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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, zUe)f~4  
    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, T!iRg=<bz  
    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? "#3p=}]  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? IK /@j  
    Ja*k |Rz~  
    ranlbxp2l  
    Z7Y+rP[l  
    _YT9zG  
    function z = zernfun(n,m,r,theta,nflag) NIzxSGk|  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. A3!xYG=+  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N WgV'T#*  
    %   and angular frequency M, evaluated at positions (R,THETA) on the #VLO6  
    %   unit circle.  N is a vector of positive integers (including 0), and ITq$8  
    %   M is a vector with the same number of elements as N.  Each element hv6w=?7  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) &ND8^lR=Y;  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, E&RiEhuv  
    %   and THETA is a vector of angles.  R and THETA must have the same ;)SWUXa;{  
    %   length.  The output Z is a matrix with one column for every (N,M) pYtvenBy  
    %   pair, and one row for every (R,THETA) pair. DaK2P;WP  
    % r N.<S[  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ^<}>]F_  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), W,g0n=2V  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral W{{{c2 .  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ]xYm@%>6  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized s--\<v  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Gp PlO]  
    % `4&a"`&$  
    %   The Zernike functions are an orthogonal basis on the unit circle. 4;_<CB  
    %   They are used in disciplines such as astronomy, optics, and 2".^Ma^D!  
    %   optometry to describe functions on a circular domain. --32kuF&(  
    % [xrM){ItW  
    %   The following table lists the first 15 Zernike functions. QIcg4\d%s  
    % _kJ?mTk  
    %       n    m    Zernike function           Normalization M<s Y_<z  
    %       -------------------------------------------------- }'FNGn.~#  
    %       0    0    1                                 1 Za&.sg3RG  
    %       1    1    r * cos(theta)                    2 B F,rZZL  
    %       1   -1    r * sin(theta)                    2 s*XwU  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) =N +Ou5D  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Wb|xEwqd`  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) \k8|3Y~g  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) rLy <3  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) |PI]v`[  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) +mr\AAFn  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Ao%;!(\I%  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) \Jcj4  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) q,W6wM;,E  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) e/h2E dY  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?]d [K>bv  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) @ k`^Z5tN  
    %       -------------------------------------------------- :=<0Z1S  
    % N~w4|q!]  
    %   Example 1: /!u#S9_B  
    % d+6 by,'  
    %       % Display the Zernike function Z(n=5,m=1) oQkY@)3.w  
    %       x = -1:0.01:1; F$;vPAxbK"  
    %       [X,Y] = meshgrid(x,x); 1o;*`  
    %       [theta,r] = cart2pol(X,Y); D>YbL0K>X~  
    %       idx = r<=1; p PF]&:&-b  
    %       z = nan(size(X)); (IrX \Y  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); c1,dT2:=  
    %       figure r RfPq  
    %       pcolor(x,x,z), shading interp rQ4i%.  
    %       axis square, colorbar (4U59<ie  
    %       title('Zernike function Z_5^1(r,\theta)') `$X|VAS2  
    % R]-$]koQO  
    %   Example 2: fO4e[g;G  
    % C&\vVNV;9  
    %       % Display the first 10 Zernike functions zW@OSKq4  
    %       x = -1:0.01:1; CD]2a@j {  
    %       [X,Y] = meshgrid(x,x); d^&F%)AT  
    %       [theta,r] = cart2pol(X,Y); iz2I4 _N  
    %       idx = r<=1; WF6'mg^^?  
    %       z = nan(size(X)); 7Y8B \B)w  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 4-?'gN_  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; s1[_Pk;!  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 4zF|}aiQ  
    %       y = zernfun(n,m,r(idx),theta(idx));  l*+"0  
    %       figure('Units','normalized') ]Tje6i F  
    %       for k = 1:10 Se o3a6o  
    %           z(idx) = y(:,k); rQncW~  
    %           subplot(4,7,Nplot(k)) I) $of9   
    %           pcolor(x,x,z), shading interp NMSpi[dr  
    %           set(gca,'XTick',[],'YTick',[]) ZU vA`   
    %           axis square A/ eZ!"Y  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) i w,F)O  
    %       end NZ\aK}?~!  
    % ~dIb>[7wy  
    %   See also ZERNPOL, ZERNFUN2. kXj%thDx  
    FmALmS  
    !n=@(bT*wT  
    %   Paul Fricker 11/13/2006 /12D >OK  
    "CEy r0h  
    W~1/vJ.*l  
    ]~,V(K  
    5-277?  
    % Check and prepare the inputs: ,_66U;T  
    % ----------------------------- :'OCQ.[{s  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) BO5gwvyI  
        error('zernfun:NMvectors','N and M must be vectors.') I5"ew=x#  
    end  c|N!ZYJI  
    iA~b[20&  
    Dm@wTt8N(  
    if length(n)~=length(m) *&j)"hX  
        error('zernfun:NMlength','N and M must be the same length.') 8ycmvpJ  
    end {__Z\D2I  
    /H)K_H#|;  
    w8t,?dY  
    n = n(:); Z=O2tR  
    m = m(:); Co2* -[R  
    if any(mod(n-m,2)) dqMR<Nl&  
        error('zernfun:NMmultiplesof2', ... *yuw8  
              'All N and M must differ by multiples of 2 (including 0).') Lw=.LN  
    end qYg4H|6  
    (89NK]2x  
    b$sw`Rsw  
    if any(m>n) S9Fg0E+J  
        error('zernfun:MlessthanN', ... t)o #!)|  
              'Each M must be less than or equal to its corresponding N.') Ejdw"P"  
    end -aiQp@^/J  
    A9! gww  
    _nR8L`l*z  
    if any( r>1 | r<0 ) H_'i.t 'SS  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') {~ yj]+Im  
    end Kp *nOZ  
    d7A08l{  
    4C(vBKl  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) "(}xIsy  
        error('zernfun:RTHvector','R and THETA must be vectors.') %3e}YQe)  
    end 0(s0<9s%  
    %zHNX4  
    4#uWj ?u  
    r = r(:); 6Q,-ZM=Z_p  
    theta = theta(:); ^e$;I8l  
    length_r = length(r); O6P0Am7s  
    if length_r~=length(theta) MN2#  
        error('zernfun:RTHlength', ... oK h#th  
              'The number of R- and THETA-values must be equal.') 09"C&X~  
    end R@``MC0  
    /)SwQgK#  
    B&0^3iKFi  
    % Check normalization: ?H7*?HV  
    % -------------------- rE"`q1b#  
    if nargin==5 && ischar(nflag) c/ wzV  
        isnorm = strcmpi(nflag,'norm'); ]GYO`,  
        if ~isnorm v*5n$UFV  
            error('zernfun:normalization','Unrecognized normalization flag.') -6MgC9]  
        end : j&M&+  
    else Wy%q9x]}  
        isnorm = false; )t{oyBT  
    end e*uaxh+7  
    SsDz>PP  
    58*s\*V` \  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wJJ4F$"b  
    % Compute the Zernike Polynomials Vg/{;uLAe  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% w[z^B&  
    _`WbR&d2Id  
    )((Jnm D  
    % Determine the required powers of r: nM  D^x  
    % ----------------------------------- SF61rm  
    m_abs = abs(m); k,M%/AXd  
    rpowers = []; v$"#9oh  
    for j = 1:length(n) ~#iRh6 ^98  
        rpowers = [rpowers m_abs(j):2:n(j)]; @D8c-`LC"*  
    end 9iWDEk  
    rpowers = unique(rpowers); ^.,pq?_  
    eX 9{wb(  
    -UkP{x)S  
    % Pre-compute the values of r raised to the required powers, o=?sMq1<  
    % and compile them in a matrix: 7/NXb  
    % ----------------------------- aksyr$d0V<  
    if rpowers(1)==0 y]9 3z!#Z  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); [AQ6ads)  
        rpowern = cat(2,rpowern{:}); l2>G +t(,  
        rpowern = [ones(length_r,1) rpowern]; e0"R7a  
    else bC?uy o"  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7f#[+i  
        rpowern = cat(2,rpowern{:});  L5""  
    end 8Cz_LyL  
    }pj>BK>  
    Z}.N4 /  
    % Compute the values of the polynomials: ./mh 9ax  
    % -------------------------------------- K8doYN  
    y = zeros(length_r,length(n)); LF <fp&C)h  
    for j = 1:length(n) z71.5n!C  
        s = 0:(n(j)-m_abs(j))/2; #gi0FXL  
        pows = n(j):-2:m_abs(j); y5iLFR3z  
        for k = length(s):-1:1 $6h:j#{JE  
            p = (1-2*mod(s(k),2))* ... 4x.'H18  
                       prod(2:(n(j)-s(k)))/              ... "T@9]>6.f  
                       prod(2:s(k))/                     ... q5?g/-_0[  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... q=ZLSBZ  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 31 4PcSc  
            idx = (pows(k)==rpowers); %5RY Ea  
            y(:,j) = y(:,j) + p*rpowern(:,idx); oAe]/j$  
        end 5Fm.] /  
         WM NcPHcj  
        if isnorm 8I$B^,N  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); W$ #FM$U  
        end 6#fl1GdH-  
    end gxpR#/(E~  
    % END: Compute the Zernike Polynomials ^kxkP}[Z.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7 }>j [  
    x;S v&  
    +4f>njARIb  
    % Compute the Zernike functions: z:bxnM2\  
    % ------------------------------ < i"U%Ds(  
    idx_pos = m>0; V"(S<o  
    idx_neg = m<0; f:\jPkf'  
    Ev%4}GwO4  
    9r@r\-  
    z = y; ~T>_}Q[M2p  
    if any(idx_pos) T[B@7$Dp*  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); -X5rGp++  
    end /]2-I_WB  
    if any(idx_neg) mZ3i#a4  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); lBh|+K N  
    end Oyhl*`-*t  
    Cq;t;qN,nQ  
    _,_>B8  
    % EOF zernfun _H>ABo  
     
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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)  x4fl=  
    tgK$}#.*  
    DDE还是手动输入的呢? Q1J./C}  
    ["|AD,$%  
    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