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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, O329Bkg  
    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, i)0*J?l=  
    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? t?v0ylN  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? '7W?VipU  
    w<awCp  
    ,7pO-:*g  
    I,AI$A  
    %t\`20-1<  
    function z = zernfun(n,m,r,theta,nflag) )*^PMf  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. SF;;4og  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N S[NV-)r=  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ZBJYpeGe  
    %   unit circle.  N is a vector of positive integers (including 0), and E<a~ `e  
    %   M is a vector with the same number of elements as N.  Each element CPGXwM=   
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 1H @GwQ|<=  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, c*_I1}l  
    %   and THETA is a vector of angles.  R and THETA must have the same HqU"i Y>b  
    %   length.  The output Z is a matrix with one column for every (N,M) j*$GP'Df3  
    %   pair, and one row for every (R,THETA) pair. X63DBF4A  
    % q]5"V>D \  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike F#iLMO&Q  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), >.#uoW4ZV  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral RH. oo&  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, tGD$cBE  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized / v;g v[  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. wLU w'Ai  
    % 5gV8=Ml"V  
    %   The Zernike functions are an orthogonal basis on the unit circle. qrNW\ME  
    %   They are used in disciplines such as astronomy, optics, and @}x)>tqD  
    %   optometry to describe functions on a circular domain. DSy,#yA  
    % [8SW0wsk  
    %   The following table lists the first 15 Zernike functions. :%A1k2  
    % s iv KXd  
    %       n    m    Zernike function           Normalization .Kq>/6  
    %       -------------------------------------------------- '8k\a{t_z  
    %       0    0    1                                 1  tB[(o%k  
    %       1    1    r * cos(theta)                    2 bK("8T\?  
    %       1   -1    r * sin(theta)                    2 r#]gAG4t\  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) q`}Q[Li  
    %       2    0    (2*r^2 - 1)                    sqrt(3) A@_F ;4X  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) &6MGPh7T  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 3 T Q#3h  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) rg_-gZl8&z  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) akBR"y:~:H  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) =}r&>|rrJ  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) c.,:r X0S  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) p0$K.f| ^  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) f;pR8  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0} liK  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) KL.{)bi  
    %       -------------------------------------------------- ahS*YeS7  
    % J}`K&DtM9  
    %   Example 1: .K}u`v T  
    % F^T7u?^)  
    %       % Display the Zernike function Z(n=5,m=1) m2{z  
    %       x = -1:0.01:1; Ps<)?q6(  
    %       [X,Y] = meshgrid(x,x); Y: KB"H  
    %       [theta,r] = cart2pol(X,Y); .(CzsupY_q  
    %       idx = r<=1; zmf5!77  
    %       z = nan(size(X)); ,`/!0Wmt  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); +5?hkQCX1^  
    %       figure u/ y`M]17  
    %       pcolor(x,x,z), shading interp 5&2=;?EO  
    %       axis square, colorbar 5:CC\!&QBV  
    %       title('Zernike function Z_5^1(r,\theta)') Ej'a G   
    % A~nq4@uj  
    %   Example 2: V[+ Pb]  
    % |mk$W$h  
    %       % Display the first 10 Zernike functions lUCdnp;w'  
    %       x = -1:0.01:1; N.xmHvPk  
    %       [X,Y] = meshgrid(x,x); kc|`VB8L  
    %       [theta,r] = cart2pol(X,Y); xfO!v>  
    %       idx = r<=1; fBD5K3  
    %       z = nan(size(X)); gA2\c5F<  
    %       n = [0  1  1  2  2  2  3  3  3  3]; A+Y>1-=JO  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; v]U[7 j  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; N;-+)=M,rf  
    %       y = zernfun(n,m,r(idx),theta(idx)); %>xW_5;Z  
    %       figure('Units','normalized') evg i\"  
    %       for k = 1:10 #hR}7K+@  
    %           z(idx) = y(:,k); ;c:vz F~Q  
    %           subplot(4,7,Nplot(k)) #5G!lbH  
    %           pcolor(x,x,z), shading interp X"iy.@7  
    %           set(gca,'XTick',[],'YTick',[]) xE;fM\7pu  
    %           axis square 79:x>i=  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) fRaVY`|wK  
    %       end MV9{>xX  
    % w|?Nq?KA  
    %   See also ZERNPOL, ZERNFUN2. U G^6I5  
    6n%^ U2H/-  
    0\o0(eHCQz  
    %   Paul Fricker 11/13/2006 ((EN&X,v  
    W1r-uR  
    }4_izKS  
    i7e{REBXb  
    a4g=cs<9}  
    % Check and prepare the inputs: ttZ!P:H2  
    % ----------------------------- SRM[IU  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) C&<f YCwG  
        error('zernfun:NMvectors','N and M must be vectors.') z56W5g2  
    end KQ3)^J_Z  
    uZmfvMr3  
    x*BfRj  
    if length(n)~=length(m) JWMIZ{/M  
        error('zernfun:NMlength','N and M must be the same length.') 1/a*8vuGh  
    end  <MvFAuAT  
    Qf>dfJ^q  
    ! ~&X1,l1*  
    n = n(:); ]jY->NsA]  
    m = m(:); I|Z5*iXqCm  
    if any(mod(n-m,2)) qx0J}6+NlU  
        error('zernfun:NMmultiplesof2', ... v8 6ls[lzu  
              'All N and M must differ by multiples of 2 (including 0).') ']Y:f)i#  
    end .o|Gk 5)  
    1__p1  
    7OC ,KgJ3  
    if any(m>n) {_^sR}%]F  
        error('zernfun:MlessthanN', ... <0R?#^XBZB  
              'Each M must be less than or equal to its corresponding N.') `Ph4!-6#  
    end [uAfE3  
    iKp4@6an  
    Sw'DS  
    if any( r>1 | r<0 ) 2!]':(8mR  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') F P mLost  
    end VEb}KFyP  
    %@H;6   
    %I6iXq#  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Q CfA3*  
        error('zernfun:RTHvector','R and THETA must be vectors.') %0:  (''  
    end  &h4(lM  
    oh& P Q{  
    *e_ /D$SC  
    r = r(:); |!57Z4X  
    theta = theta(:); !R)v2Mk|  
    length_r = length(r); )JuD !  
    if length_r~=length(theta) ^BNg^V.  
        error('zernfun:RTHlength', ... ? 76jz>;b  
              'The number of R- and THETA-values must be equal.') ~(I\O?k>H  
    end LAMTf"a  
    6wnfAli.  
    RMLs(?e  
    % Check normalization: p_P'2mf  
    % -------------------- Rfa1 v*(  
    if nargin==5 && ischar(nflag) YM1@B`yWE  
        isnorm = strcmpi(nflag,'norm'); "'6KQnpZ  
        if ~isnorm -I4@` V  
            error('zernfun:normalization','Unrecognized normalization flag.') EkOBI[`  
        end E8FS jLZ  
    else SwSBQq%h]M  
        isnorm = false; 8#7z5:_  
    end GbStqR~^#  
    h\D y(\  
    #{ `(;83  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ||qsoF5B]  
    % Compute the Zernike Polynomials aQinR"o  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% QabF(}61  
    =$b^ X?x  
    Pfi '+I`s  
    % Determine the required powers of r: 6I_W4`<VeZ  
    % ----------------------------------- LG&~#x  
    m_abs = abs(m); 8Jxo;Y  
    rpowers = []; ~p oy`h'  
    for j = 1:length(n) Qy@chN{eP  
        rpowers = [rpowers m_abs(j):2:n(j)]; ";s?#c  
    end ">CjnF2>R  
    rpowers = unique(rpowers); L6 hTz'  
    :[\}Hn=  
    ;uDH&3W  
    % Pre-compute the values of r raised to the required powers, .rN 5A+By`  
    % and compile them in a matrix: ;t"#7\  
    % ----------------------------- MlS<txFPS  
    if rpowers(1)==0 |910xd`Z  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ^U:pv0Qz  
        rpowern = cat(2,rpowern{:}); tR0o6s@v/<  
        rpowern = [ones(length_r,1) rpowern]; g4I(uEJk  
    else rf]]I#C7  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,}`II|.oB  
        rpowern = cat(2,rpowern{:}); 2hmV 1gj  
    end qrm~=yU%  
    "'II~/9  
    O1rnF3Be  
    % Compute the values of the polynomials: 3x 'BMAA+  
    % -------------------------------------- [<f\+g2ct  
    y = zeros(length_r,length(n)); 1 ,[T;pdDd  
    for j = 1:length(n) "E 8-76n  
        s = 0:(n(j)-m_abs(j))/2; p# O%<S@?  
        pows = n(j):-2:m_abs(j); GG%j+Ed  
        for k = length(s):-1:1 A[=)Zw "  
            p = (1-2*mod(s(k),2))* ... >9Ub=tZm  
                       prod(2:(n(j)-s(k)))/              ... ",`fGu )  
                       prod(2:s(k))/                     ... J%3S3C2*m  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... {gK i15t  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 7 P=1+2V  
            idx = (pows(k)==rpowers); R;'Pe>  
            y(:,j) = y(:,j) + p*rpowern(:,idx); MCL5a@BX)  
        end |2{y'?,  
         p4HX83y{  
        if isnorm ]W-:-.prh  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); xr)kHJ:v  
        end RK"dPr  
    end KuE 2a,E4  
    % END: Compute the Zernike Polynomials GfL}f9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1&Nk  
    wgzjuTqwBF  
    L;E9"7Jo  
    % Compute the Zernike functions: lj'c0k8  
    % ------------------------------ /Q})%j1S0  
    idx_pos = m>0; i nF&Pv  
    idx_neg = m<0; @6}c\z@AxM  
    Gzc{2"p  
    c,X\1yLy  
    z = y; &Q(Q/]U~  
    if any(idx_pos) t<~riFs]  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); AEOo]b*&d  
    end u{tjB/K&  
    if any(idx_neg) VO#]IXaP  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); qmtVk  
    end y%`^* E&  
    /|`;|0/2  
    8O("o7~"  
    % EOF zernfun z Sj.Y{J  
     
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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)  Hm-+1Wx  
    6pLB`1[v  
    DDE还是手动输入的呢? LJc w->  
    MPAZ%<gmD  
    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