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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, b:Kw_Q  
    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, _a$DY ,;  
    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? *"FLkC4  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛?  IB{ZE/   
    v8bl-9DQ  
    $af}+:'  
     |7zP 8  
    Treh{s  
    function z = zernfun(n,m,r,theta,nflag) 'S7@+kJ  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ^r*%BUU9]%  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 6^O?p2xpo  
    %   and angular frequency M, evaluated at positions (R,THETA) on the h5rP]dbhXU  
    %   unit circle.  N is a vector of positive integers (including 0), and QX.6~*m1  
    %   M is a vector with the same number of elements as N.  Each element qMES<UL>  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) NcBe|qxQ  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ?vn 0%e868  
    %   and THETA is a vector of angles.  R and THETA must have the same =8p+-8M[d  
    %   length.  The output Z is a matrix with one column for every (N,M) t"/"Ge#a  
    %   pair, and one row for every (R,THETA) pair. b+].Uc  
    % h Yc{ 9$  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .xkV#ol  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), BrH;(*H)8  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral I"32[?0 (;  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, xPMyG);  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized P!+nZXo  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. !Vr45l  
    % )^f9[5ee  
    %   The Zernike functions are an orthogonal basis on the unit circle. 9LO.8Jy  
    %   They are used in disciplines such as astronomy, optics, and %C`'>,t>  
    %   optometry to describe functions on a circular domain. `3y!XET  
    % cbCE $  
    %   The following table lists the first 15 Zernike functions. M=[q+A  
    % `x$}~rP&)!  
    %       n    m    Zernike function           Normalization e*2&s5 #RT  
    %       -------------------------------------------------- .\~P -{Hd  
    %       0    0    1                                 1 8#]7`o  
    %       1    1    r * cos(theta)                    2 NnLhJPh  
    %       1   -1    r * sin(theta)                    2 X!rQ@F3  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 6+ $d  
    %       2    0    (2*r^2 - 1)                    sqrt(3) %rDmW?T  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) frmqBCVJ:  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 0^y@p&;/.  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) A2|o=mOH  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ok3  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ()C^ta_]  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) <a+eF}*2  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) < [S1_2b.t  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) N=Uc=I7C  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -':"6\W  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) X4 }`>  
    %       -------------------------------------------------- Ztyv@z'/Z  
    % Lk`k>Nn)  
    %   Example 1: ! [|vx!p  
    % iijd $Tv  
    %       % Display the Zernike function Z(n=5,m=1) ~*mOt 7G  
    %       x = -1:0.01:1; ,dZ#,<  
    %       [X,Y] = meshgrid(x,x); HTUYvU*-  
    %       [theta,r] = cart2pol(X,Y); zY+t,2z  
    %       idx = r<=1; i|c`M/) h:  
    %       z = nan(size(X)); TDl!qp @  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); HTDyuqs  
    %       figure y '_V/w s  
    %       pcolor(x,x,z), shading interp Q.9Ph ~  
    %       axis square, colorbar kj{rk^x  
    %       title('Zernike function Z_5^1(r,\theta)') //X e*0  
    % uXQ7eXX  
    %   Example 2: yZ;k@t_WRD  
    % kJurUDo  
    %       % Display the first 10 Zernike functions JA?,0S  
    %       x = -1:0.01:1; y\)G7 (  
    %       [X,Y] = meshgrid(x,x); |D;"D  
    %       [theta,r] = cart2pol(X,Y); S2'`|uI  
    %       idx = r<=1; +EST58  
    %       z = nan(size(X)); B:3+',i1  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ^A *]&%(h  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; t,=@hs hN  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 4-]Do?  
    %       y = zernfun(n,m,r(idx),theta(idx)); *R_'$+  
    %       figure('Units','normalized') *Z]5!$UpC  
    %       for k = 1:10 ?AV&@EX2C  
    %           z(idx) = y(:,k); CJMaltPp&  
    %           subplot(4,7,Nplot(k)) I~p8#<4#b  
    %           pcolor(x,x,z), shading interp z-KrQx2  
    %           set(gca,'XTick',[],'YTick',[]) jiA5oX^g  
    %           axis square H _Zo@y~J  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 9UeVvH  
    %       end r@*=|0(OrK  
    % ).0V%}>  
    %   See also ZERNPOL, ZERNFUN2. tC2 )j7@  
    !j!Z%]7  
    ;[{:'^n  
    %   Paul Fricker 11/13/2006 g.[+yzuE6  
    Y<p zy8z  
    Z?~gQ $  
    N?qIpv/a.  
    O`wYMng)  
    % Check and prepare the inputs: jIAW-hc]  
    % ----------------------------- >AR Tr'B  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) fKL'/?LD]  
        error('zernfun:NMvectors','N and M must be vectors.') tA`mD>[  
    end c;c:Ea5  
    Bii6Z@kS  
    tWpl`HH  
    if length(n)~=length(m) `pP9z;/Xq  
        error('zernfun:NMlength','N and M must be the same length.') -W|*fKN`3  
    end r?64!VS;  
    `t{D7I7  
    'R^iKNPs  
    n = n(:); wzD\8_;6N  
    m = m(:); O24Jj\"  
    if any(mod(n-m,2)) -M"IVyy@  
        error('zernfun:NMmultiplesof2', ... E4Y "X  
              'All N and M must differ by multiples of 2 (including 0).') w) =eMdj\o  
    end E^b pckP  
    o;ik Z*+*  
    +VSZhg,Np8  
    if any(m>n) ?Wwh _TO  
        error('zernfun:MlessthanN', ... rs[?v*R74  
              'Each M must be less than or equal to its corresponding N.') ^F>4~68d  
    end NNwc!x)*  
    ~k9O5S{  
    F|ETug n  
    if any( r>1 | r<0 ) Cf Qf7-  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') }C=Quy%Z<  
    end jMK3T  
    Hab!qWK`  
    hZ!oRWIU%G  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ?sV[MsOsC  
        error('zernfun:RTHvector','R and THETA must be vectors.') S*4f%!  
    end q#;BhPc  
    a*V9_Px$&  
    BRe{1i 6  
    r = r(:); GA.BI"l  
    theta = theta(:); T'hml   
    length_r = length(r); doLkrEm&  
    if length_r~=length(theta) >Cvjs  
        error('zernfun:RTHlength', ... d{W}p~UbH  
              'The number of R- and THETA-values must be equal.') [u[ U_g*  
    end Z,3 CC \  
    y $:yz;  
    *]5z^> q;7  
    % Check normalization: !&W|myN^  
    % -------------------- A 6:Q<  
    if nargin==5 && ischar(nflag) USprsaj  
        isnorm = strcmpi(nflag,'norm'); 4&|C}  
        if ~isnorm 5Yl6?  
            error('zernfun:normalization','Unrecognized normalization flag.') +i+tp8T+7  
        end -)X{n?i  
    else q&Q/?g>f  
        isnorm = false; U M@naU  
    end Yr+d1(  
    V7P6zAJy  
    P Q,+hq  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !#.\QU|  
    % Compute the Zernike Polynomials "MTWjW*6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yOc|*O=]U  
    }?J~P%HpF  
    L%f;J/  
    % Determine the required powers of r: b7!UZu]IEv  
    % ----------------------------------- m*gj|1k  
    m_abs = abs(m); C,.-Q"juH  
    rpowers = []; ms7SoY bSu  
    for j = 1:length(n) ?s%v 3T  
        rpowers = [rpowers m_abs(j):2:n(j)]; ' X}7]y  
    end AQe!Sqg'  
    rpowers = unique(rpowers); XoJgs$3B  
    K}Na3}m  
    U%q:^S%#eG  
    % Pre-compute the values of r raised to the required powers, ~Zmi(Ra  
    % and compile them in a matrix: [%jxf\9jJ_  
    % ----------------------------- E`tQe5K  
    if rpowers(1)==0 N#UXP5C(  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); rCE;'? Y  
        rpowern = cat(2,rpowern{:}); dnwdFsf  
        rpowern = [ones(length_r,1) rpowern]; qC..\{z  
    else ".E5t@ }?m  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ?gN9kd)  
        rpowern = cat(2,rpowern{:}); l DnMjK\M  
    end y^G>{?Tha  
    #d% vT!Bz~  
    .uKx>YB}  
    % Compute the values of the polynomials: SW#BZ3L  
    % -------------------------------------- HUkerV  
    y = zeros(length_r,length(n)); q`[K3p   
    for j = 1:length(n) .gq(C9<B[  
        s = 0:(n(j)-m_abs(j))/2; ESIzGaM  
        pows = n(j):-2:m_abs(j); jN6b*-2  
        for k = length(s):-1:1 \yG`Sfu2  
            p = (1-2*mod(s(k),2))* ... (f~gEKcB2u  
                       prod(2:(n(j)-s(k)))/              ...  ,gmH2.  
                       prod(2:s(k))/                     ... q & b5g !  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... \vVSh  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); (Xo SG  
            idx = (pows(k)==rpowers); d=y0yq{L  
            y(:,j) = y(:,j) + p*rpowern(:,idx); sPy2/7Wqd  
        end GRIa8>  
         ^df x~C  
        if isnorm 1ef'7a7e8  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 72,"Cj  
        end q@kOTkHv)  
    end _q)!B,y-/N  
    % END: Compute the Zernike Polynomials AK*N  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4\6: \  
    9 mPIykAj8  
    ~{M@?8wi  
    % Compute the Zernike functions: jo_ sAb  
    % ------------------------------ ) * TF"  
    idx_pos = m>0; QrC/ssf}  
    idx_neg = m<0; VNj@5s  
    8;#AO8+U7)  
    -72j:nk  
    z = y; 9tk" :ld  
    if any(idx_pos) IqUp4}  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); J>%t<xYf4  
    end LeHiT>aX!  
    if any(idx_neg) FVgMmYU  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); V7C1FV2  
    end rl?7W];  
    @o#+5P  
    Uo6(|mm  
    % EOF zernfun w^{! U  
     
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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)  U5+vN[ K  
    Q`6i=mB;  
    DDE还是手动输入的呢? `&*bM0(J  
    '^}+Fv<O  
    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