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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, +#IsRiH%>  
    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, 13v#  
    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? VM[U&g<8n  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 7UzbS,$x  
    1O{x9a5Z?O  
    *'&mcEpg  
    j9XRC9   
    z/&2Se:  
    function z = zernfun(n,m,r,theta,nflag) Nt^9N #+N  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. FPB O=?H.  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1s@%q <  
    %   and angular frequency M, evaluated at positions (R,THETA) on the alB[/.1  
    %   unit circle.  N is a vector of positive integers (including 0), and AO "pm  
    %   M is a vector with the same number of elements as N.  Each element $Z8=QlG>  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) _ Uxt9 X  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Ous_269cM  
    %   and THETA is a vector of angles.  R and THETA must have the same h;(#^+LH  
    %   length.  The output Z is a matrix with one column for every (N,M) D3BNA]P\2@  
    %   pair, and one row for every (R,THETA) pair. Ka$YKY,  
    % ~c*$w O\  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike MsL*\)*s  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9N kr=/I"P  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 3TS(il9A  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, .2V`sg.!  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized :UrS@W^B  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?z]h Ysy  
    % kUp[b~  
    %   The Zernike functions are an orthogonal basis on the unit circle. rnV\O L  
    %   They are used in disciplines such as astronomy, optics, and GV aIZh<  
    %   optometry to describe functions on a circular domain. ~VqDh*0  
    % I2R" Y<  
    %   The following table lists the first 15 Zernike functions. r?l7_aBv3  
    % snW=9b)m  
    %       n    m    Zernike function           Normalization ;>z.wol  
    %       -------------------------------------------------- ~)kOO oH  
    %       0    0    1                                 1 WHM|kt  
    %       1    1    r * cos(theta)                    2 /I>o6CI  
    %       1   -1    r * sin(theta)                    2 }{(dG7G+  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) -/O_wqm#  
    %       2    0    (2*r^2 - 1)                    sqrt(3) DnZkZ;E/  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) )zR(e>VX  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 0F495'*A  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) S3G9/  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) yG`J3++ S  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 2qF ?%  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) TI9]v(  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 88GS Bg:YH  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ~_ 8X%ut y  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?C[W~m P  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) A=(<g";m  
    %       -------------------------------------------------- zP8a=Iv  
    % ~KW|<n4m  
    %   Example 1: ]hPu  
    % ka^sOC+Y  
    %       % Display the Zernike function Z(n=5,m=1) TBGN',,  
    %       x = -1:0.01:1; ey~5DY7  
    %       [X,Y] = meshgrid(x,x); $@[`v0y*  
    %       [theta,r] = cart2pol(X,Y); {7%W /C#A  
    %       idx = r<=1; a%"27 n(M  
    %       z = nan(size(X)); Cmsg'KqqT  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); R@+%~"Z  
    %       figure l. 9 i `  
    %       pcolor(x,x,z), shading interp yFYFFv\?  
    %       axis square, colorbar -Dx_:k|k  
    %       title('Zernike function Z_5^1(r,\theta)') m=hlim;P,  
    % @&AUbxoj  
    %   Example 2: i1OF @~?  
    % ?51Y&gOEZ  
    %       % Display the first 10 Zernike functions /.{q2]  
    %       x = -1:0.01:1; O)$rC  
    %       [X,Y] = meshgrid(x,x); TspuZR@2  
    %       [theta,r] = cart2pol(X,Y); q$|Wxnz  
    %       idx = r<=1; ~^{jfHTlv  
    %       z = nan(size(X)); oV(|51(f  
    %       n = [0  1  1  2  2  2  3  3  3  3]; h2b,(  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; %a_ rYrL  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 8%@![$q<g  
    %       y = zernfun(n,m,r(idx),theta(idx)); j>{Dbl:#2  
    %       figure('Units','normalized') YPV@/n[N  
    %       for k = 1:10 Em%0C@C  
    %           z(idx) = y(:,k); &tAhRMa  
    %           subplot(4,7,Nplot(k)) Mx3MNX /  
    %           pcolor(x,x,z), shading interp iB]xYfQ&@V  
    %           set(gca,'XTick',[],'YTick',[]) LkUYh3  
    %           axis square TQ/EH~Sz  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ","O8'$OC  
    %       end m ll-cp  
    % `Mh 3v@K:  
    %   See also ZERNPOL, ZERNFUN2. {Tps3{|wt  
    IpJMq^ Z  
    $j+RUelFY  
    %   Paul Fricker 11/13/2006 ji|+E`Nii  
    6ka, FjJ\  
    r~q(m>Ct6  
    ]tjQy1M  
    n0ZrgTVJ  
    % Check and prepare the inputs: z f rEM  
    % ----------------------------- 9_h  V1:  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) K~C6dy  
        error('zernfun:NMvectors','N and M must be vectors.') StuQ}  
    end a7]wPXKq  
    ,esryFRG  
    >{v,H Oxl  
    if length(n)~=length(m) ""; Bq*Y#  
        error('zernfun:NMlength','N and M must be the same length.') d7f{2  
    end rT&rv^>f  
    +;T `uOF}  
    ;*H@E(g  
    n = n(:); (R9{wGV [  
    m = m(:); ;ewqGDe'3  
    if any(mod(n-m,2)) fLtN-w6t  
        error('zernfun:NMmultiplesof2', ... vhEqHjR:  
              'All N and M must differ by multiples of 2 (including 0).') 3.t j%+  
    end }MCh$  
    1 +0-VRl  
    5 $vUdDTg  
    if any(m>n) `GBa3  
        error('zernfun:MlessthanN', ... O<RLw)nzg  
              'Each M must be less than or equal to its corresponding N.') )$> pu{o  
    end FQ3{~05T  
    )F$<-0pT  
    yx@%x?B  
    if any( r>1 | r<0 ) 2`[iTBZ=^  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 9W7 ljUg  
    end g5YDRL!Wh  
    Qf>$'C(7!a  
    7xhBdi[ dQ  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 5Al1u|;HB  
        error('zernfun:RTHvector','R and THETA must be vectors.') X0}+X'3  
    end ^%qe&Pe2  
    |#Gug('  
    0E<xzYo  
    r = r(:); fK0VFN8<I  
    theta = theta(:); *K57($F  
    length_r = length(r); J [k,S(Y  
    if length_r~=length(theta) Hdn%r<+c  
        error('zernfun:RTHlength', ... P,eP>55'K  
              'The number of R- and THETA-values must be equal.') z>6hK:27  
    end j6JK4{  
    pef)c,U$  
    pkKcTY1Fx  
    % Check normalization:  jO5,PTV  
    % -------------------- ^5GyW`a}  
    if nargin==5 && ischar(nflag) 1Mtm?3Pt  
        isnorm = strcmpi(nflag,'norm'); xpU7ZY  
        if ~isnorm pnpx`u;  
            error('zernfun:normalization','Unrecognized normalization flag.') F YLBaN  
        end  EL$"/ptE  
    else w<P$)~6  
        isnorm = false; J-k/#A4o  
    end rP7[{'%r  
    . XVW2ISv  
    vT&) 5nN  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %aB RL6  
    % Compute the Zernike Polynomials 9*<=K  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YaT6vSz  
    %0gcNk"=  
    #$^vP/"$  
    % Determine the required powers of r: &Rp/y%9  
    % ----------------------------------- dc+U #]tS  
    m_abs = abs(m); 0DB8[#i%:  
    rpowers = []; \,ko'4 8@  
    for j = 1:length(n) z|k0${iu#  
        rpowers = [rpowers m_abs(j):2:n(j)]; E5+-N  
    end l2*o@&.  
    rpowers = unique(rpowers); TS Ev^u)3  
    8{f~tPY  
    %S$+ 3q%F  
    % Pre-compute the values of r raised to the required powers, .*k$abb  
    % and compile them in a matrix: h6(\ tRd!\  
    % ----------------------------- |lG7/\A  
    if rpowers(1)==0 I)AbH<G{  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); t9\}!{<s  
        rpowern = cat(2,rpowern{:}); c]+uj q  
        rpowern = [ones(length_r,1) rpowern]; $[xS>iuD  
    else LZI[5tA"  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); sq45fRAi  
        rpowern = cat(2,rpowern{:}); 9{cpxJ  
    end )7jJ3G*  
    6>Z)w}x^  
    4/?@ %  
    % Compute the values of the polynomials: WcOnv'l,  
    % -------------------------------------- nSr_sD6"  
    y = zeros(length_r,length(n)); uf/4vz,  
    for j = 1:length(n) uz /Wbc>y  
        s = 0:(n(j)-m_abs(j))/2; 3Jh!YzI8  
        pows = n(j):-2:m_abs(j); 8:sQB% BB  
        for k = length(s):-1:1 H 2JKQm_  
            p = (1-2*mod(s(k),2))* ... 4Nl3"@<$  
                       prod(2:(n(j)-s(k)))/              ... %nV6#pr  
                       prod(2:s(k))/                     ... @9e}kiW  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 5XzN%<_h9  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); [lz#+~rOS  
            idx = (pows(k)==rpowers); Wi+}qO  
            y(:,j) = y(:,j) + p*rpowern(:,idx); eq6>C7.$  
        end tu:W1?  
         hCPyCq]  
        if isnorm A:4?Jd>  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); :%4N4| Q  
        end `Iqh\oY8-  
    end BS|$-i5L  
    % END: Compute the Zernike Polynomials _O3X;U7rc  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EpCF/i?9:  
    PSHzB! H=n  
    Pq(7lua7  
    % Compute the Zernike functions: r]=Z :  
    % ------------------------------ 7tP qez#  
    idx_pos = m>0; jYk5]2#A  
    idx_neg = m<0; <  UD90}  
    ~$jRn(2  
    P}=U #AV4  
    z = y; hlBMRx49  
    if any(idx_pos) mfx-Ja_a  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); `>Ms7G9S~e  
    end .x'?&7#(  
    if any(idx_neg) p|>m 2(|  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); nt_FqUJ  
    end ):]5WHYg  
    SajasjE!^1  
    0WSOA[R%[b  
    % EOF zernfun ] B?NDxU  
     
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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)  7eW6$$ju,N  
    Qjd]BX;  
    DDE还是手动输入的呢? "bz.nE*  
    1~E;@eK'  
    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