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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, ) Lv{  
    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, 54-x 14")  
    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? -)/>qFj )  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? }z6HxB]$  
    QaV*}W  
    /V~(!S>  
    '=xl}v  
    y!].l0e2a  
    function z = zernfun(n,m,r,theta,nflag) =W?c1EPLCx  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. a?dM8zAnc  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 08Gr  
    %   and angular frequency M, evaluated at positions (R,THETA) on the O`4X[r1LD  
    %   unit circle.  N is a vector of positive integers (including 0), and qW9|&GuZ$  
    %   M is a vector with the same number of elements as N.  Each element 2 q>4nN  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 7e4\BzCC  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, l"64w>,  
    %   and THETA is a vector of angles.  R and THETA must have the same sz5@=  
    %   length.  The output Z is a matrix with one column for every (N,M) t+ @F"[j  
    %   pair, and one row for every (R,THETA) pair. G(L*8U< UG  
    % Oc1ZIIkh\  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike qH$p]+Rk 5  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 5 m:nh<)#  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral d)@M MF  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, r+n&Pp+9  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized *Z(qk`e.b  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *)Y;`Yg$  
    % BFY~::<b  
    %   The Zernike functions are an orthogonal basis on the unit circle. "D+QT+sD  
    %   They are used in disciplines such as astronomy, optics, and =e63>*M|  
    %   optometry to describe functions on a circular domain. CwAl-o  
    % a^N/N5-Z  
    %   The following table lists the first 15 Zernike functions. g`6S*&8I  
    % @<P [z[  
    %       n    m    Zernike function           Normalization GIp?}tM  
    %       -------------------------------------------------- IkupW|}rc  
    %       0    0    1                                 1 MFVFr "  
    %       1    1    r * cos(theta)                    2 {.ph)8  
    %       1   -1    r * sin(theta)                    2 /dO&r'!:  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) `7NgQ*g.d/  
    %       2    0    (2*r^2 - 1)                    sqrt(3) HHdc[pJ0D  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 3Xy>kG}  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) >Kx l+F  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 9P]TIV.  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Z@>>ZS1Do  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) SngV<J>zR  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) :geXplTx  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *op7:o_  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) cWm.']  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f!87JE=<  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) gpPktp2  
    %       -------------------------------------------------- H_ x35|"  
    % <1ai0]  
    %   Example 1: ^b4o 0me  
    % YO=;)RA  
    %       % Display the Zernike function Z(n=5,m=1) v<O\ l~S  
    %       x = -1:0.01:1; E;N+B34  
    %       [X,Y] = meshgrid(x,x); 4;_.|!LN  
    %       [theta,r] = cart2pol(X,Y); tZ(Wh  
    %       idx = r<=1; A!NT 2YdHZ  
    %       z = nan(size(X)); +ISB"a  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); X;-,3dy  
    %       figure &c A?|(7-  
    %       pcolor(x,x,z), shading interp ^s%Qt  
    %       axis square, colorbar #GTmC|[  
    %       title('Zernike function Z_5^1(r,\theta)') b*,R9  
    % &Zov9o:gx  
    %   Example 2: v5&WW?IBQ  
    % Drg'RR><  
    %       % Display the first 10 Zernike functions aPWFb.JO4  
    %       x = -1:0.01:1; 4*'NpqC(_  
    %       [X,Y] = meshgrid(x,x); z\fk?Tj<ro  
    %       [theta,r] = cart2pol(X,Y); E_$ ST3  
    %       idx = r<=1; S6cSeRmw  
    %       z = nan(size(X)); #Qkl| h  
    %       n = [0  1  1  2  2  2  3  3  3  3]; p<Zf,F}  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; z8A`BVqI  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; EQg 6*V  
    %       y = zernfun(n,m,r(idx),theta(idx)); 1W5YS +pf  
    %       figure('Units','normalized') `OduBUI]]  
    %       for k = 1:10 B} &C h  
    %           z(idx) = y(:,k); +1e*>jE  
    %           subplot(4,7,Nplot(k)) S!rUdxO  
    %           pcolor(x,x,z), shading interp T `N(=T^*  
    %           set(gca,'XTick',[],'YTick',[]) X~lOFH;}q  
    %           axis square ao 32n  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) k/Cr ^J"  
    %       end X!r!lW  
    % Y8Mo.v  
    %   See also ZERNPOL, ZERNFUN2. <{e0 i  
    0ro)e~_@*  
    d`^j\b>5(  
    %   Paul Fricker 11/13/2006 7GKeqv  
    hd+JKh!u  
    k~Gjfo  
    gJ2R(YMF  
    w W-GBY3  
    % Check and prepare the inputs: !5x"d7  
    % ----------------------------- eQzTb91  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) InRn!~_N  
        error('zernfun:NMvectors','N and M must be vectors.') AX[/S8|6  
    end a]75z)X R  
    *,mbZE=<  
    ^ .>)*P  
    if length(n)~=length(m) @@}A\wA-  
        error('zernfun:NMlength','N and M must be the same length.') ;b(/PH!O  
    end :s*&_y  
    )TG\P,H9  
    ~KEnZa0  
    n = n(:); _)lK.5  
    m = m(:); sd Z=3)  
    if any(mod(n-m,2)) df}B:?Ew.  
        error('zernfun:NMmultiplesof2', ... vrh}X[JEw'  
              'All N and M must differ by multiples of 2 (including 0).') $yRbo '-  
    end |)1"*`z  
    i9w xP i  
    ,X+071.(  
    if any(m>n) /ZX8gR5x  
        error('zernfun:MlessthanN', ... JWM/np6  
              'Each M must be less than or equal to its corresponding N.') O`H[,+vm[  
    end :x= ZvAvo  
    bSBI[S  
    % kKtPrT  
    if any( r>1 | r<0 ) B-JgXW.\0  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') wHdq:,0-!  
    end bMf +/n  
    4{*K%pv\  
    6$2)m;| XY  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) /9W-;l{=z  
        error('zernfun:RTHvector','R and THETA must be vectors.') d7P| x  
    end 7J##IH+z35  
    .GLotc  
    t4 h5R  
    r = r(:); eRC@b^~  
    theta = theta(:); zI(b#eUF  
    length_r = length(r); #2|sS|0<  
    if length_r~=length(theta) X2Y-TE T  
        error('zernfun:RTHlength', ... N(/DC)DJg  
              'The number of R- and THETA-values must be equal.') SC"=M^E  
    end \Ui8Sgeei  
    ZJ  u\  
    8%I4jL<  
    % Check normalization: r's4-\  
    % -------------------- $:F]O$A  
    if nargin==5 && ischar(nflag) ExV>s*y  
        isnorm = strcmpi(nflag,'norm'); k2p{<SO;  
        if ~isnorm RwN*/Li  
            error('zernfun:normalization','Unrecognized normalization flag.') 6d`6=D:  
        end M {_`X  
    else : !J!l u  
        isnorm = false; e>y"V; Mj  
    end 7J7uHl`yq`  
    W/xb[w9v  
    Fp|x,-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% qzsS"=5  
    % Compute the Zernike Polynomials KGzBK:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a{,EX[~b  
    -0Y8/6](  
    tb^3-ZUb  
    % Determine the required powers of r: L0_R2E A  
    % ----------------------------------- PtwE[YDu  
    m_abs = abs(m); Z3T:R"l;  
    rpowers = []; 67')nEQ9  
    for j = 1:length(n) sf@g $  
        rpowers = [rpowers m_abs(j):2:n(j)]; dy#dug6j  
    end , B h[jb`y  
    rpowers = unique(rpowers); }=az6cLE2  
    D 0\  
    '72ZLdi}-  
    % Pre-compute the values of r raised to the required powers, qGr(MDLc  
    % and compile them in a matrix: WwPfz<I  
    % ----------------------------- ~g~z"!K  
    if rpowers(1)==0 aZ@Ke$jD  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); _hM3p  
        rpowern = cat(2,rpowern{:}); XM>ByfD{  
        rpowern = [ones(length_r,1) rpowern]; S_ e }>-  
    else &=xm>;`3  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ; :a7rN"(  
        rpowern = cat(2,rpowern{:}); x$?{)EY  
    end }I9\=jT  
    l%rwJLN1  
    CXb)k.L   
    % Compute the values of the polynomials: 7P.C~,+D%P  
    % -------------------------------------- jun>(7  
    y = zeros(length_r,length(n)); Ks{^R`O au  
    for j = 1:length(n) X-e)w  
        s = 0:(n(j)-m_abs(j))/2; Cj31'  
        pows = n(j):-2:m_abs(j); zl=RK  
        for k = length(s):-1:1 yv[ s)c}  
            p = (1-2*mod(s(k),2))* ... vn KKK.E  
                       prod(2:(n(j)-s(k)))/              ... /`Yp]l  
                       prod(2:s(k))/                     ... w f,7  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 3z!\Z[  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); + i!/J  
            idx = (pows(k)==rpowers); =k2In_  
            y(:,j) = y(:,j) + p*rpowern(:,idx); =ugxPgn  
        end /~K-0K#w  
         k]] (I<2  
        if isnorm ~ubGx  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); )?:V5UO\  
        end XA-DJ  
    end "'~'xaU!=a  
    % END: Compute the Zernike Polynomials W52AX.Nm  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % tN{  
    k"LbB#Q  
    S=n,unn#t  
    % Compute the Zernike functions: o=X6PoJ N_  
    % ------------------------------ +>@<'YI<  
    idx_pos = m>0; e@Q<hb0<eU  
    idx_neg = m<0; p%jl-CC1  
    AVyqtztQ  
    .|NF8Fj  
    z = y; :CNHN2 J  
    if any(idx_pos) SYOND>E  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ?PO~$dUc]  
    end Z}5 ;K"T/  
    if any(idx_neg) cP''  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); b?hdWQSW7  
    end kQxY"HD  
    *Sm$FMWQ  
    T9osueh4  
    % EOF zernfun =cs;avtL  
     
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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)  pt.0%3  
    XF(0>-  
    DDE还是手动输入的呢? fx(^}e  
    $o"S zy  
    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