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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, dDtFx2(R  
    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, fS A)G$b]  
    这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? &ZJgQ-Pc(m  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Q$ZHv_VLx  
    !gP0ndRJ=  
    Zb''mf\  
    z`$J_CjY  
    z.)p P'CJo  
    function z = zernfun(n,m,r,theta,nflag) N[U9d}Zv  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. nWWM2v  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N D59T?B|BdD  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ^J x$t/t  
    %   unit circle.  N is a vector of positive integers (including 0), and Ec]|p6a3  
    %   M is a vector with the same number of elements as N.  Each element onte&Ed\  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) D>sYPrf  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, hu5!ev2  
    %   and THETA is a vector of angles.  R and THETA must have the same orIQ~pF#  
    %   length.  The output Z is a matrix with one column for every (N,M) 1 W'F3  
    %   pair, and one row for every (R,THETA) pair. v{;7LXy0  
    % `UzVS>]l[+  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .T.5TMiOSq  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), NZXjE$<Vr  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral GsV4ZZ  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, <@, $hso7:  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized  7}B   
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. E9 Y\X  
    % UAYd?r  
    %   The Zernike functions are an orthogonal basis on the unit circle. c-CYdi@  
    %   They are used in disciplines such as astronomy, optics, and ;D2E_!N dt  
    %   optometry to describe functions on a circular domain. WDx Mo`zT  
    % '2^ Yw  
    %   The following table lists the first 15 Zernike functions. F u _@!K  
    % smU4jh9S  
    %       n    m    Zernike function           Normalization p25Fn`}H  
    %       -------------------------------------------------- TbhH&kG)1  
    %       0    0    1                                 1 c^.l 2Q!  
    %       1    1    r * cos(theta)                    2 LSd*| 3E}n  
    %       1   -1    r * sin(theta)                    2 p1O6+hRio  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) fA^Em)cs2  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ~&VN_;j_  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 6yIvaY$KR  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) (36K3=Qa  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Vk}49O<K/  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 3]LN;s]ac  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ,Og4 ?fS  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) <$E6oZ  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ZX.TqvK/r  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) BWq/TG=>  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) FY#!N L  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) )]Ti>RO7  
    %       -------------------------------------------------- =Hu0v}i/  
    % R>)MiHcCg  
    %   Example 1: R p.W,)i  
    % f_6`tq m%  
    %       % Display the Zernike function Z(n=5,m=1) ]]uHM}l  
    %       x = -1:0.01:1; [ygF0-3ND  
    %       [X,Y] = meshgrid(x,x); w2"]Pl  
    %       [theta,r] = cart2pol(X,Y); x:z0EYL  
    %       idx = r<=1; /iM$Tb5  
    %       z = nan(size(X)); <8o(CA\  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 1]OSWCEm*[  
    %       figure ;j} yB  
    %       pcolor(x,x,z), shading interp \,xFg w4  
    %       axis square, colorbar zCe/Kukvy  
    %       title('Zernike function Z_5^1(r,\theta)') }E&NPp>  
    % ^Udv]Wh  
    %   Example 2: +]!lS7nsW  
    % Ka-p& Uv1<  
    %       % Display the first 10 Zernike functions Vb4;-?s_  
    %       x = -1:0.01:1; )iLM]m   
    %       [X,Y] = meshgrid(x,x); 4\2V9F{s  
    %       [theta,r] = cart2pol(X,Y); dbF M,"^  
    %       idx = r<=1; _ Jc2&(;  
    %       z = nan(size(X)); vK$^y^  
    %       n = [0  1  1  2  2  2  3  3  3  3]; wD9a#AgEd  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; \C|cp|A*&  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; #Ob]]!y  
    %       y = zernfun(n,m,r(idx),theta(idx)); Wk7WK` >i  
    %       figure('Units','normalized') (Wj2?k/]  
    %       for k = 1:10 9K"JYJ q2  
    %           z(idx) = y(:,k); fC<m^%*zgA  
    %           subplot(4,7,Nplot(k)) v.g"{us  
    %           pcolor(x,x,z), shading interp X"*^l_9-v  
    %           set(gca,'XTick',[],'YTick',[]) F]=B'ZI  
    %           axis square z'MS#6|}  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) F:T GsV#  
    %       end #@//7Bf%  
    % t&RruwN_;  
    %   See also ZERNPOL, ZERNFUN2. $ |<m9CW  
    k onoI&kV|  
    .{so  
    %   Paul Fricker 11/13/2006 +!O- kd  
    's%q  
    %'ZN`XftG  
    VXKT\9g3A  
    8A2 z 5Aa  
    % Check and prepare the inputs: Ot9V< D6h  
    % ----------------------------- NGTe4Crx  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) AtHS@p  
        error('zernfun:NMvectors','N and M must be vectors.') 9){  
    end La,QB3K/  
    hnxc`VX>g  
    l5O=VqCj  
    if length(n)~=length(m) R}{GwbF_\  
        error('zernfun:NMlength','N and M must be the same length.') `a4 $lyZ  
    end +;gsRhWk  
    @.9I3E-=  
    ^ddO&!U  
    n = n(:); TSto9 $}*  
    m = m(:); lOerrP6f(  
    if any(mod(n-m,2)) Pl  
        error('zernfun:NMmultiplesof2', ... 8vD3=yK%^  
              'All N and M must differ by multiples of 2 (including 0).') ok0X<MR!I  
    end TQ'E5^  
    optBA3@e!  
    j\2[H^   
    if any(m>n) 32>x^>G=>  
        error('zernfun:MlessthanN', ... |E^|X!+9  
              'Each M must be less than or equal to its corresponding N.') 9([6d.`~  
    end P Jo  
    kC$I2[t!  
    Ft-6m%  
    if any( r>1 | r<0 ) C0m\SNR  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') BQNp$]5s  
    end 77aX-e*=E  
    1f//wk|  
    3% vis\~^  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) )%j"  
        error('zernfun:RTHvector','R and THETA must be vectors.') tOg=zXm   
    end YoSQN/Z  
    %z)EO9vtr  
    uMiyq<  
    r = r(:); a$}6:E  
    theta = theta(:); eyB_l.U7  
    length_r = length(r); nNR:cG fG  
    if length_r~=length(theta) )f*Iomp]@  
        error('zernfun:RTHlength', ... dY'Y5Th~  
              'The number of R- and THETA-values must be equal.') WU\m^!`w=F  
    end #7W.s!#}Dd  
    - 9&g[  
    pVG>A&4  
    % Check normalization: p24.bLr  
    % -------------------- O E|+R4M  
    if nargin==5 && ischar(nflag) O@,i1ha%  
        isnorm = strcmpi(nflag,'norm'); O),I[kb  
        if ~isnorm >q9{  
            error('zernfun:normalization','Unrecognized normalization flag.') JDhwN<0R  
        end Xb<)LHA~3  
    else ,nYZxYLf+  
        isnorm = false; [.3sE  
    end yq6LH   
    g :i*O^c @  
    ^ f{qJ[,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2"M_sL  
    % Compute the Zernike Polynomials :,YLx9i>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r@|ZlM@O  
    d*9j77C]  
    @:gl:mc  
    % Determine the required powers of r: 065A?KyD  
    % ----------------------------------- 9 z*(8d  
    m_abs = abs(m); <^sAY P|  
    rpowers = []; B;c=eMw  
    for j = 1:length(n) jt%WPkY:  
        rpowers = [rpowers m_abs(j):2:n(j)]; p JX, n  
    end Xz* tbW#  
    rpowers = unique(rpowers); |"\lL9CT  
    8b~7~VCk  
    Y3M','H([  
    % Pre-compute the values of r raised to the required powers, 2'dG7lLu4  
    % and compile them in a matrix: mxhW|}_-j  
    % ----------------------------- AeQC:  
    if rpowers(1)==0 /cY[at|p  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Te}IMi:  
        rpowern = cat(2,rpowern{:}); MM*-i=  
        rpowern = [ones(length_r,1) rpowern]; g TD%4V  
    else YiNo#M91  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); vGyppm[0  
        rpowern = cat(2,rpowern{:}); Tvrc%L(]  
    end c}\ d5R_L  
    %w@ig~vD'  
    ;]u1~  
    % Compute the values of the polynomials: L]NYYP-  
    % -------------------------------------- qL~Pjr>cF  
    y = zeros(length_r,length(n)); ?a8nz, zb  
    for j = 1:length(n)  qKx59  
        s = 0:(n(j)-m_abs(j))/2; !g/_ w  
        pows = n(j):-2:m_abs(j); !$XO U'n  
        for k = length(s):-1:1 GiFf0c 9  
            p = (1-2*mod(s(k),2))* ... dr>]+H=3E  
                       prod(2:(n(j)-s(k)))/              ... <H_LFrB$W  
                       prod(2:s(k))/                     ... EKJH_!%  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... C7T;;1P?  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); A1 b6Zt  
            idx = (pows(k)==rpowers); A7e_w 7?a  
            y(:,j) = y(:,j) + p*rpowern(:,idx); p+5#dbyr  
        end @OrXbG7&>#  
         BiI{8`M!$x  
        if isnorm &U8 54  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); -ca]Q|m8  
        end k0=|10bi  
    end eb(m8vLR  
    % END: Compute the Zernike Polynomials ap{{(y&R  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [)bz6\d[  
     bsD'\  
    s}<)B RZi  
    % Compute the Zernike functions: 0n7HkDo  
    % ------------------------------ c|3h|  
    idx_pos = m>0; 5auL<Pq   
    idx_neg = m<0; ?|gGsm+  
    $)Jc-V 6E  
    }.w#X   
    z = y; ^JiaR)#r  
    if any(idx_pos) EgCp:L{  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); mp muziH  
    end _TV2)  
    if any(idx_neg) pC55Ec<  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); m]  EDuW  
    end t>m8iS>  
    `W D*Q-&n  
    deHY8x5uI  
    % EOF zernfun o&*1U"6D  
     
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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)  G~mB=]  
    J6?_?XzToT  
    DDE还是手动输入的呢? 1 _:1/~R1  
    QKuc21  
    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