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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 正序阅读 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, 7'[C+/:  
    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, :x*8*@kC  
    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? (D'Z4Y  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? L3Leb%,!  
    n6gYZd  
    c<V.\y0x  
    k'N `5M)  
    ?VMj;+'tr  
    function z = zernfun(n,m,r,theta,nflag) /j]r?KAzw  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. "y>\ mC  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ]P TTI\n  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ,L+tm>I  
    %   unit circle.  N is a vector of positive integers (including 0), and #@,39!;,:O  
    %   M is a vector with the same number of elements as N.  Each element v>3)^l:=Y*  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Sti)YCXH  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Q6y883>9  
    %   and THETA is a vector of angles.  R and THETA must have the same PXGS5,  
    %   length.  The output Z is a matrix with one column for every (N,M) S;$@?vF  
    %   pair, and one row for every (R,THETA) pair. 4z-sR/d  
    % P'#m1ntxQ  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike @GGzah#  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), \ ?[#>L4  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral t"`LJE._P  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 40pGu  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized b 2n.v.$G  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. n4%|F'ma  
    % f\"Qgn  
    %   The Zernike functions are an orthogonal basis on the unit circle. J/j1Yf'9  
    %   They are used in disciplines such as astronomy, optics, and %t0Fx  
    %   optometry to describe functions on a circular domain. 'kc_OvVA  
    % ~R.8r-kD`  
    %   The following table lists the first 15 Zernike functions. *b?C%a9  
    % :Ia3yi#  
    %       n    m    Zernike function           Normalization A~Eu_m  
    %       -------------------------------------------------- @v9 PI/c  
    %       0    0    1                                 1 L0SeG:  
    %       1    1    r * cos(theta)                    2 ]RmQ*F-  
    %       1   -1    r * sin(theta)                    2 ^RG6h  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 0SV#M6`GX  
    %       2    0    (2*r^2 - 1)                    sqrt(3) :g3n [7wR  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 4NL Tt K  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) SMaC{RPQ  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) \)VV6'zih  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) CGIcuHp  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) QBa1c-Y  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) XOO!jnQu  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) vV1F|  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ]]$s"F<  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) QthHQA  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ;Jt*s  
    %       -------------------------------------------------- 38! $9)  
    % {*H&NI  
    %   Example 1: T#^   
    % s)"C~w^  
    %       % Display the Zernike function Z(n=5,m=1) _3h(R`VdWO  
    %       x = -1:0.01:1; o)'T#uK  
    %       [X,Y] = meshgrid(x,x); K1Nhz'^=D  
    %       [theta,r] = cart2pol(X,Y); i]*W t8~!  
    %       idx = r<=1; >z6 (fM`i  
    %       z = nan(size(X)); !o 2" th  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Lm\N`  
    %       figure 7X.rGJZq  
    %       pcolor(x,x,z), shading interp z %` \p  
    %       axis square, colorbar pt;E~_  
    %       title('Zernike function Z_5^1(r,\theta)') Mjq1qEi"B  
    % =,KRZqz  
    %   Example 2: |c,":R  
    % Q,Vv  
    %       % Display the first 10 Zernike functions +T=Z!2L  
    %       x = -1:0.01:1; CfQOG7e@  
    %       [X,Y] = meshgrid(x,x); ]y@8mb&  
    %       [theta,r] = cart2pol(X,Y); Ol:&cX3G  
    %       idx = r<=1; AD =@  
    %       z = nan(size(X)); i;c0X+[  
    %       n = [0  1  1  2  2  2  3  3  3  3]; -W wFUm  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; OwV>`BIwns  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; [=9-AG~}  
    %       y = zernfun(n,m,r(idx),theta(idx)); vmL% %7  
    %       figure('Units','normalized') >|!F.W  
    %       for k = 1:10 KgX~PP>  
    %           z(idx) = y(:,k); M~w =ZJ@  
    %           subplot(4,7,Nplot(k)) ji<b#YO4  
    %           pcolor(x,x,z), shading interp z`((l#(  
    %           set(gca,'XTick',[],'YTick',[]) t>f<4~%MJ  
    %           axis square <Bb $d@c  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) G!k&'{2  
    %       end k_#ra7zP  
    % cjsQm6  
    %   See also ZERNPOL, ZERNFUN2. MPA<?  
    $'dJ+@  
    Rtw^ lo  
    %   Paul Fricker 11/13/2006 eX7Ev'(H  
    ii0AhQ  
    <",4O  
    Q+)fI  
    6ND,4'6  
    % Check and prepare the inputs: s/UIo ^m  
    % ----------------------------- /8@JWK^I{  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) L=3^A'|  
        error('zernfun:NMvectors','N and M must be vectors.') sXOGIv  
    end q.FgX  
    2]C`S,)  
    X{\>TOk   
    if length(n)~=length(m) G!T)V2y  
        error('zernfun:NMlength','N and M must be the same length.') 0[TZ$<v"  
    end S9}P 5;u  
    P!:Y<p{=>  
    buX$O{43I  
    n = n(:); "2(lgxhj  
    m = m(:); #K'3` dpL  
    if any(mod(n-m,2)) G^!20`p:  
        error('zernfun:NMmultiplesof2', ... Bx0^?>  
              'All N and M must differ by multiples of 2 (including 0).') ~Y@(  
    end 5c($3Pno=  
    ~ z*  
    N_Cu%HP  
    if any(m>n) .cN\x@3-j  
        error('zernfun:MlessthanN', ... (o)nN8  
              'Each M must be less than or equal to its corresponding N.') /ZHuT=j1  
    end n c:^)G  
    sh[Yu  
    _C~e(/=z  
    if any( r>1 | r<0 ) U0t/(Jyg  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') EMzJJe{Cv  
    end Ke,UwYG2~G  
    Y>geP+ -  
    _ $PZID  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) JVf8KHDj  
        error('zernfun:RTHvector','R and THETA must be vectors.') P;bl+a'gu  
    end aAiSP+#  
    'x{g P?.  
    -q|K\>tgU  
    r = r(:); +'Pl?QyH  
    theta = theta(:); f!a[+^RB:  
    length_r = length(r); :,%~rR  
    if length_r~=length(theta) FFb`4.  
        error('zernfun:RTHlength', ... YpoO:  
              'The number of R- and THETA-values must be equal.') 6 /gh_'&  
    end eWS[|' dl  
    gN1b?_g  
    )a.Y$![  
    % Check normalization: DvHcT] l>5  
    % -------------------- F7gipCc1We  
    if nargin==5 && ischar(nflag) 7S LJLn3d  
        isnorm = strcmpi(nflag,'norm'); =($RT  
        if ~isnorm wv<D%nF2|  
            error('zernfun:normalization','Unrecognized normalization flag.') PN[ `p1F  
        end A\iDK10Q$  
    else ]#P9.c_}  
        isnorm = false; (xpj?zlmM  
    end =Ig'Aw$x  
    Ke '?  
    oGx OJyD  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `G&W%CHB  
    % Compute the Zernike Polynomials eyf\j,xP&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L22GOa0  
    0'*whhH  
    Y,?s-AB  
    % Determine the required powers of r: @y3w_;P  
    % ----------------------------------- G[n^SEY!  
    m_abs = abs(m); X> :@`}bq  
    rpowers = []; /uS(Z-@  
    for j = 1:length(n) \.y|=Ql_u  
        rpowers = [rpowers m_abs(j):2:n(j)]; 2%U)y;$m2  
    end )QEvV:\  
    rpowers = unique(rpowers); F%@( $f  
    u[9i>7}9  
    Q1 ?O~ao  
    % Pre-compute the values of r raised to the required powers, dOh'9kk3  
    % and compile them in a matrix: l4?o0;:)  
    % ----------------------------- ?9xaBWf  
    if rpowers(1)==0 X5UcemO  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 4zs1BiMG  
        rpowern = cat(2,rpowern{:}); Q1J./C}  
        rpowern = [ones(length_r,1) rpowern]; ["|AD,$%  
    else *c4uCI:0t  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); yG|^-O}L  
        rpowern = cat(2,rpowern{:}); S%gb1's  
    end *t J+!1  
    {$z)7s  
    < .&t'W  
    % Compute the values of the polynomials: YYU Di@K  
    % -------------------------------------- 6wC|/J^  
    y = zeros(length_r,length(n)); O9*cV3}H  
    for j = 1:length(n) s'' ?: +  
        s = 0:(n(j)-m_abs(j))/2; (e sTb,  
        pows = n(j):-2:m_abs(j); ^_ <jg0V  
        for k = length(s):-1:1 .WM0x{t/  
            p = (1-2*mod(s(k),2))* ... z1[2.&9D-  
                       prod(2:(n(j)-s(k)))/              ... s2A3.SN  
                       prod(2:s(k))/                     ... B5h-JON]-  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... s$`g%H>  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); D|m6gP;P  
            idx = (pows(k)==rpowers); S6CM/  
            y(:,j) = y(:,j) + p*rpowern(:,idx); YY-{&+,  
        end >yFEUD:  
         d2lOx|jt  
        if isnorm meunAEe  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); H?98^y7  
        end n B4)%  
    end S!Ue+jW  
    % END: Compute the Zernike Polynomials G0Zq:kJ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @/h_v#W  
    i0DYdUj  
    IPmSkK  
    % Compute the Zernike functions: EeGP E  
    % ------------------------------ TY~8`+bJ  
    idx_pos = m>0; ]jiM  
    idx_neg = m<0; y;A<R[|Ve  
    Uf )?sz  
    {N1Ss|6  
    z = y; Y: &?xR  
    if any(idx_pos) 0STtwfTr:  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); iTsmUq<b]l  
    end RG/M-  
    if any(idx_neg) bOjvrg;Sz\  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); H&jK|]UXoO  
    end )&:4//}a  
    T|^rFaA  
    ^$qr6+  
    % EOF zernfun :e>y= s>  
     
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    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
    离线jssylttc
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    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    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)  E/x2LYH  
    3Z?"M  
    DDE还是手动输入的呢? JP$@*F@t  
    {2u#Q 7]|  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    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。
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