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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 ~fkjn  
    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! <Yu}7klJE  
     
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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 yEIM58l  
    function z = zernfun(n,m,r,theta,nflag) )isz }?Dj  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. hu0z):>y  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N n@  lf+  
    %   and angular frequency M, evaluated at positions (R,THETA) on the .Nz2K[  
    %   unit circle.  N is a vector of positive integers (including 0), and 3:Q5dr+1_  
    %   M is a vector with the same number of elements as N.  Each element |;e K5(|  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ~kPHf_B;z  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, L#mf[a@pCn  
    %   and THETA is a vector of angles.  R and THETA must have the same <VI.A" Qk~  
    %   length.  The output Z is a matrix with one column for every (N,M) ^N#B( F  
    %   pair, and one row for every (R,THETA) pair. 6U5L>sQ  
    % IHHL. gT  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike TELN4*  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), t=o2:p6&  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral =]jc{Y%o  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, -J*BY2LU3f  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ewHk (ru  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. '4k l$I  
    %  #v+ 2W  
    %   The Zernike functions are an orthogonal basis on the unit circle. 7pf]h$2  
    %   They are used in disciplines such as astronomy, optics, and 4H'\nsM  
    %   optometry to describe functions on a circular domain. .anXsjD%W  
    % 3gtQS3$4s  
    %   The following table lists the first 15 Zernike functions. DCr&%)Ll  
    % T1AD(r\W5  
    %       n    m    Zernike function           Normalization 0N.B =j|  
    %       -------------------------------------------------- L!G]i;=:  
    %       0    0    1                                 1 ?e( y/  
    %       1    1    r * cos(theta)                    2 *YH5kX  
    %       1   -1    r * sin(theta)                    2 mU@pRjq=  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) _wMxKM  
    %       2    0    (2*r^2 - 1)                    sqrt(3) A)6xEeyR  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) :Z)a&A9v  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) %;S T7  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ;PM(q<@\  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) \Gm$hTvB&  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) iY@wg 8ry  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) xVYy`_|  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) &%eWCe+ +  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) e=uElp'%  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) G*;?&;*  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) b)ytm=7ha  
    %       -------------------------------------------------- 4z6i{n-k  
    % 96G8B62  
    %   Example 1: WEy$SN+P  
    % v *'anw&Z  
    %       % Display the Zernike function Z(n=5,m=1) yC#%fgQ r  
    %       x = -1:0.01:1; DzZEn]+zt  
    %       [X,Y] = meshgrid(x,x); xib?XzxGo  
    %       [theta,r] = cart2pol(X,Y); Aw?i6d  
    %       idx = r<=1; Yf1&"WW4  
    %       z = nan(size(X)); E3..$x-/  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 3an9Rb V  
    %       figure X=C*PWa7  
    %       pcolor(x,x,z), shading interp Qc4r?7S<  
    %       axis square, colorbar ki|KtKAu_9  
    %       title('Zernike function Z_5^1(r,\theta)') DA=#T2)p  
    % i28WgDG)5  
    %   Example 2: FR*CiaD1  
    % hSAdD!  
    %       % Display the first 10 Zernike functions {L6@d1u  
    %       x = -1:0.01:1; J!{"^^*  
    %       [X,Y] = meshgrid(x,x); 6ij L+5  
    %       [theta,r] = cart2pol(X,Y); ht>C6y  
    %       idx = r<=1; -9PJ4"H  
    %       z = nan(size(X)); 5;v_?M!UCK  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ^Pwtu  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; qStZW^lFeY  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Fh8 8DDJ  
    %       y = zernfun(n,m,r(idx),theta(idx)); DsJ ikg(J  
    %       figure('Units','normalized') nm#ISueh  
    %       for k = 1:10 ) wZ;}O  
    %           z(idx) = y(:,k); ]u5B]ZQnA  
    %           subplot(4,7,Nplot(k)) ?.{SYaS  
    %           pcolor(x,x,z), shading interp Ow" e3]}Mt  
    %           set(gca,'XTick',[],'YTick',[]) ZYS`M?Au  
    %           axis square 7Gh+EJJ3I  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) }H5~@c$  
    %       end 8n&Gn%DvX  
    % +DsdzR`Gx,  
    %   See also ZERNPOL, ZERNFUN2. pH9xyN[:a  
    F o k%  
    %   Paul Fricker 11/13/2006 7y?aw`Sw:  
    *VX"_C0Jy=  
    EjA3hHJ  
    % Check and prepare the inputs: CE5A^,EsB  
    % ----------------------------- ?d!*[Ke8  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ! V^wq]D2  
        error('zernfun:NMvectors','N and M must be vectors.') 42oW]b%P{;  
    end XJZ\ss  
    M&[bb $00j  
    if length(n)~=length(m) !{1;wC(b  
        error('zernfun:NMlength','N and M must be the same length.') #}p@+rkg2  
    end | V: 9 ][\  
    v:F_! Q  
    n = n(:); V?L8BRnV  
    m = m(:); wo+ b":  
    if any(mod(n-m,2)) =?3b3PZn  
        error('zernfun:NMmultiplesof2', ... T)Y{>wT  
              'All N and M must differ by multiples of 2 (including 0).') e S: 8Pn  
    end H8x66}  
    .vnQZ*6  
    if any(m>n) \<aR^Sj.  
        error('zernfun:MlessthanN', ... P @Jo[J<  
              'Each M must be less than or equal to its corresponding N.') $ucDz f=o  
    end gbrn'NT  
    eKUP,y;[I  
    if any( r>1 | r<0 ) 41v#|%\w  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') <GWzdj?  
    end ]*Cq'<h$  
    ^qY?x7mx1  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) V8hmfV~=]P  
        error('zernfun:RTHvector','R and THETA must be vectors.') >Jk]=_%  
    end 'NNfzh  
    ^ 'ws/(  
    r = r(:); rT|wZz9$@  
    theta = theta(:); \ z3>kvk  
    length_r = length(r); 8w$q4fg0  
    if length_r~=length(theta) J#DN2y <  
        error('zernfun:RTHlength', ... %<O0Yenu  
              'The number of R- and THETA-values must be equal.') 4KX\'K  
    end (zX75QSKV  
    %M*2j%6  
    % Check normalization: b%QcB[k[WB  
    % -------------------- Ya &\b 6  
    if nargin==5 && ischar(nflag) Z8ds`KZM  
        isnorm = strcmpi(nflag,'norm'); *.6m,QqJ(  
        if ~isnorm +-!2nk`"a  
            error('zernfun:normalization','Unrecognized normalization flag.') `F$lO2#k  
        end ]]NTvr  
    else l4> c  
        isnorm = false; m%cwhH_B  
    end S}P rgw/  
    hb<cynY  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r+!29  
    % Compute the Zernike Polynomials W6s-epsRmT  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3wMnTT"At  
    !C@+CZXLx  
    % Determine the required powers of r: mpNS}n6  
    % ----------------------------------- \4KV9wm  
    m_abs = abs(m); VfFbZds8f  
    rpowers = []; 6~-,.{Y  
    for j = 1:length(n) #}lWM%9Dy  
        rpowers = [rpowers m_abs(j):2:n(j)]; v?YxF}  
    end +!K*FU=).  
    rpowers = unique(rpowers); -%dBZW\u2  
    d"tR ?j  
    % Pre-compute the values of r raised to the required powers, ]*hH.ZBY"^  
    % and compile them in a matrix: w$Z%RF'p  
    % ----------------------------- 3T/&T`T+c  
    if rpowers(1)==0 )x<BeD  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); vSy[lB|)24  
        rpowern = cat(2,rpowern{:}); mqtYny'  
        rpowern = [ones(length_r,1) rpowern]; ?=im  ~  
    else w6h*dh$w  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); SZUo RWx  
        rpowern = cat(2,rpowern{:}); ZfXgVTJ`  
    end V KxuK0{  
    q8!]x-5$6j  
    % Compute the values of the polynomials: Ae%AG@L  
    % -------------------------------------- [1mEdtqf*  
    y = zeros(length_r,length(n)); [tRb{JsUd  
    for j = 1:length(n) BV6B:=E0  
        s = 0:(n(j)-m_abs(j))/2; CQPq5/@Y4  
        pows = n(j):-2:m_abs(j); "A> _U<Y  
        for k = length(s):-1:1 e{H(  
            p = (1-2*mod(s(k),2))* ... 8F&Y;  
                       prod(2:(n(j)-s(k)))/              ...  \EXa 9X2  
                       prod(2:s(k))/                     ... k=cDPu -  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... yJ="dEn>i"  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); y\ })C-&  
            idx = (pows(k)==rpowers); +sV~#%%  
            y(:,j) = y(:,j) + p*rpowern(:,idx); "|Kag|(qB  
        end <I#M^}`  
         1xr2x;  
        if isnorm ExM VGe  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); }>EWF E`  
        end 3~{0X-  
    end ]V)*WP#a  
    % END: Compute the Zernike Polynomials e<qfM&*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z6-ZAS(>m  
    =ym<yI<  
    % Compute the Zernike functions: !zsrORF{  
    % ------------------------------ F B:nkUR`  
    idx_pos = m>0; U^eos;:s8  
    idx_neg = m<0; |+KwyHE`9  
    '\GU(j  
    z = y; $fB j}\o  
    if any(idx_pos) UZs'H"K  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); pSI8"GwQ  
    end K&,";9c  
    if any(idx_neg) *<[zG7+&[  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); J"Fp),  
    end Qm=iCZ|E^!  
     fZ&' _  
    % 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) 7xQ:[P!G+  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. HOF=qE*p  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated jE wt1S V  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive (,tu7u{  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, #@xB ?u-0q  
    %   and THETA is a vector of angles.  R and THETA must have the same ky-nP8L}  
    %   length.  The output Z is a matrix with one column for every P-value, +jK-k_  
    %   and one row for every (R,THETA) pair. 2wDDVUwyB  
    % HTv#2WX  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike <5,|h3]-#  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) (Q @'fb9z  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) QQ_7Q^  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 vj 344B  
    %   for all p. `R>z{-@=  
    %  jr_z ?  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 u1Slu%^e  
    %   Zernike functions (order N<=7).  In some disciplines it is {y a .  
    %   traditional to label the first 36 functions using a single mode [0hahR  
    %   number P instead of separate numbers for the order N and azimuthal kY!zBk  
    %   frequency M. 4obW>  
    % :enR8MS  
    %   Example: E1tCY.N{  
    % ."=%]l 0  
    %       % Display the first 16 Zernike functions 7Xu#|k  
    %       x = -1:0.01:1; {=?(v`88  
    %       [X,Y] = meshgrid(x,x); EFljUT?&  
    %       [theta,r] = cart2pol(X,Y); beC%Tnb7  
    %       idx = r<=1; %Zbm%YaW5  
    %       p = 0:15; {wsJ1 v8!  
    %       z = nan(size(X));  oC*a;o  
    %       y = zernfun2(p,r(idx),theta(idx)); |Tc4a4jS  
    %       figure('Units','normalized') '"\'<>Be  
    %       for k = 1:length(p) 6_])(F3+w.  
    %           z(idx) = y(:,k); E5@=LS  
    %           subplot(4,4,k) CoNaGb  
    %           pcolor(x,x,z), shading interp '?mF,C o{  
    %           set(gca,'XTick',[],'YTick',[]) F]PsS(  
    %           axis square 6% ofS8 [  
    %           title(['Z_{' num2str(p(k)) '}']) :i4(cap&}F  
    %       end d1/9 A-{  
    % H@Ot77(*  
    %   See also ZERNPOL, ZERNFUN. Ie!&FQe2q  
    {_\cd.AuT  
    %   Paul Fricker 11/13/2006 d+,!p8Q  
    "mQcc }8  
    Xd5s8C/}  
    % Check and prepare the inputs: aEvbGo  
    % ----------------------------- yDKH;o  
    if min(size(p))~=1 Y`(Ri-U4  
        error('zernfun2:Pvector','Input P must be vector.') 1=C12  
    end NWvIwt{  
    7xv9v1['  
    if any(p)>35 YCh`V[0  
        error('zernfun2:P36', ...  MiIxj%,(  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Fd\uTxykp  
               '(P = 0 to 35).']) g.@[mf0r  
    end `Ucj_6&Tqs  
    H~nX! sO  
    % Get the order and frequency corresonding to the function number: +cqUp6x.  
    % ---------------------------------------------------------------- VGD~) z57  
    p = p(:); p|2GPrA]aL  
    n = ceil((-3+sqrt(9+8*p))/2); 2O Ur">_  
    m = 2*p - n.*(n+2); 1x;@BV  
    Y;_F,4H  
    % Pass the inputs to the function ZERNFUN: 8|=C/k  
    % ---------------------------------------- 4n6AK`E  
    switch nargin ,++HiYOG}e  
        case 3 t^"8M6BqC;  
            z = zernfun(n,m,r,theta); 4RB%r  
        case 4 ]"uG04"Vk  
            z = zernfun(n,m,r,theta,nflag); anbw\yh8  
        otherwise '(3 QyCD  
            error('zernfun2:nargin','Incorrect number of inputs.') eG!ma`v  
    end } SW p~3P  
    IiqqdU]  
    % 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) #J&3Zds  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. re[5lFQ~Z  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 'hU5]}=  
    %   order N and frequency M, evaluated at R.  N is a vector of zhs @ YMY  
    %   positive integers (including 0), and M is a vector with the 1bQO:n):~  
    %   same number of elements as N.  Each element k of M must be a 8Lx/ZGy  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ,4t6Cq!  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 6CHb\k  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix |{kbc0*  
    %   with one column for every (N,M) pair, and one row for every $Bz};@  
    %   element in R. M9R'ONYAa  
    % wB0vpt5f  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 81:%Z&?vRl  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is eZ(ThA*2=t  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Dh2Cj-| ~  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 .(q'7Q Z/  
    %   for all [n,m]. (Q=o 9o:b  
    % 4!!PrXE  
    %   The radial Zernike polynomials are the radial portion of the s2SV   
    %   Zernike functions, which are an orthogonal basis on the unit XJeWhk3R9  
    %   circle.  The series representation of the radial Zernike _|I8+(~)  
    %   polynomials is yPtE5"(o  
    % TYGI f4z  
    %          (n-m)/2 /}~=)QHH  
    %            __ Itr 4 Pr  
    %    m      \       s                                          n-2s uxD3+Q  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r f^1J_}cL  
    %    n      s=0 T(x@ gwc  
    % ca7Y+9< ;  
    %   The following table shows the first 12 polynomials. AH&RabH2  
    % r6uN6XCM  
    %       n    m    Zernike polynomial    Normalization G4SA u  
    %       --------------------------------------------- Fnak:R0  
    %       0    0    1                        sqrt(2) u*2?Gky  
    %       1    1    r                           2 8+|W%}  
    %       2    0    2*r^2 - 1                sqrt(6) zw15r" R  
    %       2    2    r^2                      sqrt(6) Vq]ixag2^  
    %       3    1    3*r^3 - 2*r              sqrt(8) o0`']-)*2  
    %       3    3    r^3                      sqrt(8) G8+&fn6  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ; &6 {c  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) %p  
    %       4    4    r^4                      sqrt(10) 5Z_C (5)/Y  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 85G-`T  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 2YZ>nqy  
    %       5    5    r^5                      sqrt(12) QyVAs;  
    %       --------------------------------------------- GB Yy^wjU  
    % N!~]D[D  
    %   Example: SgxrU&::  
    % dX/7n=  
    %       % Display three example Zernike radial polynomials ZtO$kK%q;  
    %       r = 0:0.01:1; kVWcf-f  
    %       n = [3 2 5]; tlp,HxlP  
    %       m = [1 2 1]; !Ea >tQ|  
    %       z = zernpol(n,m,r);  4t(/F`  
    %       figure 46NuT]6/4  
    %       plot(r,z) [yN+(^ i  
    %       grid on \_,p@r]Q  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') /J-'[Mc'D[  
    % _"Bj`5S  
    %   See also ZERNFUN, ZERNFUN2. 0ex.~S_Oj4  
    f#:3 TJV  
    % A note on the algorithm. Y}R$RDRL  
    % ------------------------ KHZ[drb6$  
    % The radial Zernike polynomials are computed using the series LcvczS T  
    % representation shown in the Help section above. For many special <9X@\uvU.<  
    % functions, direct evaluation using the series representation can Wrb[\ ?-  
    % produce poor numerical results (floating point errors), because uc+{<E3,%  
    % the summation often involves computing small differences between e%)iDt\j  
    % large successive terms in the series. (In such cases, the functions }ZVond$y4  
    % are often evaluated using alternative methods such as recurrence 4@fv%LOQo  
    % relations: see the Legendre functions, for example). For the Zernike RKzty=j4  
    % polynomials, however, this problem does not arise, because the nC,QvV  
    % polynomials are evaluated over the finite domain r = (0,1), and W{'hn&vU  
    % because the coefficients for a given polynomial are generally all rmA?Xlh\  
    % of similar magnitude. F\+AA  
    % %r1#G.2YW  
    % ZERNPOL has been written using a vectorized implementation: multiple }~zDcj_  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] yC1OeO8{  
    % values can be passed as inputs) for a vector of points R.  To achieve "dIWHfQB  
    % this vectorization most efficiently, the algorithm in ZERNPOL b3-j2`#  
    % involves pre-determining all the powers p of R that are required to /gF)msUF  
    % compute the outputs, and then compiling the {R^p} into a single 5n2!Y\  
    % matrix.  This avoids any redundant computation of the R^p, and 8WQ#)  
    % minimizes the sizes of certain intermediate variables. aXj UDu7  
    % wJ2cAX;"  
    %   Paul Fricker 11/13/2006 &v .S_Ym  
    Z(|$[GZP[  
    YSGE@  
    % Check and prepare the inputs: BH {z]a  
    % ----------------------------- ,zx{RDI  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <rgK}&q  
        error('zernpol:NMvectors','N and M must be vectors.') 6 agG*x  
    end + GQ{{B  
    z[J=WI  
    if length(n)~=length(m) 9Zl4NV&B  
        error('zernpol:NMlength','N and M must be the same length.') 9k71h`5  
    end I"czo9Yspd  
    .q MxShUU  
    n = n(:); 9*s8%pL  
    m = m(:); =nCA=-Jv  
    length_n = length(n); DDR4h"Y  
    }O~D3z4l0  
    if any(mod(n-m,2)) 4dFr~ {  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') HcIJ&".~  
    end z{OL+-OY  
    Mu`_^gG  
    if any(m<0) q-Z<.GTq  
        error('zernpol:Mpositive','All M must be positive.') R4;1LZ8XzS  
    end +I5\ `By=  
    heIys.p  
    if any(m>n) :a)RMp+^0  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') l[[`-f8j  
    end s:6K'*  
    aMe &4Q  
    if any( r>1 | r<0 ) xL_QTj  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') kIS )*_  
    end iWjNK"W  
    5(CInl  
    if ~any(size(r)==1) Ag0w8F  
        error('zernpol:Rvector','R must be a vector.') #\X)|p2  
    end Awfd0L;9  
    %52e^,//  
    r = r(:); A nl1+  
    length_r = length(r); [/hoNCH!  
    PH%t#a!j3/  
    if nargin==4 }\<=B%{  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 19GF%+L ,  
        if ~isnorm 6 DQOar>d  
            error('zernpol:normalization','Unrecognized normalization flag.') ^jL)<y4`  
        end .CEC g*f  
    else A7Y CSjB  
        isnorm = false; ' u<IS/w  
    end o0No"8DnjH  
    *%jXjTA0D  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BKTTta1mY  
    % Compute the Zernike Polynomials Cw:|(`9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yx?oxDJg  
    M/W9"N[ta  
    % Determine the required powers of r: ?84f\<"  
    % ----------------------------------- +?6]Vu&|f  
    rpowers = []; -ABj>y[  
    for j = 1:length(n) HkRvcX 5  
        rpowers = [rpowers m(j):2:n(j)]; 5u9lKno  
    end ph b ;D  
    rpowers = unique(rpowers); 1 M!4hM Q  
    r:o9:w:  
    % Pre-compute the values of r raised to the required powers, W<&/5s  
    % and compile them in a matrix: xp:I(  
    % ----------------------------- Iw[zN[oz  
    if rpowers(1)==0 %6fnL~ A  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ]EF"QLNN(  
        rpowern = cat(2,rpowern{:}); $Xo_8SX,  
        rpowern = [ones(length_r,1) rpowern]; )M7yj O!  
    else *fi`DiO  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); (&*Bl\YoX  
        rpowern = cat(2,rpowern{:}); IW n G@!  
    end BPW.&2?<  
    )2EvZn  
    % Compute the values of the polynomials: 4 w*m]D{  
    % -------------------------------------- 2IHS)kkT|  
    z = zeros(length_r,length_n); 0K"+u9D^  
    for j = 1:length_n [%LGiCU]  
        s = 0:(n(j)-m(j))/2; F ',1R"/}  
        pows = n(j):-2:m(j); p\~ a=  
        for k = length(s):-1:1 Ye|gW=FUR  
            p = (1-2*mod(s(k),2))* ... G@D8 [  
                       prod(2:(n(j)-s(k)))/          ... .3+ 8Ip#z  
                       prod(2:s(k))/                 ... o}waJN`yI  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... p79QEIbk=  
                       prod(2:((n(j)+m(j))/2-s(k))); a>#$&&oQ0  
            idx = (pows(k)==rpowers); 5<GeAW8ns]  
            z(:,j) = z(:,j) + p*rpowern(:,idx); G1X73qoHT<  
        end ZiKO|U@/  
         hUi5~;Q5Fi  
        if isnorm +{6:]  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); LmsPS.It  
        end 8$JJI( {bH  
    end -k>k<bDAI  
    4Z{R36 {  
    % 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)  y|D-W>0cX3  
    DZ;2aH  
    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。 e}"k8 ./  
    _k W:FB  
    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)