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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, [^"(%{H  
    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, E]e[Ty1  
    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? Fr Q-v]c  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? e]L3=R;  
    pC?1gc1G  
    PrYWha=c-  
    CI3_lWax%  
    +~v3D^L15  
    function z = zernfun(n,m,r,theta,nflag) 4s+J-l  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. xQ(KmP2hl  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ZkV vL4yIK  
    %   and angular frequency M, evaluated at positions (R,THETA) on the )]e d;V  
    %   unit circle.  N is a vector of positive integers (including 0), and r^ Rcjyc1  
    %   M is a vector with the same number of elements as N.  Each element 5)zj){wL  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) <45dy5!Tz  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, j2v[-N4 {J  
    %   and THETA is a vector of angles.  R and THETA must have the same G^eFS;  
    %   length.  The output Z is a matrix with one column for every (N,M) i|! 9o:  
    %   pair, and one row for every (R,THETA) pair. k=q%FlE  
    % W0,"V'C  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike -<HvhW  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 5]O LV1Xt  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral l`w|o  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ->#@rF:S  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Fn0LE~O}-8  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. o\V4qekk  
    % Vm%ux>}  
    %   The Zernike functions are an orthogonal basis on the unit circle. sMpC4E  
    %   They are used in disciplines such as astronomy, optics, and .$E~.6J %i  
    %   optometry to describe functions on a circular domain. |$*9j""u  
    % p]IhQnj2  
    %   The following table lists the first 15 Zernike functions. K &~#@I;  
    % 4lo}-@j  
    %       n    m    Zernike function           Normalization q,h.W JI  
    %       -------------------------------------------------- KcyM2hE7  
    %       0    0    1                                 1 {xb%P!o`  
    %       1    1    r * cos(theta)                    2 F#C6.`B  
    %       1   -1    r * sin(theta)                    2 U3iyuE  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ,% DAh  
    %       2    0    (2*r^2 - 1)                    sqrt(3) U~`^Y8UF  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) )k<~}wvQ0  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) RBojT   
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) j`-y"6)  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) (Y@|h%1W  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) G5@fqh6ws  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 4 Fc1 '  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) vWU4ZBT8G  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) N`GwL aF  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) vT<wd#  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ?ut juMdl  
    %       -------------------------------------------------- rVW'KN  
    % MvwJ(3  
    %   Example 1: [#h!3d|?B  
    % H {Wpf9_ K  
    %       % Display the Zernike function Z(n=5,m=1) Dve5m=  
    %       x = -1:0.01:1;  l e/#J  
    %       [X,Y] = meshgrid(x,x); &ZFAUE,[  
    %       [theta,r] = cart2pol(X,Y); @V CQ4X7T  
    %       idx = r<=1; +OP:"Q_#  
    %       z = nan(size(X)); D`@U[`Sw  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); SPm5tU  
    %       figure hl~F1"q )  
    %       pcolor(x,x,z), shading interp YU"\Wd[  
    %       axis square, colorbar |(8h:g  
    %       title('Zernike function Z_5^1(r,\theta)') "TNUw&ih  
    % ':>*=&  
    %   Example 2: S#z8H+'  
    % &l*dYzqq  
    %       % Display the first 10 Zernike functions DG FvRB  
    %       x = -1:0.01:1; QX3![;0F  
    %       [X,Y] = meshgrid(x,x); %QZ!Tb  
    %       [theta,r] = cart2pol(X,Y); 1VsEic  
    %       idx = r<=1; 9 3I9`!e  
    %       z = nan(size(X)); H%%nB  
    %       n = [0  1  1  2  2  2  3  3  3  3]; PP`n>v=n  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; u(a&x|WY  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; kO>{<$  
    %       y = zernfun(n,m,r(idx),theta(idx)); dNt|"9~&  
    %       figure('Units','normalized')  -KiS6$-  
    %       for k = 1:10 RN3D:b+  
    %           z(idx) = y(:,k); W,J,h6{F  
    %           subplot(4,7,Nplot(k)) 0'}?3/u-  
    %           pcolor(x,x,z), shading interp .T X& X  
    %           set(gca,'XTick',[],'YTick',[]) 4V3 w$:,  
    %           axis square !/Ps}.)A`  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) R?Q-@N>wE  
    %       end 3k0%H]wt  
    % $yK!Q)e:  
    %   See also ZERNPOL, ZERNFUN2. mR@Xt#  
    ><7`$2Or  
    SX,z J`"  
    %   Paul Fricker 11/13/2006 VMXXBa&  
    ml 2z  
    *s4!;2ZhsU  
    Br!&Y9  
    }w8AnaC  
    % Check and prepare the inputs: z Pc;[uHT  
    % ----------------------------- !AHm+C_=Lg  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Z.(x|Q9  
        error('zernfun:NMvectors','N and M must be vectors.') 3%a37/|~y  
    end 7rg[5hP T  
    F'*&-l  
    0G 1o3[F  
    if length(n)~=length(m) PSE| 4{'  
        error('zernfun:NMlength','N and M must be the same length.') 4BtdN-T}b  
    end #4u; `j"4=  
     8[OiG9b  
    qZ }XjL  
    n = n(:); TZ2f-KI  
    m = m(:); N9<eU!4>  
    if any(mod(n-m,2)) bm.H0rHR4  
        error('zernfun:NMmultiplesof2', ... 0wcWDE 9  
              'All N and M must differ by multiples of 2 (including 0).') bb/MnhB  
    end r&DK> H  
    +rY0/T_0,  
    {`Z)'G\`  
    if any(m>n) lhTbgM  
        error('zernfun:MlessthanN', ... X>>rvlDN  
              'Each M must be less than or equal to its corresponding N.') M3kE91  
    end x6tY _lzJ  
    9@B+$~:}7  
    }VRo:sJb  
    if any( r>1 | r<0 ) e8 .bH#  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 2ZeL  
    end 8msDJ {,X  
    Nb1lawC  
    Akf9nT  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) E>D@#I>  
        error('zernfun:RTHvector','R and THETA must be vectors.') u{,^#I}  
    end p]oo^  
    tPHiz%  
    ja2]VbB  
    r = r(:); 9g"H9)EZ^  
    theta = theta(:); TbuR?#  
    length_r = length(r); TW0^wSm  
    if length_r~=length(theta) 8hg(6 XUG  
        error('zernfun:RTHlength', ... -( G2@NG  
              'The number of R- and THETA-values must be equal.') oH0\6:S  
    end bJD"&h5  
    =Yk$Q\c  
    ez>@'yhK  
    % Check normalization: [PT_y3'%  
    % -------------------- O.dNhd$  
    if nargin==5 && ischar(nflag) KWo)}m*6  
        isnorm = strcmpi(nflag,'norm'); 4`F*] Ft  
        if ~isnorm =l+p nG  
            error('zernfun:normalization','Unrecognized normalization flag.') ^-_!:7TH]  
        end $;1~JOZh  
    else u4'Lm+&O  
        isnorm = false; d\f 5\Y  
    end D4wB &~U  
    67eo~~nUtg  
    +!(hd  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7d*<'k]{,  
    % Compute the Zernike Polynomials Yy}aQF#M  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $j/F7.S  
    wSK?mS6  
    ,3j*D+  
    % Determine the required powers of r: c#DTL/8"DO  
    % ----------------------------------- ORoraEK  
    m_abs = abs(m); {~"=6iyj  
    rpowers = []; Ow I?(ruL'  
    for j = 1:length(n) JoYzC8/r  
        rpowers = [rpowers m_abs(j):2:n(j)]; fomkwN  
    end 9maw+c!~  
    rpowers = unique(rpowers); )+G(4eIT  
    AN;?`AM;  
    QbWD&8T0O  
    % Pre-compute the values of r raised to the required powers, ){} #v&  
    % and compile them in a matrix: lV?SvXe  
    % ----------------------------- }bVyvH  
    if rpowers(1)==0 SUw{xGp  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); {e^llfj$#  
        rpowern = cat(2,rpowern{:}); ) l)5^7=W  
        rpowern = [ones(length_r,1) rpowern]; = 7?'S#  
    else 5c#L6 dA)  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);  ,Y!)V  
        rpowern = cat(2,rpowern{:}); 'e)t+  
    end ;H^!yj5H  
    ~$$V=$&  
    #/:[ho{JQ  
    % Compute the values of the polynomials: yZ{YIy~  
    % -------------------------------------- O6pjuhMx  
    y = zeros(length_r,length(n)); vcmS]$}  
    for j = 1:length(n) rcK*",>  
        s = 0:(n(j)-m_abs(j))/2; + y^s 6j}  
        pows = n(j):-2:m_abs(j); [{ pc1U-  
        for k = length(s):-1:1 4u&doSXR  
            p = (1-2*mod(s(k),2))* ... P7o6B,9  
                       prod(2:(n(j)-s(k)))/              ... ~(8A&!#,!  
                       prod(2:s(k))/                     ... c(jA"K[|b  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... cZYX[.oIB  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Rq 7ksTo  
            idx = (pows(k)==rpowers); ubL Lhf  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ,rd+ dN  
        end DXUI/C f  
         i<bs{Cu_S  
        if isnorm _D:/?=y;e  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); | ] YT6-?.  
        end sxqX R6p{  
    end >C i=H(8vN  
    % END: Compute the Zernike Polynomials &jJgAZ!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c"~TH.,d  
    "*0 szz'  
    1@y?OWC  
    % Compute the Zernike functions: y8L:nnSj  
    % ------------------------------ ZoUfQ!2*  
    idx_pos = m>0; #GF1MFkoS  
    idx_neg = m<0; qg O)@B+  
    @dXf_2Tv=  
    W1OGN4`C  
    z = y; T \AuL  
    if any(idx_pos) yH`xk%q_  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); L.R  
    end  v/.2Z(sZ  
    if any(idx_neg) 8,R]R=  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); D>mLSh  
    end Y PM>FDxDB  
    gO5;hd[ l  
    }PXWRv.gW  
    % EOF zernfun Zy#r<j]T  
     
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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)  h(L5MZs  
    Wama>dy%  
    DDE还是手动输入的呢? \,N dg*qC  
    ]'G7(Y\)f  
    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