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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 9EHhVi  
    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! AAuH}W>n  
     
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    离线phility
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    只看该作者 1楼 发表于: 2011-03-12
    可以用matlab编程,用zernike多项式进行波面拟合,求出zernike多项式的系数,拟合的算法有很多种,最简单的是最小二乘法,你可以查下相关资料,挺简单的
    离线phility
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    只看该作者 2楼 发表于: 2011-03-12
    泽尼克多项式的前9项对应象差的
    离线niuhelen
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    只看该作者 3楼 发表于: 2011-03-12
    回 2楼(phility) 的帖子
    非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 _|jEuif  
    function z = zernfun(n,m,r,theta,nflag) @js`$  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ^tF lA)  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ~~r7TPq  
    %   and angular frequency M, evaluated at positions (R,THETA) on the utzf7?nIS  
    %   unit circle.  N is a vector of positive integers (including 0), and Yj"{aFK#u@  
    %   M is a vector with the same number of elements as N.  Each element ^vw[z2"  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) dkWV/DAm  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, #W9{3JGUY  
    %   and THETA is a vector of angles.  R and THETA must have the same EQ [K  
    %   length.  The output Z is a matrix with one column for every (N,M) ls({{34NF  
    %   pair, and one row for every (R,THETA) pair. 0}mVP  
    % g|Tkl  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ZyX+V?4  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9 ;Qgby  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral J7pF*2  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, !&adO,jN+=  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized {zIcEN$ ~  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. +aQM %~  
    % 2WUl8?f2Y  
    %   The Zernike functions are an orthogonal basis on the unit circle. oM^VtH=>  
    %   They are used in disciplines such as astronomy, optics, and .^xQtnq  
    %   optometry to describe functions on a circular domain. Vd;N T$S$  
    % a)S{9q}%  
    %   The following table lists the first 15 Zernike functions. 6o.Dgt/f  
    % cv5+[;(b  
    %       n    m    Zernike function           Normalization XUVBD;"f!  
    %       -------------------------------------------------- uCHM  
    %       0    0    1                                 1 }ijFvIHV  
    %       1    1    r * cos(theta)                    2 "_0sW3rG  
    %       1   -1    r * sin(theta)                    2 9\Md.>  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) B7.<A#y2  
    %       2    0    (2*r^2 - 1)                    sqrt(3)  G){A&F  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) o&$Of  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 14`S9SL{V  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) \E1CQP-  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) .6c Bx  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) p`Ok(C_  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 6!@p$ pm)a  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ]+5Y\~I  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) G0u H6x?  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) [(; .D  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) T"DG$R,Aj  
    %       -------------------------------------------------- |JiN; O+K  
    % /Yj; '\3  
    %   Example 1: !{F\ \D/  
    % XnKf<|j6k  
    %       % Display the Zernike function Z(n=5,m=1) uHuL9Q^  
    %       x = -1:0.01:1; &,QBJx<#  
    %       [X,Y] = meshgrid(x,x); qzWnl[3  
    %       [theta,r] = cart2pol(X,Y); \I7&F82e  
    %       idx = r<=1; I@kMM12>c  
    %       z = nan(size(X)); _D{{C  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 4}t$Lf_  
    %       figure ]P2Wa   
    %       pcolor(x,x,z), shading interp /~{ fPS  
    %       axis square, colorbar YRu/KUT$ 7  
    %       title('Zernike function Z_5^1(r,\theta)') -n:;/ere7-  
    % *-3*51 jW  
    %   Example 2: Iv{uk$^7S  
    % $\aJ.N6rb  
    %       % Display the first 10 Zernike functions I K,aA;d  
    %       x = -1:0.01:1; })?KpYk  
    %       [X,Y] = meshgrid(x,x); G%dzJpC(  
    %       [theta,r] = cart2pol(X,Y); {>d\  
    %       idx = r<=1; #iT3 aou  
    %       z = nan(size(X));  Cy5M0{  
    %       n = [0  1  1  2  2  2  3  3  3  3]; `^ )oVs  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 8aY}b($*ZI  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; M1eM^m8U  
    %       y = zernfun(n,m,r(idx),theta(idx)); w x,gth*p  
    %       figure('Units','normalized')  n[7=  
    %       for k = 1:10 (Bss%\  
    %           z(idx) = y(:,k); c^~R %Bx  
    %           subplot(4,7,Nplot(k)) 6n^vG/.M  
    %           pcolor(x,x,z), shading interp ;m"R.Q9*  
    %           set(gca,'XTick',[],'YTick',[]) `pXPF}T  
    %           axis square '/fueku  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) bLC+73BjC  
    %       end Q SvgbjdE  
    % + 7nA; C  
    %   See also ZERNPOL, ZERNFUN2. yW;]J8 7*  
    } DjbVYH  
    %   Paul Fricker 11/13/2006 >L^ 2Z*  
    qdZo cTf'  
    Sr-!-eC  
    % Check and prepare the inputs: # "TL*p  
    % ----------------------------- `L"l{^cH  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) (wp?tMN5#  
        error('zernfun:NMvectors','N and M must be vectors.') gFxaUrZA  
    end Cp]q>lM"  
    T*#<p;  
    if length(n)~=length(m) O/ZyWT  
        error('zernfun:NMlength','N and M must be the same length.') `o%Ua0x2  
    end fn.}LeeS>  
    Gu%}B@4^  
    n = n(:); AE4>pzBe  
    m = m(:); Zv8G[(  
    if any(mod(n-m,2)) b\+9#)Up@  
        error('zernfun:NMmultiplesof2', ... F"a31`L>H  
              'All N and M must differ by multiples of 2 (including 0).') k&o1z'<C  
    end 9]|G-cyt  
    +|Mi lwr  
    if any(m>n) $u{ 8wF/)  
        error('zernfun:MlessthanN', ... #.<(/D+  
              'Each M must be less than or equal to its corresponding N.') ig?Tj4kD  
    end Gl5W4gW;&  
    88+J(^y>  
    if any( r>1 | r<0 ) B3yp2tncj  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') BoXGoFn  
    end 6zJ>n~&(  
    Nk shJ2  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) rY(^6[!  
        error('zernfun:RTHvector','R and THETA must be vectors.') ,IG?(CK|  
    end ^/jALA9!  
    ?N@p~ *x  
    r = r(:); 6n'XRfQp)&  
    theta = theta(:); fg8U* 7  
    length_r = length(r); x2z%J,z@4  
    if length_r~=length(theta) k&3'[&$I*,  
        error('zernfun:RTHlength', ... Sv03="&  
              'The number of R- and THETA-values must be equal.') M-NY&@Nj  
    end )-d &XN7  
    }t.VH:02y  
    % Check normalization: #zw 'H9l  
    % -------------------- u9)<i]2  
    if nargin==5 && ischar(nflag) b+mh9q'5E  
        isnorm = strcmpi(nflag,'norm'); 44_CT?t<  
        if ~isnorm f*ZIBTb 9  
            error('zernfun:normalization','Unrecognized normalization flag.') <@:LONe<  
        end +vCW${U  
    else j!QP>AM|`  
        isnorm = false; 1|%C66f^  
    end ]0)=0pc]E  
    3}X;WE `  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yX;v   
    % Compute the Zernike Polynomials B$%7U><'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0Xw3h^%  
    r2 o-/$  
    % Determine the required powers of r: GHo=)NTjy  
    % ----------------------------------- N}j^55M_]  
    m_abs = abs(m); $NhKqA`0  
    rpowers = []; qlfYX8edZ  
    for j = 1:length(n) |{H-PH*Iz  
        rpowers = [rpowers m_abs(j):2:n(j)];  e{33%5  
    end IMay`us]:8  
    rpowers = unique(rpowers); /=2  
    N5:muh \  
    % Pre-compute the values of r raised to the required powers, vd'd@T  
    % and compile them in a matrix: Mlr}v^"G  
    % ----------------------------- xYCX}bksh  
    if rpowers(1)==0 Xm}~u?$3  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); f6Io|CZWJ  
        rpowern = cat(2,rpowern{:}); T'nQj<dBt:  
        rpowern = [ones(length_r,1) rpowern]; +yd(t}H@  
    else = A;B-_c  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); QBiLH]qa  
        rpowern = cat(2,rpowern{:}); , *A',  
    end ONw;NaE,  
    {JlW1;Jc7  
    % Compute the values of the polynomials: Y l1sAf/  
    % -------------------------------------- =D2x@ank[  
    y = zeros(length_r,length(n)); O[[#\BL  
    for j = 1:length(n) yPqZ ,  
        s = 0:(n(j)-m_abs(j))/2; .OC{,f+  
        pows = n(j):-2:m_abs(j); #]!0$z|Z  
        for k = length(s):-1:1 &18CCp\3)c  
            p = (1-2*mod(s(k),2))* ... XABI2Ex  
                       prod(2:(n(j)-s(k)))/              ... .<C}/Cl  
                       prod(2:s(k))/                     ... Q}MS $[y  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... DdL0MGwX  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); l[q%1-N  
            idx = (pows(k)==rpowers); 9ZEF%&58Y  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 6T5nr  
        end s]=s|  
         &k+'TcWm  
        if isnorm $6X CHVx  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); jWd 7>1R?  
        end t%n3~i4X:  
    end {IW pI *  
    % END: Compute the Zernike Polynomials )MKzAAt~  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /f7Fv*z/  
    /v1Rn*VF!  
    % Compute the Zernike functions: jtk2>Ol   
    % ------------------------------ {1y-*@yU(  
    idx_pos = m>0; ^rc!X]C9  
    idx_neg = m<0; nKJJ7 R L  
    G2%%$7Jj  
    z = y; ~ YKBxt  
    if any(idx_pos) n(gw%w+\7  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); U. 1Vpfy  
    end O0{M3-  
    if any(idx_neg) |"Js iT  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ~$N%UQn?b#  
    end D 5qCn^R  
    c D+IMlT  
    % 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) &WIiw$@  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. `/'Hq9$F<"  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated > ln%3 =  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive wwS{V  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, vMXS%Q  
    %   and THETA is a vector of angles.  R and THETA must have the same Y?2I /  
    %   length.  The output Z is a matrix with one column for every P-value, t)LD-%F  
    %   and one row for every (R,THETA) pair. kmM1)- v  
    % m9UI3fBX  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike zxtx~XO  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2)  = uZ[  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) m<wng2`NTv  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 \FSkI0  
    %   for all p. /a%5!)NE%  
    % E ?(  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 NamBJ\2E1[  
    %   Zernike functions (order N<=7).  In some disciplines it is 5tg  
    %   traditional to label the first 36 functions using a single mode 9cAb\5c|  
    %   number P instead of separate numbers for the order N and azimuthal %_wX9Z T  
    %   frequency M. maLKUSgo  
    % ZD] ^Y}  
    %   Example:  KAmv7  
    % iK6L\'k  
    %       % Display the first 16 Zernike functions V+X>t7.Q  
    %       x = -1:0.01:1; D;It0"  
    %       [X,Y] = meshgrid(x,x); 'H2TwSbIXI  
    %       [theta,r] = cart2pol(X,Y); ^c}Z$V  
    %       idx = r<=1; RF 4u\ \  
    %       p = 0:15; ^WP`;e  
    %       z = nan(size(X)); F_=RY ]  
    %       y = zernfun2(p,r(idx),theta(idx)); o~,dkV  
    %       figure('Units','normalized') RV5X0  
    %       for k = 1:length(p) E)m{m$Hb  
    %           z(idx) = y(:,k); 7</&=lly  
    %           subplot(4,4,k) IMjnj|Fj  
    %           pcolor(x,x,z), shading interp U7.3`qd"  
    %           set(gca,'XTick',[],'YTick',[]) @@7<L  
    %           axis square @gQ{*dN  
    %           title(['Z_{' num2str(p(k)) '}']) {%xwoMVc+  
    %       end o&1ewE(O]  
    % ~k?7XF I  
    %   See also ZERNPOL, ZERNFUN. :3$WY<  
    _h!OGLec  
    %   Paul Fricker 11/13/2006 NH$a:>  
    NyI0 []z  
    yHl1:cf(y  
    % Check and prepare the inputs: }<o.VY&;.  
    % ----------------------------- @ 9D, f  
    if min(size(p))~=1 f( 5c  
        error('zernfun2:Pvector','Input P must be vector.') Y"~I(,nx!  
    end R+FBCVU&TJ  
    2e zQX2q  
    if any(p)>35 pw*<tXH!  
        error('zernfun2:P36', ... TU{^/-l  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Od70w*,  
               '(P = 0 to 35).']) ^4_)a0Kcm,  
    end 1u7Kc'.xc  
    mL`,v WL/`  
    % Get the order and frequency corresonding to the function number: [op!:K0  
    % ---------------------------------------------------------------- xz5Jli  
    p = p(:); ~;k-/Z"  
    n = ceil((-3+sqrt(9+8*p))/2); NARW3\  
    m = 2*p - n.*(n+2); zE5%l`@|o  
    W/9dT^1y4'  
    % Pass the inputs to the function ZERNFUN: a:Js i=  
    % ---------------------------------------- K O"U5v  
    switch nargin "5u*C#T2$  
        case 3 Enn7p9&  
            z = zernfun(n,m,r,theta); e5_a.c  
        case 4 ~~k_A|&  
            z = zernfun(n,m,r,theta,nflag); 6Y0k}+j|>E  
        otherwise {^2``NYM_  
            error('zernfun2:nargin','Incorrect number of inputs.') Mfe/(tlI  
    end fEE[h uG  
    NL 3ri7n  
    % 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) &2[OH}4  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. UXwnE@`F  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 9`Bmop  
    %   order N and frequency M, evaluated at R.  N is a vector of 9HrT>{@  
    %   positive integers (including 0), and M is a vector with the @igr~hJ  
    %   same number of elements as N.  Each element k of M must be a <dl:';@a-  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) S[(Tpk2_  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is U;u@\E@2  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix |l)SX\Qf`@  
    %   with one column for every (N,M) pair, and one row for every Jt5\  
    %   element in R. @dei} !e  
    % m/uBM6SXx  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- NovF?kh2  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ,Bax0p  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to =aZgq99  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Uo?g@D  
    %   for all [n,m]. _K["qm{X_  
    % H <41H;m  
    %   The radial Zernike polynomials are the radial portion of the TG 9 a1q  
    %   Zernike functions, which are an orthogonal basis on the unit =vJ:R[Ilw  
    %   circle.  The series representation of the radial Zernike S?ELFq(g  
    %   polynomials is TtTp ,If  
    % .Qk T-12  
    %          (n-m)/2 ci*rem  
    %            __ x6Zhw9RV  
    %    m      \       s                                          n-2s EYWRTh  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r t4(Z@X$  
    %    n      s=0 OQ>8Q`  
    % k?]`PUrV  
    %   The following table shows the first 12 polynomials. |8^53*f ?  
    % A) {q 7WI  
    %       n    m    Zernike polynomial    Normalization 7u7`z%  
    %       --------------------------------------------- :_9MS0  
    %       0    0    1                        sqrt(2) r Q)?Bhf  
    %       1    1    r                           2 ramYSX@  
    %       2    0    2*r^2 - 1                sqrt(6) %gUf  
    %       2    2    r^2                      sqrt(6) *|WS,  
    %       3    1    3*r^3 - 2*r              sqrt(8) [`pp[J-~7  
    %       3    3    r^3                      sqrt(8) SR)jJ=R3  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ,5}%_  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) fNR2(8;}  
    %       4    4    r^4                      sqrt(10) @GTkS!86  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) C:z+8wt  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) wJc~AP)I%z  
    %       5    5    r^5                      sqrt(12) Y$JGpeq8w  
    %       --------------------------------------------- A#NJ8_  
    % N8*6sK.  
    %   Example: J:W|2U="  
    % I_h u s  
    %       % Display three example Zernike radial polynomials y,&'nk}  
    %       r = 0:0.01:1; DzZEn]+zt  
    %       n = [3 2 5]; xib?XzxGo  
    %       m = [1 2 1]; Aw?i6d  
    %       z = zernpol(n,m,r); Yf1&"WW4  
    %       figure E3..$x-/  
    %       plot(r,z) 3an9Rb V  
    %       grid on &.W,Hh  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') l-^2>K[  
    % lL8pIcQW  
    %   See also ZERNFUN, ZERNFUN2. @6["A'h  
    >qE f991SZ  
    % A note on the algorithm. )ZZjuFQJ)  
    % ------------------------ H){}28dX  
    % The radial Zernike polynomials are computed using the series RBOb/.$  
    % representation shown in the Help section above. For many special t)qu@m?FZ)  
    % functions, direct evaluation using the series representation can vbA<=V*P  
    % produce poor numerical results (floating point errors), because ws/e~ T<c  
    % the summation often involves computing small differences between xE>jlr?  
    % large successive terms in the series. (In such cases, the functions "Yp:{e  
    % are often evaluated using alternative methods such as recurrence 3{:AG,G  
    % relations: see the Legendre functions, for example). For the Zernike 8-#_xsZ^;  
    % polynomials, however, this problem does not arise, because the I1f4u6\*X  
    % polynomials are evaluated over the finite domain r = (0,1), and Tumv0=q4wd  
    % because the coefficients for a given polynomial are generally all bF2RP8?en  
    % of similar magnitude. 3#\++h]QZ  
    % "FD`1  
    % ZERNPOL has been written using a vectorized implementation: multiple q\DN8IJ  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] -G'U\EXT  
    % values can be passed as inputs) for a vector of points R.  To achieve hZZ  
    % this vectorization most efficiently, the algorithm in ZERNPOL EKgY  
    % involves pre-determining all the powers p of R that are required to jm ORKX+)  
    % compute the outputs, and then compiling the {R^p} into a single mV>l`&K=  
    % matrix.  This avoids any redundant computation of the R^p, and P^3`znq{  
    % minimizes the sizes of certain intermediate variables. ;{L~|q J  
    % *;ehSg9  
    %   Paul Fricker 11/13/2006 SAVA6 64  
    oMNt676  
    ?8U#,qq#`  
    % Check and prepare the inputs: nsA}A~(E  
    % ----------------------------- $.+_f,tU  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) omP\qOc  
        error('zernpol:NMvectors','N and M must be vectors.') :  I q  
    end ~:JoKm`vU  
    @> |3d  
    if length(n)~=length(m) n[K LY!  
        error('zernpol:NMlength','N and M must be the same length.') d6+$[4w  
    end !jQj1QZR`  
    OH >#f6`[  
    n = n(:); 5FJ(x:k?z  
    m = m(:); 1fH2obI~X  
    length_n = length(n); 4j1$1C{  
    gf ?_tB0C  
    if any(mod(n-m,2)) @?2ES@G+Ji  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') u<['9U  
    end _|Uv7>}J^  
    tE8aL{<R  
    if any(m<0) A.9ZFFz  
        error('zernpol:Mpositive','All M must be positive.') 56?RFnZ&j  
    end ^7Rc\   
    O0^?VW$y_  
    if any(m>n) ,+4*\yI3l  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') Q2>o+G  
    end drQI@sPp  
    `nCVO;B  
    if any( r>1 | r<0 ) f6,?Yex8B  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') =OeLF  
    end gs"w 0[$  
    p:NIRs  
    if ~any(size(r)==1) OQ&'3hv{  
        error('zernpol:Rvector','R must be a vector.') "h5.^5E6  
    end e?7Oom  
    ^)E# c  
    r = r(:); F{G.dXZZ<  
    length_r = length(r); H?_wsh4J  
    i+Lqj  
    if nargin==4 Xqy9D ZIn  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); (PC)R9r5  
        if ~isnorm |jB/d@RE  
            error('zernpol:normalization','Unrecognized normalization flag.') ES)@iM?5  
        end sj3[ny;b  
    else h0&Oy52  
        isnorm = false; r>ag( ^J\  
    end Q*N{3G!  
    l4> c  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m%cwhH_B  
    % Compute the Zernike Polynomials S}P rgw/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% hb<cynY  
    FN/siw(?3  
    % Determine the required powers of r: \ZtKaEXnx  
    % ----------------------------------- Jr=XVQ(F  
    rpowers = []; c2u*<x  
    for j = 1:length(n) $-p9cyk  
        rpowers = [rpowers m(j):2:n(j)]; \4KV9wm  
    end VfFbZds8f  
    rpowers = unique(rpowers); 1+#E|YWJ  
    aH dQi,=z  
    % Pre-compute the values of r raised to the required powers, Qd/x{a8  
    % and compile them in a matrix: X4<Y5?&0  
    % ----------------------------- N/zP!%L  
    if rpowers(1)==0 sp&gw XPG  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); W]5Hc|!^^  
        rpowern = cat(2,rpowern{:}); q+BG  
        rpowern = [ones(length_r,1) rpowern]; P]O=K  
    else kEp{L  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); @A|#/]S1  
        rpowern = cat(2,rpowern{:}); g`w46X  
    end <1#hX(Q  
    )O2IEwPd.  
    % Compute the values of the polynomials: _C)\X(;  
    % -------------------------------------- N 9s+Tm  
    z = zeros(length_r,length_n); 0DFVB%JdI  
    for j = 1:length_n g2?yT ?  
        s = 0:(n(j)-m(j))/2; k;Fxr%  
        pows = n(j):-2:m(j); #;= sJ[m4  
        for k = length(s):-1:1 "d`u#YmR  
            p = (1-2*mod(s(k),2))* ... x!6<7s  
                       prod(2:(n(j)-s(k)))/          ... n1x"B>3  
                       prod(2:s(k))/                 ... =ea.+  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... p=m:^9/  
                       prod(2:((n(j)+m(j))/2-s(k))); <Uc  
            idx = (pows(k)==rpowers); ?r{hrAx  
            z(:,j) = z(:,j) + p*rpowern(:,idx); s!S_Bt):3  
        end ?AH B\S  
         %=Y=]g2  
        if isnorm z8XWp[K  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); .Q^V,[on1T  
        end D"2bgw  
    end "}Ikx tee  
    ]:;dJc'  
    % 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)  .pQ5lK(R  
    kj[box N  
    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。 =CRptk6tS  
    .ex;4( -!  
    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)