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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, G+sB/l"  
    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, g3vbskY|  
    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? kz1Z K  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? PDc4ok`)  
    X`v6gv5qj  
    :-+][ [  
    gjK: a@{  
    HW_2!t_R  
    function z = zernfun(n,m,r,theta,nflag) -$%~EY}  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. yTbtS-  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N [Z'4YXS  
    %   and angular frequency M, evaluated at positions (R,THETA) on the aB G*  
    %   unit circle.  N is a vector of positive integers (including 0), and 4E!Pxjl3a  
    %   M is a vector with the same number of elements as N.  Each element 4 }_}3.  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) S=< ]u  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, k-*k'S_  
    %   and THETA is a vector of angles.  R and THETA must have the same >>R)?24,<  
    %   length.  The output Z is a matrix with one column for every (N,M) V^.Z&7+E`_  
    %   pair, and one row for every (R,THETA) pair. Cu$`-b^y  
    % WH $*\IGJL  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike KVoi>?a   
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), |%X_<Cpk  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral u0+<[Ia'q  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, <;b  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized gi@&Mr)fS  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. U!"RfRD.<  
    % },5LrX`L  
    %   The Zernike functions are an orthogonal basis on the unit circle. @ohJ'  
    %   They are used in disciplines such as astronomy, optics, and \n#l+R23  
    %   optometry to describe functions on a circular domain. XxB%  
    % 8BS$6Pa  
    %   The following table lists the first 15 Zernike functions. r 3T#Nv  
    % MS|1Q@S9  
    %       n    m    Zernike function           Normalization TxkvHiq2  
    %       -------------------------------------------------- _cfAJ)8=  
    %       0    0    1                                 1 jP3~O  
    %       1    1    r * cos(theta)                    2 aQ 6T2bQ  
    %       1   -1    r * sin(theta)                    2 /oM&29 jy  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) {;UBW7{  
    %       2    0    (2*r^2 - 1)                    sqrt(3) .d)H2X  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) WIwGw%_~  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) aI\ >=*HF  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) $U_1e'  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) jI}{0LW&F&  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) _{i- .;K  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 5FNf)F   
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) q=BAYZ\`  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) q*J-ii  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) sD XJXJZ  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) <l9qhqHv&  
    %       -------------------------------------------------- BxxqzN+  
    % v}z{OB  
    %   Example 1: q p1rP#  
    % zgpv I~Ck  
    %       % Display the Zernike function Z(n=5,m=1) ? v@q&  
    %       x = -1:0.01:1; '&xRb*  
    %       [X,Y] = meshgrid(x,x); M^A;tPw  
    %       [theta,r] = cart2pol(X,Y); [;INVUwG^  
    %       idx = r<=1; $J:~jY/J  
    %       z = nan(size(X)); l>>, ~  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); b WZ X  
    %       figure U &W}c^#  
    %       pcolor(x,x,z), shading interp z?_5fte`  
    %       axis square, colorbar T^ah'WmNw  
    %       title('Zernike function Z_5^1(r,\theta)') o|a]Q  
    % +@oo8io  
    %   Example 2: [SLBA_d  
    % _UeIzdV9  
    %       % Display the first 10 Zernike functions h@ ?BA<'S  
    %       x = -1:0.01:1; )N&v. w  
    %       [X,Y] = meshgrid(x,x); {I_I$x_  
    %       [theta,r] = cart2pol(X,Y); 9=^4p=1J  
    %       idx = r<=1; @)wNINvD  
    %       z = nan(size(X)); W r;?t!  
    %       n = [0  1  1  2  2  2  3  3  3  3]; <wt9K2,  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; }NJ? .Y  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; j]_"MMwk$<  
    %       y = zernfun(n,m,r(idx),theta(idx)); _9zydtw  
    %       figure('Units','normalized') BcTV5Wcr  
    %       for k = 1:10 ViT$]Nv  
    %           z(idx) = y(:,k); s*pgR=dZZ  
    %           subplot(4,7,Nplot(k)) Z,Tv8;  
    %           pcolor(x,x,z), shading interp $lrq*Nf9c  
    %           set(gca,'XTick',[],'YTick',[]) 7_#i,|]58  
    %           axis square q^w3n2  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 76*5/J-  
    %       end PizPsJ|&  
    % 5zBsulRt  
    %   See also ZERNPOL, ZERNFUN2. rRZ ,X%  
    6|:]2S  
    @=[ SsS  
    %   Paul Fricker 11/13/2006 ]LhNP}c  
    I806I@ix  
    $.@)4Nu!_  
    q[SUYb;,  
    N^. !l_  
    % Check and prepare the inputs: xcYYo'U  
    % ----------------------------- =w!14@W  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) i;>Hy|  
        error('zernfun:NMvectors','N and M must be vectors.') P}QuGy[  
    end ='cr@[~i  
    #_bSWV4  
    Z*|qbu)  
    if length(n)~=length(m) ^dR5fAS  
        error('zernfun:NMlength','N and M must be the same length.') )#dP:  
    end 8BZDaiE"  
    LJMw-#61sj  
    xe6 2gaT  
    n = n(:); @GG Pw9a  
    m = m(:); Q pY:L  
    if any(mod(n-m,2)) WVf>>E^1  
        error('zernfun:NMmultiplesof2', ... 8Mq] V v  
              'All N and M must differ by multiples of 2 (including 0).') LPk85E  
    end i=<N4Vx  
    b='YCa  
    _6 ~/`_(KP  
    if any(m>n) 89'XOXl&1  
        error('zernfun:MlessthanN', ... pr[[)[]/  
              'Each M must be less than or equal to its corresponding N.') Ui46 p  
    end |!)3[<.  
    g<Sa{<0  
    ,g$N  
    if any( r>1 | r<0 ) KPUc+`cN%  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') h2Z Gh  
    end 4PEJ}B W  
    KutR l$,  
    C/+8lA6NV  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) jv]:`$}G\  
        error('zernfun:RTHvector','R and THETA must be vectors.') mYN|)QVKy  
    end #{l+I( M  
    G~e`O,+  
    y@G5I>v  
    r = r(:); *"Ipu"G5?  
    theta = theta(:); ?zNv7Bj  
    length_r = length(r); HQ+:0" B  
    if length_r~=length(theta) w8(qiU  
        error('zernfun:RTHlength', ... ]v 6u  
              'The number of R- and THETA-values must be equal.') kG+CT  
    end h2u> CXD  
    lYeot8  
    mD)_quz.sk  
    % Check normalization: 0QDm3V0n  
    % -------------------- E q.?Ga  
    if nargin==5 && ischar(nflag) %?C{0(Z{  
        isnorm = strcmpi(nflag,'norm'); %u43Pj  
        if ~isnorm UPPDs"  
            error('zernfun:normalization','Unrecognized normalization flag.') 5HioxHL  
        end HT5G HkT  
    else ;EE*#"IJ  
        isnorm = false; 5Y)!q?#H  
    end #T n~hnW  
    e4ajT  
    ?PSm) ~ Oa  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'UT 4x9&z  
    % Compute the Zernike Polynomials Vrf` :%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kN;l@>  
    /z,sM"d  
    j+J)S1  
    % Determine the required powers of r: Sz"J-3b^  
    % ----------------------------------- r 06}@7  
    m_abs = abs(m); 6lq7zi}'w  
    rpowers = []; ^&DHBx"J  
    for j = 1:length(n) NwuME/C7#  
        rpowers = [rpowers m_abs(j):2:n(j)]; Om{[ <tL  
    end Ps.O.2Z5ZB  
    rpowers = unique(rpowers); +?(2-RBd  
    q=}Lm;r  
    3U@ p  
    % Pre-compute the values of r raised to the required powers, }O@S ;[v S  
    % and compile them in a matrix: M&y!w   
    % ----------------------------- |z|5j!Nfh  
    if rpowers(1)==0 JQE^ bcr  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ]{nFB3vtB  
        rpowern = cat(2,rpowern{:}); =M 7FD  
        rpowern = [ones(length_r,1) rpowern]; #* S0d1  
    else M{:gc7%  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); < 7zyRm@S  
        rpowern = cat(2,rpowern{:}); z(>{"t<C  
    end .F?yt5{5No  
    )"jG)c^1*  
    ||}|=Sz  
    % Compute the values of the polynomials: J~DP*}~XK  
    % -------------------------------------- _$wWKJy9  
    y = zeros(length_r,length(n)); m^O:k"+!  
    for j = 1:length(n) KcfW+> W3  
        s = 0:(n(j)-m_abs(j))/2; 23y7l=.b/  
        pows = n(j):-2:m_abs(j); ,u{d@U^)3@  
        for k = length(s):-1:1 iX.=8 ~3  
            p = (1-2*mod(s(k),2))* ... nV McHN   
                       prod(2:(n(j)-s(k)))/              ... [lQp4xgxi  
                       prod(2:s(k))/                     ... Cr4shdN34  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... =1^Ru*G  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Fx0K.Q2Y0  
            idx = (pows(k)==rpowers); r1-?mMSU&  
            y(:,j) = y(:,j) + p*rpowern(:,idx); bI@+Or  
        end I4  Tc&b  
         TpZ) wC  
        if isnorm o]MQ)\ r  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 7-w +/fv  
        end }o=R7n%  
    end zScV 9,H1  
    % END: Compute the Zernike Polynomials wv ,F>5P  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *A GC[w}/  
    gUL`)t\}*  
    gO4J[_  
    % Compute the Zernike functions: 23pHB |X  
    % ------------------------------ vp4!p~C{  
    idx_pos = m>0; *0l^/jqn:  
    idx_neg = m<0; W}WGg|ug  
    2[9hl@=%  
    ?O\n!c  
    z = y; o&;+!Si@T  
    if any(idx_pos) y$*Tbzp  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); z.]t_`KuF9  
    end ]Vl * !,(i  
    if any(idx_neg) 0$}+tq+  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Kdx?s;i  
    end KTBsH;6  
    4'Vuhqk  
    _9#4  
    % EOF zernfun z:RwCd1\  
     
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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)  ]i(tou-[i  
    d&BocJ  
    DDE还是手动输入的呢? lr'h  
    i,r O3J n  
    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