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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 S'HnBn /  
    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! Ks\ NE=;5  
     
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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 2!Bd2  
    function z = zernfun(n,m,r,theta,nflag) DD44"w_9  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. N4*G{g  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N a" H WGY  
    %   and angular frequency M, evaluated at positions (R,THETA) on the z5bo_Eq  
    %   unit circle.  N is a vector of positive integers (including 0), and /CTc7.OYt  
    %   M is a vector with the same number of elements as N.  Each element (5Sivw*mP  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) R1Ye<R!Q  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Z`&4SH=j  
    %   and THETA is a vector of angles.  R and THETA must have the same kPjd_8z2n  
    %   length.  The output Z is a matrix with one column for every (N,M) p!/[K6u  
    %   pair, and one row for every (R,THETA) pair. <A9y9|>o  
    % 8?Z4-6!{V,  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike H_?o-L?+  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), B>Wu;a.:L  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 6{qIU}!  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ]m#5`zGK1|  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized -TZ p FT"  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2Dd|~{%  
    % *UW=Mdt  
    %   The Zernike functions are an orthogonal basis on the unit circle. ~r{5`;c  
    %   They are used in disciplines such as astronomy, optics, and ?`[NFqv_]  
    %   optometry to describe functions on a circular domain. Bb{!Yh].:A  
    % T}3v(6ew4  
    %   The following table lists the first 15 Zernike functions. P_u|-~|\  
    % Kq.:G%  
    %       n    m    Zernike function           Normalization rfw-^`&{  
    %       -------------------------------------------------- Db"DG(  
    %       0    0    1                                 1 kbPE "urR  
    %       1    1    r * cos(theta)                    2 U=8@@ yE  
    %       1   -1    r * sin(theta)                    2 B-d(@7,1  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) RwVaZJe)l  
    %       2    0    (2*r^2 - 1)                    sqrt(3) na^sBq?\  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) {J5JYdK  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) {7Mj P+\  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) t\v+ogbk)  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) +}Av-47`h  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ,_ag;pt9)  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) \Ey~3&x9f  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7FO'{Qq  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) IHC1G1KW=A  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) S-#q~X!yJ  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) E| :!Q8"%w  
    %       -------------------------------------------------- ZX~ _g@  
    % 6x=YQwn~  
    %   Example 1: LEECW_:  
    % H.G!A6bd  
    %       % Display the Zernike function Z(n=5,m=1) #%@MGrsK  
    %       x = -1:0.01:1; -6 sW6;Q  
    %       [X,Y] = meshgrid(x,x); $<p8TtI=YQ  
    %       [theta,r] = cart2pol(X,Y); nY $tp  
    %       idx = r<=1; UofTll)  
    %       z = nan(size(X)); (Vg}Hh?p  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); (cv!Y=]  
    %       figure yg]2erR  
    %       pcolor(x,x,z), shading interp >"3>fche  
    %       axis square, colorbar *5,c Rz  
    %       title('Zernike function Z_5^1(r,\theta)')  j<"nO(  
    % %i)B*9k  
    %   Example 2: 2i|B=D(  
    % 9N[EZhW  
    %       % Display the first 10 Zernike functions c-j_INGm  
    %       x = -1:0.01:1; +rWZ|&r%  
    %       [X,Y] = meshgrid(x,x); +CM7C%U   
    %       [theta,r] = cart2pol(X,Y); DG;y6#|p  
    %       idx = r<=1; fRTo.u  
    %       z = nan(size(X)); bl/,*Wx:4.  
    %       n = [0  1  1  2  2  2  3  3  3  3]; /NF#+bx  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; dV8iwI  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ;1DdjETr  
    %       y = zernfun(n,m,r(idx),theta(idx)); ;HOPABWz)  
    %       figure('Units','normalized') HI&kP+,y  
    %       for k = 1:10 -Cid3~mX3  
    %           z(idx) = y(:,k); Kud'pZ{P  
    %           subplot(4,7,Nplot(k)) 2k#t .-  
    %           pcolor(x,x,z), shading interp *Dr5O9Y  
    %           set(gca,'XTick',[],'YTick',[]) "Mmf6hu  
    %           axis square cjULX+h  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #G3N(wV3  
    %       end i[semo\E  
    % #f'DEo<b  
    %   See also ZERNPOL, ZERNFUN2. TOI4?D]  
    AW5iV3  
    %   Paul Fricker 11/13/2006 ]B9 ^3x[:  
    +?`b=6e(`  
    ! d9AG|  
    % Check and prepare the inputs: 'PdmI<eXQ  
    % ----------------------------- 4}KU>9YRA  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) TF+ l5fv  
        error('zernfun:NMvectors','N and M must be vectors.') "r.2]R3  
    end ^&c$[~W  
    iz}sM>^  
    if length(n)~=length(m) L&Qi@D0P  
        error('zernfun:NMlength','N and M must be the same length.') %Ny) ?B  
    end lj&>cScC  
    {,O`rW_eS  
    n = n(:);  [~Hg}-c  
    m = m(:); gp|1?L 54  
    if any(mod(n-m,2)) B94 &elu  
        error('zernfun:NMmultiplesof2', ... :h";c"  
              'All N and M must differ by multiples of 2 (including 0).') zREJ#r  
    end 9EF~l9`'U  
    F'J [y"~_  
    if any(m>n) E1>/R  
        error('zernfun:MlessthanN', ... F!KV\?eM$  
              'Each M must be less than or equal to its corresponding N.') w.kCBDL  
    end OKwOugi0  
    "2HY5 AE  
    if any( r>1 | r<0 ) q"aPJ0ni'  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') +AQDD4bu  
    end Gm=>!.p  
    Sw! j=`O  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) w4MwD?i]R  
        error('zernfun:RTHvector','R and THETA must be vectors.') ehO:')XF  
    end -a) T6:e  
    Q2~5"  
    r = r(:); ?=|kC*$/G  
    theta = theta(:); Ht=$] Px  
    length_r = length(r); gAE!a Ky  
    if length_r~=length(theta) + Oobb-v  
        error('zernfun:RTHlength', ... k7bl'zic  
              'The number of R- and THETA-values must be equal.') ,@Z_{,b  
    end {tzxA_  
    Mz|L-62  
    % Check normalization: ! sYf<  
    % -------------------- Q(\ wx  
    if nargin==5 && ischar(nflag) 2Ug.:![  
        isnorm = strcmpi(nflag,'norm'); VbxAd 2')  
        if ~isnorm >riq98Us/  
            error('zernfun:normalization','Unrecognized normalization flag.') V;[p438o  
        end +0#JnqH"  
    else ch,|1}bi  
        isnorm = false;  ?f2G?Y  
    end { R*Y=Ie  
    X!0kK8v  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R# 6H'TVE  
    % Compute the Zernike Polynomials _.f@Y`4d  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 41;)-(1  
    .98.G4J>  
    % Determine the required powers of r: Lpm?# g uR  
    % ----------------------------------- *h,3}\  
    m_abs = abs(m); #Go(tS~o  
    rpowers = []; B8 2,.?  
    for j = 1:length(n) !`C?nY  
        rpowers = [rpowers m_abs(j):2:n(j)]; h;n\*[fDc  
    end +L6" vkz  
    rpowers = unique(rpowers); a @SUi~+3  
    )q(:eoLDm  
    % Pre-compute the values of r raised to the required powers, ?N#[<kd  
    % and compile them in a matrix: Es:6  
    % ----------------------------- U(3(ZqP  
    if rpowers(1)==0 +v1-.z  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 9qeZb%r&  
        rpowern = cat(2,rpowern{:}); 4hNwKe"Ki  
        rpowern = [ones(length_r,1) rpowern]; /W9 &Ke  
    else %AgA -pBp  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9UmBm#"  
        rpowern = cat(2,rpowern{:}); ;vUxO<cKFq  
    end z+6QZQk  
    D% @KRcp^b  
    % Compute the values of the polynomials: _sm;HH7'*  
    % -------------------------------------- yam}x*O\xn  
    y = zeros(length_r,length(n)); $F1_^A[  
    for j = 1:length(n) : ~'Z(-a  
        s = 0:(n(j)-m_abs(j))/2; # `58F.  
        pows = n(j):-2:m_abs(j); x)\V lR  
        for k = length(s):-1:1 g =x"cs/[  
            p = (1-2*mod(s(k),2))* ... SEU\}Ni{  
                       prod(2:(n(j)-s(k)))/              ... Xv*}1PZH  
                       prod(2:s(k))/                     ... (. H ]|  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... {tmKCG  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); =f4< ({9  
            idx = (pows(k)==rpowers); |<2 *v-a  
            y(:,j) = y(:,j) + p*rpowern(:,idx); %ph"PR/t?  
        end r+TK5|ke  
         e7's)C>/'  
        if isnorm _y-B";Vmm  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ~%KM3Vap  
        end EJ8I[(  
    end rV U:VL`2  
    % END: Compute the Zernike Polynomials w #<^RKk  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kyK'  
    1(# RN9   
    % Compute the Zernike functions: CnQg*+  
    % ------------------------------ U%n,XOJ  
    idx_pos = m>0; p~FQcW'a~  
    idx_neg = m<0; 9[,s4sxH  
    p}f-c  
    z = y; qTS @D  
    if any(idx_pos) 5Fr;  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Y@ObwKcG  
    end m6eFXP1U  
    if any(idx_neg) "kU>~~y,  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); (`F|nG=X  
    end ?Oqzd$-  
    1ThwvF%Qo  
    % 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) T G_bje  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. p %L1uwLG  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated .<HC[ls  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive #n=A)#'my  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, pFEZDf}:  
    %   and THETA is a vector of angles.  R and THETA must have the same YsZ{1W  
    %   length.  The output Z is a matrix with one column for every P-value, .uBO  
    %   and one row for every (R,THETA) pair. ](_{,P  
    % {:,_A  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike =w?M_[&K)  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) b~jIv:9T  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) :Jjw"}SfK#  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 FOc|*>aKP  
    %   for all p. %3rTQ:X  
    % GKiq0*/M  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Q js2hj-$  
    %   Zernike functions (order N<=7).  In some disciplines it is W=UqX{-j)  
    %   traditional to label the first 36 functions using a single mode  oHOW5  
    %   number P instead of separate numbers for the order N and azimuthal sw[oQ!f  
    %   frequency M. H_Iim[v#  
    % UlnyTz~  
    %   Example: KUF$h Er  
    % o>@=N2n  
    %       % Display the first 16 Zernike functions .\Ul!&y  
    %       x = -1:0.01:1; g})6V  
    %       [X,Y] = meshgrid(x,x); <b6s&"%=  
    %       [theta,r] = cart2pol(X,Y); >_-!zjO8u  
    %       idx = r<=1; jYrym-  
    %       p = 0:15; P87ld._  
    %       z = nan(size(X)); L'13BRu`  
    %       y = zernfun2(p,r(idx),theta(idx)); d[) _sa  
    %       figure('Units','normalized') `'*F 1F  
    %       for k = 1:length(p) y+?=E g  
    %           z(idx) = y(:,k); CdDH1[J  
    %           subplot(4,4,k) nDyvX1]  
    %           pcolor(x,x,z), shading interp Vu_&~z7h  
    %           set(gca,'XTick',[],'YTick',[]) ;&If9O 1  
    %           axis square UHr {  
    %           title(['Z_{' num2str(p(k)) '}']) PP!l  
    %       end jo<>Hc{g>  
    % ri"?, }(  
    %   See also ZERNPOL, ZERNFUN. wTHK=n\i  
    {EOn r1  
    %   Paul Fricker 11/13/2006 qo6 1O\qm  
    5woIGO3X  
    U&,r4>V@h>  
    % Check and prepare the inputs: ^uC"dfH  
    % ----------------------------- `@4 2jG}*  
    if min(size(p))~=1 Sc%aJ1  
        error('zernfun2:Pvector','Input P must be vector.') Uc\|X;nkRk  
    end ooomi"u  
    b;Hm\aK  
    if any(p)>35 6lN?)<uQ  
        error('zernfun2:P36', ... ^Vl^,@  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... N{ : [/  
               '(P = 0 to 35).']) 9@( O\xr  
    end yEE|e&#>  
    =ZCH1J5"  
    % Get the order and frequency corresonding to the function number: tU9rCL:P  
    % ---------------------------------------------------------------- ?nB he lW^  
    p = p(:); P=f<#l"v  
    n = ceil((-3+sqrt(9+8*p))/2); n wY2BIB  
    m = 2*p - n.*(n+2); & 6`  
    $/4Wod*l  
    % Pass the inputs to the function ZERNFUN: U#8\#jo  
    % ---------------------------------------- v>JB rIb$  
    switch nargin `K$:r4/[  
        case 3 %2^['8t#NH  
            z = zernfun(n,m,r,theta); JOA%Y;`<#  
        case 4 U,oD44  
            z = zernfun(n,m,r,theta,nflag); |hu"5*  
        otherwise @uQ%o%Ru6  
            error('zernfun2:nargin','Incorrect number of inputs.') w;lx:j!Vp$  
    end j%6|:o3G(  
    ~+nS)4 (  
    % 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) @)aXNQY  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. /V#MLPA  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ^% ~Et>C  
    %   order N and frequency M, evaluated at R.  N is a vector of ==jkp U*=  
    %   positive integers (including 0), and M is a vector with the Jm{As*W>  
    %   same number of elements as N.  Each element k of M must be a F!z! :yp  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) V/QTYy1  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ,gAr|x7_  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix OGSEvfW  
    %   with one column for every (N,M) pair, and one row for every eLHa9R{)B  
    %   element in R. o`<h=+a\  
    % J,dG4.ht  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ')5jllxv  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is v :'P"uU;4  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ')C _An>X6  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 S&4w`hdD>~  
    %   for all [n,m]. &%_y6}xIw  
    % UD Iac;vT  
    %   The radial Zernike polynomials are the radial portion of the w]]x[D]L  
    %   Zernike functions, which are an orthogonal basis on the unit :ofE8]  
    %   circle.  The series representation of the radial Zernike ,g<>`={kK+  
    %   polynomials is @Z3[ c[D)9  
    % +1JZB* W  
    %          (n-m)/2 -9tXv+v?  
    %            __ GI'&g@?u  
    %    m      \       s                                          n-2s 30gZ_ 8C>}  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r `4"y#Z  
    %    n      s=0 D{&+7C:8.  
    % 0ER6cTo-t  
    %   The following table shows the first 12 polynomials. uK"$=v6|  
    % (&N$W&  
    %       n    m    Zernike polynomial    Normalization &gKP6ANx2  
    %       --------------------------------------------- I&Eg-96@  
    %       0    0    1                        sqrt(2) Tko CyD9  
    %       1    1    r                           2 @=aq&gb  
    %       2    0    2*r^2 - 1                sqrt(6) +e{djp@m  
    %       2    2    r^2                      sqrt(6) `9G$p|6  
    %       3    1    3*r^3 - 2*r              sqrt(8) OTy 4"%  
    %       3    3    r^3                      sqrt(8) K>DnD0  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) `*uuB;  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) sKE*AGFL d  
    %       4    4    r^4                      sqrt(10) eAenkUBz6,  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) cwDD(j  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) V, "AG  
    %       5    5    r^5                      sqrt(12) FZ}C;yUPD  
    %       --------------------------------------------- $fU/9jTa  
    % R - ?0k:  
    %   Example: J^<j=a|D  
    % HzO0K=Z=R0  
    %       % Display three example Zernike radial polynomials -mWw.SfEZ  
    %       r = 0:0.01:1; -*]9Ma<wa  
    %       n = [3 2 5]; j:vD9sdQ  
    %       m = [1 2 1]; Do{*cSd  
    %       z = zernpol(n,m,r); 8Vg`;_-  
    %       figure "_% 0|;  
    %       plot(r,z) RIVN>G[;L  
    %       grid on wtgO;w  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') `[W)6OUCx}  
    % K\%"RgF@&  
    %   See also ZERNFUN, ZERNFUN2. ZP G8q  
    ud~VQXZo  
    % A note on the algorithm. 0,i+  
    % ------------------------ Y9(i}uTi  
    % The radial Zernike polynomials are computed using the series 1J!tcj1(  
    % representation shown in the Help section above. For many special hzf}_1  
    % functions, direct evaluation using the series representation can Z!5m'yZO  
    % produce poor numerical results (floating point errors), because Zqe[2()  
    % the summation often involves computing small differences between h.+,*9T\  
    % large successive terms in the series. (In such cases, the functions Wf&G9Be?8  
    % are often evaluated using alternative methods such as recurrence b^=8%~?%4  
    % relations: see the Legendre functions, for example). For the Zernike Lu$:,^ C  
    % polynomials, however, this problem does not arise, because the ",,qFM!  
    % polynomials are evaluated over the finite domain r = (0,1), and %/,Uk+3p  
    % because the coefficients for a given polynomial are generally all   !AD,  
    % of similar magnitude. FL_ arhrqD  
    % 14)kKWG  
    % ZERNPOL has been written using a vectorized implementation: multiple ^ 8Nr %NJ  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] & /FA>  
    % values can be passed as inputs) for a vector of points R.  To achieve Ml_:Q]kl^  
    % this vectorization most efficiently, the algorithm in ZERNPOL Yhv`IV-s  
    % involves pre-determining all the powers p of R that are required to 0aq-drl5\  
    % compute the outputs, and then compiling the {R^p} into a single mm9S#Ya  
    % matrix.  This avoids any redundant computation of the R^p, and TlZlE^EE<  
    % minimizes the sizes of certain intermediate variables. ;lPhSkD  
    %  3kzGL  
    %   Paul Fricker 11/13/2006 5'}!v  
    E4fvYV_ra  
    #| e5  
    % Check and prepare the inputs: t6U+a\-<  
    % ----------------------------- CI ]U)@\U  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +Y%I0.?&5  
        error('zernpol:NMvectors','N and M must be vectors.') Z~R/ p;@  
    end Z( clw  
    XS~w_J#q  
    if length(n)~=length(m)  9%hB   
        error('zernpol:NMlength','N and M must be the same length.') ]KII?{ <k  
    end IU"!oM^  
    _\V{X}ftqa  
    n = n(:); kTe<1^,m  
    m = m(:); hQRc,d6x5  
    length_n = length(n); 3 mMdq*X5  
    iegPEb  
    if any(mod(n-m,2)) <zWQ[^  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Z-r0 D  
    end !QzMeN;D  
    }t{^*(  
    if any(m<0) ViC76aJ  
        error('zernpol:Mpositive','All M must be positive.') :zk.^q  
    end {>qrf:  
    `1}yB  
    if any(m>n) <@2?2l+`X  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') gA8 u E  
    end |@4h z9~3  
    a\.?{/  
    if any( r>1 | r<0 ) "+OMo-<K7  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 91'^--N  
    end >L3p qK   
    ZX RN?b  
    if ~any(size(r)==1) .6]cu{K(  
        error('zernpol:Rvector','R must be a vector.') ntUVhIE0  
    end TuPxyB  
    O&1p2!Bk4  
    r = r(:); ]7ZC>.t  
    length_r = length(r); .oOt(K +  
    R(#;yn  
    if nargin==4 /IR5[67  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 8&AorYw[  
        if ~isnorm kxiyF$ 9  
            error('zernpol:normalization','Unrecognized normalization flag.') +c2>j8e6  
        end VY26 Cf"  
    else ? 9M+fi  
        isnorm = false; 6#-; ,2i  
    end EG=>F1&M  
    )4O`%9=M&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M%LwC/h:,  
    % Compute the Zernike Polynomials w4/)r-Z4I  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {;gWn' aq  
    `9)2nkJk'z  
    % Determine the required powers of r: Fgq*3t  
    % ----------------------------------- w'j]Y%  
    rpowers = []; d:ajD  
    for j = 1:length(n) \YyU5f7';  
        rpowers = [rpowers m(j):2:n(j)]; gI$`d?[0{  
    end ZjID<5#  
    rpowers = unique(rpowers); ) x $Vy=  
    ^Rh~+  
    % Pre-compute the values of r raised to the required powers, S`-IQ,*}  
    % and compile them in a matrix: >I ; #BE3  
    % ----------------------------- 9Ei#t FMc  
    if rpowers(1)==0 H`..)zL|  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ,S K6*tpI  
        rpowern = cat(2,rpowern{:}); 6@36 1f[  
        rpowern = [ones(length_r,1) rpowern]; JVCgYY({KQ  
    else KAaeaiD  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); :H k4i%hGk  
        rpowern = cat(2,rpowern{:}); ago t (  
    end R9HS%O6b6  
    T+( A7Qrx%  
    % Compute the values of the polynomials: '\qr=0aW  
    % -------------------------------------- ;Q 6e&Ips/  
    z = zeros(length_r,length_n); vK/Z9wR*05  
    for j = 1:length_n TUCp mj  
        s = 0:(n(j)-m(j))/2; *38\&"s4_  
        pows = n(j):-2:m(j); X~b+LG/  
        for k = length(s):-1:1 ZPFTNwf  
            p = (1-2*mod(s(k),2))* ... K~N[^pF  
                       prod(2:(n(j)-s(k)))/          ... W u{nC  
                       prod(2:s(k))/                 ... mjc:0hH  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... p =O1aM  
                       prod(2:((n(j)+m(j))/2-s(k))); {[iQRYD0|  
            idx = (pows(k)==rpowers); !7|9r$  
            z(:,j) = z(:,j) + p*rpowern(:,idx); b8Sl3F?-~  
        end Sv",E@!f  
         uQ)]g  
        if isnorm LZ$!=vg4  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); xrd ^vE  
        end [# H8Mb+7  
    end )^3655mb  
    1.CYs<  
    % 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)  ;O CYx[|  
    p=zjJ~DVd  
    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。 Y<B| e91C  
    l!b#v`  
    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)