切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9279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, )m'_>-`^:  
    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, c:6w >:  
    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? ?hmb"^vlG  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ,9pi9\S  
    0K2[E^.WN  
    5&kR1Bp#-  
    <R(2 9QN  
    Tk!b`9  
    function z = zernfun(n,m,r,theta,nflag) ~ZNhU;%YW  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 4bBxZY  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N EsWszpRqb  
    %   and angular frequency M, evaluated at positions (R,THETA) on the CS{9|FNz  
    %   unit circle.  N is a vector of positive integers (including 0), and \l2 s^7G_  
    %   M is a vector with the same number of elements as N.  Each element [>:gwl _\  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) bn`zI~WS  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 4eEs_R  
    %   and THETA is a vector of angles.  R and THETA must have the same =_H39)|T  
    %   length.  The output Z is a matrix with one column for every (N,M) :mrGB3x{  
    %   pair, and one row for every (R,THETA) pair. kwL) &@  
    % &wuV}S 7  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike )q^vitkjup  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), q"Md)?5N  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral y|dXxd9  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, d<Os TA  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 8<kme"% s  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. '=H^m D+gl  
    % ?>y-5B[K/(  
    %   The Zernike functions are an orthogonal basis on the unit circle. uJ`&hX  
    %   They are used in disciplines such as astronomy, optics, and )1vojp 4Za  
    %   optometry to describe functions on a circular domain. ~ YK <T+  
    % #{M -3  
    %   The following table lists the first 15 Zernike functions. vVW=1(QWI#  
    % @5y(>>C}8%  
    %       n    m    Zernike function           Normalization z~ H Gc"~  
    %       -------------------------------------------------- Wj:QC<5 v  
    %       0    0    1                                 1 )^\='(s  
    %       1    1    r * cos(theta)                    2 tTt3D]h(  
    %       1   -1    r * sin(theta)                    2 3+-(;>>\  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) %5G BMMn  
    %       2    0    (2*r^2 - 1)                    sqrt(3) lnDDFsA  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) [&CM-` N  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Q 8rtZ  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Oi0;.< kX  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) +V@=G &Ou0  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) kYB <FwwB  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) /;rN/ot2o  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )DmiN^:  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) r?!xL\C\  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) m-89nOls  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ,}tdfkZFYl  
    %       -------------------------------------------------- tA-B3 ]  
    % 9oP{Al  
    %   Example 1: skz]@{38  
    % f~FehN7  
    %       % Display the Zernike function Z(n=5,m=1) (U_Q7hja?  
    %       x = -1:0.01:1; 'pY;]^M  
    %       [X,Y] = meshgrid(x,x); ~ ZL`E  
    %       [theta,r] = cart2pol(X,Y); Z0eBx  
    %       idx = r<=1; mi<D bnou  
    %       z = nan(size(X)); 9|?Lz  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); !B lk=L+p  
    %       figure wYA/<0'yH  
    %       pcolor(x,x,z), shading interp  |{)xC=  
    %       axis square, colorbar el?V2v[  
    %       title('Zernike function Z_5^1(r,\theta)') 5j,qAay9  
    % ,-3(^d\1F  
    %   Example 2: ;q; C ^l  
    % `4H9f&8(  
    %       % Display the first 10 Zernike functions A+*oT(`  
    %       x = -1:0.01:1; \83A|+k  
    %       [X,Y] = meshgrid(x,x); 8 tygs  
    %       [theta,r] = cart2pol(X,Y); 50ew/fZj|  
    %       idx = r<=1; %^=!s  
    %       z = nan(size(X));  - 1  
    %       n = [0  1  1  2  2  2  3  3  3  3]; <cv1$ x ~P  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; '$XHRS/q]  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Bh,)5E^m  
    %       y = zernfun(n,m,r(idx),theta(idx)); +MZO%4  
    %       figure('Units','normalized') /iy*3P,`  
    %       for k = 1:10 5^K#Tj ;2  
    %           z(idx) = y(:,k); ~H|LWCU)K8  
    %           subplot(4,7,Nplot(k)) loUwR z  
    %           pcolor(x,x,z), shading interp SP*JleQN  
    %           set(gca,'XTick',[],'YTick',[]) d?M!acB  
    %           axis square bmVgTm&  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) qH Ga  
    %       end ; Uqx&5P}  
    % 'e>sHL  
    %   See also ZERNPOL, ZERNFUN2. DRW.NL o  
    2c~?UK[1  
    s#4ew}  
    %   Paul Fricker 11/13/2006 !mxh]x<e  
    O;0<^M/0G  
    y)/$ge _U  
    ]jVSsSv  
    mvA xx`jc  
    % Check and prepare the inputs: bepYeT  
    % ----------------------------- QHzX 5$IM  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) k,R~oSA'n  
        error('zernfun:NMvectors','N and M must be vectors.') '<D`:srV  
    end to!W={S<ol  
    <,pLW~2-"  
    FPMSaN P  
    if length(n)~=length(m) $',GkK{NX  
        error('zernfun:NMlength','N and M must be the same length.') \+Rwm:lI  
    end Kt 90mA  
    R_Gq8t$  
    8<-oJs_o+  
    n = n(:); (L0 hS'  
    m = m(:); JXY!c\,  
    if any(mod(n-m,2)) a^XTW7]r  
        error('zernfun:NMmultiplesof2', ... ;WS7.  
              'All N and M must differ by multiples of 2 (including 0).') \ ~LU 'j  
    end 5'kTe=  
    @b!R2Yq  
    3 2 1={\X  
    if any(m>n) I4<{R  
        error('zernfun:MlessthanN', ... HcBH!0  
              'Each M must be less than or equal to its corresponding N.') {{]=zt|69  
    end @x=BJuUuX  
    PF'5z#] NP  
    `F2*o47|t  
    if any( r>1 | r<0 ) f1Yv hvWL  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') siRnH(^ J  
    end h#@4@x{  
    Ie2w0Cs28  
    gl9pgY1ni  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) I^M#[xA  
        error('zernfun:RTHvector','R and THETA must be vectors.') 11B{gUv.]  
    end {wp Mg  
    V8nz-DL{  
    6t_ 3%{  
    r = r(:); !k:zLjtp  
    theta = theta(:); T^'*_*m  
    length_r = length(r); %89" A'g  
    if length_r~=length(theta) {V%%^Zhwy  
        error('zernfun:RTHlength', ... 8L7Y A)u  
              'The number of R- and THETA-values must be equal.') EFRZ% Y  
    end 0r0\b*r  
    Lz9$,Y[  
    )l!J$X+R  
    % Check normalization: hB P$9GR  
    % -------------------- )4!CR/ao  
    if nargin==5 && ischar(nflag) rysP)e  
        isnorm = strcmpi(nflag,'norm'); + 9\:$wMN  
        if ~isnorm NoJnchiU  
            error('zernfun:normalization','Unrecognized normalization flag.') +H[}T ]  
        end Ok}{jwJ%W;  
    else FI?gT  
        isnorm = false; >J^7}J  
    end NIGB[2V(  
    V6@*\+:3)  
    J|gdO+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yN9$gfJC^  
    % Compute the Zernike Polynomials -uv 9(r\P  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M.xhVgFf)  
    OX)#F'Sl}  
    7m|`tjQ1  
    % Determine the required powers of r: L,C? gd@"  
    % ----------------------------------- Tn4W\?R  
    m_abs = abs(m); !paN`Fz\a  
    rpowers = []; m4P hn~>Gg  
    for j = 1:length(n) 6\,DnO   
        rpowers = [rpowers m_abs(j):2:n(j)]; ,zAK3d&hj  
    end }zkL[qu;  
    rpowers = unique(rpowers); >W`S(a Mn  
    CN\|_y  
    C2+{U  
    % Pre-compute the values of r raised to the required powers, mP9cBLz  
    % and compile them in a matrix: 22)0zY%\  
    % ----------------------------- Jh37pI  
    if rpowers(1)==0 b$d J?%W  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); E fP>O  
        rpowern = cat(2,rpowern{:}); 3)6+1Yc  
        rpowern = [ones(length_r,1) rpowern]; uSABh ^  
    else B;xZ% M]  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 0V<Aub[${  
        rpowern = cat(2,rpowern{:}); $Cz1C  
    end c=b+g+*xd  
    rnnX|}J  
    [NHg&R H  
    % Compute the values of the polynomials: |. J,8~x  
    % -------------------------------------- - *:p.(c  
    y = zeros(length_r,length(n)); SX1X< 9  
    for j = 1:length(n) t2=a(N-/,  
        s = 0:(n(j)-m_abs(j))/2; 8O~0RYk  
        pows = n(j):-2:m_abs(j); gW%pM{PW  
        for k = length(s):-1:1 TA Ftcs:  
            p = (1-2*mod(s(k),2))* ... {V}t'x`4c  
                       prod(2:(n(j)-s(k)))/              ... &?SX4c~?u  
                       prod(2:s(k))/                     ... KKLR'w,A>  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... /Jh1rck  
                       prod(2:((n(j)+m_abs(j))/2-s(k)));  7]p>XAb  
            idx = (pows(k)==rpowers); <e;jW K  
            y(:,j) = y(:,j) + p*rpowern(:,idx); EfFz7j&X  
        end Gx.P ]O3  
         {I4%   
        if isnorm v2Dt3$@H6  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 4cott^K.  
        end +J"'  'cZ  
    end [dl+:P:zc  
    % END: Compute the Zernike Polynomials Xl#Dw bx  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M0RRmW@f.a  
    ]kRI}Om2  
    bRWIDPh  
    % Compute the Zernike functions: 3Bejp+xX  
    % ------------------------------ fZ[kh{|  
    idx_pos = m>0; Vd,'  s  
    idx_neg = m<0; py]KTRzy  
    gh TcB  
    [-4KY4R  
    z = y; -M6L.gi)oJ  
    if any(idx_pos) wAw42{M  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 8s<^]sFP  
    end A'Gl Cp  
    if any(idx_neg) 92ZWU2"  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); w'A tf  
    end :d.1;st  
    B1E$v(P3M  
    N*Yy&[  
    % EOF zernfun O]t\B *%}  
     
    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)  hw=GR_,  
    dv!r.  
    DDE还是手动输入的呢? ~3CVxbB^<  
    G1d(,4Xp  
    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