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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, 8:Z@lp^  
    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, |^YzFrc  
    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? Zkf 3t>[  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? G> f^ 2  
    &{X{36  
    4JX`>a{<  
    LEYWH% y  
    *ah>-}-  
    function z = zernfun(n,m,r,theta,nflag) kWzuz#  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ap Fs UsE  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ygmv_YLjm  
    %   and angular frequency M, evaluated at positions (R,THETA) on the .OVW4svX  
    %   unit circle.  N is a vector of positive integers (including 0), and $sU5=,  
    %   M is a vector with the same number of elements as N.  Each element >wz;}9v  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 6Cz7A  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, _myg._[  
    %   and THETA is a vector of angles.  R and THETA must have the same )e4WAlg8c  
    %   length.  The output Z is a matrix with one column for every (N,M) tQNk=}VR7r  
    %   pair, and one row for every (R,THETA) pair. u Y?/B~  
    % Y|{r vBKjf  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike YD/B')/ s  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), &/b? I `  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral @4 zi]v  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 8>C; >v  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized \zk?$'d  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 6{JR0  
    % *u|lmALs  
    %   The Zernike functions are an orthogonal basis on the unit circle. -/ (DP x  
    %   They are used in disciplines such as astronomy, optics, and _hAj2%SL  
    %   optometry to describe functions on a circular domain. M/::`yJQu  
    % p)?qJ2c|  
    %   The following table lists the first 15 Zernike functions. yt/20a  
    % sD LVYD  
    %       n    m    Zernike function           Normalization ]`#xR *a  
    %       -------------------------------------------------- S5:"_U  
    %       0    0    1                                 1 m.\ >95!  
    %       1    1    r * cos(theta)                    2 `c qH}2s#  
    %       1   -1    r * sin(theta)                    2 ^l]]qdNr  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) N<#S3B?.  
    %       2    0    (2*r^2 - 1)                    sqrt(3) "E@NZ*"u  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) e3yorQ][  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) )bB"12Z|8  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) O:oU`vE  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 1kB'sc3N!  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) {pcf;1^t  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) !SLP8|Cd  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) d-6sC@PB  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) P?Gd}mdX?m  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ql#{=oGDnA  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ]Ks]B2Osz  
    %       -------------------------------------------------- VTy,43<  
    % "4Vi=*2V  
    %   Example 1: p0}+071o%  
    % J[j/aDdP  
    %       % Display the Zernike function Z(n=5,m=1) ~6@c]:  
    %       x = -1:0.01:1; p^pQZ6-  
    %       [X,Y] = meshgrid(x,x); EuKrYY]g  
    %       [theta,r] = cart2pol(X,Y); #hy5c,}>  
    %       idx = r<=1; TnvHO_P,  
    %       z = nan(size(X)); _/QKWk&j  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ?MV[=LPL  
    %       figure I,],?DQX2)  
    %       pcolor(x,x,z), shading interp Gx(KN57D  
    %       axis square, colorbar 7 SjF9x  
    %       title('Zernike function Z_5^1(r,\theta)') OBKC$e6I  
    % %8Z|/LGg  
    %   Example 2: C|.$L<`  
    % bik] JIM  
    %       % Display the first 10 Zernike functions ND9 n1WZ&x  
    %       x = -1:0.01:1; K,lK\^y  
    %       [X,Y] = meshgrid(x,x); 2-"Lxe65f  
    %       [theta,r] = cart2pol(X,Y); K]H"qG.K  
    %       idx = r<=1; O" X!S_R  
    %       z = nan(size(X)); G:h;C].  
    %       n = [0  1  1  2  2  2  3  3  3  3]; gqO%^b)6  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; KV^:sxU  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 7})!>p )  
    %       y = zernfun(n,m,r(idx),theta(idx)); eLDL  "L  
    %       figure('Units','normalized') .v #0cQX+.  
    %       for k = 1:10 jt'Y(u]2  
    %           z(idx) = y(:,k); 3&a*]  
    %           subplot(4,7,Nplot(k)) ,%)WT>  
    %           pcolor(x,x,z), shading interp WQIM2_=M  
    %           set(gca,'XTick',[],'YTick',[]) @,y FY  
    %           axis square eu}:Wg2  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ?j&~vy= T  
    %       end (?*mh?  
    % 26('V `N  
    %   See also ZERNPOL, ZERNFUN2. $.r}g\43P  
    dFI.`pB  
    n7`.<*:  
    %   Paul Fricker 11/13/2006 Gpxb_}P  
    2kp|zX(  
    _Ssv:x c,  
    hIzPy3  
    .^9/ 0.g8t  
    % Check and prepare the inputs: lk+=2 6>  
    % ----------------------------- /\3XARt  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) B Z\EqB  
        error('zernfun:NMvectors','N and M must be vectors.') AT8B!m   
    end fr,CH{Uq  
    {R1Cxt}  
    +X%fcoc  
    if length(n)~=length(m) ?VOs:sln  
        error('zernfun:NMlength','N and M must be the same length.') 654jS!  
    end e%@[d<Ta\  
    eHnei F  
    )K\k6HC.  
    n = n(:); QX.F1T 2e?  
    m = m(:); Be14$7r  
    if any(mod(n-m,2)) x%:> Ol  
        error('zernfun:NMmultiplesof2', ... RqX4ep5j  
              'All N and M must differ by multiples of 2 (including 0).') ?^G$;X7B  
    end X/;"CM  
    F"@'(b  
    e~2*> 5\:  
    if any(m>n) HZr/0I?  
        error('zernfun:MlessthanN', ... j_ywG{Jk  
              'Each M must be less than or equal to its corresponding N.') G.q^Zd#.T  
    end g6a3MJV`  
    w_z^5\u0  
    i_ODgc`H  
    if any( r>1 | r<0 ) =|{,5="  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') B'BbTI,  
    end fY\tvo%  
    ->"h5h  
    ae(]9VW  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) xz+`]Q  
        error('zernfun:RTHvector','R and THETA must be vectors.') PEQvEruZ}  
    end - 5-SlQu  
    zC>(!fJqq  
    b8b PK<  
    r = r(:); Y ,pS/  
    theta = theta(:); , LPFb6o  
    length_r = length(r); j n&9<"W  
    if length_r~=length(theta) pvwnza1  
        error('zernfun:RTHlength', ... iN9!?Ov_  
              'The number of R- and THETA-values must be equal.') X9`C2fyVd  
    end ;0X|*w1JO  
    {^19.F  
    +sbacMfq  
    % Check normalization: >dC(~j{  
    % -------------------- R\Ynn^w  
    if nargin==5 && ischar(nflag) 2'^OtM,  
        isnorm = strcmpi(nflag,'norm'); ;51!a C  
        if ~isnorm E,?aBRxy  
            error('zernfun:normalization','Unrecognized normalization flag.') fD}]Mi:V  
        end  !+VN   
    else N*oJ$:#  
        isnorm = false; ,'{B+CHoS  
    end _j<M}  
    /g- X=|?F  
    3$G25=eN  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V^5k> `A  
    % Compute the Zernike Polynomials <.B > LU  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M,U=zNPnk  
    cZ2, u,4  
    "=TTsxyM6P  
    % Determine the required powers of r: #w?%&,Kp  
    % ----------------------------------- l#f]KLv4N_  
    m_abs = abs(m); jJQfCOD$  
    rpowers = []; { rJF)\2  
    for j = 1:length(n) `e;Sjf<  
        rpowers = [rpowers m_abs(j):2:n(j)]; ^p%+rB.j[  
    end 0$h$7'a  
    rpowers = unique(rpowers); (S 3kP5:F  
    ' g!_Flk  
    Jj!tRZT  
    % Pre-compute the values of r raised to the required powers, <1%XN  
    % and compile them in a matrix: NbMH@6%E  
    % ----------------------------- 8r|  
    if rpowers(1)==0 Pw{{+PBu R  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); t4W0~7   
        rpowern = cat(2,rpowern{:}); |2` $g  
        rpowern = [ones(length_r,1) rpowern]; YZu# 0)  
    else 1ucUnNkcV  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); JV{!Ukuyp+  
        rpowern = cat(2,rpowern{:}); EGO@`<"h  
    end BqAwo  
    R,Uy3N  
    dOgM9P  
    % Compute the values of the polynomials: j`M<M[C*4N  
    % -------------------------------------- wm[d5A4  
    y = zeros(length_r,length(n)); =U|SK"oO  
    for j = 1:length(n) 3/<^R}w\  
        s = 0:(n(j)-m_abs(j))/2; ?bi^h/ f  
        pows = n(j):-2:m_abs(j); 3nGK674;z  
        for k = length(s):-1:1 bv"({:x  
            p = (1-2*mod(s(k),2))* ... l_IX+4(@b|  
                       prod(2:(n(j)-s(k)))/              ... !Bbwl-e`  
                       prod(2:s(k))/                     ... f3|=T8"t  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... {%}6 d~Bg  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); I9&<:`  
            idx = (pows(k)==rpowers); 'B:De"_(N  
            y(:,j) = y(:,j) + p*rpowern(:,idx); KAEpFobYo  
        end J=bOw//  
         + a@SdWf  
        if isnorm P?ol]MwaB  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); *M5C*}dl  
        end .b)(_*  
    end oK[,xqyA  
    % END: Compute the Zernike Polynomials o : DnZN  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ds[~Cp   
    9Dkgu ^`  
    \"j1fAD!  
    % Compute the Zernike functions: t $%}*@x7  
    % ------------------------------ \dbtd hT;Z  
    idx_pos = m>0; S(xA}0]  
    idx_neg = m<0; }Ec"&  
    Qp Vm  
    </Lqk3S-!  
    z = y; *xKR;?.  
    if any(idx_pos) _~<TAFBr  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ^<b.j.$<z  
    end ^el:)$  
    if any(idx_neg) EFOQ;q  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); M,lu)~H  
    end x&p=vUuukP  
    |%9~W^b  
    6?~pjMV  
    % EOF zernfun y['icGU6  
     
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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)  x+&&[>-P  
    q&.!*rPD  
    DDE还是手动输入的呢? t%%I.zIV7  
    5D#*lMSP"'  
    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