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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, ^Pk-<b4}  
    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, ;nbUbRb  
    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? \)pT+QxZ  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? /M;A)z  
    SDTX3A1  
    W c"f  
    p Rn vd|  
    g6kVHxh-  
    function z = zernfun(n,m,r,theta,nflag) od\Q<Jm}  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. %usy`4 2  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ]_yk,}88d  
    %   and angular frequency M, evaluated at positions (R,THETA) on the eVZ/3o  
    %   unit circle.  N is a vector of positive integers (including 0), and [C]u!\(IF  
    %   M is a vector with the same number of elements as N.  Each element &?=UP4[oif  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) b[3K:ot+  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, jMvWS71  
    %   and THETA is a vector of angles.  R and THETA must have the same b=!G3wVw<  
    %   length.  The output Z is a matrix with one column for every (N,M) 1} {bHj  
    %   pair, and one row for every (R,THETA) pair. W`KRaL0^  
    % XO*62 >Ed  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike S/? KC^JP  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ptXLWv`  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral (dxkDS-G  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, h-Q3q:  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized c:Tw.WA  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. RSLMO8  
    % q1Vh]d  
    %   The Zernike functions are an orthogonal basis on the unit circle. %{*}KsS`p  
    %   They are used in disciplines such as astronomy, optics, and 8lo /BGxS>  
    %   optometry to describe functions on a circular domain. .FS`Fh;  
    % w6M EY"<L  
    %   The following table lists the first 15 Zernike functions. YY (,H!  
    % h^h!OQKQ  
    %       n    m    Zernike function           Normalization 777N0,o(  
    %       -------------------------------------------------- 6_a42#  
    %       0    0    1                                 1 E}aTH  
    %       1    1    r * cos(theta)                    2 ceDe!Iu  
    %       1   -1    r * sin(theta)                    2 ]:B|_| H  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) -t, .A/?  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ?3wEO>u  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) @3/.W+  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) [.O 3z*[9#  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) OchIEF "N  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) _ 13M  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) E4^zW_|xE  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) yp=(wcJ  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) v*+.;60_  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) lS.*/u*5  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,4hQ#x  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) S.bB.<  
    %       -------------------------------------------------- MXWCYi  
    % 9|Cu2  
    %   Example 1: b$kCyOg  
    % Tti]H9g_  
    %       % Display the Zernike function Z(n=5,m=1) IG?044Y  
    %       x = -1:0.01:1; Re3vW re  
    %       [X,Y] = meshgrid(x,x); v Dgf}  
    %       [theta,r] = cart2pol(X,Y); -MrEJ  
    %       idx = r<=1; P>/n!1c  
    %       z = nan(size(X)); P%hi*0pwZ  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); +@wa?"  
    %       figure ln#Jb&u  
    %       pcolor(x,x,z), shading interp _@[M0t}g_  
    %       axis square, colorbar ^zPa^lo-  
    %       title('Zernike function Z_5^1(r,\theta)') d Ybb>rlu  
    % / lh3.\|  
    %   Example 2: PT7L65  
    % w,(e,8#:  
    %       % Display the first 10 Zernike functions 0GW(?7ZC  
    %       x = -1:0.01:1; a $pxt!6  
    %       [X,Y] = meshgrid(x,x); L 0?-W%$>  
    %       [theta,r] = cart2pol(X,Y); :jB8Q$s  
    %       idx = r<=1; |tC`rzo  
    %       z = nan(size(X)); `<>Emc8Z  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ZzA4iT=KO  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 9/[3xhB4  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; HE911 lc:  
    %       y = zernfun(n,m,r(idx),theta(idx)); mAkR<\?iTF  
    %       figure('Units','normalized') f!;4 -.p`  
    %       for k = 1:10 RkVU^N"  
    %           z(idx) = y(:,k); &D, gKT~  
    %           subplot(4,7,Nplot(k)) "V!y"yQ  
    %           pcolor(x,x,z), shading interp rWKc,A[  
    %           set(gca,'XTick',[],'YTick',[]) zG|}| //}  
    %           axis square ;W6P$@'zs  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 'ojI_%9<  
    %       end 4R5+"h:  
    % 1?\ #hemL  
    %   See also ZERNPOL, ZERNFUN2. 6 <JiHVP7  
    hC@oyC(4  
    y#HDJ=2  
    %   Paul Fricker 11/13/2006 V3O<l}ak  
    sr!m   
    d&n&_>  
    G"3)\FEM  
    r'7>J:cy=  
    % Check and prepare the inputs: +qsNz*@p"  
    % ----------------------------- _idTsd:\  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) tZR%s  
        error('zernfun:NMvectors','N and M must be vectors.') Ie(vTP1Cj  
    end  FVOR~z  
    .b*%c?e  
    n!5 :I#B  
    if length(n)~=length(m) F+r3~T%  
        error('zernfun:NMlength','N and M must be the same length.') Td%[ -  
    end 8 ;oU{  
    F.i%o2P3  
    :K{!@=o  
    n = n(:); Bi?+e~R  
    m = m(:); /7Z;/|oU  
    if any(mod(n-m,2)) .JIn(  
        error('zernfun:NMmultiplesof2', ... W|_^Oe<  
              'All N and M must differ by multiples of 2 (including 0).') ,TY&N-  
    end C<Q;3w`#1j  
    j}NGyS" =  
    Jwzkd"D  
    if any(m>n) FZTBvdUYp  
        error('zernfun:MlessthanN', ... SB R=  
              'Each M must be less than or equal to its corresponding N.')  S^;D\6(r  
    end S<"T:Y &  
    A<esMDX  
    Q%6Lc.i  
    if any( r>1 | r<0 ) s,Uc cA@  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ^W-03  
    end [ix45xu7  
    M$j]VZ  
    ajFSbi)l  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) S~auwY,<  
        error('zernfun:RTHvector','R and THETA must be vectors.') V$O{s~@ti  
    end 6@_Vg~=S  
    VHhW_ya1g{  
    zRDBl02v$T  
    r = r(:); n~xh %r;  
    theta = theta(:); "NqB_?DT  
    length_r = length(r); {bB;TO<b`  
    if length_r~=length(theta) V<f76U)  
        error('zernfun:RTHlength', ... .s7Cr0^k,|  
              'The number of R- and THETA-values must be equal.') T^9k,J(rM  
    end xB=~3  
    /8 /2#`3R  
    M5DW!^  
    % Check normalization: :Z0m "  
    % -------------------- >W%tEc  
    if nargin==5 && ischar(nflag) ?ysC7 ((  
        isnorm = strcmpi(nflag,'norm'); S0+nQM%  
        if ~isnorm j_2-  
            error('zernfun:normalization','Unrecognized normalization flag.') Dk&@AjJga  
        end 8jyg1NN D  
    else qF!oP  
        isnorm = false; *G|w#-\.c  
    end e-vwve  
    z)$X/v  
    v{7Jzjd  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Cf#[E~24  
    % Compute the Zernike Polynomials `em}vdY  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% J)R;NYl  
    >gNVL (  
    0. _)X  
    % Determine the required powers of r: sYlA{Z"  
    % ----------------------------------- % j4  
    m_abs = abs(m); 5e^t;  
    rpowers = []; U2  0@B`<  
    for j = 1:length(n)  +c@s  
        rpowers = [rpowers m_abs(j):2:n(j)]; uH'n.d"WG  
    end f>d aK9$(  
    rpowers = unique(rpowers); 1^<R2x  
    ~3YN;St-  
    Y0`=h"g  
    % Pre-compute the values of r raised to the required powers, R{zAs?j  
    % and compile them in a matrix: }F'B!8n  
    % ----------------------------- 5c*kgj:x  
    if rpowers(1)==0 'urn5[i  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); dD _(MbTt  
        rpowern = cat(2,rpowern{:}); uh`W} n  
        rpowern = [ones(length_r,1) rpowern]; \bJ,8J1C  
    else >U/ m/H'  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); fh rS7f'Zd  
        rpowern = cat(2,rpowern{:}); /ekeU+j  
    end gWcl@|I;\  
    s&-m!|P  
    a#i;*J  
    % Compute the values of the polynomials: mx`C6G5  
    % -------------------------------------- HFV4S]U=  
    y = zeros(length_r,length(n)); V[&4Km9C  
    for j = 1:length(n) (7 i@ @  
        s = 0:(n(j)-m_abs(j))/2; 1_}* aQ  
        pows = n(j):-2:m_abs(j); I"/p^@IX  
        for k = length(s):-1:1 yHS=8!  
            p = (1-2*mod(s(k),2))* ... U&W{;myt  
                       prod(2:(n(j)-s(k)))/              ... _&0_@  
                       prod(2:s(k))/                     ... YcJZG|[  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... pF~[  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 3 K Y-+ k  
            idx = (pows(k)==rpowers); r q2]u  
            y(:,j) = y(:,j) + p*rpowern(:,idx); [se J'Io  
        end 0<3)K[m~H  
         &%."$rC/0b  
        if isnorm 5&}~W)"9  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ^{L/) Xy5  
        end j*uc$hC"  
    end wvH=4TT=w"  
    % END: Compute the Zernike Polynomials EA@p]+P  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Jb. V4  
    DIx!Sw7EC  
    JO\F-xO  
    % Compute the Zernike functions: ILsw'  
    % ------------------------------ q/I':a[1  
    idx_pos = m>0; =7&2-'(@  
    idx_neg = m<0; 1=fP68n  
    =pQ'wx|>|  
    ~N{ 7  
    z = y; D[d+lq#p  
    if any(idx_pos) ]w2nVC 3  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); //9M~qHa"  
    end <[7 bUB  
    if any(idx_neg) AcF6p)@_  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); i vy+e-)  
    end ANuIPF4NxP  
    $LxfdSa  
    qo2/?]  
    % EOF zernfun 07L >@Gf  
     
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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)  #Tr>[ZC  
    &p."` C  
    DDE还是手动输入的呢? X=sC8Edx  
    WcG!6.U>  
    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