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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, *0eU_*A^zO  
    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, Fr  
    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? 7dACbqba  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? (?JdiY/  
    *`Vmncv3  
    /9I/^i~  
    }(ORh2Ri  
    )hj:Xpj9#  
    function z = zernfun(n,m,r,theta,nflag) )rG4Nga5}  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Cgh84 2%  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1nskf*Z  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Y-YuY  
    %   unit circle.  N is a vector of positive integers (including 0), and ja';NIO-  
    %   M is a vector with the same number of elements as N.  Each element ` K {k0_{  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) x6s|al  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, IY#:v%U  
    %   and THETA is a vector of angles.  R and THETA must have the same 'D ?o^  
    %   length.  The output Z is a matrix with one column for every (N,M) FC, =g`Q!  
    %   pair, and one row for every (R,THETA) pair. ZDR@VYi+~  
    % Hy[: _E  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike sAL ]N][Y  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), W_Y8)KxG:L  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral BrwC9:  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, x}?<9(nE c  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 5j1d=h  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. AO|9H`6U6F  
    % 6xJffl  
    %   The Zernike functions are an orthogonal basis on the unit circle. &EQhk9j  
    %   They are used in disciplines such as astronomy, optics, and X(nyTR8  
    %   optometry to describe functions on a circular domain. F1L[3D^-  
    % 4#03x:/<\  
    %   The following table lists the first 15 Zernike functions. y-1e(:GF  
    % o" ,8   
    %       n    m    Zernike function           Normalization >dt*^}*  
    %       -------------------------------------------------- M[YFyM(  
    %       0    0    1                                 1 \{lv~I  
    %       1    1    r * cos(theta)                    2 !V37ePFje  
    %       1   -1    r * sin(theta)                    2 ?s^3 o{!<W  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) [c 8=b,EI  
    %       2    0    (2*r^2 - 1)                    sqrt(3) &S*~EM.l8  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 1w^wa_qx  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) =W.}&  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8)  V>'  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) lZcNio  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ZLv/otf:|"  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) &P|[YP37_  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) E s5: S#  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) == 5F[UX  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) A>yU0\A  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) \@:,A]  
    %       -------------------------------------------------- cj8cV|8@  
    % 1jl !VU6  
    %   Example 1: p%"dYH%]&0  
    % U4pIRa)S  
    %       % Display the Zernike function Z(n=5,m=1) .z`70ot?  
    %       x = -1:0.01:1; @%R<3!3v  
    %       [X,Y] = meshgrid(x,x); ;[sW\Ou  
    %       [theta,r] = cart2pol(X,Y); /8h=6"  
    %       idx = r<=1; ^hC'\09=c  
    %       z = nan(size(X)); LSJ?;Zg(=z  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 6@J=n@J$p  
    %       figure c0@8KW[,  
    %       pcolor(x,x,z), shading interp ~.m<`~u  
    %       axis square, colorbar #dA$k+3  
    %       title('Zernike function Z_5^1(r,\theta)') vjGQ!xF  
    % )#}>,,S  
    %   Example 2: %X{EupiFA  
    % ' [ 4;QYw  
    %       % Display the first 10 Zernike functions DYD<?._I  
    %       x = -1:0.01:1; V0\[|E;F  
    %       [X,Y] = meshgrid(x,x); Iry$z^  
    %       [theta,r] = cart2pol(X,Y); :o' XE|N  
    %       idx = r<=1; 6Dq4Q|C  
    %       z = nan(size(X)); \2i7\U  
    %       n = [0  1  1  2  2  2  3  3  3  3]; e<L@QNX  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; u*l|MIi6J  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; $1an#~  
    %       y = zernfun(n,m,r(idx),theta(idx)); /~[Lr   
    %       figure('Units','normalized') TC\+>LXiZ  
    %       for k = 1:10 Z4j6z>qE  
    %           z(idx) = y(:,k); t;&XIG~  
    %           subplot(4,7,Nplot(k)) SiratkP9n7  
    %           pcolor(x,x,z), shading interp yw3"jdcl  
    %           set(gca,'XTick',[],'YTick',[]) g{65QP  
    %           axis square ,fVD`RR(W?  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 11[lc2  
    %       end :S+K\  
    % #< im?  
    %   See also ZERNPOL, ZERNFUN2. %BqaVOKJ"f  
    :HkBP90o  
    7RAB"T;?Q  
    %   Paul Fricker 11/13/2006 5'~_d@M  
    0lfK} a  
    f!Q\M1t)  
    n|SV)92o1  
    *T|B'80  
    % Check and prepare the inputs: )O]T}eI  
    % ----------------------------- Hcq.Lq;2:  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) nM )C^$3<t  
        error('zernfun:NMvectors','N and M must be vectors.') xt{'Be&Ya+  
    end Ccf/hA#mb  
    tli.g  
    bLgH3[{  
    if length(n)~=length(m) rz?Cn X.t  
        error('zernfun:NMlength','N and M must be the same length.') kI\m0];KnQ  
    end nV;'UpQw  
    &|>+LP@8  
    {f!/:bM  
    n = n(:); *sho/[~_  
    m = m(:); `BPTcL<W  
    if any(mod(n-m,2)) a^|DD#5  
        error('zernfun:NMmultiplesof2', ... <AHpk5Sn{  
              'All N and M must differ by multiples of 2 (including 0).') -EjXVn! vQ  
    end \{,TpK.  
    Ac7^JXh%  
    ]rmBM  
    if any(m>n) 1gEH~Jmj  
        error('zernfun:MlessthanN', ... Y Y:Bw W:  
              'Each M must be less than or equal to its corresponding N.') c8qr-x1HG  
    end ( ?V`|[+u  
    L+%"e w  
    THA9OXP  
    if any( r>1 | r<0 ) !QUY (  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') v0^9 "V:y  
    end &J[a.:..  
    #Ondhy%h[  
    E_HB[ 9  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) E*_^+ %  
        error('zernfun:RTHvector','R and THETA must be vectors.') DT1gy:?L  
    end "cH RGJG#  
    ]|;+2@kDR  
    ) "#'   
    r = r(:); TQ Vk;&A  
    theta = theta(:); 85{m+1O~  
    length_r = length(r); ?Cq7_rq  
    if length_r~=length(theta) |Lq8cA)|y  
        error('zernfun:RTHlength', ... prBLNZp  
              'The number of R- and THETA-values must be equal.') l{3B }_,  
    end j)1yv.  
    6Z/`p~e  
    q!K :N?  
    % Check normalization: rCyb3,W  
    % -------------------- R+sT &d  
    if nargin==5 && ischar(nflag) r;cDYg  
        isnorm = strcmpi(nflag,'norm'); 5:Qz  
        if ~isnorm ."K>h3(&V  
            error('zernfun:normalization','Unrecognized normalization flag.') X@nBj;   
        end _Fb}zPU!  
    else _MBa&XEM  
        isnorm = false; <J[ le=  
    end C \ Cc[v  
    Fc[KIG3@  
    -H1=N  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% E3L?6Qfx>  
    % Compute the Zernike Polynomials a(Y'C`x  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |J`EM7qMK  
    J=V  
    lwnO  
    % Determine the required powers of r: iP/v "g"g  
    % ----------------------------------- BEZ~<E&0H  
    m_abs = abs(m); !Jg;%%E3:i  
    rpowers = []; )!y>2$20 r  
    for j = 1:length(n) ^({)t  
        rpowers = [rpowers m_abs(j):2:n(j)]; a"~o'W7  
    end T.q2tC[bR  
    rpowers = unique(rpowers); ?}||?2=P  
    eK8H5YE  
    B|(g?  
    % Pre-compute the values of r raised to the required powers, ^df wWP  
    % and compile them in a matrix: PN}+LOD<t  
    % ----------------------------- ,OZ  
    if rpowers(1)==0 &K[*vyD  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); :I"CQ C[Z  
        rpowern = cat(2,rpowern{:}); ROO*/OOd  
        rpowern = [ones(length_r,1) rpowern]; dQut8>0&  
    else *0WVrM06?  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Z:b?^u4.  
        rpowern = cat(2,rpowern{:}); OhF55,[  
    end 3CUQQ_  
    Z[vx0[av&  
    M,Gy.ivz  
    % Compute the values of the polynomials: gv!8' DKn  
    % -------------------------------------- !}*N';  
    y = zeros(length_r,length(n)); 6fwNlC/9  
    for j = 1:length(n) yUoR6w  
        s = 0:(n(j)-m_abs(j))/2; 0'q4=!l  
        pows = n(j):-2:m_abs(j); ,5'o>Y  
        for k = length(s):-1:1 Y#U.9>h  
            p = (1-2*mod(s(k),2))* ... Q G) s  
                       prod(2:(n(j)-s(k)))/              ... N#w5}It  
                       prod(2:s(k))/                     ... G  hM  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... jKSj);  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); d[9,J?'OQ  
            idx = (pows(k)==rpowers); MVatV[G  
            y(:,j) = y(:,j) + p*rpowern(:,idx); QE<Z@/V*a  
        end mY|c7}>V;  
         a[ {qb  
        if isnorm UhB +c  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); KbJ6U75|f  
        end rcnH^P  
    end PZ[-a-p40  
    % END: Compute the Zernike Polynomials ZvY"yl?e  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U#<d",I  
    fif;n[<  
    +]l?JKV  
    % Compute the Zernike functions: YOxgpQ:i  
    % ------------------------------ q|5WHB  
    idx_pos = m>0; SH*'<  
    idx_neg = m<0; 7:`XE&Z  
    ;H$ Cq' I  
    O{:{P5  
    z = y; j }~?&yB  
    if any(idx_pos) =dm9+ff  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); G/ x6zdk  
    end |N,^*xP(6  
    if any(idx_neg) UrniJB]  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); bGh&@&dHr  
    end g pciv  
    2 BY|Cp4R  
    zD_5TG M=  
    % EOF zernfun 3Vu}D(PJ  
     
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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)  -X*.scw  
    7|Vpk&.>  
    DDE还是手动输入的呢? p;[.&o J  
    5i$P$ 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