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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, yToT7 X7F7  
    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, vN0L( B  
    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? g~Nij~/  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? cu479VzPx:  
    0K$WSGB?6j  
    ^;)SFmjg%  
    (Y'UvZlM%P  
    3)C6OF>7  
    function z = zernfun(n,m,r,theta,nflag) K{= r.W  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. {m+S{dWp  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N lrmt)BLoh  
    %   and angular frequency M, evaluated at positions (R,THETA) on the []=FZ`4  
    %   unit circle.  N is a vector of positive integers (including 0), and )WP]{ W)r  
    %   M is a vector with the same number of elements as N.  Each element %qNj{<&  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) F;?TR[4!k  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 1&8j3"  
    %   and THETA is a vector of angles.  R and THETA must have the same 2[8fFo>  
    %   length.  The output Z is a matrix with one column for every (N,M) ,<;l"v(  
    %   pair, and one row for every (R,THETA) pair. %;=IMMK  
    % 9{9#AI.G  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike (:&&;]sI  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), u-%r~ }  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral bG5^h  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, q#Yg0w~  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized I5TQ>WJbf  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. r|\5'ZMx  
    % 7E!";HT  
    %   The Zernike functions are an orthogonal basis on the unit circle. ;Xfd1    
    %   They are used in disciplines such as astronomy, optics, and 0,1L e$)6  
    %   optometry to describe functions on a circular domain. fXF=F,!t  
    % _ bXVg3oDt  
    %   The following table lists the first 15 Zernike functions. ONr?.MJ6j  
    % nxn[ ~~  
    %       n    m    Zernike function           Normalization 1kvPiV=X>  
    %       -------------------------------------------------- 3P+4S|@q(4  
    %       0    0    1                                 1 Ks49$w<  
    %       1    1    r * cos(theta)                    2 jpYw#]Q  
    %       1   -1    r * sin(theta)                    2 DU/9/ I?~  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) tAb;/tM3I  
    %       2    0    (2*r^2 - 1)                    sqrt(3) z`86-Ov  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) IKMs Y5i  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 9D{u,Q V  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) LT,iS)dY+  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) vWqyZ-p,q  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) r!=]Q}`F  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 8Z9MD<RLw  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) v{mv*`~nA\  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Q-! i$#-  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;_?zB NW  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 4cXAT9  
    %       -------------------------------------------------- D_l/Gxdpr  
    % .iOw0z  
    %   Example 1:  /gqqKUx  
    % AI^AK0.L  
    %       % Display the Zernike function Z(n=5,m=1) q;~R:}?@  
    %       x = -1:0.01:1; 8FO1`%8Oe  
    %       [X,Y] = meshgrid(x,x); T8,k7 7  
    %       [theta,r] = cart2pol(X,Y); ]6a/0rg:t  
    %       idx = r<=1; 6T^N!3p_  
    %       z = nan(size(X)); t/v@vJ`vSH  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); \ &eY)^vw  
    %       figure 7%:??*"~  
    %       pcolor(x,x,z), shading interp ) >>u|#@z  
    %       axis square, colorbar fap|SMGt  
    %       title('Zernike function Z_5^1(r,\theta)') 4&FNU)tt  
    % %-h7Z3YcN  
    %   Example 2: %-@'CNP  
    % #W>x\  
    %       % Display the first 10 Zernike functions &_Cxv8  
    %       x = -1:0.01:1; +L`V[;  
    %       [X,Y] = meshgrid(x,x); SjZd0H0  
    %       [theta,r] = cart2pol(X,Y); kN'|,eKH4  
    %       idx = r<=1; B]'e$uyL7  
    %       z = nan(size(X)); M,b<B_$  
    %       n = [0  1  1  2  2  2  3  3  3  3]; E0sbU<11  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; K%Usjezv&  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Mq+viU&   
    %       y = zernfun(n,m,r(idx),theta(idx)); tpv?`(DDU  
    %       figure('Units','normalized') ox(*  
    %       for k = 1:10 pu\b`3C(  
    %           z(idx) = y(:,k); $s e !8s"  
    %           subplot(4,7,Nplot(k)) 3mpP| b"  
    %           pcolor(x,x,z), shading interp ?,WUJH?^  
    %           set(gca,'XTick',[],'YTick',[]) N+*(Y5TU  
    %           axis square Z 7`5x  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) +<)tql*  
    %       end TZ^{pvBy  
    % 1P5*wNF  
    %   See also ZERNPOL, ZERNFUN2. i FC"!23f  
    5T!&r  
    mcvDxjk,h  
    %   Paul Fricker 11/13/2006 NY~ dM\  
    YEg .  
    dj?G.-  
    *Zc9yZl2  
    /DLr(  
    % Check and prepare the inputs: 8&?^XcJ*x  
    % ----------------------------- qv.[k<~a>  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2;&mkc K'  
        error('zernfun:NMvectors','N and M must be vectors.') c}YJqhk0J  
    end $`^H:Djr  
    0v;ve  
    =fY lzZh  
    if length(n)~=length(m) uEBQoP2  
        error('zernfun:NMlength','N and M must be the same length.') cYsR0#  
    end G"}qV%"6"  
    _Mlhum t  
    5r'=O2AZX  
    n = n(:); w#W5}i&x  
    m = m(:); RwUW;hU  
    if any(mod(n-m,2)) Y3D3.T6Q  
        error('zernfun:NMmultiplesof2', ... HTxB=Q|  
              'All N and M must differ by multiples of 2 (including 0).') #X4LLS]VV  
    end ozVpfs  
    7}gA0fP9  
    55LgBD  
    if any(m>n) [`q.A`Fd  
        error('zernfun:MlessthanN', ... t9ER;.e  
              'Each M must be less than or equal to its corresponding N.') O ,l\e 3;  
    end 3 Q@9S  
    cvxIp#FbW  
    ]OUD5T  
    if any( r>1 | r<0 ) TV<Aj"xw  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') C2NzP& FD  
    end 4 uShM0qa  
    ,K T<4  
    : g&>D#{  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) eDuX"/kHA  
        error('zernfun:RTHvector','R and THETA must be vectors.') O)l%OOv   
    end 9_eS`,'  
    Lg8 ]dBXu  
     KvGbDG  
    r = r(:); %@>YNPD`E  
    theta = theta(:); DQcWq'yY^  
    length_r = length(r); /\~l1.6`  
    if length_r~=length(theta) @sN^BX`z  
        error('zernfun:RTHlength', ... S=4R5igrC  
              'The number of R- and THETA-values must be equal.') ?b5H 2 W  
    end FWIih5 3`  
    /=bSt  
    rYbCOazr  
    % Check normalization: #0(fOHPQ  
    % -------------------- V):`&@  
    if nargin==5 && ischar(nflag) Kf|0*c  
        isnorm = strcmpi(nflag,'norm'); `nKJR'QC  
        if ~isnorm $kv@tzO  
            error('zernfun:normalization','Unrecognized normalization flag.') _'&k#Q  
        end 0Qt~K#mr/  
    else y`({ .L  
        isnorm = false; f]c <9Q>*  
    end 9g96 d-  
    l4zw]AYk+X  
    5|5=Y/   
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r-*l1([eW  
    % Compute the Zernike Polynomials |"_)zQ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )x)gHY8;  
    @Zj& `/  
    CNq[4T'~A  
    % Determine the required powers of r: KA?v.s  
    % ----------------------------------- !h?=Wv ==]  
    m_abs = abs(m); Q~8y4=|#CY  
    rpowers = []; TKd6MZhT  
    for j = 1:length(n) v3~FR,Kl  
        rpowers = [rpowers m_abs(j):2:n(j)]; Y^yG/F  
    end  C[R`Ml  
    rpowers = unique(rpowers); {|Bd?U;  
    0Lx3]"v  
    %oR>Uo  
    % Pre-compute the values of r raised to the required powers, h+5 @I%WX  
    % and compile them in a matrix: }Iip+URG  
    % ----------------------------- j|k @MfA  
    if rpowers(1)==0 ^zHRSO  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); y>)MAzz~\  
        rpowern = cat(2,rpowern{:}); 4aA9\\hfGY  
        rpowern = [ones(length_r,1) rpowern]; Jb9F=s+  
    else \x(.d.l/  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); K|Om5 p  
        rpowern = cat(2,rpowern{:}); qZ&a76t  
    end dt<~sOT3s  
    t|<FA#  
    2Sjt=LOc="  
    % Compute the values of the polynomials: >GmN~"iJ  
    % -------------------------------------- tGC2 ^a#~  
    y = zeros(length_r,length(n)); )Y~xIj >  
    for j = 1:length(n) lf6|.  
        s = 0:(n(j)-m_abs(j))/2; "U*5Z:8?9  
        pows = n(j):-2:m_abs(j); B2Qp}  
        for k = length(s):-1:1 vkuc8 li  
            p = (1-2*mod(s(k),2))* ... dGU8+)2cn  
                       prod(2:(n(j)-s(k)))/              ... HZ{n&iJ  
                       prod(2:s(k))/                     ... :,47rN,qa  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ]H>+m 9  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ![).zi+m  
            idx = (pows(k)==rpowers); ?|lIXz  
            y(:,j) = y(:,j) + p*rpowern(:,idx); M}u1qXa  
        end Fav^^vf*1  
         K I`11lJW~  
        if isnorm SD^E7W$?  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); F(;jM(  
        end l 1|~  
    end o(zTNk5d  
    % END: Compute the Zernike Polynomials /z#F,NB  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ld95[cTP  
    mbGcDG[HQ  
    TOrMXcn!/  
    % Compute the Zernike functions: _F^$aZt?e  
    % ------------------------------ Ox|TMSb^  
    idx_pos = m>0; Li]k7w?H  
    idx_neg = m<0; 6< >SHw  
    ^&-a/'D$,  
    >J@egIKzP  
    z = y; @+:4J_N  
    if any(idx_pos) %<AS?Ry  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); |Q5+l.%  
    end r ^ Y~mq  
    if any(idx_neg) $o"g73`3  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); JtFiFaCxY  
    end ~$Y|ca  
    ewym 1}o  
    Za0gs @$  
    % EOF zernfun ~#q;bS  
     
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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)  +nQ!4  
    a.,i.2  
    DDE还是手动输入的呢? Wj OH/$(  
    2LK]Q/WG,+  
    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