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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 #e8CuS  
    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! \2NiI]t]  
     
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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 zF{ z_c#3@  
    function z = zernfun(n,m,r,theta,nflag) ,$*IJeKx  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 54'z"S:W  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N W5yqnjK $4  
    %   and angular frequency M, evaluated at positions (R,THETA) on the `[:f;2(@  
    %   unit circle.  N is a vector of positive integers (including 0), and sxuYwQ  
    %   M is a vector with the same number of elements as N.  Each element ^(6.M\Q  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) P"xP%zqo  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, :_?>3c}L  
    %   and THETA is a vector of angles.  R and THETA must have the same s\ ~r 8  
    %   length.  The output Z is a matrix with one column for every (N,M) N*$Q(K  
    %   pair, and one row for every (R,THETA) pair. VTV-$Du[}  
    % h\20  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike CF$^we  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )D#*Q~   
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral i4uUvZ f  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, f-23.]`v  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Qb SX'mx<  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. U9;AU] A  
    % aIm\tPbb  
    %   The Zernike functions are an orthogonal basis on the unit circle. Put +<o <  
    %   They are used in disciplines such as astronomy, optics, and zx\?cF  
    %   optometry to describe functions on a circular domain. QU\|RX   
    % N 2x\O~7  
    %   The following table lists the first 15 Zernike functions. hx:x5L>  
    % gMgbqGF)  
    %       n    m    Zernike function           Normalization yCm iW %L4  
    %       -------------------------------------------------- IJs` 3?  
    %       0    0    1                                 1 hsVWD,w  
    %       1    1    r * cos(theta)                    2 G8<,\mg+  
    %       1   -1    r * sin(theta)                    2 >S!QvyM(V  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) PR.?"$!D{  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 5$jKw\FF=  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) //AS44^IS  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ;up89a-,9  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) }b~ZpUL!  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8)  C9*'.~  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Mb+cXdZb  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) :PjHsNp;^  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0A|.ch  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) -,p(PK  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) QDyL0l{C  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) jMZ{>l.v  
    %       -------------------------------------------------- a[t2T jB  
    % N|8TE7- F|  
    %   Example 1: :,:r  
    % :~g=n&x  
    %       % Display the Zernike function Z(n=5,m=1) 7]G3yt->  
    %       x = -1:0.01:1; $7lI Dt  
    %       [X,Y] = meshgrid(x,x); iGm[fxQ|  
    %       [theta,r] = cart2pol(X,Y); qf+I2 kyS  
    %       idx = r<=1; gwT"o  
    %       z = nan(size(X)); ,qt9S0 QS  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); up`!r;5-  
    %       figure LiiQ;x  
    %       pcolor(x,x,z), shading interp ~u-mEdu3C  
    %       axis square, colorbar @@_f''f$  
    %       title('Zernike function Z_5^1(r,\theta)') KLlW\MF1  
    % >Ei_##  
    %   Example 2: JZN'U<R  
    % R~;<}!Gtx  
    %       % Display the first 10 Zernike functions %5a>@K]  
    %       x = -1:0.01:1; HPm12&8,  
    %       [X,Y] = meshgrid(x,x); =3l%ZL/  
    %       [theta,r] = cart2pol(X,Y); ~x`OCii  
    %       idx = r<=1; kcI3pmgj  
    %       z = nan(size(X)); b6Dve]  
    %       n = [0  1  1  2  2  2  3  3  3  3]; AEhh 6v  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; LbvnV~S  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 0I& !a$:  
    %       y = zernfun(n,m,r(idx),theta(idx)); b`fPP{mG  
    %       figure('Units','normalized') _KC()OIeC  
    %       for k = 1:10 (*BQd1Z  
    %           z(idx) = y(:,k); 05.^MU?^U  
    %           subplot(4,7,Nplot(k)) &+d>xy\^/  
    %           pcolor(x,x,z), shading interp M-"%4^8_  
    %           set(gca,'XTick',[],'YTick',[]) j8L!miv6  
    %           axis square DNC2]kS<  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) R/xeC [r  
    %       end n<uF9N<   
    % f9F@G&&Ugg  
    %   See also ZERNPOL, ZERNFUN2. 5fA<I _ D  
    JZ]4?_l  
    %   Paul Fricker 11/13/2006 PW~+=,  
    E9YR *P4$  
    C /\)-^  
    % Check and prepare the inputs: -]\UFR  
    % ----------------------------- ^ ,d!K2`  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) u-$(TyDEl|  
        error('zernfun:NMvectors','N and M must be vectors.') V*2uW2\}  
    end a4Fe MCvV9  
    :B6hYx  
    if length(n)~=length(m) db'Jl^  
        error('zernfun:NMlength','N and M must be the same length.') xJ:15eDC  
    end ,dLh`t<\  
    nK)U.SZ  
    n = n(:); %l( qyH)*  
    m = m(:); -O:+?gG  
    if any(mod(n-m,2)) # 4L[8(+V  
        error('zernfun:NMmultiplesof2', ... )xy1 DA  
              'All N and M must differ by multiples of 2 (including 0).') kG^DHEne  
    end nm_]2z O  
    ,|<2wn#q  
    if any(m>n) 2Xys;Dwx  
        error('zernfun:MlessthanN', ...  pQKR  
              'Each M must be less than or equal to its corresponding N.') 6*J`2U9Q  
    end 1>r7s*  
    ~k'KS 7c  
    if any( r>1 | r<0 ) I6,'o)l{_  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') */;[ -9  
    end m-dyvW+  
    PbvRh~n  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 7F!_gj p  
        error('zernfun:RTHvector','R and THETA must be vectors.') TL-sxED,,D  
    end oi^2Pvauh  
    !`LaX!bmp  
    r = r(:); i<@6f'Kir  
    theta = theta(:); dbQUW#<Q  
    length_r = length(r); ]h3<r8D_#  
    if length_r~=length(theta) D6=Z%h\*  
        error('zernfun:RTHlength', ... !o1{. V9q  
              'The number of R- and THETA-values must be equal.') o{f|==<t3#  
    end G1=GzAd$5  
    B"rnSui  
    % Check normalization: ) jv]Oz  
    % -------------------- RB`Emp&T  
    if nargin==5 && ischar(nflag) {EE/3e@  
        isnorm = strcmpi(nflag,'norm'); z-$bce9*  
        if ~isnorm DN3#W w2[r  
            error('zernfun:normalization','Unrecognized normalization flag.') RY3ANEu+  
        end uLX5khQ  
    else :8\!;!  
        isnorm = false; \ 'G%%%;4  
    end ~w_4 nE  
    ,7&`V=C  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?f<JwF<  
    % Compute the Zernike Polynomials 5 0uYU[W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +[C dd{2  
    rH Et]Xa  
    % Determine the required powers of r: (C>FM8$J  
    % ----------------------------------- Y /$`vgqs  
    m_abs = abs(m); <Z GEmQ  
    rpowers = []; `@1y|j:m  
    for j = 1:length(n) l$N b1&  
        rpowers = [rpowers m_abs(j):2:n(j)]; ;T0F1  
    end D;VQoO  
    rpowers = unique(rpowers); *5?a% p  
    &D0suK#  
    % Pre-compute the values of r raised to the required powers, zO8`xrN!  
    % and compile them in a matrix: ~b;l08 <  
    % ----------------------------- &~gqEl6RF  
    if rpowers(1)==0 itClCEOA  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); R1OC7q  
        rpowern = cat(2,rpowern{:}); {@, } M  
        rpowern = [ones(length_r,1) rpowern]; RP 'VEJ   
    else 3 r4QB  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); hiO:VA  
        rpowern = cat(2,rpowern{:}); ^PA[fL"  
    end \9k$pC+l  
    DID&fj9m  
    % Compute the values of the polynomials: 8fA9yQ 8  
    % -------------------------------------- &U q++f6  
    y = zeros(length_r,length(n));  t9T3e  
    for j = 1:length(n) ;Yo9e~  
        s = 0:(n(j)-m_abs(j))/2; WvSh i=  
        pows = n(j):-2:m_abs(j); 5(e?,B }  
        for k = length(s):-1:1 Z)}q=NjA  
            p = (1-2*mod(s(k),2))* ... Xvu|ss  
                       prod(2:(n(j)-s(k)))/              ... E)z[@Np  
                       prod(2:s(k))/                     ... Pl^-]~  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 7LMad%  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ;ELQIHnD"  
            idx = (pows(k)==rpowers); Y8!T4dkn  
            y(:,j) = y(:,j) + p*rpowern(:,idx); uMOm<kn  
        end Cx$C+  
         6&V4W"k  
        if isnorm AdBF$nn[  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ]yu,YZ@7  
        end +W|MAJtg  
    end 3?|gBiX  
    % END: Compute the Zernike Polynomials .C=&` ;Vs  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0=Jf93D5  
    Cw;&{jY  
    % Compute the Zernike functions: St/<\Y,wr  
    % ------------------------------ &X0/7)*"v  
    idx_pos = m>0; a,X=!oJ  
    idx_neg = m<0; X&qRanOP;z  
    [P#^nyOh(  
    z = y; s)Sa KE*d  
    if any(idx_pos) Yc;cf% c1  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); !g:UkU\J  
    end DDxNqVVt4  
    if any(idx_neg) ^pz3L'4n  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); z{T2! w~[  
    end N{Og; roGD  
    "h.}o DS  
    % 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) TnET1$@qr*  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Sd'Meebu  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated lh`inAt)"  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive kfb+OE:7  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, #$V`%2>  
    %   and THETA is a vector of angles.  R and THETA must have the same $ hB;r  
    %   length.  The output Z is a matrix with one column for every P-value, ;aYPv8s~,:  
    %   and one row for every (R,THETA) pair. J{72%S  
    % 'R6D+Vk/  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike }i_[wq{E&  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ~D@pk>I  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) #r 1 $=GY  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 %Fb"&F^7  
    %   for all p. b!hxx Z  
    % G-oC A1UdN  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 +T[3wL~  
    %   Zernike functions (order N<=7).  In some disciplines it is s%]-Sw9  
    %   traditional to label the first 36 functions using a single mode UNocm0!N'  
    %   number P instead of separate numbers for the order N and azimuthal 0 O4'Ts ?  
    %   frequency M. IBl}.o&]B#  
    % A`u04Lm7  
    %   Example: ;}IF'ANA  
    % 77/y{#Sk  
    %       % Display the first 16 Zernike functions yAEOn/.~  
    %       x = -1:0.01:1; `9Ngax=_  
    %       [X,Y] = meshgrid(x,x); MnFem $ @  
    %       [theta,r] = cart2pol(X,Y); >rlQY>5pH  
    %       idx = r<=1; LXIlrZ9D5  
    %       p = 0:15; Mr:*l`b_  
    %       z = nan(size(X)); |w[}\#2  
    %       y = zernfun2(p,r(idx),theta(idx)); Zx25H"5j  
    %       figure('Units','normalized') $V?zJ:a>L  
    %       for k = 1:length(p) [$?S9)Xd  
    %           z(idx) = y(:,k); S}e*~^1J  
    %           subplot(4,4,k) '.~vN L+ O  
    %           pcolor(x,x,z), shading interp DMcvu*A  
    %           set(gca,'XTick',[],'YTick',[]) ,IuO;UV#)  
    %           axis square lsW.j#yE!  
    %           title(['Z_{' num2str(p(k)) '}']) tZ>>aiI3  
    %       end l>"gO9j  
    % hX)r%v:  
    %   See also ZERNPOL, ZERNFUN. LYh5f#  
    =B1t ?( "  
    %   Paul Fricker 11/13/2006 ]Ke|wRQD  
    &.kg8|s{  
    f&`v-kiAn=  
    % Check and prepare the inputs: {114 [  
    % ----------------------------- m'k`p5[=h  
    if min(size(p))~=1 mUr@w*kq|p  
        error('zernfun2:Pvector','Input P must be vector.') eHv~?b5l  
    end bXq,iX  
    9YHSL[  
    if any(p)>35 EN.yU!N.4  
        error('zernfun2:P36', ... UkNC|#l)  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... H?40yu2m5  
               '(P = 0 to 35).']) sLbz@54  
    end ?O4Dhu  
    i[3$Wi$  
    % Get the order and frequency corresonding to the function number: %9mB4Fc6b)  
    % ---------------------------------------------------------------- 0x^$q? \A  
    p = p(:); d;lp^K M  
    n = ceil((-3+sqrt(9+8*p))/2); TOMvJ>bF  
    m = 2*p - n.*(n+2); b{s E#m%r  
    ?0[%+AD hM  
    % Pass the inputs to the function ZERNFUN: b,T=0W  
    % ---------------------------------------- >jl"Yr#  
    switch nargin ieBW 0eMi  
        case 3 (~Zg\(5#  
            z = zernfun(n,m,r,theta); "G6d'xkP  
        case 4 &[*<>  
            z = zernfun(n,m,r,theta,nflag); =q xcM+OX1  
        otherwise WS(@KN  
            error('zernfun2:nargin','Incorrect number of inputs.') jn}6yXB  
    end *kyy''r  
    D{BH~IM  
    % 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) |?8CV\D!  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Wnf`Rf)1z  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of !uO|T'u0a  
    %   order N and frequency M, evaluated at R.  N is a vector of STOE=TC>  
    %   positive integers (including 0), and M is a vector with the ae!_u \$  
    %   same number of elements as N.  Each element k of M must be a '!1lK  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) '.kbXw0}  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is  %;W8;  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix $^ >n@Q@&L  
    %   with one column for every (N,M) pair, and one row for every xD1wHp!+  
    %   element in R. um8ZhXq  
    % nQ~q -=,L  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- H`io|~Q  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 5<a<!]|C  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to uo%O\} #u9  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 (U.&[B  
    %   for all [n,m]. `7 B [<  
    % KPO((G0&  
    %   The radial Zernike polynomials are the radial portion of the m",bfZ  
    %   Zernike functions, which are an orthogonal basis on the unit 3QR-8  
    %   circle.  The series representation of the radial Zernike aPb!-o{  
    %   polynomials is z.H`a+cl  
    % 'Gt`3qG  
    %          (n-m)/2 V&}Z# 9Dx  
    %            __ >=Rb:#UM  
    %    m      \       s                                          n-2s }oU&J81  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Sv>aZ  
    %    n      s=0 Z$hxo )|  
    % Xs?>6i@$$  
    %   The following table shows the first 12 polynomials. ftH 0aI  
    % Sqge5v  
    %       n    m    Zernike polynomial    Normalization <kM%z{p  
    %       --------------------------------------------- q;AQ6k(  
    %       0    0    1                        sqrt(2) m76]INq  
    %       1    1    r                           2 'V>+G>U  
    %       2    0    2*r^2 - 1                sqrt(6)  OB^  
    %       2    2    r^2                      sqrt(6) -OW$  
    %       3    1    3*r^3 - 2*r              sqrt(8) {Y/  
    %       3    3    r^3                      sqrt(8) . LS.Z 4@  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) T(}da**X  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Ud9\;Qse  
    %       4    4    r^4                      sqrt(10) V\xQM;  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ~G$OY9UC  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) \C6m.%%={R  
    %       5    5    r^5                      sqrt(12) G^OSXf5  
    %       --------------------------------------------- JH5])i0  
    % xAO ]u[J  
    %   Example: r\1*N.O3|O  
    % {38aaf|'/  
    %       % Display three example Zernike radial polynomials *>#cs#)  
    %       r = 0:0.01:1; z&:[.B   
    %       n = [3 2 5]; ynd}w G'  
    %       m = [1 2 1]; wb?hfe  
    %       z = zernpol(n,m,r); D|BN_ai9  
    %       figure ZN1p>+oY!  
    %       plot(r,z) 8]L.E  
    %       grid on <w A_2S Y  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 0jS/U|0  
    % x?F{=\z/o  
    %   See also ZERNFUN, ZERNFUN2. %2+]3h>g  
    e8#3Y+Tc  
    % A note on the algorithm. sg y  
    % ------------------------ JQCwI`%i  
    % The radial Zernike polynomials are computed using the series g\Z k*5(  
    % representation shown in the Help section above. For many special =1(BKk>  
    % functions, direct evaluation using the series representation can kr/h^e  
    % produce poor numerical results (floating point errors), because 1W8W/Y=hT  
    % the summation often involves computing small differences between |q>Mw-=  
    % large successive terms in the series. (In such cases, the functions X>4`{x`  
    % are often evaluated using alternative methods such as recurrence b(VU{cf2d  
    % relations: see the Legendre functions, for example). For the Zernike GwycSb1  
    % polynomials, however, this problem does not arise, because the +u2Co_FJ&  
    % polynomials are evaluated over the finite domain r = (0,1), and tvUvd(8 w  
    % because the coefficients for a given polynomial are generally all >tzXbmFp;  
    % of similar magnitude. E.3}a>f  
    % d7P @_jO6  
    % ZERNPOL has been written using a vectorized implementation: multiple ,+RO 5n  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] r?TK@^z  
    % values can be passed as inputs) for a vector of points R.  To achieve f#t^<`7  
    % this vectorization most efficiently, the algorithm in ZERNPOL m0,9yY::wj  
    % involves pre-determining all the powers p of R that are required to MD)"r>k  
    % compute the outputs, and then compiling the {R^p} into a single X3nhqQTZ  
    % matrix.  This avoids any redundant computation of the R^p, and -T8 gV1*(<  
    % minimizes the sizes of certain intermediate variables. 1`t?5|s>  
    % Uu+C<j&-  
    %   Paul Fricker 11/13/2006 7y\g~?5N  
    x(`$D  
    \2,18E  
    % Check and prepare the inputs: :I('xVNPz  
    % ----------------------------- ss<'g@R  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 8hfh,v5(  
        error('zernpol:NMvectors','N and M must be vectors.') %},gE[N!J  
    end k ))*z FV  
    %np#Bv-L  
    if length(n)~=length(m) lo:~~l  
        error('zernpol:NMlength','N and M must be the same length.') Om  
    end m =}X$QF`^  
    \sd"iMEi  
    n = n(:); q@9 i3*q;  
    m = m(:); `[CJtd2\  
    length_n = length(n); }hYE6~pr  
    q1Sm#_7  
    if any(mod(n-m,2)) O| ]Ped9  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') gxM8IQ  
    end `1@[uWl  
    [u80-x<  
    if any(m<0) B6\VxSX4{  
        error('zernpol:Mpositive','All M must be positive.') H\mVK!](D  
    end =fG8YZ(  
    LDeVNVM  
    if any(m>n) %ZyPK,("  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') hH}/v0_jb  
    end S$52KOo  
    b<AE}UK  
    if any( r>1 | r<0 ) @3c#\jx  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') dA/o4co  
    end 4d G-  
    "}p?pF<'0  
    if ~any(size(r)==1) e0HP~&BRs  
        error('zernpol:Rvector','R must be a vector.') :, [ !8QP  
    end +3>/,w(x  
    ; ZV^e  
    r = r(:); HDyZzjgG  
    length_r = length(r); 03WRj+w  
    ~4MjJKzA  
    if nargin==4 7RE6y(V1  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); xm5FQ) T  
        if ~isnorm P &0cF{  
            error('zernpol:normalization','Unrecognized normalization flag.') [V, ;X  
        end lMez!qx,=  
    else *u'`XRJU/  
        isnorm = false; [; ?{BB  
    end Z4ov  
    Q 02??W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *.%)rm  
    % Compute the Zernike Polynomials G!Oq>7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% OW}j4-~wL  
    h) PB  
    % Determine the required powers of r: 9~'Ip7X,!  
    % ----------------------------------- 5qQ(V)ah  
    rpowers = []; n UCk0:{  
    for j = 1:length(n) rqe_zyc&  
        rpowers = [rpowers m(j):2:n(j)]; 5z w23!  
    end |uwteG5?$s  
    rpowers = unique(rpowers); n3g WM C  
    OXX(OCG>  
    % Pre-compute the values of r raised to the required powers, j_uY8c>3\q  
    % and compile them in a matrix: Z?v6pjZ?  
    % ----------------------------- e=)* O  
    if rpowers(1)==0 n;^k   
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); , gr&s+  
        rpowern = cat(2,rpowern{:}); k,iV$,[TF  
        rpowern = [ones(length_r,1) rpowern]; :'rZZeb'  
    else F"f}vl  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ?Wz(f{Hm  
        rpowern = cat(2,rpowern{:}); 7K]U |K#  
    end |DPpp/  
    X:-bAu}D  
    % Compute the values of the polynomials: }:l%,DBw  
    % -------------------------------------- +6)kX4  
    z = zeros(length_r,length_n); %%,hR'+|  
    for j = 1:length_n pF*~)e  
        s = 0:(n(j)-m(j))/2; \v P2B  
        pows = n(j):-2:m(j); $17 v,  
        for k = length(s):-1:1  !QW 0  
            p = (1-2*mod(s(k),2))* ... l)PFzIz=V  
                       prod(2:(n(j)-s(k)))/          ... h:Mn$VR,  
                       prod(2:s(k))/                 ... ($ B ]9*  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 6dR-HhF  
                       prod(2:((n(j)+m(j))/2-s(k))); Nw$[a$^n  
            idx = (pows(k)==rpowers); OK2/k_jXN'  
            z(:,j) = z(:,j) + p*rpowern(:,idx); KYmWfM3^  
        end C`~4q<W'  
         2:HP5   
        if isnorm $s(4?^GP  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); y7IbE   
        end !_z<W~t"  
    end @1SKgbt>  
    i&'^9"Z)O  
    % 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
    在线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)  !cEbz b  
    W.#}q K" q  
    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。 T}L^CU0  
    ?Y? gzD  
    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)