切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9247阅读
    • 5回复

    [讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

    上一主题 下一主题
    离线jssylttc
     
    发帖
    25
    光币
    13
    光券
    0
    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, |zq4*  5  
    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, ahA{B1M)n  
    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? |QVr `tE<  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? QBoFpxh=  
    <FUon  
    iU5P$7.p  
    }taLk@T  
    ocF>LR%P  
    function z = zernfun(n,m,r,theta,nflag) IU|kNBo  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. O~27/  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N G}VDEC  
    %   and angular frequency M, evaluated at positions (R,THETA) on the  `?|Rc  
    %   unit circle.  N is a vector of positive integers (including 0), and :\b|dvI<  
    %   M is a vector with the same number of elements as N.  Each element ~^&R#4J  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) $Jp~\_X  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, &~ *.CQa  
    %   and THETA is a vector of angles.  R and THETA must have the same .k@^KY  
    %   length.  The output Z is a matrix with one column for every (N,M) ~-_i  
    %   pair, and one row for every (R,THETA) pair. =q+R   
    % BFWi(58q  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike wiJRCH  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Vr/Bu4V"  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral _({@B`N}  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ZQAO"huk]  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized R_1qn  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. H_w%'v&  
    % <~{du ?4n  
    %   The Zernike functions are an orthogonal basis on the unit circle. SO;N~D1Z6  
    %   They are used in disciplines such as astronomy, optics, and #6=MKpR  
    %   optometry to describe functions on a circular domain. NQX>Qh 2  
    % Kb&V!#o)  
    %   The following table lists the first 15 Zernike functions. <sX VW  
    % j13DJ.xu  
    %       n    m    Zernike function           Normalization !`&\Lx_  
    %       --------------------------------------------------  l{$[}<  
    %       0    0    1                                 1 f+x ;:  
    %       1    1    r * cos(theta)                    2 mnjs(x<m  
    %       1   -1    r * sin(theta)                    2 sN~\+_  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) PcC/_+2  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Vr=OYI'A  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) J;}3t!  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) j*400  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Qz,|mo+  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) m%QSapV  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) }D*yr3b  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) >&U @f  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) n2f6 p<8A  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) gL3iw!7  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9b"MQ[B4#a  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) pKT2^Q}-h  
    %       -------------------------------------------------- w0w1PE-V=  
    % bgF^(T35  
    %   Example 1: +G*JrwJ&=  
    % WsI>n  
    %       % Display the Zernike function Z(n=5,m=1) Ez+Z[*C  
    %       x = -1:0.01:1; .Z\Q4x#!Z  
    %       [X,Y] = meshgrid(x,x); .cDOl_z<:G  
    %       [theta,r] = cart2pol(X,Y); Xg7|JS!  
    %       idx = r<=1; 0uvzxmN  
    %       z = nan(size(X)); +=BAslk  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); "OmD@ EMT  
    %       figure $ s-Y%gc  
    %       pcolor(x,x,z), shading interp `~# < &w  
    %       axis square, colorbar <a( }kk}  
    %       title('Zernike function Z_5^1(r,\theta)') S($Su7g%_  
    % J2VTo: In  
    %   Example 2: A+getdr  
    % F;q#&  
    %       % Display the first 10 Zernike functions lg$zGa?  
    %       x = -1:0.01:1; % 0T+t.  
    %       [X,Y] = meshgrid(x,x); o!c] (  
    %       [theta,r] = cart2pol(X,Y); i-" p)2d=#  
    %       idx = r<=1; !w39FfU{  
    %       z = nan(size(X)); YA:nOvd@O  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ~" i0x  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; r(h`XMsU  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 9?<{_'  
    %       y = zernfun(n,m,r(idx),theta(idx)); L| hx arJ  
    %       figure('Units','normalized') bBc[bc>R  
    %       for k = 1:10 `aC){&AP(  
    %           z(idx) = y(:,k); 5PT5#[  
    %           subplot(4,7,Nplot(k)) T>`74B:  
    %           pcolor(x,x,z), shading interp vBcq_sbo  
    %           set(gca,'XTick',[],'YTick',[]) 2c5-)Dt)T  
    %           axis square _hu")os  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) jszK7$]^  
    %       end ?9{~> 4@  
    % @p6<Lw_E  
    %   See also ZERNPOL, ZERNFUN2. Z?5V4F:f  
    'o_:^'c  
    V[o`\|<  
    %   Paul Fricker 11/13/2006 n+;6=1d7ZW  
    QX4ai3v  
    7&qy5 y-Ap  
    /U&Opo {aO  
    Uu 8,@W+  
    % Check and prepare the inputs: `-h8vj5uG  
    % ----------------------------- hrGM|_BE  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) c2t=_aAIPQ  
        error('zernfun:NMvectors','N and M must be vectors.') pi<TFe@eG  
    end q@t0NvNSu  
    H,nec<Jp  
    hCjR&ZA  
    if length(n)~=length(m) i.D3'l  
        error('zernfun:NMlength','N and M must be the same length.') nw>8GivO  
    end npJt3 Y_I  
    J &pO%Q=b  
    SCvVt  
    n = n(:); E;JsBH  
    m = m(:); }J">}j]/  
    if any(mod(n-m,2)) p2Zo  
        error('zernfun:NMmultiplesof2', ... n!p<A.O7@  
              'All N and M must differ by multiples of 2 (including 0).') VCXJwVb  
    end .A sv%p[W  
    S}p4iE"n  
    MjlP+; !  
    if any(m>n) #]nx!*JNZ  
        error('zernfun:MlessthanN', ... i;LXu%3\  
              'Each M must be less than or equal to its corresponding N.') OQW#a[=WQ  
    end 1N7Kv4,  
    1$M@]7e+!+  
    n2)q}_d  
    if any( r>1 | r<0 ) X.hm s?]  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') + s- lCz  
    end Tb3J9q+ya  
    kY*rb_2j  
    &?mD$Eo  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Zt.'K(]2h  
        error('zernfun:RTHvector','R and THETA must be vectors.') DxUKUE  
    end QUu}Xg:  
    "@?? Fw!  
    {}3${  
    r = r(:);  =-IbS}3  
    theta = theta(:); C(00<~JC  
    length_r = length(r); e,t(q(L  
    if length_r~=length(theta) uc;8 K,[t  
        error('zernfun:RTHlength', ... +=O5YR!{  
              'The number of R- and THETA-values must be equal.') MyT q  
    end 8 7D*-Gw  
    bbrXgQ`s+w  
    0q&<bV:D  
    % Check normalization: b )B? F  
    % --------------------  o4|M0  
    if nargin==5 && ischar(nflag) R8ZK]5{o  
        isnorm = strcmpi(nflag,'norm'); ;kY(<{2  
        if ~isnorm Ney/[3 A  
            error('zernfun:normalization','Unrecognized normalization flag.') j'A_'g'^  
        end mV3cp rRqv  
    else S:h{2{  
        isnorm = false; ILGMMA_2  
    end 9I&xfvD,  
    ;M)QwF1  
    9I}-[|`u  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,P;Pm68V  
    % Compute the Zernike Polynomials Tj:B!>>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% D)L+7N0D~  
    U4d:] z  
    Qk:Y2mL  
    % Determine the required powers of r: XD.)Dl8  
    % ----------------------------------- e 9;~P}  
    m_abs = abs(m); gt@m?w(  
    rpowers = []; uG,5BV.M  
    for j = 1:length(n) f|\onHI)>  
        rpowers = [rpowers m_abs(j):2:n(j)]; f&Gt|  
    end KrQ1GepJ  
    rpowers = unique(rpowers); E=nIRG|g  
    bbE!qk;hEP  
    '@_d(N1jTw  
    % Pre-compute the values of r raised to the required powers, 4 o Fel.o  
    % and compile them in a matrix: aDU<wxnSvO  
    % ----------------------------- E|iQc8gr&  
    if rpowers(1)==0 qm/)ku0  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); N sXHO  
        rpowern = cat(2,rpowern{:}); Q+[n91ey**  
        rpowern = [ones(length_r,1) rpowern]; 4K\G16'$v  
    else ~E17L]ete  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); -XB/lnG  
        rpowern = cat(2,rpowern{:}); fdFo#P  
    end ]'&LGA`  
    ;ub;l h3  
    Z?h~{Mg  
    % Compute the values of the polynomials: Q'=x|K#xj  
    % -------------------------------------- b,7k)ND1F  
    y = zeros(length_r,length(n)); c2l@6<Ww  
    for j = 1:length(n) |fK1/<sz#  
        s = 0:(n(j)-m_abs(j))/2; ,Lr. 9I.  
        pows = n(j):-2:m_abs(j); NPy&OcRl  
        for k = length(s):-1:1 v[1aW v:  
            p = (1-2*mod(s(k),2))* ... H\ F :95  
                       prod(2:(n(j)-s(k)))/              ... Cd#(X@n  
                       prod(2:s(k))/                     ... wW>A_{Y  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... J')o|5S1N  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); !fE`4<|?  
            idx = (pows(k)==rpowers); akp-zn&je  
            y(:,j) = y(:,j) + p*rpowern(:,idx); o#3ly-ht  
        end >mwlsL~X  
         0"<H;7K#W  
        if isnorm 'DP1,7  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); $V -~Bu-  
        end wr$("A(  
    end M\uiq38  
    % END: Compute the Zernike Polynomials 6]K_m(F  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'j#*6xD  
    dqU~`b9  
    "g5^_UP  
    % Compute the Zernike functions: 9+Np4i@  
    % ------------------------------ |jGf<Bf5  
    idx_pos = m>0; -_=nDH  
    idx_neg = m<0; f,U.7E  
    !|S(Ms  
    }bb;~  
    z = y; L +b6!2O,  
    if any(idx_pos) (S>C#A=E\  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); G/)O@Ugp  
    end WlOmJtt4)  
    if any(idx_neg) B-*+r`@Bd  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); R`NYEptJ  
    end 3Z>Ux3[  
    YL!P0o13r  
    (nQ^  
    % EOF zernfun xG~P+n7t5$  
     
    分享到
    离线phoenixzqy
    发帖
    4352
    光币
    5478
    光券
    1
    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
    发帖
    959
    光币
    1087
    光券
    1
    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
    发帖
    25
    光币
    13
    光券
    0
    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)  @NR>{Eg  
    7g^]:3f!   
    DDE还是手动输入的呢? !aUs>1i  
    &$+AXzn  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    离线jssylttc
    发帖
    25
    光币
    13
    光券
    0
    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    离线18257342135
    发帖
    51
    光币
    1518
    光券
    0
    只看该作者 5楼 发表于: 2016-12-13
    支持一下,慢慢研究