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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, + jLy>=u  
    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, 8 x{Owj:Q  
    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? ">H*InF  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ? #K|l*  
    28MMH Q  
    Z vysLHj  
    P'W} ]mCD  
    4V+bE$Wu  
    function z = zernfun(n,m,r,theta,nflag) B~Sj#(WEa  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ^? fOccfQ{  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N f"MID6  
    %   and angular frequency M, evaluated at positions (R,THETA) on the fQ<sq0' e\  
    %   unit circle.  N is a vector of positive integers (including 0), and -&Rv=q>  
    %   M is a vector with the same number of elements as N.  Each element Blpk n1  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 2dn^K3  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, WTSh#L  
    %   and THETA is a vector of angles.  R and THETA must have the same S$mv(C  
    %   length.  The output Z is a matrix with one column for every (N,M) >ahDc!Jyu  
    %   pair, and one row for every (R,THETA) pair. z0 "DbZ;d  
    % 8D*7{Q  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike l]*RiK2AC  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )x.%PUA  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral n Bu!2c  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, (,|,j(=]  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized +3v)@18B1  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. u$nzpw0=H  
    % y=3 dGOFB  
    %   The Zernike functions are an orthogonal basis on the unit circle. _7c3=f83  
    %   They are used in disciplines such as astronomy, optics, and p Cz6[*kC  
    %   optometry to describe functions on a circular domain. @C;1e7  
    % JF=R$!5  
    %   The following table lists the first 15 Zernike functions. :qzg?\(  
    % R"nB4R0Uh  
    %       n    m    Zernike function           Normalization !>`Q]M`  
    %       -------------------------------------------------- bLc5$U$!I  
    %       0    0    1                                 1 WgNA%.|,  
    %       1    1    r * cos(theta)                    2 "HOZ2_(o  
    %       1   -1    r * sin(theta)                    2 6= ?0&Bx&  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ]!hjKu"  
    %       2    0    (2*r^2 - 1)                    sqrt(3) WogUILB  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ;UdM8+^/V]  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) oF%m  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) \/ bd  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) hx$]fvDevD  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) .D*Qu}  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) eg[EFI.h  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) wkg4I.  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) MAa9JA8kw)  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (Y1*Bs[l  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 4_#$k{  
    %       -------------------------------------------------- |x}TpM;ni  
    % ~f<'] zXv  
    %   Example 1: =G-OIu+H!U  
    % !3b& S4  
    %       % Display the Zernike function Z(n=5,m=1) !0{SVsc)  
    %       x = -1:0.01:1; x9lA';})  
    %       [X,Y] = meshgrid(x,x); &;PxDlY5  
    %       [theta,r] = cart2pol(X,Y); /}:{(Go  
    %       idx = r<=1; N_Us6 X  
    %       z = nan(size(X)); q"d9C)Md  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); {yn,u)@r9S  
    %       figure kWzp*<lWe  
    %       pcolor(x,x,z), shading interp o;8$#gyNY  
    %       axis square, colorbar &L6Ivpj-  
    %       title('Zernike function Z_5^1(r,\theta)') \0\O/^W0  
    % ~Ztn(1N  
    %   Example 2: UP]( 1lAf  
    % I9?\Jbqg  
    %       % Display the first 10 Zernike functions @Q1!xA^S  
    %       x = -1:0.01:1; 2?,Jn&i5  
    %       [X,Y] = meshgrid(x,x); !6/UwPs  
    %       [theta,r] = cart2pol(X,Y); S_lGr k\j  
    %       idx = r<=1; ,np=m17  
    %       z = nan(size(X)); AR| 4^  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Ah2@sp,z  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; %\'=Y/yP  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; fUw:jE xz  
    %       y = zernfun(n,m,r(idx),theta(idx)); [$dVs16K  
    %       figure('Units','normalized') U,rI/'  
    %       for k = 1:10 J*@pM  
    %           z(idx) = y(:,k); HUKrp*Hv  
    %           subplot(4,7,Nplot(k)) =!TUf/O-  
    %           pcolor(x,x,z), shading interp Y9.3`VX  
    %           set(gca,'XTick',[],'YTick',[]) M5bE5C  
    %           axis square .; MS 78BR  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) *@rA7zPFf  
    %       end %" iX3  
    % P(T-2Ux6  
    %   See also ZERNPOL, ZERNFUN2. >}SEU-7&\  
    W8W7<ml0A  
    N{Sp-J>  
    %   Paul Fricker 11/13/2006 @pH2"k| @  
    7L~ *%j  
    [6VB&   
    4Kj.o  
    0cm34\*  
    % Check and prepare the inputs: b++r#Q g  
    % ----------------------------- xe@e#9N$  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) |\(uO|)ju  
        error('zernfun:NMvectors','N and M must be vectors.') 9#DXA}  
    end X,Ql6uO  
    "uH>S+%|b  
    ')xOL =w  
    if length(n)~=length(m) & bTCTDZh  
        error('zernfun:NMlength','N and M must be the same length.') !5,C"r  
    end 1l-5H7^w2?  
    }aWy#Oe  
    Q[OwP  
    n = n(:); *fN+wiPD  
    m = m(:); 93*csO?Db  
    if any(mod(n-m,2)) o U=vl!\J  
        error('zernfun:NMmultiplesof2', ... lY,^  
              'All N and M must differ by multiples of 2 (including 0).') (.-4Jn  
    end /k'7j*t Z  
    Zc7;&cz  
    j\C6k  
    if any(m>n) J[ }H^FR  
        error('zernfun:MlessthanN', ... (Yewd/T  
              'Each M must be less than or equal to its corresponding N.') oN032o?S  
    end '/O:@P5qY  
    %`]+sg[i  
    x/,;:S  
    if any( r>1 | r<0 ) Y j oe|  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') '5h` ="  
    end '#6e Ub  
    Lx,=Up.  
    aUw-P{zp%  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) OnTe_JML  
        error('zernfun:RTHvector','R and THETA must be vectors.') eiK_JPFA-  
    end  7)2K6<q  
    Pb&tWv\ql  
    iil<zEic  
    r = r(:); um,f!ho-U  
    theta = theta(:); Lt*P&  
    length_r = length(r); aAjl 58  
    if length_r~=length(theta) D:T]$<=9  
        error('zernfun:RTHlength', ... !q\8`ss  
              'The number of R- and THETA-values must be equal.') +a5F:3$  
    end H )ej]DXy  
    dlYpbw}W&<  
    \_Bj"K  
    % Check normalization: &g|[/~dIr  
    % -------------------- BqNsW (+  
    if nargin==5 && ischar(nflag) wn"}<ka  
        isnorm = strcmpi(nflag,'norm'); nCYkUDnZ  
        if ~isnorm P%2v(  
            error('zernfun:normalization','Unrecognized normalization flag.') Znb={hh  
        end zu d_BOq{f  
    else >9H^r\  
        isnorm = false; i$NlS}W  
    end J*;RL`  
    Z"8lW+r *  
    ,@ '^3u  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9nR\7!_  
    % Compute the Zernike Polynomials TUfj\d,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZJ3g,dc  
    A-!e$yz>  
    7ws<' d7/  
    % Determine the required powers of r: #8Bh5L!SJ1  
    % ----------------------------------- ~nA k-toJ  
    m_abs = abs(m); *2h%dT:,%  
    rpowers = []; httywa^  
    for j = 1:length(n) }Ulxt:}   
        rpowers = [rpowers m_abs(j):2:n(j)]; :8`A  
    end 1'&.6{)P  
    rpowers = unique(rpowers); 0:V /z3?  
    JLu$1A@ '  
    KW:N 6w  
    % Pre-compute the values of r raised to the required powers, t]HY@@0g  
    % and compile them in a matrix: 5m1J&TZ0  
    % ----------------------------- nQc,^A)I  
    if rpowers(1)==0 D7hTn@I  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 0GUJc}fgvN  
        rpowern = cat(2,rpowern{:}); z$J m1l  
        rpowern = [ones(length_r,1) rpowern]; q%sZV>  
    else ;FqmZjm  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); JV~ Dly>  
        rpowern = cat(2,rpowern{:}); T8+[R2_  
    end 7(84j5zb  
    D /QLp3+o  
    }5+^  
    % Compute the values of the polynomials: sa'1hX^@  
    % -------------------------------------- gKh*q.  
    y = zeros(length_r,length(n)); =mYf] PIX  
    for j = 1:length(n) `vBBJ@f4)  
        s = 0:(n(j)-m_abs(j))/2; #QwkRzVoy  
        pows = n(j):-2:m_abs(j); owIpn=8|Q  
        for k = length(s):-1:1 C~2!@<y  
            p = (1-2*mod(s(k),2))* ... j!4{+&Laq  
                       prod(2:(n(j)-s(k)))/              ... c,@Vz 7c  
                       prod(2:s(k))/                     ... 9"P+K.%  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... X$!fR >Zc  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); >M#@vIo?<6  
            idx = (pows(k)==rpowers); E+\?ptw  
            y(:,j) = y(:,j) + p*rpowern(:,idx); )Q=u[ p  
        end 4kL6aSqT  
         {4S UG o>  
        if isnorm -_ 9k+AV  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); \WiCI:  
        end >` s"C  
    end t=Oq<r  
    % END: Compute the Zernike Polynomials E ASnh   
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Pxl,"  
    Z:{| ?4  
    `Abd=1nH  
    % Compute the Zernike functions: J.UNw8z  
    % ------------------------------ 9G[ DuYJI  
    idx_pos = m>0; k7U.]#5V  
    idx_neg = m<0; IP`lx  
    4 {M   
    M8-8 T  
    z = y; KtaoOe  
    if any(idx_pos) L-Q8iFW'  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ?-j/X6(\(  
    end `"=Hk@E  
    if any(idx_neg) 7{<v$g$  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); <{W{ Y\_A>  
    end myj/93p}`b  
    Sn[xI9}O  
    |Z>-<]p9g  
    % EOF zernfun fizW\f8ai  
     
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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)  FnVW%fh  
    q8ZxeMqx%  
    DDE还是手动输入的呢? ^%%Rf  
    5X3JQ"z  
    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