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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, iF5'ygR-Z  
    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, 8kE]_t  
    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? ulT8lw='  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? `J<*9dq%  
    `=+^|Y}  
    TlL^7f}  
    _!;Me )C  
    kNqS8R|  
    function z = zernfun(n,m,r,theta,nflag) qs\2Z@;  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. J2q,7wI#  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N c5q9 LQ/  
    %   and angular frequency M, evaluated at positions (R,THETA) on the DBLk!~IF  
    %   unit circle.  N is a vector of positive integers (including 0), and #?MY&hdU9  
    %   M is a vector with the same number of elements as N.  Each element >F jR9B  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) (z7vl~D  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 7*Qk`*Ii  
    %   and THETA is a vector of angles.  R and THETA must have the same X)SDG#&+bF  
    %   length.  The output Z is a matrix with one column for every (N,M) rD?L  
    %   pair, and one row for every (R,THETA) pair. 682Z}"I0  
    % Wc3kO'J  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike a)Q!'$"'  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), <99M@ cF  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral @WH@^u  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 7g=2Z[o  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized iUMY!eqp  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2Y}?P+:%>  
    % ZN"j%E{d  
    %   The Zernike functions are an orthogonal basis on the unit circle. hc (e$##  
    %   They are used in disciplines such as astronomy, optics, and AIb2k  
    %   optometry to describe functions on a circular domain. dca ;'$  
    % CO-_ea U(  
    %   The following table lists the first 15 Zernike functions. 4p%A8%/q  
    % "gd=J_Yw  
    %       n    m    Zernike function           Normalization WPZ?*Sx  
    %       -------------------------------------------------- T@}|zDC#  
    %       0    0    1                                 1 UT~a &u  
    %       1    1    r * cos(theta)                    2 R(.}C)q3  
    %       1   -1    r * sin(theta)                    2 IcP)FB 4  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) G 6VF>2  
    %       2    0    (2*r^2 - 1)                    sqrt(3) (%iRaw7hp  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) AE: Z+rM*  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ^@P1 JNe  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) XxHx:mi  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 2._X|~0a  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Vz14j_  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) RMO,ZVq  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 86@c't@  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) U$oduY#  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Z, BC*  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) B1]bRxwn?  
    %       -------------------------------------------------- 80A.<=(=.  
    % Y|8v O  
    %   Example 1: gTRF^knrY  
    % Z*G(5SqUh"  
    %       % Display the Zernike function Z(n=5,m=1) imQUR C  
    %       x = -1:0.01:1; (E,T#uc{  
    %       [X,Y] = meshgrid(x,x); R+g z<H.Q  
    %       [theta,r] = cart2pol(X,Y); Q1V9PRZX  
    %       idx = r<=1; sNun+xsf^  
    %       z = nan(size(X)); XdH\OJ  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); rt JtK6t  
    %       figure +_-bJo2a  
    %       pcolor(x,x,z), shading interp 4|A>b})H  
    %       axis square, colorbar hdTzCfeZ5@  
    %       title('Zernike function Z_5^1(r,\theta)') t|t#vcB  
    % aq7~QX_0G  
    %   Example 2: !w BJ,&E  
    % #plY\0E@  
    %       % Display the first 10 Zernike functions $mF_,|  
    %       x = -1:0.01:1; j }b\Z9)!  
    %       [X,Y] = meshgrid(x,x); a>\vUv*  
    %       [theta,r] = cart2pol(X,Y); 8* Jw0mSw  
    %       idx = r<=1; E2)h ?cs  
    %       z = nan(size(X)); 8[6o (  
    %       n = [0  1  1  2  2  2  3  3  3  3]; @p\}pY$T  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Dk48@`l2  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; \EseGgd21  
    %       y = zernfun(n,m,r(idx),theta(idx)); Vh>Z,()>>@  
    %       figure('Units','normalized') bLt.O(T}  
    %       for k = 1:10 % `Z! 4L  
    %           z(idx) = y(:,k); P2 Vg4   
    %           subplot(4,7,Nplot(k)) fNGZo  
    %           pcolor(x,x,z), shading interp `y+tf?QN  
    %           set(gca,'XTick',[],'YTick',[]) Ov<NsNX]  
    %           axis square 9@+5LZR  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Z|}G6]h  
    %       end @k&qb!Qah  
    % |Ph3#^rM?  
    %   See also ZERNPOL, ZERNFUN2. 'vNG(h#%d  
    }@.|?2b +  
    8QMPY[{   
    %   Paul Fricker 11/13/2006 :1Sl"?xU  
    }1IpON  
    [9:9Ql_h  
    +}jJ&Z9 )  
    },|M9 I0  
    % Check and prepare the inputs: V59(Z  
    % ----------------------------- hlt[\LP=$  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) s(W|f|R  
        error('zernfun:NMvectors','N and M must be vectors.') =-p$jXVW%  
    end [z 7bixN  
    ID/ F  
    $ImrOf^qt  
    if length(n)~=length(m) \l~*PG2  
        error('zernfun:NMlength','N and M must be the same length.') 1^gl}^|B  
    end Bj7gQ%>H4  
     T Q,?>6n  
    @IXsy  
    n = n(:); 7 [0L9\xm  
    m = m(:); %.Q !oYehj  
    if any(mod(n-m,2)) !>:?rSg*  
        error('zernfun:NMmultiplesof2', ... 2 3gPbtq/  
              'All N and M must differ by multiples of 2 (including 0).') '(/7[tJ  
    end  O#I1V K  
    kZ"BBJ6w  
    kBN+4Dr/$  
    if any(m>n) 5e WwgA  
        error('zernfun:MlessthanN', ... <F04GO\  
              'Each M must be less than or equal to its corresponding N.') zP<pEI  
    end 08*v~(T  
    P9 Z}H(?C  
    }\C-} Q  
    if any( r>1 | r<0 ) {5  sO  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') iQ"XLrpl  
    end Vx-7\NB  
    i&n'N8D@  
    a0Zv p>Ft  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) |ZQ@fmvL/p  
        error('zernfun:RTHvector','R and THETA must be vectors.') U,LTVYrO  
    end ?Q&yEGm(  
    #Skv(IL  
    !^ _ "~  
    r = r(:); i7ly[6{^pr  
    theta = theta(:); N?.%?0l  
    length_r = length(r); A%^ILyU6c  
    if length_r~=length(theta) {^N[("`  
        error('zernfun:RTHlength', ... )RcL/n  
              'The number of R- and THETA-values must be equal.') &ot/nQQ  
    end LCQE_}Mh  
    9H%X2#:fH  
    a`0=AQ  
    % Check normalization: :Lz\yARpk  
    % -------------------- I"`M@ %  
    if nargin==5 && ischar(nflag) &zo|Lfe  
        isnorm = strcmpi(nflag,'norm'); \Acqr@D  
        if ~isnorm vI@%Fg+D  
            error('zernfun:normalization','Unrecognized normalization flag.') D:K4H+ch  
        end )YqXRm  
    else ,#8e_3Z$  
        isnorm = false; c ;'[W60  
    end  Sr?#S  
    `HBf&Z  
    xL" |)A =  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +[tP_%/r'^  
    % Compute the Zernike Polynomials dc rSz4E|>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KSrx[q  
    x]33LQ1]  
    w[ ~#av9  
    % Determine the required powers of r: ^53r/V}%  
    % ----------------------------------- x@Hc@R<!  
    m_abs = abs(m); RH|XxH*  
    rpowers = []; LWt&3  
    for j = 1:length(n) &ZQJ>#~j^  
        rpowers = [rpowers m_abs(j):2:n(j)]; } GiHjzsR  
    end u#@Q:tnN_  
    rpowers = unique(rpowers); Tq~=TSD  
    zi3\63D3eO  
    H8On<C=  
    % Pre-compute the values of r raised to the required powers, O &-wxJ]S  
    % and compile them in a matrix: Nj 00W1  
    % ----------------------------- fX|Y;S-@+  
    if rpowers(1)==0 Tq4-wE+  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); mN19WQ(r  
        rpowern = cat(2,rpowern{:}); DX|# gUAm  
        rpowern = [ones(length_r,1) rpowern]; tmtT (  
    else (zFi$  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); _ :VB}>  
        rpowern = cat(2,rpowern{:}); -bU oCF0  
    end 1&U>,;]*  
    s4uhsJL V$  
    @#-q^}3  
    % Compute the values of the polynomials: XSZ k%_  
    % -------------------------------------- # m *J&  
    y = zeros(length_r,length(n)); RnDt)3  
    for j = 1:length(n) ih;]nJ]+-  
        s = 0:(n(j)-m_abs(j))/2; #&7}-"Nd  
        pows = n(j):-2:m_abs(j); r#Oo nZ  
        for k = length(s):-1:1 `kJ^zw+  
            p = (1-2*mod(s(k),2))* ... ]:~OG@(  
                       prod(2:(n(j)-s(k)))/              ... wg]j+r@  
                       prod(2:s(k))/                     ... t0T"@t#c  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Ez-[ )44/  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); .uk>QM s1  
            idx = (pows(k)==rpowers); 4tS.G  
            y(:,j) = y(:,j) + p*rpowern(:,idx); =>! Y{: y(  
        end I}vmU^Y>  
         1[vi.  
        if isnorm BV[5}  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); B[4KX  
        end =Yt R`  
    end ;{%\9nS  
    % END: Compute the Zernike Polynomials :B:"NyPA  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% heK7pH7;d  
    )6J9J+%bi  
    ?28N ^  
    % Compute the Zernike functions: 2j[&=R/.  
    % ------------------------------ 81i655!Z  
    idx_pos = m>0; McQe1  
    idx_neg = m<0; Z7?~S2{c  
    vt9)pMs  
    C-H@8p?T  
    z = y;  W0]gLw9*  
    if any(idx_pos) ?C A,  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); F;<xnC{[  
    end #: [<iSk  
    if any(idx_neg) W!>.$4Q9  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); oT>(V]*5  
    end fL=~NC"  
    |YY_^C`"-  
    fu]s/'8B  
    % EOF zernfun $ . 9V&  
     
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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)  uj R_"r|l  
    XkXHGDEf1  
    DDE还是手动输入的呢? b`~p.c%(  
    MbZJ;,e?  
    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