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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, 5rmQ:8_5  
    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, l dp$jrNLr  
    这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? q$"?P  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? p,!IPWo  
    e X6o 7a  
    pl$wy}W-  
    mq(-L  
    Cq'{ %  
    function z = zernfun(n,m,r,theta,nflag) `g4N]<@z  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. o-JB,^TE  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Rt5pl,Nf  
    %   and angular frequency M, evaluated at positions (R,THETA) on the eu":\ks  
    %   unit circle.  N is a vector of positive integers (including 0), and <":83RCS  
    %   M is a vector with the same number of elements as N.  Each element hT`&Xb  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) b"nkF\P@Fj  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, C ](djkA$  
    %   and THETA is a vector of angles.  R and THETA must have the same wQ[!~>A  
    %   length.  The output Z is a matrix with one column for every (N,M) 9+/D\|"{  
    %   pair, and one row for every (R,THETA) pair. \HG4i/V:h  
    % 1_l)$"  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /a)^)  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), N(3Bzd)   
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 'Gamb+[  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, PZO.$'L|7  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Cl3L)  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. t=|}?lN<  
    % Qvel#*-4  
    %   The Zernike functions are an orthogonal basis on the unit circle. L\5:od[EP  
    %   They are used in disciplines such as astronomy, optics, and h:sf?X[  
    %   optometry to describe functions on a circular domain. QpRk5NeLe  
    % Q laoa)d#  
    %   The following table lists the first 15 Zernike functions. dBS_N/  
    % GG-b)64h`  
    %       n    m    Zernike function           Normalization Dy 8H(_  
    %       -------------------------------------------------- ?P4y$P  
    %       0    0    1                                 1 f.bwA x  
    %       1    1    r * cos(theta)                    2 2aX$7E?  
    %       1   -1    r * sin(theta)                    2 D,|TQ Q  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Q7{{r&|t&  
    %       2    0    (2*r^2 - 1)                    sqrt(3) C'{B  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) wXZ9@(^  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) gm =C0Sp?  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) yeBfzKI{b  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ZS=;)  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ]6s/y  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ,4 q^(  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) hJ8% r_  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) N U+PG`Vb  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )X:Sfk  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) T 1_B0H2  
    %       -------------------------------------------------- hl] y):  
    % o iC@ /  
    %   Example 1: /m,i,NX07  
    % G N=8;Kq%  
    %       % Display the Zernike function Z(n=5,m=1) t0kZFU  
    %       x = -1:0.01:1; !VsdKG)  
    %       [X,Y] = meshgrid(x,x); >[wB|V5  
    %       [theta,r] = cart2pol(X,Y); g0;;+z  
    %       idx = r<=1; {P\Ob0)q  
    %       z = nan(size(X)); {'B(S/Z 7  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Gpcordt/  
    %       figure A f!`7l-  
    %       pcolor(x,x,z), shading interp q?)5yukeF  
    %       axis square, colorbar M?Q\ Hw  
    %       title('Zernike function Z_5^1(r,\theta)') >{-rl@^H:  
    % !'IZr{Y>  
    %   Example 2: Uovna:"  
    % b'` XFB#V  
    %       % Display the first 10 Zernike functions y4aT-^C'  
    %       x = -1:0.01:1; (l9jczi  
    %       [X,Y] = meshgrid(x,x); Pn4jI(  
    %       [theta,r] = cart2pol(X,Y); o4@d,uIw^  
    %       idx = r<=1; YC<FKWc  
    %       z = nan(size(X)); 2V$Jn8v,`{  
    %       n = [0  1  1  2  2  2  3  3  3  3]; l-!"   
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; wZbT*rU  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; g\?07@Zd|  
    %       y = zernfun(n,m,r(idx),theta(idx)); rc7c$3#X  
    %       figure('Units','normalized') Eza^Tbq%j?  
    %       for k = 1:10 *~cNUyd  
    %           z(idx) = y(:,k); Ov4 [gHy&  
    %           subplot(4,7,Nplot(k)) %[ *+  
    %           pcolor(x,x,z), shading interp Xc^(e?L4  
    %           set(gca,'XTick',[],'YTick',[]) U3v~R4  
    %           axis square "LW\osjen  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) zV;NRf) 9.  
    %       end V$;`#J$\b  
    % w40*vBz  
    %   See also ZERNPOL, ZERNFUN2. W<[7LdAB  
    Ol<LL#<j4  
    H4{7,n  
    %   Paul Fricker 11/13/2006 (^sb('"  
    $Fy~xMA8O  
    pU,\ &3N  
    $P#+Y,r~\  
    3,{;wJ Z  
    % Check and prepare the inputs: qoZAZ&|HI  
    % ----------------------------- K|6}g7&X  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) [nX{ sM%  
        error('zernfun:NMvectors','N and M must be vectors.') SrOv* D3  
    end JHVndK4L  
    hp}rCy|01  
    #BS!J&a  
    if length(n)~=length(m) )cZ KB0*+  
        error('zernfun:NMlength','N and M must be the same length.') f`\J%9U_O  
    end mz;ExV16  
    Z/v )^VR  
    <5CQ#^ cK  
    n = n(:); sk0/3X*Q%  
    m = m(:); gh"_,ZhZt  
    if any(mod(n-m,2)) m9jjKu]|  
        error('zernfun:NMmultiplesof2', ... %?qzP '  
              'All N and M must differ by multiples of 2 (including 0).') W=|'&UU Ul  
    end QV*la=j/  
    >SYOtzg%  
    @cm[]]f'l  
    if any(m>n) !VrBoU4<d  
        error('zernfun:MlessthanN', ... c\tw#;\9  
              'Each M must be less than or equal to its corresponding N.') ?6I`$ &OA  
    end rfZg  
    ?9 `T_,  
    `$3P@SO"  
    if any( r>1 | r<0 ) \S~<C[P  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') CaoQPb*  
    end 5VfpeA `  
    o5Knot)Oy  
    (.{."  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) "e29j'u!*  
        error('zernfun:RTHvector','R and THETA must be vectors.') m^)\P?M5|  
    end Th~pju  
    [!ZYtp?Hf  
    td#m>S  
    r = r(:); b~8&P_  
    theta = theta(:); =aehhs>  
    length_r = length(r); }ASBP:c"t  
    if length_r~=length(theta) K:pG<oV|}  
        error('zernfun:RTHlength', ... MU N:}S  
              'The number of R- and THETA-values must be equal.') >4#\ U!  
    end otP2qAI  
    )*o) iN 7l  
    5=4-IO6W[]  
    % Check normalization: [FWB  
    % -------------------- z:{R4#(Q  
    if nargin==5 && ischar(nflag) -**fT?n  
        isnorm = strcmpi(nflag,'norm'); 2Paw*"U  
        if ~isnorm [Kbna>`  
            error('zernfun:normalization','Unrecognized normalization flag.') SC2g5i`  
        end Ew9 MWlk  
    else \nQEvcH  
        isnorm = false; )9!ZkZbv_m  
    end M49Hm[0(  
    i\MW'b  
    +.hJ[|F1&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% D[Ld=e8t  
    % Compute the Zernike Polynomials `R$bx 64  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wp-3U}P2(  
    6(HJYa  
    RWn#"~  
    % Determine the required powers of r: jqoU;u`  
    % ----------------------------------- HsK5 2<  
    m_abs = abs(m); "n<u(m8E  
    rpowers = []; a6o p  
    for j = 1:length(n) 8EI&}I  
        rpowers = [rpowers m_abs(j):2:n(j)]; z &[[4[  
    end  R"PO@v  
    rpowers = unique(rpowers); W8!8/ IZbN  
    8@I.\u)0  
    6r,zOs-I]  
    % Pre-compute the values of r raised to the required powers, Szlww  
    % and compile them in a matrix: )v.\4Q4  
    % ----------------------------- /B  
    if rpowers(1)==0 It^_?oiK  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); rX&?Xi1JeV  
        rpowern = cat(2,rpowern{:}); Y+~>9-S  
        rpowern = [ones(length_r,1) rpowern]; ]}A yDy6C  
    else k${F7I(Tb  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %M05& <  
        rpowern = cat(2,rpowern{:}); N{zou?+  
    end 0'*'%Iga  
    t]pJt  
    .ZH5^Sv$vp  
    % Compute the values of the polynomials: Xec U&  
    % -------------------------------------- yAVt[+0  
    y = zeros(length_r,length(n)); DzCb'#   
    for j = 1:length(n) ~bJ*LM?wOP  
        s = 0:(n(j)-m_abs(j))/2; YA^g[,  
        pows = n(j):-2:m_abs(j); `#N7ym;s@  
        for k = length(s):-1:1 N&lKo}hk  
            p = (1-2*mod(s(k),2))* ... Ad`jV_z  
                       prod(2:(n(j)-s(k)))/              ... z3-AYQ.H  
                       prod(2:s(k))/                     ... Z7R+'OC  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... '~'3x4Bo  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); j-etEWOTr  
            idx = (pows(k)==rpowers); h%@#jvh?4  
            y(:,j) = y(:,j) + p*rpowern(:,idx); b ~FmX  
        end /d-7n|#E  
         ,cFp5tV$  
        if isnorm K3t^y`z  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); rW3fd.;kss  
        end yh Ymbu  
    end LHP?!rO0  
    % END: Compute the Zernike Polynomials ]7{-HuQ8>}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v|mZcAz  
    bga2{<VF  
    x;R9Gc[5  
    % Compute the Zernike functions: zHCz[jlrMq  
    % ------------------------------ -f:uNF]Ls  
    idx_pos = m>0; 3bPvL/\Lb  
    idx_neg = m<0; /c1FFkq|K  
    I*K~GXWs#  
    !xK`:[B  
    z = y; = 8%+$vX  
    if any(idx_pos) VN8ao0^d;d  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 1 {V*(=Tp  
    end "2bCq]I0  
    if any(idx_neg) I2'UC) 0  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); C,D~2G  
    end w~g)Dz2G  
     `#lNur\x  
    4<&`\<jZ  
    % EOF zernfun [e'Ts#($A  
     
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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)  Iw8;",e2  
    a5:Q%F<!  
    DDE还是手动输入的呢? JNRG [j  
    f7}/ {}g  
    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