切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9157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, C<XDQ>?  
    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, ^mQfXfuL  
    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? qw1J{xoHW  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 2s%M,Nb  
    !*6z=:J  
    =:eE!  
    f*Js= hvO  
    Al}PJz\  
    function z = zernfun(n,m,r,theta,nflag) /|AuI qW  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. IIiN1 Lu,5  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N F-0PmO~3+W  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 5V!XD9P'  
    %   unit circle.  N is a vector of positive integers (including 0), and _xt(II   
    %   M is a vector with the same number of elements as N.  Each element x$DJ  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Uiw7Y\Im|  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, MX,0gap  
    %   and THETA is a vector of angles.  R and THETA must have the same b%j:-^0V  
    %   length.  The output Z is a matrix with one column for every (N,M) BxYA[#fd}  
    %   pair, and one row for every (R,THETA) pair. V}+;b bUc-  
    % krc!BK`V  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Ypj)6d  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), mC(t;{  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral b0 `9wn  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, |Eu~= J7@  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 3 ?~+5DU  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. _1Gut"!{\  
    % "\?G  
    %   The Zernike functions are an orthogonal basis on the unit circle. *wcoDQ b;  
    %   They are used in disciplines such as astronomy, optics, and ,>v9 Y#U  
    %   optometry to describe functions on a circular domain. v*'\w#  
    % ,5*xE\9G  
    %   The following table lists the first 15 Zernike functions. :exuTn  
    % E,yK` mPp^  
    %       n    m    Zernike function           Normalization (OQ @!R&  
    %       -------------------------------------------------- q.{/{9  
    %       0    0    1                                 1 #)}bUNc'  
    %       1    1    r * cos(theta)                    2 m]q!y3  
    %       1   -1    r * sin(theta)                    2 2tm-:CPG  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) \zL7 j 4  
    %       2    0    (2*r^2 - 1)                    sqrt(3) I.1l  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) v=-3 ,C  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ,s&~U<Z  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Uy|=A7Ad c  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) -wMW@:M_  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 2!?z%s-S  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) #2ASzCe  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) [qMdOY%jx  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ER1mA:8>E  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) [;YBX] t  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) BM~niW;k  
    %       -------------------------------------------------- pu*u[n  
    % kA=~ 8N  
    %   Example 1: E?U]w0g  
    % 0.+eF }'H  
    %       % Display the Zernike function Z(n=5,m=1) fO!O" D5  
    %       x = -1:0.01:1; ]GKx[F{)  
    %       [X,Y] = meshgrid(x,x); q%Jy>IXt  
    %       [theta,r] = cart2pol(X,Y); 4,ynt&  
    %       idx = r<=1; Al=? j#J6p  
    %       z = nan(size(X)); |ZlT>u  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); YKOO(?lv  
    %       figure ?$4R <  
    %       pcolor(x,x,z), shading interp .|`=mx  
    %       axis square, colorbar (ul-J4E\O  
    %       title('Zernike function Z_5^1(r,\theta)') qpqz. {\  
    % 9Ru%E>el-  
    %   Example 2: 8'WMspX  
    % q)xl$*g  
    %       % Display the first 10 Zernike functions ;Jn0e:x`E  
    %       x = -1:0.01:1; ^|i\d \  
    %       [X,Y] = meshgrid(x,x); mX.3R+t  
    %       [theta,r] = cart2pol(X,Y); E816 YS='  
    %       idx = r<=1; yXo0z_ G  
    %       z = nan(size(X)); G_N-}J>EP  
    %       n = [0  1  1  2  2  2  3  3  3  3]; yx w27~  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; $"{3yLg  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; B~g05`s  
    %       y = zernfun(n,m,r(idx),theta(idx)); #Y>%Dr&  
    %       figure('Units','normalized') wW! r}I#  
    %       for k = 1:10 &W<>^C2v  
    %           z(idx) = y(:,k); j*~dFGl)  
    %           subplot(4,7,Nplot(k)) 6aZt4Lw2\  
    %           pcolor(x,x,z), shading interp n!eqzr{  
    %           set(gca,'XTick',[],'YTick',[]) <*Kh=v  
    %           axis square 'BdmFKy1  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) eGe[sv"k  
    %       end M:UB>-`bW  
    % 2*q: ^  
    %   See also ZERNPOL, ZERNFUN2. V*7Z,nA  
    G1;'nwf}  
    Xm=^\K3  
    %   Paul Fricker 11/13/2006 nB@iQxcz  
    nHA`B.:B  
    j_'rhEdLP  
    h$7Fe +#I#  
    H"q`k5R  
    % Check and prepare the inputs: hp]ng!I{\u  
    % ----------------------------- {.3  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) =Q8H]F  
        error('zernfun:NMvectors','N and M must be vectors.') `\F%l?aY  
    end '0_j{ig  
    Op/79 ]$  
    f{^M.G@  
    if length(n)~=length(m) L_lDFF  
        error('zernfun:NMlength','N and M must be the same length.') <[y$D=n  
    end 0fPHh>u  
    /#qs(! d  
    Rw/JPC"  
    n = n(:); _L4<^Etfm  
    m = m(:); jq("D,  
    if any(mod(n-m,2)) FSU%?PxO  
        error('zernfun:NMmultiplesof2', ... )}Rfa}MD  
              'All N and M must differ by multiples of 2 (including 0).') P7wqZ?  
    end wsJ%* eYf  
    /y9J)lx  
    I)XOAf$6  
    if any(m>n) EAD0<I<>  
        error('zernfun:MlessthanN', ... .mT#%ex  
              'Each M must be less than or equal to its corresponding N.') G_^iR-  
    end 9o`7Kc/g  
    s !hI:$J.  
    ]/o12pI  
    if any( r>1 | r<0 ) x!C8?K =|  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 2B9 i R  
    end RrO0uadmn  
    '6o`^u>  
    1qLl^DW  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) i+)}aA  
        error('zernfun:RTHvector','R and THETA must be vectors.') [*9YIjn  
    end !]rETP_  
    :>P4L,Da]  
    U R1JbyT  
    r = r(:); S$jV|xK B  
    theta = theta(:); r:c@17  
    length_r = length(r); *^@#X-NG  
    if length_r~=length(theta) 2JiAd*WK  
        error('zernfun:RTHlength', ... <'}b*wUB  
              'The number of R- and THETA-values must be equal.') n^iNo  
    end :Su#xI  
    <?LfOSdMs^  
    *2,e=tY>  
    % Check normalization: #+K Kvk  
    % -------------------- &2io^A P  
    if nargin==5 && ischar(nflag) >bfYy=/  
        isnorm = strcmpi(nflag,'norm'); ([,vX"4  
        if ~isnorm OU,PO2xX9  
            error('zernfun:normalization','Unrecognized normalization flag.') n5Nan  
        end b^[W_y  
    else ~K~b`|1  
        isnorm = false; 'yPCZ`5H(  
    end m]FaEQVoE  
    N5 SLF4R1  
    E0"10Qbi  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j+DE|Q&]I  
    % Compute the Zernike Polynomials cOSxg=~>u  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RzA2*]%a  
    4M @ oj  
    $!YKZ0)B'0  
    % Determine the required powers of r: -{X<*P4p  
    % ----------------------------------- \{c,,th  
    m_abs = abs(m); iNod</+"K  
    rpowers = []; nu&_gF,{  
    for j = 1:length(n) }P<Qz^sr_  
        rpowers = [rpowers m_abs(j):2:n(j)]; f._l105.  
    end RAIVdQ}.Z  
    rpowers = unique(rpowers); L`9TB"0R+  
    <y@,3DD3A9  
    ]\ CU9J|H8  
    % Pre-compute the values of r raised to the required powers, , CJAzGBS  
    % and compile them in a matrix: YfE>Pn'r  
    % ----------------------------- -DTB6}kw  
    if rpowers(1)==0 XR*Q|4  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); tHrK~|  
        rpowern = cat(2,rpowern{:}); ic%?uWN  
        rpowern = [ones(length_r,1) rpowern]; d"#gO,H0  
    else Ua):y) A  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); j?EskT6  
        rpowern = cat(2,rpowern{:}); .z=U= _e  
    end 3gb|x?  
    U't E^W  
    )O,wRd>5  
    % Compute the values of the polynomials: 3`8dii  
    % -------------------------------------- >qR7'QwP  
    y = zeros(length_r,length(n)); 8g\wVKkTQp  
    for j = 1:length(n) OnZF6yfN=3  
        s = 0:(n(j)-m_abs(j))/2; nD7|8,'  
        pows = n(j):-2:m_abs(j); a%Uw;6|{  
        for k = length(s):-1:1  )|v^9  
            p = (1-2*mod(s(k),2))* ... &!ED# gs  
                       prod(2:(n(j)-s(k)))/              ... HbcOTd)=5  
                       prod(2:s(k))/                     ... !7}IqSs  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... o4$Ott%Wm  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); \[:PykS  
            idx = (pows(k)==rpowers);  s[3e=N  
            y(:,j) = y(:,j) + p*rpowern(:,idx); led))qd@V-  
        end 2ck 4C/ h  
         4|`Yz%'  
        if isnorm i=YXKe6fD  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); YRPm^kW  
        end MWiMUTZg3  
    end /D]Kkm)  
    % END: Compute the Zernike Polynomials / /'Tck  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {9L5Q  
    yQ9ZhdQS  
    rah,dVE]  
    % Compute the Zernike functions: :M06 ;:e  
    % ------------------------------ %m9CdWb=w  
    idx_pos = m>0; l71 gf.4g  
    idx_neg = m<0; z"lqrSJ:  
    *l{yW"Su  
    Guh%eR'Wt  
    z = y; "< v\M85&  
    if any(idx_pos) 'Y.Vn P&H  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Mi ; glm  
    end b/t  
    if any(idx_neg) ({4]  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); !22yvT.;[  
    end 6xoq;=o  
    %JtbRs(~q  
    VU|;:  
    % EOF zernfun 4[TR0bM%  
     
    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)
    在线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)  }MbH3ufC  
    v%{.A)  
    DDE还是手动输入的呢? TXXy\$  
    6 sxffJt  
    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