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    [讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

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    离线jssylttc
     
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    只看楼主 倒序阅读 楼主  发表于: 2012-04-23
    下面这个函数大家都不会陌生,计算zernike函数值的,并根据此可以还原出图像来, +'"NKZ.>TT  
    我输入10阶的n、m,r,theta为38025*1向量,最后得到的z是29525*10阶的矩阵, )~{8C:  
    这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? rNl%I@G  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? S^:7V[=EgI  
    \B Uno6  
    6[3>[ej:x  
    ,c&%/"i:w  
    FwpTQix!  
    function z = zernfun(n,m,r,theta,nflag) m0,TH[HWGF  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 7Ml OBPh  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N }Ryrd!3bY  
    %   and angular frequency M, evaluated at positions (R,THETA) on the G<FB:?|  
    %   unit circle.  N is a vector of positive integers (including 0), and N+zR7`AG8  
    %   M is a vector with the same number of elements as N.  Each element G\B:iyKl  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ehV}}1>O  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, G4,.kK  
    %   and THETA is a vector of angles.  R and THETA must have the same n%d7`?tm4  
    %   length.  The output Z is a matrix with one column for every (N,M) S^7u`-  
    %   pair, and one row for every (R,THETA) pair. THcX.%ToT  
    % Kwo0%2Onkd  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Is(ZVI  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 4Jk[X>I~  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral :OD-L)Or  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, =kBWY9 :$,  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized jMP;$w  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ,xg(F0q  
    % [u;>b?[{  
    %   The Zernike functions are an orthogonal basis on the unit circle. DoFF<LXBt  
    %   They are used in disciplines such as astronomy, optics, and ,D93A  
    %   optometry to describe functions on a circular domain. S.*.nv  
    % xsRu~'f  
    %   The following table lists the first 15 Zernike functions. 9)S,c =z83  
    % =PmIrvr'[5  
    %       n    m    Zernike function           Normalization ,F?O} ijk  
    %       -------------------------------------------------- 3z!^UA>q  
    %       0    0    1                                 1 rds0EZ4W  
    %       1    1    r * cos(theta)                    2 4Ep6vm X  
    %       1   -1    r * sin(theta)                    2 7xfN}iHG  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ?Vc/mO2X  
    %       2    0    (2*r^2 - 1)                    sqrt(3) '&F Pk T:5  
    %       2    2    r^2 * sin(2*theta)             sqrt(6)  Eikt,  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8)  <xwaFZ  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) -f=4\3y3p  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) $sb `BS  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) @WuG8G  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 4=ZN4=(_[  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) N3&n"w _d  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Z#flu Q%V  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) uE&2M>2  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) _MzdbUb5,  
    %       -------------------------------------------------- V ee;&  
    % `m\l#r 2C  
    %   Example 1: BF(Kaf;<t.  
    % ZWy,NN1  
    %       % Display the Zernike function Z(n=5,m=1) 1zIrU6H2;_  
    %       x = -1:0.01:1; ke5_lr(  
    %       [X,Y] = meshgrid(x,x); l/6(V:  
    %       [theta,r] = cart2pol(X,Y); {AO`[  
    %       idx = r<=1; 2-DJ3OL]k  
    %       z = nan(size(X)); Vv.q{fRvYB  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); sXR}#*8p  
    %       figure -3Auo0  
    %       pcolor(x,x,z), shading interp "kg?Or.  
    %       axis square, colorbar b-)3MR:4  
    %       title('Zernike function Z_5^1(r,\theta)') #W[C;f|,  
    % !kWx'tJ$  
    %   Example 2: oU)HxV  
    % W%P0X5YQ  
    %       % Display the first 10 Zernike functions 6a*OQ{8  
    %       x = -1:0.01:1; Kz9h{ Tu4  
    %       [X,Y] = meshgrid(x,x); h2mU  
    %       [theta,r] = cart2pol(X,Y); r]O8|#P,Z$  
    %       idx = r<=1; ~n9-  
    %       z = nan(size(X)); ~w}Zv0  
    %       n = [0  1  1  2  2  2  3  3  3  3]; B{-+1f4  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; e(EXQP2P>  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; x#TWZ;  
    %       y = zernfun(n,m,r(idx),theta(idx)); H^0`YQJ3  
    %       figure('Units','normalized') "(^1Dm$(  
    %       for k = 1:10 =f-.aq(G/  
    %           z(idx) = y(:,k); mx")cGGQ  
    %           subplot(4,7,Nplot(k)) nuLxOd*n  
    %           pcolor(x,x,z), shading interp 6l?\iE  
    %           set(gca,'XTick',[],'YTick',[]) mc}r15:<  
    %           axis square 7Hp~:i30  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) he1OLk  
    %       end e(;nhU3a*,  
    % 7|$ H}$  
    %   See also ZERNPOL, ZERNFUN2. q NE( @at  
    bx%P-r31  
    7Jvb6V<R  
    %   Paul Fricker 11/13/2006 pk2}]jx"  
    +}@6V4BRn  
    ,L,?xvWG  
    @Z%I g  
    h]#bPb  
    % Check and prepare the inputs: "\u_gk{g  
    % ----------------------------- o&vODs  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) E/N*n!sV  
        error('zernfun:NMvectors','N and M must be vectors.') xDTDfhA  
    end !mtX*;b(e  
    H:&|q+K=#  
    $ h<l  
    if length(n)~=length(m) Y]!{ n W  
        error('zernfun:NMlength','N and M must be the same length.') V;t8v\  
    end %$.]g  
    @Zd/>'  
    ILq"/S.  
    n = n(:); ]@UJ 8hDy  
    m = m(:); tr $~INe  
    if any(mod(n-m,2)) 84$#!=v  
        error('zernfun:NMmultiplesof2', ... ;~5w`F)  
              'All N and M must differ by multiples of 2 (including 0).') ^qD@qJ  
    end )./'`Mx?  
    nkvkHh  
    X6lR?6u%|  
    if any(m>n) FtL{ f=  
        error('zernfun:MlessthanN', ... %T:7I[f  
              'Each M must be less than or equal to its corresponding N.') |6}:n,KA.  
    end $Q!J.}P@  
    /\&Wk;u3  
    t/9,JG  
    if any( r>1 | r<0 ) V`MV_zA2  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') I%<,JRAV  
    end  'WW['  
    Q~p[jQ,4wZ  
    |p7k2wzN  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) \.7O0Q{  
        error('zernfun:RTHvector','R and THETA must be vectors.') E6NrBPm  
    end R^=)Ucj  
    "L p"o  
    G~\ SI.  
    r = r(:); )FfJ%oT}  
    theta = theta(:); H _%yh,L  
    length_r = length(r); Ltt+BUJc  
    if length_r~=length(theta) /6%<97/d  
        error('zernfun:RTHlength', ... (YJ]}J^  
              'The number of R- and THETA-values must be equal.') uBe1{Z  
    end mVBF2F<4  
    Rr'^l ]  
    _(<D*V[  
    % Check normalization: mjd9]HgN  
    % -------------------- -bHfo%"^TT  
    if nargin==5 && ischar(nflag) 68^5X"OGF  
        isnorm = strcmpi(nflag,'norm'); >{dj6Wo  
        if ~isnorm dU~DlaEy(  
            error('zernfun:normalization','Unrecognized normalization flag.') ]k (n_+!  
        end MFyMo  
    else = yH#Iil  
        isnorm = false; "c  S?t  
    end !y>MchNv  
    (HUGgX"=  
    +I?T|Iin  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !1H\*VM "  
    % Compute the Zernike Polynomials qOKC2WD  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @YEdN}es  
    _/)?GXwLn  
    j aj."v  
    % Determine the required powers of r: 7Lr}Y/1=  
    % ----------------------------------- ^'|\8  
    m_abs = abs(m); 1z\>>N$7B  
    rpowers = []; xCd9b:jG  
    for j = 1:length(n) +C{ %pF  
        rpowers = [rpowers m_abs(j):2:n(j)]; l|[8'*]r!  
    end OudD1( )W  
    rpowers = unique(rpowers); c !ybz{L  
    7x%0 ^~/n  
    ]byj[Gd  
    % Pre-compute the values of r raised to the required powers, ^%v<I"<Uq5  
    % and compile them in a matrix: 3huT T"G  
    % ----------------------------- jF'azlT  
    if rpowers(1)==0 6' M"-9?G  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); U~SOHfZ%(  
        rpowern = cat(2,rpowern{:}); nJTV@m XVq  
        rpowern = [ones(length_r,1) rpowern]; $?OuY*ZeY9  
    else HHbkR2H1  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); R,hX *yVq  
        rpowern = cat(2,rpowern{:}); ?D#]g[6  
    end 7^bO`  
    9oteQN{9  
    RN?z)9!  
    % Compute the values of the polynomials: W`C&$v#  
    % -------------------------------------- &8Cuu$T9)  
    y = zeros(length_r,length(n)); 7CGKm8T  
    for j = 1:length(n) K/ q:aMq  
        s = 0:(n(j)-m_abs(j))/2; x@I@7Pvo3  
        pows = n(j):-2:m_abs(j); fN8|4  
        for k = length(s):-1:1 K%<Z"2!+  
            p = (1-2*mod(s(k),2))* ... _|MY/SN4A  
                       prod(2:(n(j)-s(k)))/              ... c7jft|4S  
                       prod(2:s(k))/                     ... ,=tVa])  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... '|e5cW6z  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); V(3udB@K  
            idx = (pows(k)==rpowers); *xs8/?  
            y(:,j) = y(:,j) + p*rpowern(:,idx); p&F=<<C  
        end q_8qowu"  
         _Y*: l7  
        if isnorm GA6)O-^G  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 0\AYUa?RM  
        end TA=Ij,z~  
    end R Nr=M^Zn  
    % END: Compute the Zernike Polynomials (r,RwWYm  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >RxZ-.,a  
    Rww"Z=F  
    mp~\ioI*d  
    % Compute the Zernike functions: 6^zuRY;  
    % ------------------------------ 5I{YsM  
    idx_pos = m>0; FuaGr0]  
    idx_neg = m<0; YTq>K/  
    xH\'gli/  
    ;w?zmj<Dm  
    z = y; hHoc7  
    if any(idx_pos) WKpHb:H  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); K/Axojo  
    end K:P gkc  
    if any(idx_neg) yPm)r2Ck  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 8T 6jM+ h  
    end }6eWdm!B  
    A0S6 4(  
    lp?geav  
    % EOF zernfun f7XmVCz1  
     
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    离线phoenixzqy
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    只看该作者 1楼 发表于: 2012-04-23
    慢慢研究,这个专业性很强的。用的人又少。
    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)
    离线sansummer
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    只看该作者 2楼 发表于: 2012-04-27
    这个太牛了,我目前只能把zygo中的zernike的36项参数带入到zemax中,但是我目前对其结果的可信性表示质疑,以后多交流啊
    离线jssylttc
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    只看该作者 3楼 发表于: 2012-05-14
    回 sansummer 的帖子
    sansummer:这个太牛了,我目前只能把zygo中的zernike的36项参数带入到zemax中,但是我目前对其结果的可信性表示质疑,以后多交流啊 (2012-04-27 10:22)  t_x \&+W  
    g"hm"m}i  
    DDE还是手动输入的呢? K\5@yqy5  
    K.",=\53  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    离线jssylttc
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    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    离线18257342135
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    只看该作者 5楼 发表于: 2016-12-13
    支持一下,慢慢研究