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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, :[7.YQ   
    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, P[C03a!lXg  
    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? HCKj8-*  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? kc70HrG  
    v"G)G)*z  
    1\+d 5Q0  
    p*]nCUs}n  
    +46?+kKt  
    function z = zernfun(n,m,r,theta,nflag) C\p _  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Ndr4e?Xa,  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N B":u5_B  
    %   and angular frequency M, evaluated at positions (R,THETA) on the zAdZXa[MRY  
    %   unit circle.  N is a vector of positive integers (including 0), and uPtS.j=  
    %   M is a vector with the same number of elements as N.  Each element Og~3eL[1%C  
    %   k of M must be a positive integer, with possible values M(k) = -N(k)  6,;7iA]  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, >0qe*4n|M  
    %   and THETA is a vector of angles.  R and THETA must have the same ]pP [0 S  
    %   length.  The output Z is a matrix with one column for every (N,M) lVQy {`Ns  
    %   pair, and one row for every (R,THETA) pair. vS'5Lm  
    % z gDc=  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike A VbGJ+  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), VVyms7 VN  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral )X-/0G=N-  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, YE\s<$  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized AjA.="3  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 73OYHp_j  
    % x4vowF  
    %   The Zernike functions are an orthogonal basis on the unit circle. B7!dp`rPp  
    %   They are used in disciplines such as astronomy, optics, and Bys_8x}  
    %   optometry to describe functions on a circular domain. l TRQ/B  
    % Qcf5* ]V  
    %   The following table lists the first 15 Zernike functions. !q_fcd^c  
    % 1#<KZN =$  
    %       n    m    Zernike function           Normalization Z,jK(7D(  
    %       -------------------------------------------------- L H`z '7&/  
    %       0    0    1                                 1 Xi!`+N4  
    %       1    1    r * cos(theta)                    2 '+ cPx\4  
    %       1   -1    r * sin(theta)                    2 :F`yAB3  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 9!sR}  
    %       2    0    (2*r^2 - 1)                    sqrt(3)  rVo?I  
    %       2    2    r^2 * sin(2*theta)             sqrt(6)  9> k-";  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) E|fQbkfw  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 9xm'0 '  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) >AT T<U=  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Gv3AJ'NL  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 9c_h+XN?y  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) c={bunnz#  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ^|1)6P}6  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .;xt{kK  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ZBAtRs  
    %       -------------------------------------------------- P@z,[,sy"$  
    % y=)xo7 (  
    %   Example 1:  1ZF>e`t8  
    %  e ):rr*  
    %       % Display the Zernike function Z(n=5,m=1) H_CX5=Nq^  
    %       x = -1:0.01:1; i>`!W|=_  
    %       [X,Y] = meshgrid(x,x); g/ict 2!  
    %       [theta,r] = cart2pol(X,Y); $h( B2  
    %       idx = r<=1; eBW=bK~[VP  
    %       z = nan(size(X)); xi =\]  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); h#>%\Pvt;  
    %       figure Tp7slKc0p  
    %       pcolor(x,x,z), shading interp aA-gl9  
    %       axis square, colorbar Cg! ]x o  
    %       title('Zernike function Z_5^1(r,\theta)') /{9"O y7E  
    % n rpxZA  
    %   Example 2: &m>sGCZ  
    % VTt{ 0 ~  
    %       % Display the first 10 Zernike functions ,{br6*E  
    %       x = -1:0.01:1; WTcrfs)T  
    %       [X,Y] = meshgrid(x,x); GrB+Y!{{  
    %       [theta,r] = cart2pol(X,Y); *uq}jlD`!  
    %       idx = r<=1; @m=xCg.Z  
    %       z = nan(size(X)); 0cwb^ffN  
    %       n = [0  1  1  2  2  2  3  3  3  3]; #&cNR_"w  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; fv",4L  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; %fyah}=  
    %       y = zernfun(n,m,r(idx),theta(idx)); *"pf3x6  
    %       figure('Units','normalized') XOe8(cXa9  
    %       for k = 1:10 8sG0HI$f+  
    %           z(idx) = y(:,k); };:+0k/  
    %           subplot(4,7,Nplot(k)) $C;)Tlh  
    %           pcolor(x,x,z), shading interp d}.*hgk  
    %           set(gca,'XTick',[],'YTick',[]) $# /-+>  
    %           axis square h8Bs=T  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ;L gxL Qy;  
    %       end 2V 1|b`b#4  
    % dt -=7mz#  
    %   See also ZERNPOL, ZERNFUN2. A80r@)i  
    gJ8+HV  
    n8.W$&-ia  
    %   Paul Fricker 11/13/2006 n!r<\4I  
    /Y>$w$S  
    vncak  
    cBO.96ZHE  
    ]d}U68$T+  
    % Check and prepare the inputs: ue5C ]  
    % ----------------------------- ,p,$(V  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 'TF5CNX  
        error('zernfun:NMvectors','N and M must be vectors.') )\bA'LuFy  
    end e7(iMe  
    ?J<V-,i  
    cF[L6{Oe  
    if length(n)~=length(m) )NoNgU\7!  
        error('zernfun:NMlength','N and M must be the same length.') 7$l!f  
    end 8<Y*@1*j  
    =q%Q^  
    }'y=JV>l  
    n = n(:); <QUjhWxDb  
    m = m(:); f8T6(cA  
    if any(mod(n-m,2)) CBqeO@M  
        error('zernfun:NMmultiplesof2', ... O]>FNsh!  
              'All N and M must differ by multiples of 2 (including 0).') UkE  fuH  
    end w$X"E*~>8  
    Y~P1r]piB  
    w&vZ$n-|  
    if any(m>n) <}@*i  
        error('zernfun:MlessthanN', ... yl>V '  
              'Each M must be less than or equal to its corresponding N.') o1m+4.-  
    end |# _F  
    ']N1OVw^vf  
    3N(5V;ti  
    if any( r>1 | r<0 ) E^)>9f7  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') aDV~T24  
    end +:a#+]g  
    M x/G^yO9  
    Y+,ii$Ce~  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) KlMSkdmW  
        error('zernfun:RTHvector','R and THETA must be vectors.') ^dR="N  
    end qHZ!~Kq,"'  
    m#\I&(l+  
    9vQI ~rz?  
    r = r(:); ZU=om Rh5  
    theta = theta(:); 4jOq.j  
    length_r = length(r); X=8CZq4  
    if length_r~=length(theta) (R.l{(A  
        error('zernfun:RTHlength', ... hu ]l{TXi  
              'The number of R- and THETA-values must be equal.') !O`aaLc  
    end ;;^OKrzWW  
    8=GgTpO5  
    Io|3zE*<  
    % Check normalization: t",=]k  
    % -------------------- ~rUcko8  
    if nargin==5 && ischar(nflag) | ODi[~y  
        isnorm = strcmpi(nflag,'norm'); V0rS^SAF  
        if ~isnorm I@$cw3  
            error('zernfun:normalization','Unrecognized normalization flag.') b"DV8fdX  
        end {Wi)/B}  
    else $s2Y,0>I6  
        isnorm = false; I"=a:q  
    end XF6ed  
    $ $=N'Q  
    'ie+/O@G  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _d[4EY  
    % Compute the Zernike Polynomials .T>^bLuFy  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U#qs^f7R  
    hT=6XO od4  
    bAUruTn  
    % Determine the required powers of r: 6m~N2^z  
    % ----------------------------------- sp-){k  
    m_abs = abs(m); T 5AoBUw  
    rpowers = []; =tKb7:KU  
    for j = 1:length(n) m0}1P]dc  
        rpowers = [rpowers m_abs(j):2:n(j)]; ~7G@S&<PK(  
    end Z\\'0yuY(  
    rpowers = unique(rpowers); !_No\O  
    "f Ni3 <x]  
    I+!?~]AUuq  
    % Pre-compute the values of r raised to the required powers, &OM e'P  
    % and compile them in a matrix: $:RP tG  
    % ----------------------------- < Z>p1S  
    if rpowers(1)==0 ;VS\'#{e  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Wx`| u  
        rpowern = cat(2,rpowern{:}); Ft[)m#Dj`  
        rpowern = [ones(length_r,1) rpowern]; _Nx#)(x  
    else ?V{AP&#M$x  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); G-DvM6T  
        rpowern = cat(2,rpowern{:}); 1v?|n8  
    end ,S%DHT  
    ?BEO(;'  
    -~|E(ys  
    % Compute the values of the polynomials: 'QP~uK  
    % -------------------------------------- smJ#.I6/L  
    y = zeros(length_r,length(n)); < %t$0'  
    for j = 1:length(n) @hG]Gs[,o  
        s = 0:(n(j)-m_abs(j))/2; GGWdMGI/  
        pows = n(j):-2:m_abs(j); 67{3/(`x  
        for k = length(s):-1:1 Qp5YS  
            p = (1-2*mod(s(k),2))* ... 9i?Q=Vuc~<  
                       prod(2:(n(j)-s(k)))/              ... 'KU)]v  
                       prod(2:s(k))/                     ... rIhe}1  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... /7o{%~O  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Jg2*$gL;_  
            idx = (pows(k)==rpowers); &~ .n}h&  
            y(:,j) = y(:,j) + p*rpowern(:,idx); "%?$BoJR0  
        end S#|dmg;p  
         P'Gf7sQt7  
        if isnorm fJdTVs@  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); |/M^q{h&7s  
        end ~snYf7  
    end +FGw)>g8'm  
    % END: Compute the Zernike Polynomials s~)I1G  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V-N`R-FSr  
    z7PmyU >  
    3yXSv1  
    % Compute the Zernike functions: DZ*m"Bi  
    % ------------------------------ "/~KB~bB  
    idx_pos = m>0; t91z<Y|  
    idx_neg = m<0; tDQo1,(oY  
    6$ \69   
    b&_u+g  
    z = y; $psPNJG  
    if any(idx_pos) Y *?hA'  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); r1R\cor  
    end }[O/u <Z  
    if any(idx_neg) l(j._j~p  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 7+c}D>/`:  
    end P6 ~& ,a  
    ~ ~U,  
    E8Y(C_:s  
    % EOF zernfun zAA3bgaa  
     
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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)  ?Rg8u  
    f[$9k}.  
    DDE还是手动输入的呢? ^"hsbk&Yu  
    T.R(  
    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