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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, -U[`pUY?f  
    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, P%HyIODS  
    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? PM!t"[@&  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? }#5roNH~Z  
    \3?;[xD  
    4^_Au^8R(  
    bCUh^#]x  
    $ywh%OEH  
    function z = zernfun(n,m,r,theta,nflag) ^) ^|;C\`  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. O \8G~V 5"  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N y7EX&  
    %   and angular frequency M, evaluated at positions (R,THETA) on the yc=#Jn?S  
    %   unit circle.  N is a vector of positive integers (including 0), and @]wem  
    %   M is a vector with the same number of elements as N.  Each element ?9@Af{b t2  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) c G!2Iy~lA  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, )wv[!cYyW  
    %   and THETA is a vector of angles.  R and THETA must have the same T ) f_W  
    %   length.  The output Z is a matrix with one column for every (N,M) L$c%u  
    %   pair, and one row for every (R,THETA) pair. Ds,"E#?  
    % {<4?o? 1 g  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike l'". }6S  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), J*KBG2+13  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 4eL54).1O  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 8;f<qu|w  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized IYg3ve`x  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ,xe@G)a  
    % dQ`ZrWd_U  
    %   The Zernike functions are an orthogonal basis on the unit circle. !_H8Q}a  
    %   They are used in disciplines such as astronomy, optics, and wDMB  
    %   optometry to describe functions on a circular domain. <ZC^H  
    % um2s^G  
    %   The following table lists the first 15 Zernike functions. )k] !u  
    % ;6 6_G Sjz  
    %       n    m    Zernike function           Normalization 5@t uo`k  
    %       -------------------------------------------------- JKi@Kw  
    %       0    0    1                                 1 :F w"u4WI  
    %       1    1    r * cos(theta)                    2 xc<eU`-' b  
    %       1   -1    r * sin(theta)                    2 5J\|gZQF  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) n.6 0$kR`  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ~7 L)n  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) dzE Q$u/I  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) cc(r,ij~4  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 1L=Qg4 H  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 6O@ ^`T  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) +IO1ipc4cE  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) hB$Y4~T%  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #JR,C -w  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) (6#yw`\  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) U[e8K  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) vV\F^  
    %       -------------------------------------------------- LVFsd6:h  
    % FDd>(!>  
    %   Example 1: G9y12HV  
    % L8w76|  
    %       % Display the Zernike function Z(n=5,m=1) ]1|Ql*6y,  
    %       x = -1:0.01:1; kl3S~gE4@  
    %       [X,Y] = meshgrid(x,x); 6n6VEwYj  
    %       [theta,r] = cart2pol(X,Y); m`\i+  
    %       idx = r<=1; <,4R2'  
    %       z = nan(size(X)); CX ]\Q-y  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ^Im%D(MY  
    %       figure Rp`_Grcd  
    %       pcolor(x,x,z), shading interp JfP\7  
    %       axis square, colorbar :OQ:@Yk  
    %       title('Zernike function Z_5^1(r,\theta)') 2hwXWTSu  
    % Ux)p%-  
    %   Example 2: ,+f0cv4  
    % T ^%n!t  
    %       % Display the first 10 Zernike functions l@Eq|y,  
    %       x = -1:0.01:1; M$]O=2h+2  
    %       [X,Y] = meshgrid(x,x); _]D#)-uv}C  
    %       [theta,r] = cart2pol(X,Y); Vyt~OTI\  
    %       idx = r<=1; *n*N|6 +  
    %       z = nan(size(X)); kF+}.x%  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 0n<(*bfW  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; o ,Tr^e$  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; qzH qj;  
    %       y = zernfun(n,m,r(idx),theta(idx)); <jRFN&"h}  
    %       figure('Units','normalized') e:GgA  
    %       for k = 1:10 5e/qgI)M5  
    %           z(idx) = y(:,k); |DFvZ6}  
    %           subplot(4,7,Nplot(k)) Hr<C2p^a  
    %           pcolor(x,x,z), shading interp kToVBU$  
    %           set(gca,'XTick',[],'YTick',[]) g<rKV+$6  
    %           axis square `Ge+(1x  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) )p!*c,  
    %       end [C+Gmu  
    % ;la#Vf:]  
    %   See also ZERNPOL, ZERNFUN2. e\A(#l@g  
    N7~)qqb  
    2i7i\?<.  
    %   Paul Fricker 11/13/2006 i (%tHa37  
    F[7Kw"~J  
    Yt/SnF  
    `j}_BW_  
    }l}yn@hYC  
    % Check and prepare the inputs: sk7rU+<  
    % ----------------------------- ie$`pyj!x  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 4j=<p@  
        error('zernfun:NMvectors','N and M must be vectors.') Q_QKm0!  
    end Y[ iDX#  
    %su}Ru  
    C?[a3rNH(  
    if length(n)~=length(m) ?y>Y$-v/C  
        error('zernfun:NMlength','N and M must be the same length.') uOG-IHuF  
    end %R.xS} Q  
    SY$J+YBLM  
    `sk!C7%  
    n = n(:); |%'6f}fnE  
    m = m(:); {*?sVAvj  
    if any(mod(n-m,2)) 2<[ eD`u  
        error('zernfun:NMmultiplesof2', ... d`9W  
              'All N and M must differ by multiples of 2 (including 0).') FpdDIa  
    end 2/v35| ?  
    RHwaJ;:)#  
    *3_f &Y  
    if any(m>n) `%t$s,TiP  
        error('zernfun:MlessthanN', ... I #M%%5e  
              'Each M must be less than or equal to its corresponding N.') ? o~:'Z  
    end Ic[}V0dk  
    WCuzV7tw  
    ;m@1Ec@* p  
    if any( r>1 | r<0 ) fJ)N:q`  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') MvFM ,  
    end ET,Q3X\Oe  
    Dd!Sr8L[  
    %6NO0 F^  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) xl+DRPzl  
        error('zernfun:RTHvector','R and THETA must be vectors.') 0$eyT-:d  
    end -ajM5S=d*  
    V&Xi> X8  
    On{~St'V  
    r = r(:); )p$\gwr=2  
    theta = theta(:); .O5LI35,  
    length_r = length(r); <91t`&aWW  
    if length_r~=length(theta) 1Yc%0L(  
        error('zernfun:RTHlength', ... tmO;:n<N  
              'The number of R- and THETA-values must be equal.') M"=8O>NZ2  
    end G1kaF/`O  
    (;;J,*NP  
    ^ckj3Y#;  
    % Check normalization: e#>tM  
    % -------------------- ,M\j%3  
    if nargin==5 && ischar(nflag) cPpu  
        isnorm = strcmpi(nflag,'norm'); hc-lzYS  
        if ~isnorm HQq`pG%m6  
            error('zernfun:normalization','Unrecognized normalization flag.') 1<xcMn0et  
        end j~M#Ss-H8  
    else Gs[Vu@*  
        isnorm = false; 0o=!j3RjH  
    end s~S?D{!  
    +yzcx3<  
    q1YLq(e  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {5T0RL{\N  
    % Compute the Zernike Polynomials Q`H# fS~  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% blJIto '  
    x!'7yx  
    nIfN"  
    % Determine the required powers of r: TE3*ktB{N  
    % ----------------------------------- pG/ NuImA  
    m_abs = abs(m); '@'B>7C#  
    rpowers = []; l iw,O 6  
    for j = 1:length(n) Vy]A,Rn7  
        rpowers = [rpowers m_abs(j):2:n(j)]; ]#F q>E  
    end "Dyym<J  
    rpowers = unique(rpowers); $bk>kbl P  
    |<sf:#YzY&  
    m"n.Dz/S  
    % Pre-compute the values of r raised to the required powers, m&Lt6_vi  
    % and compile them in a matrix: UM<@t%|>  
    % ----------------------------- Q Q@9_[N  
    if rpowers(1)==0 ya:sW5fk  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); {5>3;.  
        rpowern = cat(2,rpowern{:}); d-~vR(tU  
        rpowern = [ones(length_r,1) rpowern]; vCj4;P g  
    else 7'Lp8  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); l1&5uwuF  
        rpowern = cat(2,rpowern{:}); ~%`EeJwT  
    end d+tj%7  
    ASR-a't6  
    Y+FP   
    % Compute the values of the polynomials: C5Xof|#p|  
    % -------------------------------------- ;v_ls)_,-  
    y = zeros(length_r,length(n)); 1YFeVMc  
    for j = 1:length(n) ]3}feU+  
        s = 0:(n(j)-m_abs(j))/2; ~]&B >q  
        pows = n(j):-2:m_abs(j); @d&g/ccMxd  
        for k = length(s):-1:1 z OtkC3hY  
            p = (1-2*mod(s(k),2))* ... 8/Mx5~ R  
                       prod(2:(n(j)-s(k)))/              ... +kM\ D~D1  
                       prod(2:s(k))/                     ... Vn'?3Eb<  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... aVP5%  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); J;~E<_"Hn  
            idx = (pows(k)==rpowers); 0C]4~F x~  
            y(:,j) = y(:,j) + p*rpowern(:,idx);  =^Th[B  
        end r&SO:#rOSM  
         QP:9%f>=  
        if isnorm Lx%:t YZ  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); bhYU5I 9  
        end *6XRjq^#  
    end pajy#0 U  
    % END: Compute the Zernike Polynomials mbyih+amCr  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ABcBEv3  
    L?HF'5o  
    0(8gQ 2n  
    % Compute the Zernike functions: Ah (iE  
    % ------------------------------ zrrz<dW  
    idx_pos = m>0; FuuS"G,S  
    idx_neg = m<0; 7,h3V=^)Q  
    PK+ x6]x  
    S;8.yj-  
    z = y; Oxv+1Ub<Dv  
    if any(idx_pos) =5ug\S  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 2SciB*5  
    end J?IC~5*2  
    if any(idx_neg) VD/&%O8n  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); =:gjz4}_8  
    end |<rfvsQ.  
    B7!;]'&d  
    9:-T@u  
    % EOF zernfun ]YsR E>  
     
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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)  [ H,u)8)  
     YqU/\f+  
    DDE还是手动输入的呢? }-p,iTm  
    0JXqhc9'  
    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