"O  
    %       n    m    Zernike polynomial    Normalization /_(Dq8^g@  
    %       --------------------------------------------- Zt=X %M|aw  
    %       0    0    1                        sqrt(2) gf>5xf{M  
    %       1    1    r                           2 }y/t~f+  
    %       2    0    2*r^2 - 1                sqrt(6) *?'T8yf^  
    %       2    2    r^2                      sqrt(6) \H(,'w7H  
    %       3    1    3*r^3 - 2*r              sqrt(8) ~h.B\Sc]Q  
    %       3    3    r^3                      sqrt(8) _ji%BwJ  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) S22; g  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) :b-(@a7>  
    %       4    4    r^4                      sqrt(10) ~?r6Ax-R  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 9-SXu lgu  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ,{at?y*  
    %       5    5    r^5                      sqrt(12) O}V2> W$  
    %       --------------------------------------------- mqw.v$>  
    % `#bcoK5  
    %   Example: J-c7ZcTt  
    % hT#mM*`  
    %       % Display three example Zernike radial polynomials Q0-~&e_'  
    %       r = 0:0.01:1; N h%8;  
    %       n = [3 2 5]; >MH@FnUL  
    %       m = [1 2 1]; kA0 ^~  
    %       z = zernpol(n,m,r); )-oNy-YL  
    %       figure 1[ Pbsb  
    %       plot(r,z) Ek0.r)Nw  
    %       grid on j!dklQh0  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 70~]J8T+u  
    % 3]P=co@  
    %   See also ZERNFUN, ZERNFUN2. g9JtWgu  
    3u<2~!sR  
    % A note on the algorithm. ?9"glzxr  
    % ------------------------ R,lr&;a8  
    % The radial Zernike polynomials are computed using the series &s5*akG  
    % representation shown in the Help section above. For many special -6\9B>qa  
    % functions, direct evaluation using the series representation can WYL.J5O  
    % produce poor numerical results (floating point errors), because I%Z &i-33y  
    % the summation often involves computing small differences between Iq*7F5B  
    % large successive terms in the series. (In such cases, the functions [<hiOB  
    % are often evaluated using alternative methods such as recurrence JAK*HA  
    % relations: see the Legendre functions, for example). For the Zernike ,D1QJPM  
    % polynomials, however, this problem does not arise, because the "+AD+D  
    % polynomials are evaluated over the finite domain r = (0,1), and "cx" d:  
    % because the coefficients for a given polynomial are generally all +zvK/Fj2q  
    % of similar magnitude. hSmM OS{  
    % B!0[LlF+  
    % ZERNPOL has been written using a vectorized implementation: multiple -}s?!Pg>  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Aj_}B.  
    % values can be passed as inputs) for a vector of points R.  To achieve sr4K-|@  
    % this vectorization most efficiently, the algorithm in ZERNPOL k?,g:[4!  
    % involves pre-determining all the powers p of R that are required to e_SlM=_ u  
    % compute the outputs, and then compiling the {R^p} into a single w&H7S{  
    % matrix.  This avoids any redundant computation of the R^p, and 9]iDNa/D  
    % minimizes the sizes of certain intermediate variables. tq[C"| dH  
    % 0Sk{P>A  
    %   Paul Fricker 11/13/2006 )? xg=o/?  
    cP,jC(<N  
    p%BO:%v  
    % Check and prepare the inputs: f 36rU  
    % ----------------------------- P+xZaf H  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) TocqoYX{{  
        error('zernpol:NMvectors','N and M must be vectors.') 4(NI-|q0  
    end Q5ux**(Wr  
    G>b1No3%k  
    if length(n)~=length(m) lc^%:#@  
        error('zernpol:NMlength','N and M must be the same length.') =g ]C9'I3  
    end m(Oup=\%b}  
    !**q20-aP  
    n = n(:); Y*6*;0Kx  
    m = m(:); q:( K^  
    length_n = length(n); V+Z22  
    kDrGl{U}  
    if any(mod(n-m,2)) 1{*x+GC^/  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') h W-[omr0  
    end G} p~VLf  
    wBf bpoE7  
    if any(m<0) *+G K ?Ga  
        error('zernpol:Mpositive','All M must be positive.') /cg!Ap5  
    end {VFp fo  
    W$y?~2  
    if any(m>n) OA8pao~H  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') R$\ieNb  
    end 2b<0g@~X  
    *oF{ R^  
    if any( r>1 | r<0 ) 8/=2N  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') =LC5o2bLy  
    end '{|87kI  
    ?h5Y^}8Qg  
    if ~any(size(r)==1) `bi5#xR  
        error('zernpol:Rvector','R must be a vector.') ]b[ 3 th*  
    end B::vOg77  
    Y=5hm  
    r = r(:); :F_U^pyG  
    length_r = length(r); BjShK+Y  
    XjV7Ew^7  
    if nargin==4 {*: C$"L  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); giPyo"SD  
        if ~isnorm }"szL=s  
            error('zernpol:normalization','Unrecognized normalization flag.') >uVG]  
        end I;No++N0  
    else ;gF"o5/Q  
        isnorm = false; aW"BN 5eM>  
    end qRr;&M &t_  
    yVW)DQ 4?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -v]7}[ .[  
    % Compute the Zernike Polynomials y(%6?a @  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -1@kt<Es  
    Ft07>E$/Q^  
    % Determine the required powers of r: //`X+[bMG  
    % ----------------------------------- 3o1j l2n  
    rpowers = []; |{$Vk%cUE  
    for j = 1:length(n) $(GXlhA  
        rpowers = [rpowers m(j):2:n(j)]; y+c+/L8  
    end :/u EPki  
    rpowers = unique(rpowers); M;V2O;  
    H:CwUFL  
    % Pre-compute the values of r raised to the required powers, LE Y$St  
    % and compile them in a matrix: bk V_ ^8  
    % -----------------------------  G=wJz  
    if rpowers(1)==0 5YRa2#d  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); #zfBNkk&@  
        rpowern = cat(2,rpowern{:}); F&ux9zP  
        rpowern = [ones(length_r,1) rpowern]; EayZ*e ]  
    else &L_(yJ~-  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); VLRW,lR9O  
        rpowern = cat(2,rpowern{:}); d5h:py5  
    end |_Vlw&qu+  
    tYVmB:l  
    % Compute the values of the polynomials: 1B2>8 N  
    % -------------------------------------- m'Ran3rp  
    z = zeros(length_r,length_n); O Qd,.m  
    for j = 1:length_n 6L8wsz CW  
        s = 0:(n(j)-m(j))/2; Z[ys>\_To  
        pows = n(j):-2:m(j); X'O3)Yg  
        for k = length(s):-1:1 M4`qi3I  
            p = (1-2*mod(s(k),2))* ... 1X]?-+',.  
                       prod(2:(n(j)-s(k)))/          ... <5=^s%H  
                       prod(2:s(k))/                 ... : x W.(^(d  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... .|!Kv+yD  
                       prod(2:((n(j)+m(j))/2-s(k))); GP1b/n3F1  
            idx = (pows(k)==rpowers); h(ZZ7(ue  
            z(:,j) = z(:,j) + p*rpowern(:,idx); yH irm|o  
        end ]5wc8Kh"  
         $)6y:t"  
        if isnorm u`g|u:(r  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); (r.[b  
        end 2Lm.;l4YO  
    end Nv w'[?m  
    Ean #>h  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  =&Z#QD"vl  
    0oQJ}8t  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 ?U+nR/H:6  
    I= 2jQ>$Q  
    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)