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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 I 6O  
    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! H*6W q  
     
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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 =">NQ)98u  
    function z = zernfun(n,m,r,theta,nflag) nDW9NQ  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 4\i[m:e=@  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N f!"w5qC^  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 7o4\oRGV  
    %   unit circle.  N is a vector of positive integers (including 0), and fR|A(u#9  
    %   M is a vector with the same number of elements as N.  Each element Ep}s}Stlr}  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) cNH7C"@GVu  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, g=rbPbu  
    %   and THETA is a vector of angles.  R and THETA must have the same s@C}P  
    %   length.  The output Z is a matrix with one column for every (N,M) `{Ul!  
    %   pair, and one row for every (R,THETA) pair. -HuA \0J  
    % 7 d vnupLh  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike j<x_&1  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Mfs?x a  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral t^L]/$q  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, j#6.Gq  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 9VT;ep  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2?x4vI np;  
    % a9G8q>h]O  
    %   The Zernike functions are an orthogonal basis on the unit circle. UI#h&j5pW  
    %   They are used in disciplines such as astronomy, optics, and w(rE`IgW  
    %   optometry to describe functions on a circular domain. P%zK;#8V  
    % Y0>y8U V  
    %   The following table lists the first 15 Zernike functions. D]}G.v1  
    % rGO8!X 3d  
    %       n    m    Zernike function           Normalization fIF8%J ^3  
    %       -------------------------------------------------- kP"9&R`E  
    %       0    0    1                                 1 "}!G!k:  
    %       1    1    r * cos(theta)                    2 HV.t6@\};  
    %       1   -1    r * sin(theta)                    2 =Uh$&m  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Jb(H %NJ  
    %       2    0    (2*r^2 - 1)                    sqrt(3) #S(Hd?34,  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) KSvE~h[#+  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) <q SC#[xu  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) wbHb;]  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) OCUr{Nh  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 0mnw{fE8_  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) pFXEu= $3  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;fJ.8C  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) (?c-iKGc  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) G9lUxmS<  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) $k?>DP 4  
    %       -------------------------------------------------- :0ep( <|;  
    % IU[ [ H#  
    %   Example 1: <!+Az,-  
    % .h[:xYm  
    %       % Display the Zernike function Z(n=5,m=1) *Uh!>Iv;  
    %       x = -1:0.01:1; g*Phv|kI  
    %       [X,Y] = meshgrid(x,x); B6"0OIDY"  
    %       [theta,r] = cart2pol(X,Y); `gJ(0#ac  
    %       idx = r<=1; 74u&%Rj  
    %       z = nan(size(X)); aYeR{Y]  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); GmG 5[?)  
    %       figure g\U-VZ6;p  
    %       pcolor(x,x,z), shading interp JVJMgim)0  
    %       axis square, colorbar >Q/Dk7#  
    %       title('Zernike function Z_5^1(r,\theta)') XkqCZHYkS  
    % ;*N5Y}?j'  
    %   Example 2: :Al!1BJQ  
    % @,}UWU  
    %       % Display the first 10 Zernike functions u y+pP!<  
    %       x = -1:0.01:1; =vPj%oLp'a  
    %       [X,Y] = meshgrid(x,x); So;<6~  
    %       [theta,r] = cart2pol(X,Y); XG?8s &  
    %       idx = r<=1; GVz6-T~\>  
    %       z = nan(size(X)); ibw;}^m(  
    %       n = [0  1  1  2  2  2  3  3  3  3]; )1z@  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; q| 7(  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ':q p05t  
    %       y = zernfun(n,m,r(idx),theta(idx)); G B^Br6  
    %       figure('Units','normalized') edD)TpmE,  
    %       for k = 1:10 so; ]&  
    %           z(idx) = y(:,k); pdMc}=K  
    %           subplot(4,7,Nplot(k)) ye97!nIg@  
    %           pcolor(x,x,z), shading interp Lr+$_ t}r  
    %           set(gca,'XTick',[],'YTick',[]) Y@v>FlqI{  
    %           axis square =%7-ZH9  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) +mPx8P&%  
    %       end t7pFW^&  
    % Fu~j8K  
    %   See also ZERNPOL, ZERNFUN2. df=f62  
    x38 QD;MT  
    %   Paul Fricker 11/13/2006 ]iWRo'  
    @ZJS&23E  
    *i,%,O96Nz  
    % Check and prepare the inputs: rjP/l6 ~'  
    % ----------------------------- "7 yD0T)2  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) > !JS:5|  
        error('zernfun:NMvectors','N and M must be vectors.') JC"z&ka  
    end QP x^_jA  
    maZ)cW?  
    if length(n)~=length(m) y7{?Ip4[  
        error('zernfun:NMlength','N and M must be the same length.') 0J|3kY-n>  
    end :m;p:l|W  
    _aphkeqd  
    n = n(:); \wZe] G%S  
    m = m(:); +3gp%`c4  
    if any(mod(n-m,2)) ("@!>|H  
        error('zernfun:NMmultiplesof2', ... *a)n62  
              'All N and M must differ by multiples of 2 (including 0).') !Cs_F&l"j  
    end X2_=agEP  
    y5r4&~04  
    if any(m>n) km(Po}  
        error('zernfun:MlessthanN', ... s~>}a  
              'Each M must be less than or equal to its corresponding N.') B~mj 8l4  
    end n<,BmVQ  
    &m3lXl  
    if any( r>1 | r<0 ) do_[&  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') =]t|];c%  
    end x2EUr,7  
    .`lCWeHN  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) f3;5Am  
        error('zernfun:RTHvector','R and THETA must be vectors.') mw!F{pw  
    end 7pd$\$  
    3]>|  i  
    r = r(:); /z!%d%"  
    theta = theta(:); F2WKd1U  
    length_r = length(r); sK{e*[I>W  
    if length_r~=length(theta) [ 3Gf2_  
        error('zernfun:RTHlength', ... \m,PA'nd/  
              'The number of R- and THETA-values must be equal.') XSDpRo  
    end }EPY^VIw  
    Ba,`TJ%y  
    % Check normalization: |>Vb9:q9Po  
    % -------------------- Wzh`or  
    if nargin==5 && ischar(nflag) j.Hf/vi`z  
        isnorm = strcmpi(nflag,'norm'); hM{bavd  
        if ~isnorm w(/S?d  
            error('zernfun:normalization','Unrecognized normalization flag.') p?!/+  
        end =(Mch~  
    else 3Ul*QN{6  
        isnorm = false; = &]L00u.  
    end BLttb  
    ]'}L 1r  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8Wx=p#_  
    % Compute the Zernike Polynomials .]u /O`c]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pb}*\/s  
    DF= *_,2/  
    % Determine the required powers of r: %A`+WYeuX  
    % ----------------------------------- uYN`:b8  
    m_abs = abs(m); Q?vlfZR`8  
    rpowers = []; Wc#24:OKe3  
    for j = 1:length(n) ~a:  
        rpowers = [rpowers m_abs(j):2:n(j)]; E fDH6  
    end Nc`L;CP  
    rpowers = unique(rpowers); j_AACq {.  
    $I=~S[p  
    % Pre-compute the values of r raised to the required powers, V&5wRz+`W  
    % and compile them in a matrix: wj,=$RX  
    % ----------------------------- 3n _htgcv  
    if rpowers(1)==0 ,prf;|e?  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); A&VG~r$  
        rpowern = cat(2,rpowern{:}); *pq\MiD/  
        rpowern = [ones(length_r,1) rpowern]; J zl6eo[;  
    else (HVGlw'`  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ";F'~}bDA  
        rpowern = cat(2,rpowern{:}); aOp\91  
    end G[=c Ss,  
    t0S 1QC+  
    % Compute the values of the polynomials: _b 0& !l<  
    % -------------------------------------- )pa]ui\t  
    y = zeros(length_r,length(n)); w{KavU5W  
    for j = 1:length(n) Da|z"I x  
        s = 0:(n(j)-m_abs(j))/2; oU8q o-J1H  
        pows = n(j):-2:m_abs(j); I,tud!p`  
        for k = length(s):-1:1 ^!d3=}:0  
            p = (1-2*mod(s(k),2))* ... .nJz G  
                       prod(2:(n(j)-s(k)))/              ... Y4-t7UlS;  
                       prod(2:s(k))/                     ... +>,I1{u%&  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... _)8s'MjA:&  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ;u JMG  
            idx = (pows(k)==rpowers); P0@,fd<  
            y(:,j) = y(:,j) + p*rpowern(:,idx);  R&&4y 7  
        end V!Uc(  
         & 21%zPm  
        if isnorm .Mbz3;i0  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); aN?zmkPpov  
        end 9;{C IMg&  
    end )`:UP~)H  
    % END: Compute the Zernike Polynomials  ?9/G[[(  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c{|p.hd  
    %J(:ADu]  
    % Compute the Zernike functions: e ,(mR+a8  
    % ------------------------------ _>+Ld6.T6  
    idx_pos = m>0; T)/eeZ$  
    idx_neg = m<0; fhiM U8(&  
    vXs"Dst  
    z = y; 1}x%%RD_  
    if any(idx_pos) N8jIMb'<  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); #mdc[.  
    end +7Gwg  
    if any(idx_neg) Ud?Q%) X  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); u`W2 +S  
    end K- v#.e4  
    q V =!ORuj  
    % 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) [agMfn  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 3";q[&F9y  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Rcuz(yS8  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive rq{$,/6.  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 5P2K5,o|n~  
    %   and THETA is a vector of angles.  R and THETA must have the same 6ujW Nf  
    %   length.  The output Z is a matrix with one column for every P-value, vM={V$D&  
    %   and one row for every (R,THETA) pair. vx =&QavL  
    % 2 ?C)&  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike E .h*g8bXe  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) }f ?y* H  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) F59 TZI  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 KNl$3nX  
    %   for all p. _`X:jj>  
    % WJi]t93  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 >P(.:_ ^p  
    %   Zernike functions (order N<=7).  In some disciplines it is mFeP9MfJ  
    %   traditional to label the first 36 functions using a single mode q.vIc ?a  
    %   number P instead of separate numbers for the order N and azimuthal kJU2C=m@e2  
    %   frequency M. P}iE+Z 3  
    % R2NZ{"h  
    %   Example: sO Y:e/_F  
    % Iu{V,U  
    %       % Display the first 16 Zernike functions 9r9NxKuAO  
    %       x = -1:0.01:1; (7Qo  
    %       [X,Y] = meshgrid(x,x); DU^loB+  
    %       [theta,r] = cart2pol(X,Y); ceA9) {  
    %       idx = r<=1; SbZ6t$"  
    %       p = 0:15; y_,bu^+*  
    %       z = nan(size(X)); MV"=19]  
    %       y = zernfun2(p,r(idx),theta(idx)); +ZYn? #IQ  
    %       figure('Units','normalized') ]e3Ax(i)  
    %       for k = 1:length(p) "@kaHIf[  
    %           z(idx) = y(:,k); KvS G;  
    %           subplot(4,4,k) {g6%(X\r.r  
    %           pcolor(x,x,z), shading interp M|-)GvR$J  
    %           set(gca,'XTick',[],'YTick',[]) M5B# TAybC  
    %           axis square reVgqYp{{-  
    %           title(['Z_{' num2str(p(k)) '}']) )u">it+  
    %       end *Ex|9FCt$  
    % =Qq+4F)MD  
    %   See also ZERNPOL, ZERNFUN. [aS*%Heu  
    %y@AA>x!  
    %   Paul Fricker 11/13/2006 }u|q0>^8  
    ,Q B<7a+I  
    <3iMRe  
    % Check and prepare the inputs: E^PB)D(.  
    % ----------------------------- ?%86/N>  
    if min(size(p))~=1 ^.tg7%dJ  
        error('zernfun2:Pvector','Input P must be vector.') 0x7'^Z>-oe  
    end dx]>(e@(t{  
    ^8tEach  
    if any(p)>35 R]dg_Da  
        error('zernfun2:P36', ... t) +310w  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Cw%{G'O   
               '(P = 0 to 35).']) fM}#ON>Z  
    end ?"FbsMk.d  
    .hiSw  
    % Get the order and frequency corresonding to the function number: tkhCw/  
    % ---------------------------------------------------------------- ;jPXs  
    p = p(:); VL^EHb7  
    n = ceil((-3+sqrt(9+8*p))/2); +(*DT9s+  
    m = 2*p - n.*(n+2); a?.=V  
    *"kM{*3:v  
    % Pass the inputs to the function ZERNFUN: H]!"Zq k  
    % ---------------------------------------- h![#;>(  
    switch nargin GfG|&VNlz  
        case 3 !BI;C(,RL  
            z = zernfun(n,m,r,theta); l,: F  
        case 4 x"(KBEK~  
            z = zernfun(n,m,r,theta,nflag); *VeRVaBl  
        otherwise 4YHY7J  
            error('zernfun2:nargin','Incorrect number of inputs.') p'fYULYE  
    end AS,%RN^.  
    P4?glh q#  
    % 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) b\ PgVBf9  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. q =Il|Nb>  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of S$k&vc(0  
    %   order N and frequency M, evaluated at R.  N is a vector of 2(nlJ7R  
    %   positive integers (including 0), and M is a vector with the !+njS  
    %   same number of elements as N.  Each element k of M must be a >MK98(F  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ]{kPrey  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 8[>zG2  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 6Iw\c  
    %   with one column for every (N,M) pair, and one row for every - DCbko  
    %   element in R. 3[&Cg  
    % Ha ]YJ}  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 0Qd:`HF[  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is _FEF x  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to gJhiGYx  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 a: S -  
    %   for all [n,m]. iO[<1?  
    % p8Q1-T3v  
    %   The radial Zernike polynomials are the radial portion of the %UM *79  
    %   Zernike functions, which are an orthogonal basis on the unit %bfZn9_m  
    %   circle.  The series representation of the radial Zernike "mN q&$  
    %   polynomials is c)tfAD(N8x  
    % Wl Sm  
    %          (n-m)/2 ZB&6<uw  
    %            __ e|9 A716x  
    %    m      \       s                                          n-2s wAd9  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r $SE^S   
    %    n      s=0 X jX2]  
    % L-\GHu~)  
    %   The following table shows the first 12 polynomials. D-4f.Tq4#  
    % O~QB!<Q+  
    %       n    m    Zernike polynomial    Normalization = f i$}>\  
    %       --------------------------------------------- 'QIqBU'~  
    %       0    0    1                        sqrt(2) fX+O[j  
    %       1    1    r                           2 L6LZC2N+2  
    %       2    0    2*r^2 - 1                sqrt(6) 4&f3%eTi  
    %       2    2    r^2                      sqrt(6) G9 :l'\  
    %       3    1    3*r^3 - 2*r              sqrt(8) $kKjgQ S(  
    %       3    3    r^3                      sqrt(8) yZ`wfj$Jj  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) MS]r:X6  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) BUR*n;V`  
    %       4    4    r^4                      sqrt(10) A9JdU&  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 9K&:V(gmw  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) _y3Xb`0a  
    %       5    5    r^5                      sqrt(12) JG,%qFlk  
    %       --------------------------------------------- ;\l,5EG  
    % _~ &iq1  
    %   Example: Yrn)VV[)h  
    % N!|wo:  
    %       % Display three example Zernike radial polynomials Yuc> fFA  
    %       r = 0:0.01:1; (~en (  
    %       n = [3 2 5]; TU7' J  
    %       m = [1 2 1]; ""D 4s  
    %       z = zernpol(n,m,r); <o= 8 FO  
    %       figure H4JTGt1"  
    %       plot(r,z) 4{l,  
    %       grid on (khL-F  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') N"1B/u  
    % B+0hzkPY  
    %   See also ZERNFUN, ZERNFUN2. 4g7)iL^#~  
    =>dGL|  
    % A note on the algorithm. $pudoAO  
    % ------------------------ V/;B3t~f  
    % The radial Zernike polynomials are computed using the series xjUtl  
    % representation shown in the Help section above. For many special U3:j'Su4H?  
    % functions, direct evaluation using the series representation can 6i*sm.SDw  
    % produce poor numerical results (floating point errors), because 'Qo*y%{@5  
    % the summation often involves computing small differences between B~du-Z22IZ  
    % large successive terms in the series. (In such cases, the functions XS BA$y  
    % are often evaluated using alternative methods such as recurrence ))i}7 chc  
    % relations: see the Legendre functions, for example). For the Zernike BRYHX.}h\A  
    % polynomials, however, this problem does not arise, because the W"3ph6[eW  
    % polynomials are evaluated over the finite domain r = (0,1), and u?{H}V  
    % because the coefficients for a given polynomial are generally all {91nL'-'  
    % of similar magnitude. 1>&]R=  
    % TNr :pE<  
    % ZERNPOL has been written using a vectorized implementation: multiple T"}vAG( .O  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 7`hP?a=  
    % values can be passed as inputs) for a vector of points R.  To achieve ~| 6[j<ziL  
    % this vectorization most efficiently, the algorithm in ZERNPOL C{XmVc.  
    % involves pre-determining all the powers p of R that are required to -7(@1@1  
    % compute the outputs, and then compiling the {R^p} into a single EUgs6[w 4  
    % matrix.  This avoids any redundant computation of the R^p, and 6B ?twh)  
    % minimizes the sizes of certain intermediate variables. 9RI-Lq`  
    % o7LuKRl   
    %   Paul Fricker 11/13/2006 @jlw_ob2g  
    c\V7i#u[d;  
    uc"P3,M  
    % Check and prepare the inputs: 'oC) NpnH  
    % ----------------------------- wIBO ^w\J  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) wuJ4kW$  
        error('zernpol:NMvectors','N and M must be vectors.') U~l$\ c  
    end [R7Y}k:9U  
    r{%qf;  
    if length(n)~=length(m) M+9gL3W  
        error('zernpol:NMlength','N and M must be the same length.') (DP &B%Sf  
    end KFkoS0M5|  
    w(TJ*::T  
    n = n(:); H1(Uw:V8  
    m = m(:); q=qcm`ce  
    length_n = length(n); Q'mM3pq4r  
    = +?7''{>  
    if any(mod(n-m,2)) d6sye^P  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ;?g6QIN9  
    end ; p{[1  
    M|(Q0 _8  
    if any(m<0) fLm*1S|%\  
        error('zernpol:Mpositive','All M must be positive.') _aMPa+D=P  
    end H_<C!OgR  
    hzbw>g+  
    if any(m>n) Y,e B|  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') h@WhNk7"xa  
    end }t1a* z  
    nSAdCJ;4  
    if any( r>1 | r<0 ) y e? 'Ze  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') M6-&R=78K  
    end yU}qOgXx  
    %zw1}|s#z  
    if ~any(size(r)==1) ;K &o-y  
        error('zernpol:Rvector','R must be a vector.') v(D;PS3r 7  
    end zeC RK+-  
    @Sbe^x  
    r = r(:); c+nq] xOs'  
    length_r = length(r); t=O8f5Pf{  
    hJ#xB6  
    if nargin==4 2WVka  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); gH7|=W  
        if ~isnorm pKrN:ExB"\  
            error('zernpol:normalization','Unrecognized normalization flag.') \3aoM{ztD  
        end 2nIw7>.}f  
    else W+X6@/BO  
        isnorm = false; 4l45N6"  
    end :#?5X|Gz  
    <=0 u2~E  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Um-[~-  
    % Compute the Zernike Polynomials o/Q;f@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $.rhRKs  
    xzZ38xIhV  
    % Determine the required powers of r: [ )dXIIM  
    % ----------------------------------- C"T;Qp~B  
    rpowers = []; r_6ZO&  
    for j = 1:length(n) 6@0OQb  
        rpowers = [rpowers m(j):2:n(j)]; hUMf"=q+  
    end A/KJqiag  
    rpowers = unique(rpowers); pF Rg?-  
    T7u%^xm  
    % Pre-compute the values of r raised to the required powers, \(Y\|zC'0$  
    % and compile them in a matrix: TS9|a{j3!  
    % ----------------------------- MgrLSKLT  
    if rpowers(1)==0 d]6#m'U  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); h*$y[}hDuv  
        rpowern = cat(2,rpowern{:}); g^{@'}$  
        rpowern = [ones(length_r,1) rpowern]; ?fjuh}Q5h  
    else ssRbhlD/*1  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); T<p !5`B1  
        rpowern = cat(2,rpowern{:}); ?>rW>U6:P  
    end r^paD2&}  
    DBD%6o>]K  
    % Compute the values of the polynomials: &*G #H~\  
    % -------------------------------------- <Fc;_GG  
    z = zeros(length_r,length_n); +M$Q =6/  
    for j = 1:length_n fNt`?pW H  
        s = 0:(n(j)-m(j))/2; 3ojlB|Z  
        pows = n(j):-2:m(j); ^o1*a&~J@  
        for k = length(s):-1:1 @jSYB+D  
            p = (1-2*mod(s(k),2))* ... R:k5QD9/&p  
                       prod(2:(n(j)-s(k)))/          ... DYxCQ D  
                       prod(2:s(k))/                 ... Z}l3l`h!  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... OFv%B/O  
                       prod(2:((n(j)+m(j))/2-s(k))); vchm"p?9)  
            idx = (pows(k)==rpowers); <_tT<5'[$u  
            z(:,j) = z(:,j) + p*rpowern(:,idx); \6<=$vD  
        end khrb-IY@  
         W$OG( m!W>  
        if isnorm L3--r  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); U4-g^S[  
        end \$\ENQ;Nk  
    end TbGn46!:  
    /ZPyN<@  
    % 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)  aCxE5$~$  
    d k<XzO~g  
    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。 |xvy')(b  
    &"mzwQX  
    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)