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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, `0rRKlbj4  
    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, Ik Qe~;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? tvGlp)?.  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? kutJd{68  
    -x{&an=  
    ' Rc#^U*n  
    Lc%xc`n8B  
    0p `")/  
    function z = zernfun(n,m,r,theta,nflag) WFem#hq   
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. r8,om^N6  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N TM-Fu([LMV  
    %   and angular frequency M, evaluated at positions (R,THETA) on the kM;o0wi  
    %   unit circle.  N is a vector of positive integers (including 0), and Mb.4J2F?  
    %   M is a vector with the same number of elements as N.  Each element `BjR.xMv  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) +?Ez} BP  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, $ser+Jt=  
    %   and THETA is a vector of angles.  R and THETA must have the same r**f,PDZ  
    %   length.  The output Z is a matrix with one column for every (N,M) :3O5ET'1  
    %   pair, and one row for every (R,THETA) pair. <h@]Ri  
    % vY_eDJ~'  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike %J!NL0x_  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ot }6D  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral @Z q[e   
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 0m A(:"  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized (hN?:q?'  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *VDVC0R  
    % X3yS5wh d(  
    %   The Zernike functions are an orthogonal basis on the unit circle. r^5jh1  
    %   They are used in disciplines such as astronomy, optics, and (;ADW+.`J  
    %   optometry to describe functions on a circular domain. n}q$f|4!  
    % zN")elBi  
    %   The following table lists the first 15 Zernike functions. V^sc1ak1Q  
    % i ?-Y  
    %       n    m    Zernike function           Normalization n"Z |e tZ4  
    %       -------------------------------------------------- ;A"\?i Q  
    %       0    0    1                                 1 *HeVACxo  
    %       1    1    r * cos(theta)                    2 kP^*h O!%  
    %       1   -1    r * sin(theta)                    2 `ET& VV  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) #c:kCZt#  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ``4?a7!!  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) i*CnoQH  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ^{[[Z.&R?  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) #U"1 9@|}  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) J@Yj\9U  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) J>h;_jA  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) BIj   
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) wE6A 7\k%  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) F0.zi>5  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Mk5RHDh  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) hKN6y%  
    %       -------------------------------------------------- ) rpq+~b  
    % b#='^W3  
    %   Example 1: %b?uW] j:  
    % 6$RpV'xz  
    %       % Display the Zernike function Z(n=5,m=1) taDQ65  
    %       x = -1:0.01:1; .iT4-  
    %       [X,Y] = meshgrid(x,x); [K:29N9~4  
    %       [theta,r] = cart2pol(X,Y); |,sM ST%  
    %       idx = r<=1; &*gbK6JB  
    %       z = nan(size(X)); &,MFB  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Ct!S Tk[2  
    %       figure FYl3c   
    %       pcolor(x,x,z), shading interp !\x?R6K  
    %       axis square, colorbar {[^#h|U  
    %       title('Zernike function Z_5^1(r,\theta)') Nfb`YU=  
    % PeNF+5s/K  
    %   Example 2: :<utq|#s  
    % ir&.Z5=  
    %       % Display the first 10 Zernike functions [r9d<Zi}{  
    %       x = -1:0.01:1; B*79qq  
    %       [X,Y] = meshgrid(x,x);  zy>}L #  
    %       [theta,r] = cart2pol(X,Y); wS$46M<  
    %       idx = r<=1; u)~s4tP4  
    %       z = nan(size(X)); vYnftJK&  
    %       n = [0  1  1  2  2  2  3  3  3  3]; A*i_|]Q  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; .NnGVxc5*  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; rQj~[Y.c  
    %       y = zernfun(n,m,r(idx),theta(idx)); BIfi:7I;Q  
    %       figure('Units','normalized') vgThK9{m;  
    %       for k = 1:10 9@y3IiZ"}  
    %           z(idx) = y(:,k); XU9'Rfp  
    %           subplot(4,7,Nplot(k)) %VJW@S>j/  
    %           pcolor(x,x,z), shading interp Ue7 6py9  
    %           set(gca,'XTick',[],'YTick',[]) %?=)!;[  
    %           axis square RL&lKHA  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) OKPJuV`y6  
    %       end % rcFT_  
    % {ERjeuDm]  
    %   See also ZERNPOL, ZERNFUN2. m =k%,J_  
    r/PKrw sC  
    .@k*p>K  
    %   Paul Fricker 11/13/2006 &t_h'JX&  
    \ja `c)x  
    ny1 \4C  
    [hnK/4!  
    P4 6,o  
    % Check and prepare the inputs: jdlG#j-\  
    % ----------------------------- rBfg*r`)  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) j-32S!  
        error('zernfun:NMvectors','N and M must be vectors.') _9kIRmT{  
    end j@v-|  
    D9o*8h2$  
    n(R_#,Hs  
    if length(n)~=length(m) o](.368+4  
        error('zernfun:NMlength','N and M must be the same length.') h=[-Er'B  
    end ~6d5zI4\  
    :hP58 }Q$  
    } yq  
    n = n(:); T2|:nC)@  
    m = m(:); _}ele+  
    if any(mod(n-m,2)) ,sI35I J  
        error('zernfun:NMmultiplesof2', ... %6i=lyH-  
              'All N and M must differ by multiples of 2 (including 0).') fU ={a2  
    end oMc1:=EG  
    W ~NYU  
    4B$bj `h  
    if any(m>n) P) 1 EA;  
        error('zernfun:MlessthanN', ... kl<g;3  
              'Each M must be less than or equal to its corresponding N.') 2AK}D%jfc  
    end (\& 62B1  
    !Uy>eji}  
    6~@5X}^<0  
    if any( r>1 | r<0 ) Z4@y?f v7s  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 7K:FeW'N  
    end \ V?I+Gc  
    qZbHMTnT6  
    [YE?OQ7#  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) )X%oXc&C|  
        error('zernfun:RTHvector','R and THETA must be vectors.') 0jTMZ<&zZ  
    end a}yR p  
    KPggDKS  
    Cuv|6t75'  
    r = r(:); tJm{I)G  
    theta = theta(:); ^c'f<<z|7r  
    length_r = length(r); u){S$</  
    if length_r~=length(theta) 3:AU:  
        error('zernfun:RTHlength', ... 61,O%lV  
              'The number of R- and THETA-values must be equal.') kf K[u/<i  
    end E9R]sXf8  
    ^A#x<J+  
    w4A#>;Qu*  
    % Check normalization: `^e*T'UPl  
    % -------------------- \(bj(any  
    if nargin==5 && ischar(nflag) yHOqzq56  
        isnorm = strcmpi(nflag,'norm'); dEET}s\  
        if ~isnorm 4if\5P:j  
            error('zernfun:normalization','Unrecognized normalization flag.') UR,?!rJ^B  
        end Z@oKz:U  
    else ^O \q3HA_4  
        isnorm = false; )Ga8`t"  
    end u\3ZIb  
    UM\}aq=,  
    xT=ySa$|>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KBj@V6Q  
    % Compute the Zernike Polynomials 0%H24N 9.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |0]YA  
    hXTYTbTX  
    kQ[Jo%YT?E  
    % Determine the required powers of r: `u=oeM :  
    % ----------------------------------- #G~wE*VR$  
    m_abs = abs(m); tvCcyD%w  
    rpowers = []; X TM$a9)  
    for j = 1:length(n) t%HI1eO7h  
        rpowers = [rpowers m_abs(j):2:n(j)]; b=G4MZQ  
    end ogp{rY  
    rpowers = unique(rpowers); ]_\AHnJ  
    Hh\ 4MNl  
    Iu%^*K%  
    % Pre-compute the values of r raised to the required powers, S*s:4uf  
    % and compile them in a matrix: l.uN$B  
    % ----------------------------- ->3uOF!q  
    if rpowers(1)==0 &t_A0z  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); yWmrdvL  
        rpowern = cat(2,rpowern{:}); $r):d  
        rpowern = [ones(length_r,1) rpowern]; ?(>k,[n  
    else HoL~j({  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); z6 a,0&;-L  
        rpowern = cat(2,rpowern{:}); WV@X@]U  
    end i0b.AA  
    1y~L8!: L  
    7|{ B#  
    % Compute the values of the polynomials: VZTmzIk.Y  
    % -------------------------------------- "&Gw1.p  
    y = zeros(length_r,length(n)); #)FDl70S8  
    for j = 1:length(n) @Jm.HST#S8  
        s = 0:(n(j)-m_abs(j))/2; yYM_lobn  
        pows = n(j):-2:m_abs(j); hAlPl<BO#V  
        for k = length(s):-1:1 G LoiH#R  
            p = (1-2*mod(s(k),2))* ... S7Znz@  
                       prod(2:(n(j)-s(k)))/              ... brj[c>ID  
                       prod(2:s(k))/                     ... OgQntj:%lN  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ovB=Zm  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); L,WkJe3  
            idx = (pows(k)==rpowers);  w"BIv9N  
            y(:,j) = y(:,j) + p*rpowern(:,idx); D(!;V KH  
        end ygMd$0:MN  
         b]"2 VN  
        if isnorm 3Fgz)*Gu]  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); JV&Zwbu  
        end )=y.^@UT@  
    end ?THa5%8f  
    % END: Compute the Zernike Polynomials O/(3 87=U  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LNaeB(z"  
    dV)Y,Yx0${  
    z}iSq$  
    % Compute the Zernike functions: (X*'y*:  
    % ------------------------------ n%n'1AUP:  
    idx_pos = m>0; hN[X 1*  
    idx_neg = m<0; A 0 S8Dh$  
    (v]P<3%  
    bB y'v/  
    z = y; PBCb0[\  
    if any(idx_pos) kp'b>&9r  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); )4@M`8  
    end q)NXyy4BT  
    if any(idx_neg) ,tau9>!  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); j,\tejl1  
    end Wa(W&]  
    bAN10U  
    3'.! +#  
    % EOF zernfun JIVo=5c}  
     
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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)  `_NnQ%  
    u=p ;A1oy  
    DDE还是手动输入的呢? >i^y;5  
    R`0foSq \M  
    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