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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, d~ lB4  
    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, 8A|{jH74  
    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? ?52{s"N0>  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? X<6Ro es2  
    Y+ZQN>  
    LdSBNg#3  
    %TO=]>q  
     ppwjr +  
    function z = zernfun(n,m,r,theta,nflag) %]~XbO  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ,d^ze=  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Cd>GY  
    %   and angular frequency M, evaluated at positions (R,THETA) on the i{['18Q$F3  
    %   unit circle.  N is a vector of positive integers (including 0), and . kv/db  
    %   M is a vector with the same number of elements as N.  Each element D/T& 0  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) X)-9u8  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ~j1.;WId[  
    %   and THETA is a vector of angles.  R and THETA must have the same bzI!;P1&  
    %   length.  The output Z is a matrix with one column for every (N,M) qNhV zx  
    %   pair, and one row for every (R,THETA) pair. &) '5_#S  
    % jGM+  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike t>W^^'=E  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), XDtr{r6z  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral lkj^<%N"r  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, NT qtr="  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 3$]SP1Mc(  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. M"q]jeaM  
    % rZ.,\ X_  
    %   The Zernike functions are an orthogonal basis on the unit circle. fx W,S  
    %   They are used in disciplines such as astronomy, optics, and h)O<bI8  
    %   optometry to describe functions on a circular domain. 6usy0g D  
    % ^uU'Qc4S=  
    %   The following table lists the first 15 Zernike functions. /EJwO3MW  
    % _h@s)"  
    %       n    m    Zernike function           Normalization sd (I@ &y  
    %       --------------------------------------------------  QuJ~h}k  
    %       0    0    1                                 1 e ]@Ex  
    %       1    1    r * cos(theta)                    2 /F>\-    
    %       1   -1    r * sin(theta)                    2 n?@3+wG  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) )gO=5_^u*o  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Z*/*P4\  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) M<f=xY2$v  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) r_sZw@lqJ  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) c1v,5c6d j  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) F TB@70  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) os\"(*dix  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) uYh6q1@"~  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) fz)i9D@  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) _}_lrg}U  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,zCrix 3  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) \ 2Jr( ?U  
    %       -------------------------------------------------- lX)RG*FlTC  
    % Tk9/1C{8  
    %   Example 1: \u|8MEB  
    % 8QFn/&Ql$B  
    %       % Display the Zernike function Z(n=5,m=1) 9fWr{fx  
    %       x = -1:0.01:1; B{ i5UhxD  
    %       [X,Y] = meshgrid(x,x); 5kwDmJy  
    %       [theta,r] = cart2pol(X,Y); !&~8j7{  
    %       idx = r<=1; >[4;K&$B  
    %       z = nan(size(X)); 7l-` k  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); (#w8/@JxF  
    %       figure ?}QHEk:H  
    %       pcolor(x,x,z), shading interp o=!3=2@dh  
    %       axis square, colorbar @ 2mJh^cj  
    %       title('Zernike function Z_5^1(r,\theta)') /]/3)@wT  
    % !fFmQ\|)4S  
    %   Example 2: +6vm4(3?  
    % :#M(,S"Qq  
    %       % Display the first 10 Zernike functions R:*I>cRs  
    %       x = -1:0.01:1; Ga4Ru  
    %       [X,Y] = meshgrid(x,x); fo"dX4%}  
    %       [theta,r] = cart2pol(X,Y); )^&,[Q=i  
    %       idx = r<=1; )N{Qpbh  
    %       z = nan(size(X)); l8n}&zX  
    %       n = [0  1  1  2  2  2  3  3  3  3]; &Wj %`T{  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; E6);\SJG}  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; $qN+BKd]3  
    %       y = zernfun(n,m,r(idx),theta(idx)); nwd 02tu  
    %       figure('Units','normalized') 8G?{S.%.  
    %       for k = 1:10 *+p9u 1B5  
    %           z(idx) = y(:,k); .Gq)@{o>  
    %           subplot(4,7,Nplot(k)) U=<E,tM  
    %           pcolor(x,x,z), shading interp ~lx5RTkp  
    %           set(gca,'XTick',[],'YTick',[]) 5a9PM(  
    %           axis square "%+C@>`(  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) aX=  
    %       end 1=DUFl.  
    % &`7tX.iMlh  
    %   See also ZERNPOL, ZERNFUN2. ~o:lh],~  
    0T!_;IQ  
    Sr_]R<?  
    %   Paul Fricker 11/13/2006 f1Ruaz-  
    5 ^}zysY`  
    f"h{se8C  
    >6XGF(G   
    +p =n-  
    % Check and prepare the inputs: A"S{W^iL  
    % ----------------------------- }U$Yiv  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) `0+zF-  
        error('zernfun:NMvectors','N and M must be vectors.') clz6; P  
    end 6:i(<7  
    6Lw34R  
    EEQW$W1@  
    if length(n)~=length(m) Pms"YhyZ7  
        error('zernfun:NMlength','N and M must be the same length.') H*_:IfI!  
    end wK@k}d  
    XW6>;:4k  
    4/S% eZB  
    n = n(:); clQN@1] M  
    m = m(:); 3_(fisvx  
    if any(mod(n-m,2)) EfY|S3Av  
        error('zernfun:NMmultiplesof2', ... 8W?/Sg`  
              'All N and M must differ by multiples of 2 (including 0).') h?2qX  
    end Q4 Mp[  
    (3C6'Wt  
    8D eRs#  
    if any(m>n) 2<Pi2s'  
        error('zernfun:MlessthanN', ... ))}w;w   
              'Each M must be less than or equal to its corresponding N.') f>nj9a5  
    end bit&H  
    |`9POl=  
    Ip]-OVg  
    if any( r>1 | r<0 ) pR2QS  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ml 7]s N(  
    end W?8 |h  
    G S-@drZp_  
    fU_itb(  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ^w4FqdGM  
        error('zernfun:RTHvector','R and THETA must be vectors.') Klh7&HzR  
    end xtL_,ug  
    XA} !  
    7g3vh%G.  
    r = r(:); A P><l@  
    theta = theta(:); !+& "y K@J  
    length_r = length(r); uWR\#D'  
    if length_r~=length(theta) P: &XtpP  
        error('zernfun:RTHlength', ... {:c*-+?  
              'The number of R- and THETA-values must be equal.') 6/B"H#rN  
    end 92+LY]jS  
    %qRbl4  
    F*rU=cu  
    % Check normalization: n= yT%V. l  
    % -------------------- s"`uE$6N  
    if nargin==5 && ischar(nflag) \?&P|7N  
        isnorm = strcmpi(nflag,'norm'); !"B0z+O>  
        if ~isnorm j}Lt"r2F  
            error('zernfun:normalization','Unrecognized normalization flag.') p=jD "lq  
        end &; 5QB  
    else ~p<o":k+Lv  
        isnorm = false; FQ>KbZh  
    end OOS(YP@b  
    x2+%.$'  
    ext`%$ U7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% qsn6i%VH  
    % Compute the Zernike Polynomials )~;=0O |X  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a5C%OI<  
    fb[f >1|  
    c.m8~@O5+  
    % Determine the required powers of r: a+ZP]3@ 7  
    % ----------------------------------- %CJgJ,pk>  
    m_abs = abs(m); B25@6   
    rpowers = []; ~{'.9  
    for j = 1:length(n) #p}I 84Q  
        rpowers = [rpowers m_abs(j):2:n(j)]; Ej>5PXp'2  
    end {tMpI\>S  
    rpowers = unique(rpowers); M~7gUb|  
    5mdn77F_  
    +{N LziO  
    % Pre-compute the values of r raised to the required powers, "P`V|g  
    % and compile them in a matrix: azKbGS/X  
    % ----------------------------- Se+sgw_"  
    if rpowers(1)==0 wMNtN3   
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); &yN@(P)  
        rpowern = cat(2,rpowern{:}); LL@VR#n"V  
        rpowern = [ones(length_r,1) rpowern]; tKgPKWP   
    else j#r|t+{"C  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); V|xK vH  
        rpowern = cat(2,rpowern{:}); UbKdB  
    end / 2>\Z(  
    1?sR1du,  
    AGkk|`  
    % Compute the values of the polynomials: ) D:M_T2  
    % -------------------------------------- KW|X\1H  
    y = zeros(length_r,length(n)); w?]k$  
    for j = 1:length(n) H5uWI  
        s = 0:(n(j)-m_abs(j))/2; nBv|5$w:  
        pows = n(j):-2:m_abs(j); z(L\I  
        for k = length(s):-1:1 7sZVN  
            p = (1-2*mod(s(k),2))* ... 9{_D"h}}  
                       prod(2:(n(j)-s(k)))/              ... 1wSJw  
                       prod(2:s(k))/                     ... Rf2$k/lZ  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... nAv@^G2  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); v8p-<N)  
            idx = (pows(k)==rpowers); .q#2 op  
            y(:,j) = y(:,j) + p*rpowern(:,idx); YFgQ!\&59  
        end VXlTA>a }  
         e8O[xM  
        if isnorm VE1 B"s</  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); z%5i^P  
        end i{TErJ{}e  
    end fM,U|  
    % END: Compute the Zernike Polynomials  N)G.^9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3D70`u  
    7bOL,S  
    S*~v9+  
    % Compute the Zernike functions: QWG?^T fi  
    % ------------------------------ f@Mm{3&.  
    idx_pos = m>0; ,y@` =  
    idx_neg = m<0; 10xo<@l  
    ?zp@HS a9  
    ciO^2X  
    z = y; SOQm>\U'i  
    if any(idx_pos) C*Avu  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); r!+-"hS!  
    end . OA_)J7  
    if any(idx_neg) !/O c)Yk  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Q| > \{M  
    end L/9f"%kZ  
    LQ pUyqR  
    |r_S2)zH9m  
    % EOF zernfun OO5k _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)  n<47#-  
    PScq-*^  
    DDE还是手动输入的呢? It.G-(  
    \]pRu"  
    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