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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, <9T,J"y  
    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, 'I:_}q  
    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? )*Wz5x  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? J|@D @\?7  
    hegH^IN M  
    "xn,'`a  
    _;:_ !`  
    s,l*=<  
    function z = zernfun(n,m,r,theta,nflag) R<%{I)  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. KC%&or  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N "z= ~7g  
    %   and angular frequency M, evaluated at positions (R,THETA) on the RD;A  
    %   unit circle.  N is a vector of positive integers (including 0), and V#R; -C  
    %   M is a vector with the same number of elements as N.  Each element 4vND ~9d  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) "KSdC8MS  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, s6#e?5J  
    %   and THETA is a vector of angles.  R and THETA must have the same C5jt(!pi  
    %   length.  The output Z is a matrix with one column for every (N,M) e@S\7Ks  
    %   pair, and one row for every (R,THETA) pair. xMa9o  
    % t:v>W8N53  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 9[lk=1.qN  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), DF'~ #G8  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 9e}%2,  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3(gOF&Uf9  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 9l:[jsk<d  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *P&lAyt6  
    % J3B+WD]  
    %   The Zernike functions are an orthogonal basis on the unit circle. .ud&$-[a  
    %   They are used in disciplines such as astronomy, optics, and N9M",(WTt}  
    %   optometry to describe functions on a circular domain. rFUd  
    % zAev@+.ld  
    %   The following table lists the first 15 Zernike functions. 4 Lz[bI  
    % wF59g38[z$  
    %       n    m    Zernike function           Normalization =h+-1zp{M^  
    %       -------------------------------------------------- oa[O~z{~  
    %       0    0    1                                 1 kV8qpw}K  
    %       1    1    r * cos(theta)                    2  +ZFN8  
    %       1   -1    r * sin(theta)                    2 KTAQ6k  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) '(ZT }N  
    %       2    0    (2*r^2 - 1)                    sqrt(3) m9 ]Ge]  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 2L51 H(  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 4vkqe6  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) DJqJ6z:'  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) QIJ/'72  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) L"0?g(< 5  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Ll VbY=EX7  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Fq%NY8KNE  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ;lt8~ea  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ]86*k %A  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Vn\jUEC  
    %       -------------------------------------------------- 563ExibH  
    % @hrIu" '!  
    %   Example 1: fKtlfQG  
    % L|;sB=$'{  
    %       % Display the Zernike function Z(n=5,m=1) `DM)tm3&m  
    %       x = -1:0.01:1; Dd-a*6|x  
    %       [X,Y] = meshgrid(x,x); H^vA}F`  
    %       [theta,r] = cart2pol(X,Y); bQ&%6'ck  
    %       idx = r<=1; )h{+pK  
    %       z = nan(size(X)); s?4nR:ZC}  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 73SH[f[g  
    %       figure @xBO[v  
    %       pcolor(x,x,z), shading interp +oHbAPs8  
    %       axis square, colorbar [$:L| V!{  
    %       title('Zernike function Z_5^1(r,\theta)') o` dQ  
    % ;>F1?5P{  
    %   Example 2: -"^xg"  
    % 2uV5hSHYe  
    %       % Display the first 10 Zernike functions {+3g*s/HI  
    %       x = -1:0.01:1; | h+vdE8  
    %       [X,Y] = meshgrid(x,x); 1TF S2R n  
    %       [theta,r] = cart2pol(X,Y); a`?Vc}&  
    %       idx = r<=1; 4X+I2CD  
    %       z = nan(size(X)); BN&}g}N  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ;:>q;%  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; '$J M2 u  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; FJxb!- 0&  
    %       y = zernfun(n,m,r(idx),theta(idx)); nHp(,'R/  
    %       figure('Units','normalized') t~44ub6GN`  
    %       for k = 1:10 YD{N)v  
    %           z(idx) = y(:,k); 8U4In[4  
    %           subplot(4,7,Nplot(k)) H<P d&  
    %           pcolor(x,x,z), shading interp yNU}1_oK  
    %           set(gca,'XTick',[],'YTick',[]) S/RChg_L5  
    %           axis square e ~cg  (.  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) U6y`:G;.  
    %       end Sq:J'%/z  
    % /Ei e5p  
    %   See also ZERNPOL, ZERNFUN2. 'C#[iRG4  
    N.ZuSkRM  
    }7P[%(T5  
    %   Paul Fricker 11/13/2006 9wO2`e )  
    S1m5z,G  
    f/4DFs{  
    aygK$.wos  
     !$!%era`  
    % Check and prepare the inputs: ]<c\+9  
    % ----------------------------- ^\Q%VTM  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <HIM k  
        error('zernfun:NMvectors','N and M must be vectors.') "V`DhOG&  
    end i->G {_gH  
    ?[Ma" l>  
    i&DUlmt)f  
    if length(n)~=length(m) rR#wbDr5  
        error('zernfun:NMlength','N and M must be the same length.') >J)4e~9EJ2  
    end eV }H  
    ?du*ITim  
    |zd5P  
    n = n(:); XdOntP*a  
    m = m(:); P:3o}CB1I  
    if any(mod(n-m,2)) _ sy]k A  
        error('zernfun:NMmultiplesof2', ... m| 7v76(  
              'All N and M must differ by multiples of 2 (including 0).') gFfKK`)}D'  
    end ~,xso0  
    ,q{~lf -  
    )e6sg]#  
    if any(m>n) }m7$,'C%P  
        error('zernfun:MlessthanN', ... v$5D&Tv  
              'Each M must be less than or equal to its corresponding N.') jc#gn& 4C  
    end =En1?3?  
    Ae"|a_>fMI  
    lIO#)>  
    if any( r>1 | r<0 ) NmF8BmIj  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 3*(><<ZC  
    end t=s.w(3t  
    |+>U91!  
    s'IB{lJ9  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) /g!Xe]Ss  
        error('zernfun:RTHvector','R and THETA must be vectors.') sb?!U"v.'  
    end aH8]$e8_,\  
    t}OzF cyqN  
    =wD&hDn4  
    r = r(:); :_,3")-v  
    theta = theta(:); y|3("&)"S  
    length_r = length(r); kX:1=+{xg  
    if length_r~=length(theta) EVA&By6_k  
        error('zernfun:RTHlength', ... 5Nbq9YY  
              'The number of R- and THETA-values must be equal.') 6VJS l%X  
    end l7IF9b$c  
    HnsLYY\  
    U%;E:|  
    % Check normalization: J6rWe  
    % -------------------- CteNJBm  
    if nargin==5 && ischar(nflag) [8oX[oP  
        isnorm = strcmpi(nflag,'norm'); r>CBp$  
        if ~isnorm soX^$l  
            error('zernfun:normalization','Unrecognized normalization flag.') %5@> nC?`[  
        end ltNY8xrdGN  
    else :()K2<E  
        isnorm = false; |)*!&\Ch  
    end kV!1k<f  
    0(&Rm R  
    s%6L94\t  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7-\wr^ll3  
    % Compute the Zernike Polynomials `G:hC5B  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0~W6IGE~  
    Wvl'O'R  
    s ;]"LD@  
    % Determine the required powers of r: F6:LH,~8   
    % ----------------------------------- MfKru,LSh  
    m_abs = abs(m); %e|UA-(  
    rpowers = []; &4l!2  
    for j = 1:length(n) JRAU|gr  
        rpowers = [rpowers m_abs(j):2:n(j)]; 1Oak8 \G  
    end 1(% 6X*z  
    rpowers = unique(rpowers); ejbtdU8N<  
    r/HG{XH`  
    Q7/Jyx|  
    % Pre-compute the values of r raised to the required powers, R$+"'N6p  
    % and compile them in a matrix: :/RvtmW  
    % ----------------------------- e:_[0#  
    if rpowers(1)==0 N.SV*G @  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); uigzf^6,  
        rpowern = cat(2,rpowern{:}); n,_9Eh#WD  
        rpowern = [ones(length_r,1) rpowern]; o? K>ji!  
    else .SSPJY(  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); <dz_7hR"  
        rpowern = cat(2,rpowern{:}); f2v~: u  
    end 54RexB o  
    O<dCvH  
    m2ph8KC  
    % Compute the values of the polynomials: #]^M/y h  
    % -------------------------------------- 2^U?Ztth6  
    y = zeros(length_r,length(n)); %?8.UW\m  
    for j = 1:length(n) %+UTs'I  
        s = 0:(n(j)-m_abs(j))/2; z(>:LX"xz  
        pows = n(j):-2:m_abs(j); k RSY;V  
        for k = length(s):-1:1 gI@nE:(m  
            p = (1-2*mod(s(k),2))* ... t$R0UprK  
                       prod(2:(n(j)-s(k)))/              ... /1=x8Sb  
                       prod(2:s(k))/                     ... v`:!$U* H=  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... `q1-yH0~4  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); m93{K7O2e  
            idx = (pows(k)==rpowers); H$ :BJ$x@  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ^?0?*  
        end %0 U@k!lP  
         H;Gs0Qi;  
        if isnorm $d&7q5[  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); *0r!eD   
        end k9VWyq__  
    end 2j1HN  
    % END: Compute the Zernike Polynomials ww'B!Ml>F  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {`Mb),G  
    VjZb\ d4  
    L%pAEoSG  
    % Compute the Zernike functions: sp0_f;bC  
    % ------------------------------ :cP u  
    idx_pos = m>0; Z1 (!syg  
    idx_neg = m<0; K;TTGK  
    X [?E{[@Z  
    EFu>  
    z = y; Us>  
    if any(idx_pos) jX t5.9 t  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); `1FNs?j  
    end |;U3pq)  
    if any(idx_neg) +hH7|:JQ  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); V {}TG]  
    end RGY#0.Z}  
    a"k,x-EL(  
    *_a jb:  
    % EOF zernfun ma`sv<f4-!  
     
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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)
    在线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)  ~*RBMHs  
    V}q=!zz  
    DDE还是手动输入的呢? Yg]!`(db  
    }[By N).  
    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