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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, [|oG}'Xz  
    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, JAd .\2%Y  
    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? QUn!& 55  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? LYECX  
    slPr^)  
    npltsK):  
    qsW&kW~  
    2|lR@L sr  
    function z = zernfun(n,m,r,theta,nflag) 2PyuM=(Wt  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. +bLP+]7oZ  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N H`)eT6:|/  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Rf8Obk<  
    %   unit circle.  N is a vector of positive integers (including 0), and En9J7es_  
    %   M is a vector with the same number of elements as N.  Each element f}(4v1 T  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) NMK$$0U  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, LF!KP  
    %   and THETA is a vector of angles.  R and THETA must have the same S/)),~`4  
    %   length.  The output Z is a matrix with one column for every (N,M) e8("G[P >  
    %   pair, and one row for every (R,THETA) pair. PL&> p M  
    % \Hrcf+`  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 8(Te^] v#  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 8|)!E`TKSV  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral O U7OX]h  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, aC2Vz9e  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ]vz6DJs  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. JseKqJ?g  
    % x}K|\KXy  
    %   The Zernike functions are an orthogonal basis on the unit circle. 7V::P_aUY  
    %   They are used in disciplines such as astronomy, optics, and }Y.YJXum  
    %   optometry to describe functions on a circular domain. w/o^OjwQ  
    % xbhHP2F |  
    %   The following table lists the first 15 Zernike functions. sx=1pnP9`  
    % `oikSx$vB.  
    %       n    m    Zernike function           Normalization - @>]iBl  
    %       -------------------------------------------------- vw!7f|Pg ~  
    %       0    0    1                                 1 }C_g;7*  
    %       1    1    r * cos(theta)                    2 1gK^x^l*f  
    %       1   -1    r * sin(theta)                    2 5*Zz_ .  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) eK1l~W%  
    %       2    0    (2*r^2 - 1)                    sqrt(3) A+M4=  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) A4@z+ebb l  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) {z_cczJ-  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) L]z8'n,  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) dNf9,P_}  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) !`=iKe&%E  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) N\ Mdia  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :j3'+% '2  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) u-iQ  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) @4*eH\3  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) H if| z[0$  
    %       -------------------------------------------------- *(yw6(9%  
    % [DjlkA/Zg  
    %   Example 1: H4 O"^#5  
    % 2*w`l|Sx  
    %       % Display the Zernike function Z(n=5,m=1) }GURq#  
    %       x = -1:0.01:1; nw/g[/<;  
    %       [X,Y] = meshgrid(x,x); hk5!$#^  
    %       [theta,r] = cart2pol(X,Y); jG `PyIgw  
    %       idx = r<=1; .jP|b~  
    %       z = nan(size(X)); 1VFCK&  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); +sn0bi/rG  
    %       figure "" ^n^$  
    %       pcolor(x,x,z), shading interp 1N$OXLu  
    %       axis square, colorbar W#g!Usf:/  
    %       title('Zernike function Z_5^1(r,\theta)') ',[AKXJ  
    % 5Xxdm-0  
    %   Example 2: ?E!M%c@,  
    % >wqWIw.w>  
    %       % Display the first 10 Zernike functions uaP5(hUI  
    %       x = -1:0.01:1; .R` _"7  
    %       [X,Y] = meshgrid(x,x); ck `td%  
    %       [theta,r] = cart2pol(X,Y); [^a7l$fmi  
    %       idx = r<=1; }KUK|p5  
    %       z = nan(size(X)); j-J/yhWO&  
    %       n = [0  1  1  2  2  2  3  3  3  3]; )UU`uzU;u  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; \bF<f02P  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; <e 9d5-2  
    %       y = zernfun(n,m,r(idx),theta(idx)); u8?$W%eW  
    %       figure('Units','normalized') m=h/A xW  
    %       for k = 1:10 =jm\8sl~~  
    %           z(idx) = y(:,k); m\&99-j:@b  
    %           subplot(4,7,Nplot(k)) ?Mo)&,__  
    %           pcolor(x,x,z), shading interp w$&;s<0  
    %           set(gca,'XTick',[],'YTick',[]) e`LvHU_0  
    %           axis square #o~C0`8!B=  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) S3HyB b  
    %       end e@O]c "  
    % eW<NDI&b  
    %   See also ZERNPOL, ZERNFUN2. NoF|j57?u'  
    3dZj<(.  
    3jfAv@I~  
    %   Paul Fricker 11/13/2006  l2M(  
    um/F:rp  
    mBye)q$  
    mmjWLrhlu  
    *7*cWO=  
    % Check and prepare the inputs: X<Xiva85  
    % ----------------------------- 2H8\P+  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) TT;ls<(Lg  
        error('zernfun:NMvectors','N and M must be vectors.') dhP")@3K;p  
    end g*_n|7pB  
    =s.0 f:(  
    $P8AU81  
    if length(n)~=length(m) ,qB@agjvo<  
        error('zernfun:NMlength','N and M must be the same length.') ?)<zzL",  
    end _'y`hKeI[  
    W4P\HM>2  
    +,7vbs3  
    n = n(:); Fku<|1}&y  
    m = m(:); 8yOhKEPX  
    if any(mod(n-m,2)) uTO%O}D N  
        error('zernfun:NMmultiplesof2', ... !%(kMN  
              'All N and M must differ by multiples of 2 (including 0).') XLYGhM  
    end /Trbr]lWy  
    @!ja/Y^  
    G[`2Nd<  
    if any(m>n) sc-hO9~k  
        error('zernfun:MlessthanN', ... }=|{"C  
              'Each M must be less than or equal to its corresponding N.') 8ZjRMr}  
    end ($UUgjv F  
    3 Sf':N`u  
    3,hu3"@k  
    if any( r>1 | r<0 ) u+6L>7t88I  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') /Wl8Jf7'  
    end  (t@!0_5  
    vaVV 1  
    L|b[6[XTHL  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) hhU\$'0B-  
        error('zernfun:RTHvector','R and THETA must be vectors.') j-i>Jd7  
    end S5H}   
    C(>g4.-p8  
    T~ XKV`LQ  
    r = r(:); `|92!Ej  
    theta = theta(:); TZg1,Z  
    length_r = length(r); 5D7k[+6  
    if length_r~=length(theta) i&)([C0z$  
        error('zernfun:RTHlength', ... ZifDU@J$t  
              'The number of R- and THETA-values must be equal.') i3L2N~:V  
    end 2zv:j7  
    JXt_  
    ^e\$g2).  
    % Check normalization: Ne3YhCC>  
    % -------------------- )@tHS-Jf  
    if nargin==5 && ischar(nflag) Ui1s ]R  
        isnorm = strcmpi(nflag,'norm'); d|W=_7 z  
        if ~isnorm r1=j$G  
            error('zernfun:normalization','Unrecognized normalization flag.') y G mFi  
        end m' |wlI[lq  
    else `q  | )_  
        isnorm = false; fceO|mSz_  
    end MlS5/9m@^  
    3-Xd9ou  
    "So "oT1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4Xwb`?}-  
    % Compute the Zernike Polynomials /Q89y[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7dE.\#6r  
    ?h| DeD!s  
    m9D Tz$S.  
    % Determine the required powers of r: ]vV)$xMX  
    % ----------------------------------- x",ktE>9  
    m_abs = abs(m); +`$$^x  
    rpowers = []; BT$Oh4y4  
    for j = 1:length(n) 68<W6z  
        rpowers = [rpowers m_abs(j):2:n(j)]; 1IT(5Mleb  
    end '|Lv -7  
    rpowers = unique(rpowers); U1 rr=h g  
    k f|J  
    F$:UvW@e1  
    % Pre-compute the values of r raised to the required powers, @W==)S%O  
    % and compile them in a matrix: QOPh3+.5  
    % ----------------------------- \;Q!}_ K  
    if rpowers(1)==0 5'`DrTOA  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); *.D{d0A  
        rpowern = cat(2,rpowern{:}); -Oz! GX  
        rpowern = [ones(length_r,1) rpowern]; !\Cu J5U  
    else utn,`v   
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); $S2 /*  
        rpowern = cat(2,rpowern{:}); A9J{>f  
    end 0G Q8} r  
    -QBM^L  
    LN5q_ZvR  
    % Compute the values of the polynomials: -r~9'aEs  
    % -------------------------------------- <F-IF7>a  
    y = zeros(length_r,length(n)); B| M@o^Tf  
    for j = 1:length(n) Dk2Zl  
        s = 0:(n(j)-m_abs(j))/2; jJ'NYG  
        pows = n(j):-2:m_abs(j); m*i,|{UZ  
        for k = length(s):-1:1 E7w^A  
            p = (1-2*mod(s(k),2))* ... *1:kIi7_  
                       prod(2:(n(j)-s(k)))/              ... #e@[{s7  
                       prod(2:s(k))/                     ... g 4 $  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... >f1fvv6  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); vD/l`Ib:  
            idx = (pows(k)==rpowers); C58B(Ndo  
            y(:,j) = y(:,j) + p*rpowern(:,idx); \TDn q!)?  
        end Ri::Ek3qu  
         nT}i&t!q8@  
        if isnorm p=i6~   
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); =`C K`x  
        end TXs&*\  
    end o,0 Z^"|  
    % END: Compute the Zernike Polynomials LFYSur8  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9d=\BBNZ  
    \-[ >bsg  
    1C*mR%Q  
    % Compute the Zernike functions: f%[xl6VE;  
    % ------------------------------ *7L1SjZw  
    idx_pos = m>0; x>A[~s"|N  
    idx_neg = m<0; Y OvhMi  
    +<B"g{dLuX  
    ]zIIi%  
    z = y; bhsCeH  
    if any(idx_pos) 0Xn,q]@Z  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Z\n^m^Z =  
    end l!\~T"-7;:  
    if any(idx_neg) dAOJ: @y  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 8R(l~  
    end @ @(O##(7  
    cqm:[0Xf5>  
    |X6R 2I  
    % EOF zernfun ,WW=,P  
     
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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)  _p^Wc.[~M  
    9Q 7342  
    DDE还是手动输入的呢? vV"I}L  
    tC.etoh  
    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