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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, Nd'+s>d0  
    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, !`"@!  
    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? Wew'bj  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 7ZarXv z  
    QH@?.Kb_qU  
    f1w&D ]|S+  
    Zz}Wg@&  
    Bd jo3eX  
    function z = zernfun(n,m,r,theta,nflag) BOClMeA4  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 1Z%^U ?  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N d/5i4g[q  
    %   and angular frequency M, evaluated at positions (R,THETA) on the z +,l"#Vv  
    %   unit circle.  N is a vector of positive integers (including 0), and 12qX[39/  
    %   M is a vector with the same number of elements as N.  Each element Gx /sJ(  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Z^ynw8k"  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, uJ<n W%}  
    %   and THETA is a vector of angles.  R and THETA must have the same pxP,cS  
    %   length.  The output Z is a matrix with one column for every (N,M) hr3RC+ y  
    %   pair, and one row for every (R,THETA) pair. f'&30lF  
    % (3a]#`Q  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike u`?MV2jU2  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), nAIV]9RAZ%  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral le60b@2G0  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, M"# >?6{  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized {=mf/3.r  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?.Mw  
    % s |B  
    %   The Zernike functions are an orthogonal basis on the unit circle. 7i^7sT8t  
    %   They are used in disciplines such as astronomy, optics, and Ua0fs|t1v  
    %   optometry to describe functions on a circular domain. JL gk?  
    %  Age  
    %   The following table lists the first 15 Zernike functions. $>Md]/I8  
    % r9nH6 Md\  
    %       n    m    Zernike function           Normalization *nJy  
    %       -------------------------------------------------- V&nTf100  
    %       0    0    1                                 1 z H$^.1  
    %       1    1    r * cos(theta)                    2 Mj&`Y gW5a  
    %       1   -1    r * sin(theta)                    2 "<['W(  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) BkywYCWZ )  
    %       2    0    (2*r^2 - 1)                    sqrt(3) S:!gj2q9|  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ;Z>u]uK4+  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) r\nKJdh;ka  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) NqyKR&;  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ueI1O/Mi  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) MI8f(ZJK5  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) +9mE1$C  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =AEl:SY+  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) t6-He~  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) <X@XbM  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 7G6XK   
    %       -------------------------------------------------- WRa1VU&f  
    % uWm,mGd9  
    %   Example 1: yTt,/+I%gJ  
    % <zd_-Ysn  
    %       % Display the Zernike function Z(n=5,m=1) <X>lA  
    %       x = -1:0.01:1; ~)J]`el,Q  
    %       [X,Y] = meshgrid(x,x); "rxhS; R1>  
    %       [theta,r] = cart2pol(X,Y); H}v.0R  
    %       idx = r<=1; )v\zaz  
    %       z = nan(size(X)); &n6'r^[D  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Ek'~i  
    %       figure f@JMDJ  
    %       pcolor(x,x,z), shading interp A'A5.\UN  
    %       axis square, colorbar b!(ew`Y;  
    %       title('Zernike function Z_5^1(r,\theta)') BY*{j&^  
    % Oz8"s4Y7  
    %   Example 2: Vo1,{"k  
    % z= \y)'b  
    %       % Display the first 10 Zernike functions #fB&Hv #s7  
    %       x = -1:0.01:1; ;/-v4  
    %       [X,Y] = meshgrid(x,x); I^}q;L![\  
    %       [theta,r] = cart2pol(X,Y); ~H<oqk:O-  
    %       idx = r<=1; 0)<\jo1 F  
    %       z = nan(size(X)); d,%e? 8x5  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ^a>3U l{  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ?E>(zV1D/  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; \!-IY  
    %       y = zernfun(n,m,r(idx),theta(idx)); 5hxG\f#}?  
    %       figure('Units','normalized') o )\\(^ld  
    %       for k = 1:10 \\ZR~f!<  
    %           z(idx) = y(:,k); g5",jTn#  
    %           subplot(4,7,Nplot(k)) =2Vs))>Y  
    %           pcolor(x,x,z), shading interp 6YErF|  
    %           set(gca,'XTick',[],'YTick',[]) $] ])FM"b  
    %           axis square pJg'$iR!/  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 5Z8Zb.  
    %       end F!k3/z  
    % bQ%6z}r  
    %   See also ZERNPOL, ZERNFUN2. c<k=8P   
    #|92 +  
    ~wejy3|@0  
    %   Paul Fricker 11/13/2006 cWp5' e]A  
    dM-qd`  
    d+caGpaR  
    u"$=:GK  
    i}tBB~]  
    % Check and prepare the inputs: \C{Dui) F  
    % ----------------------------- k<&zVV '  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) yr;~M{{4  
        error('zernfun:NMvectors','N and M must be vectors.') z_i (o  
    end D,3Kx ^  
    %>];F~z  
    `7D]J*?`  
    if length(n)~=length(m) cVV@MC  
        error('zernfun:NMlength','N and M must be the same length.') a- \M)}T  
    end z`Jcpt  
    m{v*\e7 P  
    g)3HVAT  
    n = n(:); 9V'ok.B.x  
    m = m(:); p&s~O,Bw$  
    if any(mod(n-m,2)) |>Ld'\i8  
        error('zernfun:NMmultiplesof2', ... B5A/Iv)2  
              'All N and M must differ by multiples of 2 (including 0).') ;c/|LXc\  
    end {+3 `{34e  
    ~|:U"w\[=  
    0I v(ioB=  
    if any(m>n) a<NZC  
        error('zernfun:MlessthanN', ... "  jBc5*  
              'Each M must be less than or equal to its corresponding N.') &g.do?  
    end |#b]e|aP  
    cj64.C  
    ?5IF;vk  
    if any( r>1 | r<0 ) >fq]c  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') \PzJ66DL!  
    end '5)PYjMnH  
    )K}-z+$)k  
    X7~^D[ X  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) XsEo tW  
        error('zernfun:RTHvector','R and THETA must be vectors.') [yhK4A  
    end FUO9jX  
    j&N {j_ M  
    d:vuRK4+  
    r = r(:); c:[8ng 2v  
    theta = theta(:); #FhgKwx  
    length_r = length(r); "- ?uB Mz  
    if length_r~=length(theta) p9y@5z  
        error('zernfun:RTHlength', ... /prR;'ks  
              'The number of R- and THETA-values must be equal.') j[RY  
    end &}rmDx  
    1a]P+-@u[  
    &v/>P1Z G  
    % Check normalization: e~ZxDAd  
    % -------------------- )z_5I (?&  
    if nargin==5 && ischar(nflag) 3 ,f3^A  
        isnorm = strcmpi(nflag,'norm'); |V&E q>G  
        if ~isnorm b[2 #t  
            error('zernfun:normalization','Unrecognized normalization flag.') | 9 <+!t\  
        end *}'3|e4w}  
    else b{Bef*`/  
        isnorm = false; so>jz@!EE  
    end xFzaVjjP  
    20 Z/Y\  
     u*m|o8  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =x@v{cP  
    % Compute the Zernike Polynomials 4J{W8jX  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =.]{OT  
    IcA]B?+  
    3De(:c)@  
    % Determine the required powers of r: '!"rE1e  
    % ----------------------------------- %D49A-R  
    m_abs = abs(m); ~='}(Fg:  
    rpowers = []; 9]^q!~u  
    for j = 1:length(n) F|&%Z(@a  
        rpowers = [rpowers m_abs(j):2:n(j)]; GD1L6kVd1  
    end (XNd]G  
    rpowers = unique(rpowers); B.4Or]  
    o&)v{q  
    aQj"FUL  
    % Pre-compute the values of r raised to the required powers, j 6dlAe  
    % and compile them in a matrix: +62}//_?  
    % ----------------------------- +,zV [\  
    if rpowers(1)==0 Rjn%<R2nW  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); F*J bTEOn  
        rpowern = cat(2,rpowern{:}); ~^J9v+  
        rpowern = [ones(length_r,1) rpowern]; N *,[(q  
    else jG%J.u^k  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); X2mZ~RB(p  
        rpowern = cat(2,rpowern{:}); ZfibHivz  
    end XG!^[ZDs  
    +fN2%aC  
    ge]Z5E(1  
    % Compute the values of the polynomials: -HvJ&O.V$  
    % -------------------------------------- K?u:-QX^  
    y = zeros(length_r,length(n)); >?jmeD3u  
    for j = 1:length(n) iSNbbu#  
        s = 0:(n(j)-m_abs(j))/2; r-_-/O"l  
        pows = n(j):-2:m_abs(j); @o6!  
        for k = length(s):-1:1 Flaqgi/j  
            p = (1-2*mod(s(k),2))* ... qu0 q LM  
                       prod(2:(n(j)-s(k)))/              ... 3$3%W<&^  
                       prod(2:s(k))/                     ... Xdh@ ^`  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... mGo NT  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); blUS6"kV}  
            idx = (pows(k)==rpowers); F$S/zh$)0  
            y(:,j) = y(:,j) + p*rpowern(:,idx); nK`H;k  
        end t!59upbN}3  
         d*$x|B|V  
        if isnorm `_x#`%!#2  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); b_)SMAsO7  
        end I:WPP'L4o  
    end lNMJcl3  
    % END: Compute the Zernike Polynomials 0x # V   
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Gh}* <X;N  
    TA+#{q+a  
    !1mAq+q!  
    % Compute the Zernike functions: iV:\,<8d  
    % ------------------------------ y\:,.cZ+TQ  
    idx_pos = m>0; .uB[zJc  
    idx_neg = m<0; ]dT]25V  
    y!x-R !3  
    Hp@cBj_@P2  
    z = y; Ch]q:o4  
    if any(idx_pos) Mo]iVj8~  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); +&* >FeJY  
    end ppu<k N  
    if any(idx_neg) mhF@S@  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); O]l-4X#8F  
    end ~Fo`Pr_  
    '.e 5Ku  
    PPh1y;D  
    % EOF zernfun Ok phbAX  
     
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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)  IN]bAd8"  
    ,@ Cru=  
    DDE还是手动输入的呢? >Y< y]vM:  
    2=NYBOE  
    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