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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, +"VXw2R_e  
    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, \Bl`;uXb  
    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? S\@U3|Q5  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 4B Jw+EV8  
    r3~YGY  
    [XD3}'Aa  
    7C~g?1  
    ;Hu`BFXyD  
    function z = zernfun(n,m,r,theta,nflag) 1HeE$  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. YF)c.Q0  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N \*30E<;C_  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 0He^r &c3  
    %   unit circle.  N is a vector of positive integers (including 0), and &[[Hfs2:-]  
    %   M is a vector with the same number of elements as N.  Each element PC& (1kJ  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) (_Rl f$D  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, S|_"~Nd=  
    %   and THETA is a vector of angles.  R and THETA must have the same KtaoU2s  
    %   length.  The output Z is a matrix with one column for every (N,M) b2hXFwPe  
    %   pair, and one row for every (R,THETA) pair. S\6.vw!'  
    % S8;5|ya  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike |p*s:*TJp  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), AN+S6t  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral vgKdhN2kI  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, bqQR";  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized v(Q-RR  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 3Sn# M{wH  
    % Ym9~/'%]  
    %   The Zernike functions are an orthogonal basis on the unit circle. f<Y g_TG  
    %   They are used in disciplines such as astronomy, optics, and E7@m& R  
    %   optometry to describe functions on a circular domain. }IV=qW,  
    % ^x}k1F3  
    %   The following table lists the first 15 Zernike functions. 4R9y~~+  
    % 77%I%<#  
    %       n    m    Zernike function           Normalization OJ<V<=MYZ  
    %       -------------------------------------------------- {br6*  
    %       0    0    1                                 1 ?rQIUP{D7  
    %       1    1    r * cos(theta)                    2 P:m6:F@hO  
    %       1   -1    r * sin(theta)                    2 +\25ynM  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Ji0FHa_  
    %       2    0    (2*r^2 - 1)                    sqrt(3) nZ# 0L`@"Y  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) *NoixV1>  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) h:<?)g~U  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) eJ60@N\A  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) jJe?pT]o  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) J|DY /v  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) R-1C#R[  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) n?8xRaEf  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Z<[:v2  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) X 3(*bj>P  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) {w<"jw&2  
    %       -------------------------------------------------- /(DnMHn\  
    % ]Tn""3#1g  
    %   Example 1: Ev0=m;@_  
    % !5>PZ{J  
    %       % Display the Zernike function Z(n=5,m=1) uQz!of%x  
    %       x = -1:0.01:1; 4.q^r]m*  
    %       [X,Y] = meshgrid(x,x); *Jg&:(#}<J  
    %       [theta,r] = cart2pol(X,Y); $SdpF-'  
    %       idx = r<=1; >ui;B$=  
    %       z = nan(size(X)); 0uJ??4N9  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Z^#u n  
    %       figure (E7C9U*  
    %       pcolor(x,x,z), shading interp +*x9$LSD  
    %       axis square, colorbar B$_-1^L e  
    %       title('Zernike function Z_5^1(r,\theta)') jXYjs8Iy  
    % gh.+}8="  
    %   Example 2: y`J8hawp  
    % mIv}%hD  
    %       % Display the first 10 Zernike functions |eP5iy wg  
    %       x = -1:0.01:1; V6fJaZ  
    %       [X,Y] = meshgrid(x,x); O+ xzM[[  
    %       [theta,r] = cart2pol(X,Y); ]+T$ D  
    %       idx = r<=1; )Qh*@=$-  
    %       z = nan(size(X)); m Q^SpK #  
    %       n = [0  1  1  2  2  2  3  3  3  3]; q;QE(}.g  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; z(1`Iy M  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; {ukQBu#}<  
    %       y = zernfun(n,m,r(idx),theta(idx)); #S"s8wdD  
    %       figure('Units','normalized') -b=A j8h  
    %       for k = 1:10 t/h,-x  
    %           z(idx) = y(:,k); Sn[/'V^$a  
    %           subplot(4,7,Nplot(k)) @oQ"FLF.  
    %           pcolor(x,x,z), shading interp =!IoL7x  
    %           set(gca,'XTick',[],'YTick',[]) (9v%66y  
    %           axis square deCi\n  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ` pfRY!  
    %       end ^n*:zmD  
    % Dfy=$:Q  
    %   See also ZERNPOL, ZERNFUN2. W;|%)D)y  
    UD ;UdehC  
    K<M WiB&  
    %   Paul Fricker 11/13/2006 {pC$jd>T  
    [I}xR(a@n  
    #q6#nfi"  
    ;3+_aoY  
    i-R}O6  
    % Check and prepare the inputs: 0e(4+:0  
    % ----------------------------- 3(_:"?xA  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) z[0tM&pv  
        error('zernfun:NMvectors','N and M must be vectors.') $0Un'"`S  
    end k zC4V  
    #?'@?0<6  
    .H Pa\b\L>  
    if length(n)~=length(m) \Yh*ywwP#  
        error('zernfun:NMlength','N and M must be the same length.') s \0,@A   
    end 2Mj_wc   
    t\f[->f  
    Av!xI  
    n = n(:); wxy@XN"/i+  
    m = m(:); EF'8-*  
    if any(mod(n-m,2)) vK$wc~  
        error('zernfun:NMmultiplesof2', ... 2Q;rSe._`  
              'All N and M must differ by multiples of 2 (including 0).') A+(+Pf U  
    end \s7/`  
    Jv?EV,S/e  
    12tk$FcY8*  
    if any(m>n) l YpoS  
        error('zernfun:MlessthanN', ... A[m<xtm5K  
              'Each M must be less than or equal to its corresponding N.') %JI*)K1WI  
    end <7`U1DR=  
    0bteI*L  
    {+V ]@sz  
    if any( r>1 | r<0 ) AOe f1^S=  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') :KS"&h{SY  
    end .9vt<<Kwh  
    15d'/f  
    k t+h\^g  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) K9 +\Z  
        error('zernfun:RTHvector','R and THETA must be vectors.') hx ^l  
    end _} K3}}  
    K$O2 Fq@y  
    QwL*A `@  
    r = r(:); FcyF E~>2  
    theta = theta(:); . Ctd$  
    length_r = length(r); `cPZsL  
    if length_r~=length(theta) t :~,7  
        error('zernfun:RTHlength', ... {u4AOM=)  
              'The number of R- and THETA-values must be equal.') @U9`V&])F[  
    end =,8nfJ+x  
    LMuDda  
    tl`x/   
    % Check normalization: q>.C5t'Qx  
    % -------------------- -Ua&/Yd/}  
    if nargin==5 && ischar(nflag) )&l5I4CIf  
        isnorm = strcmpi(nflag,'norm'); aLlHR_  
        if ~isnorm z<gII~%  
            error('zernfun:normalization','Unrecognized normalization flag.') ]GD&EQ  
        end $LiBJ~vV<  
    else Wl }J=  
        isnorm = false; IkO [R1K  
    end J0B*V0'zR  
    N:~4>p44[  
    t%Bh'HkG  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k{U[ U1j  
    % Compute the Zernike Polynomials E&f/*V^  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r_kaS als  
    `U&'71B^  
    2#N?WlYw<S  
    % Determine the required powers of r: A (H2Gt D  
    % ----------------------------------- `G%h=rr^c  
    m_abs = abs(m); 2sp4Mm  
    rpowers = []; 8U}+9  
    for j = 1:length(n) AQ,"):ofvT  
        rpowers = [rpowers m_abs(j):2:n(j)]; C_yNSD  
    end 8dC RSU  
    rpowers = unique(rpowers); Wr-I~>D%_  
    ~(B%E'  
    |;&I$'i  
    % Pre-compute the values of r raised to the required powers, }$g"|;<ha  
    % and compile them in a matrix: \:+ NVIN  
    % ----------------------------- fIJX5)D  
    if rpowers(1)==0 M^Tm{`O!  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); db&!t!#,  
        rpowern = cat(2,rpowern{:}); WD! " $  
        rpowern = [ones(length_r,1) rpowern]; /U-+ClZi@  
    else |<O^M q  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); <{@D^L6h  
        rpowern = cat(2,rpowern{:}); ^Cvt^cI  
    end vP=H 2P  
    XVb9)a  
    Z#D*HAd`  
    % Compute the values of the polynomials: U@D\+T0  
    % -------------------------------------- 57O|e/2  
    y = zeros(length_r,length(n)); $4qM\3x0,  
    for j = 1:length(n) B I=57  
        s = 0:(n(j)-m_abs(j))/2; fRq+pUx U  
        pows = n(j):-2:m_abs(j); s_^N=3Si   
        for k = length(s):-1:1 o{QV'dgu  
            p = (1-2*mod(s(k),2))* ... sB$ "mJ  
                       prod(2:(n(j)-s(k)))/              ... Q)lD2  
                       prod(2:s(k))/                     ... Z  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... %UhLCyC/  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); e/#6qCE  
            idx = (pows(k)==rpowers); J^S!GG'gb  
            y(:,j) = y(:,j) + p*rpowern(:,idx); kD7'BP/#  
        end TjI&8#AWBA  
         '-Oh$hqCx|  
        if isnorm ?%#no{9  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); K\zb+  
        end k8@bQ"#b  
    end AEDBr<  
    % END: Compute the Zernike Polynomials Zg0nsNA   
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `^ a:1^  
    9U4[o<G]=  
    )>U"WZ'<  
    % Compute the Zernike functions: Q7{{r&|t&  
    % ------------------------------ C'{B  
    idx_pos = m>0; wXZ9@(^  
    idx_neg = m<0; gm =C0Sp?  
    1ox#hQBoS  
    O(v>\MV  
    z = y; f`_{SU"3  
    if any(idx_pos) "] Uj _d  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); />pAZa  
    end :>Qu;Z1P  
    if any(idx_neg) IXlk1tHN4I  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ~4O3~Y_+GN  
    end 5rc3jIXc{|  
    (I(U23A~  
    MPn/"Fij$  
    % EOF zernfun -B! a O65^  
     
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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)  Bxj4rC[  
    _+}hId  
    DDE还是手动输入的呢? I<xcVY9L  
    !VrBoU4<d  
    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