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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, S;$@?vF  
    我输入10阶的n、m,r,theta为38025*1向量,最后得到的z是29525*10阶的矩阵, A>6 b 6  
    这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? 9l+`O0.@  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? \ ?[#>L4  
    d\`A ^  
    ^4Ra$<  
     :GC <U|p  
    8T'=lTJ  
    function z = zernfun(n,m,r,theta,nflag) N2_j[Pe  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. W[o~AbU  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N BRP9j y  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 7?K?-Oj  
    %   unit circle.  N is a vector of positive integers (including 0), and e{/(NtKf  
    %   M is a vector with the same number of elements as N.  Each element ?;.j)  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ?@9kVB*|  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, sXI_!)H  
    %   and THETA is a vector of angles.  R and THETA must have the same - Z"w  
    %   length.  The output Z is a matrix with one column for every (N,M) ZVpMR0!  
    %   pair, and one row for every (R,THETA) pair. >Dpz0v  
    % cA"',N8!5  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike W|@EKE.k  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 4-[L^1%S[  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral KO(+%>^R  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, QP|Ou*Qm)  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized chsjY]b  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. OiX>^_iDt  
    % RqW ZhHI1M  
    %   The Zernike functions are an orthogonal basis on the unit circle. [7?K9r\#  
    %   They are used in disciplines such as astronomy, optics, and BQv+9(:fQB  
    %   optometry to describe functions on a circular domain. S\GC^ FK  
    % 5O&6 (Gaf  
    %   The following table lists the first 15 Zernike functions. * B,D#;6  
    % 9^J8V]X  
    %       n    m    Zernike function           Normalization ]{V q;  
    %       -------------------------------------------------- 4VPL -":6  
    %       0    0    1                                 1 @L^2VVWk^  
    %       1    1    r * cos(theta)                    2 >#B%gxff  
    %       1   -1    r * sin(theta)                    2 D%umL/[]  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) cTm oz.0  
    %       2    0    (2*r^2 - 1)                    sqrt(3) EA%(+tJ^0  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) eX 9{wb(  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) -UkP{x)S  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 5n1;@Vr  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 7/NXb  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) aksyr$d0V<  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) y]9 3z!#Z  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Z{`;Ys:zk  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) km'3[}8o&  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =)m2u2c M  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) .pQH>;k]K  
    %       -------------------------------------------------- ZAzn-n  
    % HDYr?t~V  
    %   Example 1: 8 s!0Z1Roc  
    % |$#u~<r_ w  
    %       % Display the Zernike function Z(n=5,m=1) zu<b#Wv  
    %       x = -1:0.01:1; 4)+MvKxjS  
    %       [X,Y] = meshgrid(x,x); X>2_G ol!  
    %       [theta,r] = cart2pol(X,Y); (E59)z -  
    %       idx = r<=1; < i*v  
    %       z = nan(size(X)); ex7zg!  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 7+JQaYO`"  
    %       figure !7@IWz(, "  
    %       pcolor(x,x,z), shading interp tYiK#N7  
    %       axis square, colorbar 2V_C_5)1  
    %       title('Zernike function Z_5^1(r,\theta)')  ^ruS  
    % Bv \ihUg/  
    %   Example 2: p!>FPS  
    % m v%fX2.  
    %       % Display the first 10 Zernike functions Qn(e[ C6\  
    %       x = -1:0.01:1; ;rJR+wpNa  
    %       [X,Y] = meshgrid(x,x); -EFtk\/  
    %       [theta,r] = cart2pol(X,Y); \%=\_"^?  
    %       idx = r<=1; x)l}d3   
    %       z = nan(size(X));  Ek(. ["  
    %       n = [0  1  1  2  2  2  3  3  3  3]; _KC)f'Cx  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; qI\qpWS\  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Z+)R%Z'aL  
    %       y = zernfun(n,m,r(idx),theta(idx)); Q.#@xaX'{`  
    %       figure('Units','normalized') u_s  
    %       for k = 1:10 w-};\]I  
    %           z(idx) = y(:,k);  y$7Fq'  
    %           subplot(4,7,Nplot(k)) LGKkT?fcSC  
    %           pcolor(x,x,z), shading interp X|t?{.p  
    %           set(gca,'XTick',[],'YTick',[]) e~=fo#*2?@  
    %           axis square /4+M0Pl  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) !YSAQi;I  
    %       end ~F^=7oq  
    % -}@3,G  
    %   See also ZERNPOL, ZERNFUN2. $-YS\R\9x  
    GrjL9+|x  
    L.>tJ.ID  
    %   Paul Fricker 11/13/2006 pa Uh+"y>  
    Q*Per;%J  
    23@e?A=C  
    DtG><g}[]  
    =K .'x  
    % Check and prepare the inputs: Kf2*|ZHj  
    % ----------------------------- <Rob.x3  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) EC$wi|i  
        error('zernfun:NMvectors','N and M must be vectors.') cW;to Q!P  
    end Lw\ANku  
    j':Ybr>BR  
    UOSa`TZbZ  
    if length(n)~=length(m) Q7UFF  
        error('zernfun:NMlength','N and M must be the same length.') tiSN amvG1  
    end }"wWSPD  
    7g}4gX's  
    ,Y=r] fk  
    n = n(:); OJ\IdUZ   
    m = m(:); a{^[<  
    if any(mod(n-m,2)) 55MsF}p  
        error('zernfun:NMmultiplesof2', ... PF*<_p"j  
              'All N and M must differ by multiples of 2 (including 0).') k-xh-&  
    end 5_ -YF~  
    R<{bb'  
    BusD}9QqB  
    if any(m>n) VlRN  
        error('zernfun:MlessthanN', ... SdBv?`u|g  
              'Each M must be less than or equal to its corresponding N.') cOcF VPQ  
    end //]g78]=O  
    zm) ]cq  
    &Z/aM?  
    if any( r>1 | r<0 ) |8PUmax  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') A-1Wn^,> *  
    end %&5 !vK  
    \k/ N/&;  
    f1(V~{N,+  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) W,5A|Q~  
        error('zernfun:RTHvector','R and THETA must be vectors.') F&.iY0Pt  
    end <:0649ZB  
    )9MmL-7K  
    &$tBD@7  
    r = r(:); rlk0t159  
    theta = theta(:); ZQ9!k* ^  
    length_r = length(r); 3P~I' FQ  
    if length_r~=length(theta) *,5V;7OR  
        error('zernfun:RTHlength', ... n3t1'_/TU}  
              'The number of R- and THETA-values must be equal.') U*`7   
    end 0b+OB pqN  
    iM+K&\{_h  
    [>QV^2'Z  
    % Check normalization: h!ZEZ|{  
    % -------------------- #Mw|h^ Wm  
    if nargin==5 && ischar(nflag) $, 4;_4t  
        isnorm = strcmpi(nflag,'norm'); E</Um M+ R  
        if ~isnorm Vd~{SS 2>  
            error('zernfun:normalization','Unrecognized normalization flag.') \?7)oFNz  
        end =)vmX0vL  
    else #-dfG.*  
        isnorm = false; F%@( $f  
    end u[9i>7}9  
    Q1 ?O~ao  
    dOh'9kk3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l4?o0;:)  
    % Compute the Zernike Polynomials ?9xaBWf  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,o7aIg&_H  
    EM!#FJh  
    (G $nN*rlu  
    % Determine the required powers of r: {Ak{ ct\t  
    % -----------------------------------  {I+   
    m_abs = abs(m); n_\V G[f  
    rpowers = []; R}njFQvS)  
    for j = 1:length(n) }VxbO8\b(  
        rpowers = [rpowers m_abs(j):2:n(j)]; Yn4c6K  
    end Ac;rMwXk#  
    rpowers = unique(rpowers); c9imfA+e  
    LWE[]1=  
    H6(kxpOI\  
    % Pre-compute the values of r raised to the required powers, ,g2|8>sJP  
    % and compile them in a matrix: B2t.;uz(,  
    % ----------------------------- ga&l.:lo  
    if rpowers(1)==0 :=vB|Ch:~  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); JF!?i6V  
        rpowern = cat(2,rpowern{:}); R2WEPMH%  
        rpowern = [ones(length_r,1) rpowern]; Ry>c]\a]  
    else P5/K?I~/So  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 48dIh\TH"  
        rpowern = cat(2,rpowern{:}); 6, \i0y5n  
    end hV|pH)Nu{  
    e@'rY#:u  
    d2lOx|jt  
    % Compute the values of the polynomials: oR (hL4Dc  
    % -------------------------------------- {Ts@#V=:  
    y = zeros(length_r,length(n)); ._@Scd  
    for j = 1:length(n) BE. v+'c"  
        s = 0:(n(j)-m_abs(j))/2; 1];rW`Bw  
        pows = n(j):-2:m_abs(j); P\ \4 w)C  
        for k = length(s):-1:1 4]9+   
            p = (1-2*mod(s(k),2))* ... c#sPM!!  
                       prod(2:(n(j)-s(k)))/              ... 'U ',9  
                       prod(2:s(k))/                     ... amq]&.M  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... @w&VI6  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); hZ2!UW4'  
            idx = (pows(k)==rpowers); YBn"9w\#  
            y(:,j) = y(:,j) + p*rpowern(:,idx); `&$"oW{HW  
        end JwWW w1  
         ?:l3O_U 5  
        if isnorm ?95^&4Oh0  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); }Kc[pp|9<  
        end <>$`vuU  
    end W5,e;4/hL  
    % END: Compute the Zernike Polynomials DpjiE/*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %7=B?c |  
    YW55iyM  
    ] -"~?  
    % Compute the Zernike functions: W^; wr#  
    % ------------------------------ RM\it"g  
    idx_pos = m>0; 0,+RF "R  
    idx_neg = m<0; V5sH:A7GJ  
    h|OqM:J;  
    G)5w_^&%  
    z = y;  z}\TS.  
    if any(idx_pos) q[p+OpA  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 5W"&$6vj  
    end K6<@DP+/  
    if any(idx_neg) i5wXT  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ,l`4)@{G  
    end 1A\Jh3;Q  
    (|%YyRaX  
    bM7y}P5`1  
    % EOF zernfun ~]X4ru5,4  
     
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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)  =`VA_xVu  
    W zM9{c  
    DDE还是手动输入的呢? &(H;Bin'  
    ~G0\57;h  
    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