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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, ^GrkIh0nL  
    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, JM-ce8U  
    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? @ZkAul0@  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? -iDEh_pts  
    n *i'vtQ8  
    T$^>Fiz{Se  
    X ' #$e{  
    -j`!(IJ  
    function z = zernfun(n,m,r,theta,nflag) Ls>u` hG  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. blfE9Oy  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N QPe9s[Y  
    %   and angular frequency M, evaluated at positions (R,THETA) on the mo#0q&ZQ  
    %   unit circle.  N is a vector of positive integers (including 0), and 8gbm"!  
    %   M is a vector with the same number of elements as N.  Each element \l]pe|0EW  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) +dgo-)kP(_  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, /iTUex7T  
    %   and THETA is a vector of angles.  R and THETA must have the same @nx}6?p\,  
    %   length.  The output Z is a matrix with one column for every (N,M) 8PoHBOxpc  
    %   pair, and one row for every (R,THETA) pair. K Z!N{.Jk  
    % ;o)=XEh8P  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike U+*oI*  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), HZDaV&)@  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral }(+=/$C"#  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, =(.mf  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ;c X^8;F0  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. d{2 y/  
    % YBtq0c  
    %   The Zernike functions are an orthogonal basis on the unit circle. J+@MzkpK  
    %   They are used in disciplines such as astronomy, optics, and {\svV 0)~  
    %   optometry to describe functions on a circular domain. c}IX"  
    % D/S>w(=  
    %   The following table lists the first 15 Zernike functions. =XMD+  
    % [+%d3+27  
    %       n    m    Zernike function           Normalization UH 47e  
    %       -------------------------------------------------- AB2mt:^  
    %       0    0    1                                 1 KUp   
    %       1    1    r * cos(theta)                    2 pkXfsi-Nu  
    %       1   -1    r * sin(theta)                    2 *(d6Z#  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 8tLT'2+H#  
    %       2    0    (2*r^2 - 1)                    sqrt(3) \!UF|mD^tG  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) rnn2u+OG   
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) <.r ]dCf  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ASKAgU"h  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) $u; >hk  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) [y|^P\D  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ]pOYVf *$  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) S,*{q(   
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) !2zo]v4?  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {PODisl>\D  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) [$( sUc(%  
    %       -------------------------------------------------- ( zn_8s  
    % I&TTr7  
    %   Example 1: Wl& >6./{  
    % (s}Rj)V[^  
    %       % Display the Zernike function Z(n=5,m=1) DKy >]Hca  
    %       x = -1:0.01:1; :DtZ8$I`]C  
    %       [X,Y] = meshgrid(x,x); Io$w|~x  
    %       [theta,r] = cart2pol(X,Y); .cg=  
    %       idx = r<=1; MzEm*`<  
    %       z = nan(size(X)); [x;(cISK1  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); jlu`lG*e&  
    %       figure f >mhFy  
    %       pcolor(x,x,z), shading interp rc;7W:  
    %       axis square, colorbar K1?Z5X(b  
    %       title('Zernike function Z_5^1(r,\theta)') S}oG.r 9  
    % %`xV'2H  
    %   Example 2: Qg'c?[~W@  
    % ZYE' C  
    %       % Display the first 10 Zernike functions oLgg  
    %       x = -1:0.01:1; L;/9L[s,  
    %       [X,Y] = meshgrid(x,x); J[ e}  
    %       [theta,r] = cart2pol(X,Y); xS= _yO9-  
    %       idx = r<=1; O&`U5w  
    %       z = nan(size(X)); k2EHco0BG  
    %       n = [0  1  1  2  2  2  3  3  3  3]; b8rp8'M)  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ]Nnxnp  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; E rr4 %-  
    %       y = zernfun(n,m,r(idx),theta(idx)); UZzNVIXA%  
    %       figure('Units','normalized') N]B)Fb  
    %       for k = 1:10 EzR%w*F>Q  
    %           z(idx) = y(:,k); =yl4zQmg$  
    %           subplot(4,7,Nplot(k)) \Dn&"YG7  
    %           pcolor(x,x,z), shading interp WMW1B }Z3  
    %           set(gca,'XTick',[],'YTick',[]) fuq( 2&^  
    %           axis square FoE|Js  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) %tT"`%(+  
    %       end CPVzX%=  
    % sW }<zGYd  
    %   See also ZERNPOL, ZERNFUN2. 3(1 ]FKZtt  
    Z}|TW~J=  
    8]S,u:E:N  
    %   Paul Fricker 11/13/2006 x>}B#  
    d)N^PJ/  
    AT"!{Y "H  
    }7K@e;YUg  
    &|)hCJu  
    % Check and prepare the inputs: >xT^RYS  
    % ----------------------------- 8EOh0gk7  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) >9ob*6q,  
        error('zernfun:NMvectors','N and M must be vectors.') TI}}1ScA'  
    end lK0s=4c{  
    Vzpt(_><  
    <"<Mbbp  
    if length(n)~=length(m) UcgG  
        error('zernfun:NMlength','N and M must be the same length.') 5?Bc Y ;  
    end )D;*DUtMVm  
    VM7 !0  
    )CJES!! W  
    n = n(:); T'E ] i!$  
    m = m(:); Bp :~bHf  
    if any(mod(n-m,2)) Z.quh;  
        error('zernfun:NMmultiplesof2', ... T=WNBqKo]  
              'All N and M must differ by multiples of 2 (including 0).') HN{zT&  
    end j.DHqHx  
    qP7G[%=v  
    c7\VTYT  
    if any(m>n) z 5'ZN+  
        error('zernfun:MlessthanN', ... `ruNA>M  
              'Each M must be less than or equal to its corresponding N.') mb&lCd ^-  
    end +IrZ ;&oy  
    w!\3ICB  
    Av o|v>  
    if any( r>1 | r<0 ) Wbe0ZnM]  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 9RH"d[%yc}  
    end $xT1 1 ^  
    joqWh!kv7U  
    K1OkZ6kl  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) n#4Gv|{XMD  
        error('zernfun:RTHvector','R and THETA must be vectors.') [>f]@>  
    end # Q}_e7t  
    #;]2=@  
    &R,9+c  
    r = r(:); yY Y Nu`  
    theta = theta(:);  ;\qXbL7  
    length_r = length(r); YGp)Oy}:  
    if length_r~=length(theta) zzJja/mp  
        error('zernfun:RTHlength', ... Fi4UaJ3K  
              'The number of R- and THETA-values must be equal.') \:Za[6  
    end 7NJFWz!  
    _1aGtX|W  
    dQD$K|aUp  
    % Check normalization: 'Dath>Y=  
    % -------------------- ='}#`',  
    if nargin==5 && ischar(nflag) $ KB  
        isnorm = strcmpi(nflag,'norm'); %?9Ok  
        if ~isnorm *)'Vvu<  
            error('zernfun:normalization','Unrecognized normalization flag.') &$=!dA  
        end "UG K8x  
    else e"k/d<  
        isnorm = false; _okWQvdH  
    end "$|Zr  
    ?qmp_2:WU  
    +Rwx% =  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% E"qFXA>  
    % Compute the Zernike Polynomials X:EEPGE  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% };b1ahaG  
    Qs9OC9X1  
    \S ."?!U  
    % Determine the required powers of r: Mzkkc QLK  
    % ----------------------------------- .WX,Nd3@  
    m_abs = abs(m); [&#/|zH'j:  
    rpowers = []; G-u]L7t&1  
    for j = 1:length(n) F}Srn;V  
        rpowers = [rpowers m_abs(j):2:n(j)]; [=uIb._Wv  
    end *jITOR!uF`  
    rpowers = unique(rpowers); I4t*?  
    =-#G8L%Q  
    z-r2!^q27  
    % Pre-compute the values of r raised to the required powers, p?# pT}1  
    % and compile them in a matrix: hH>``gK  
    % ----------------------------- D-&a n@  
    if rpowers(1)==0 94/BG0  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); )d`mvZBn1  
        rpowern = cat(2,rpowern{:}); !<<AzLVL  
        rpowern = [ones(length_r,1) rpowern]; #_JYh?  
    else UzG[:ic%  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); eE(b4RCM  
        rpowern = cat(2,rpowern{:}); F #`=oM $5  
    end <RXwM6G2  
    =qu(~]2(  
    HrM$NRhu  
    % Compute the values of the polynomials: FX}Gt=  
    % -------------------------------------- 8b(!k FxD  
    y = zeros(length_r,length(n)); >IfV\ w32  
    for j = 1:length(n) *O~e T  
        s = 0:(n(j)-m_abs(j))/2; G~,:2 o3  
        pows = n(j):-2:m_abs(j); vXE0%QE'Q  
        for k = length(s):-1:1 iE].&>w  
            p = (1-2*mod(s(k),2))* ... F3M aqr y  
                       prod(2:(n(j)-s(k)))/              ... j;0vAf  
                       prod(2:s(k))/                     ... EGVM)ur  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... A8r^)QJP{  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); H t(n%;<  
            idx = (pows(k)==rpowers); q v*7K@  
            y(:,j) = y(:,j) + p*rpowern(:,idx); JRaq!/[(  
        end 1q7tiMvV-  
         0#_'o ,  
        if isnorm fmX!6Kv  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); O`[aU%4b  
        end EgjR^A1W2  
    end a_~=#]a  
    % END: Compute the Zernike Polynomials pCi#9=?N  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~`ny @WD9  
    p>w]rE:}  
    <AH1i@4  
    % Compute the Zernike functions: i1lBto[  
    % ------------------------------ AIYmS#V1W2  
    idx_pos = m>0; R%Y`=pK>}  
    idx_neg = m<0; ]6r;}1c  
    ]`g@UtD9`  
    CusF/>  
    z = y; 6w8" >~)Z  
    if any(idx_pos) MGS-4>Q#  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); w o bgu  
    end B;>{0 s  
    if any(idx_neg) ED>prE0  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 8,P- 7^  
    end l7H qo)  
    b?X.U}62_  
    \JEXX4%  
    % EOF zernfun }@ Z56  
     
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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)
    在线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)  }Ho Qwy|&  
    5nsq[Q`  
    DDE还是手动输入的呢? 4;2< ^[M  
    HttiX/2~  
    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