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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, y(Gn+  
    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, ad: qOm  
    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? !l0]IX` F  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Lmte ~oBi  
    3@I0j/1#k1  
    60!%^O =  
    7?=^0?a  
    \~hrS/$[$  
    function z = zernfun(n,m,r,theta,nflag) N x&/p$d  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. OKMdyyO<l  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N gPKf8{#%e  
    %   and angular frequency M, evaluated at positions (R,THETA) on the r&E gP  
    %   unit circle.  N is a vector of positive integers (including 0), and "V>}-G&  
    %   M is a vector with the same number of elements as N.  Each element #-;W|ib%z  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 6]?%1HSi  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, eT".psRiC  
    %   and THETA is a vector of angles.  R and THETA must have the same fwz:k]vk  
    %   length.  The output Z is a matrix with one column for every (N,M) ,~d0R4)  
    %   pair, and one row for every (R,THETA) pair. ?.VKVTX^  
    % y8~OkdlN#  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike g{yw&q[B=  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 4$KDf;m@  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ]#]Z]9w  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Dds-;9  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized wN!\$i@E:  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. M 5sk&>  
    % Gw<D'b)!  
    %   The Zernike functions are an orthogonal basis on the unit circle.  1c0' i  
    %   They are used in disciplines such as astronomy, optics, and Zt!#KSF7%  
    %   optometry to describe functions on a circular domain. x 7by|G(  
    % H[~ D]RG}'  
    %   The following table lists the first 15 Zernike functions. h:8P9WhWF  
    % d-~V.  
    %       n    m    Zernike function           Normalization 6j|Ncv  
    %       -------------------------------------------------- g{]6*`/Z  
    %       0    0    1                                 1 S $p>sItO  
    %       1    1    r * cos(theta)                    2 U80=f2  
    %       1   -1    r * sin(theta)                    2 ;_bRq:!j;  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 0~ho/_  
    %       2    0    (2*r^2 - 1)                    sqrt(3) J 4gtm"2)  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) j?N<40z  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) '. "_TEIF  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) x fb .Z(  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) TGF$zvd  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) a yoC]rE  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) B r#{  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) dun`/QKV  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) wG,"X'1  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) qf x*a88  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 2#.s{Bv  
    %       -------------------------------------------------- QOXo(S  
    % KHAc!4lA  
    %   Example 1: 1cK'B<5">]  
    % n2mO-ZXud  
    %       % Display the Zernike function Z(n=5,m=1) aoey 5hts  
    %       x = -1:0.01:1; n&:ohOH%  
    %       [X,Y] = meshgrid(x,x); sjyr9AF  
    %       [theta,r] = cart2pol(X,Y); EQ$k^Y8 "  
    %       idx = r<=1; Ok_}d&A  
    %       z = nan(size(X)); 3xy2ZYw  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); +F)-n2Bi  
    %       figure |HmY`w6*z  
    %       pcolor(x,x,z), shading interp Vg NB^w  
    %       axis square, colorbar A r!0GwE+  
    %       title('Zernike function Z_5^1(r,\theta)') c7XBZ%D  
    % RzqgN*]lY  
    %   Example 2: i3w~&y-  
    % 9`*ST(0/  
    %       % Display the first 10 Zernike functions v.(dOIrX  
    %       x = -1:0.01:1; %aNm j)L  
    %       [X,Y] = meshgrid(x,x); eNd&47lJ  
    %       [theta,r] = cart2pol(X,Y); *tUOTA 3L  
    %       idx = r<=1; f'=u`*(b7  
    %       z = nan(size(X)); JVIFpN"`  
    %       n = [0  1  1  2  2  2  3  3  3  3]; SZKYq8ZA)V  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; [Qnf]n\FJ  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; '[WL8,.Q  
    %       y = zernfun(n,m,r(idx),theta(idx)); lOt7 ij(,L  
    %       figure('Units','normalized') Tgz=I4g  
    %       for k = 1:10 g=t`3X#d  
    %           z(idx) = y(:,k); INA3^p'w  
    %           subplot(4,7,Nplot(k)) v[Q)L!J1  
    %           pcolor(x,x,z), shading interp r? /Uu &  
    %           set(gca,'XTick',[],'YTick',[]) -P}A26qB  
    %           axis square %M iv8  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 1 sHjM %  
    %       end +*8su5:[&@  
    % ,>-jZtm  
    %   See also ZERNPOL, ZERNFUN2. ..JRtuM-v  
    I> ;{BYPV  
    xh2r?K@k>  
    %   Paul Fricker 11/13/2006 iN&oSpQ  
    D./{f8  
    !5} }mf  
    "9_$7.q<y  
    S"&Gutu3o  
    % Check and prepare the inputs: KUJLx  
    % ----------------------------- 1b%Oi.;  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) EnWv9I<  
        error('zernfun:NMvectors','N and M must be vectors.') <[[yV  
    end VO0:4{-  
    2fTuIS<yr  
    MKfK9>a  
    if length(n)~=length(m) %&6Q Uv^  
        error('zernfun:NMlength','N and M must be the same length.') @:?[R&`  
    end "SMJ:g",  
    >=0]7k;  
    5?^#v  
    n = n(:); vxZ'-&;t  
    m = m(:); &x1A {j_  
    if any(mod(n-m,2)) p-i Fe\+  
        error('zernfun:NMmultiplesof2', ... 67(s\  
              'All N and M must differ by multiples of 2 (including 0).') NF&Sv  
    end \ivxi<SR  
    5REFz  
    t1w]L  
    if any(m>n) DC h !Z{I  
        error('zernfun:MlessthanN', ... \#,2#BmO"E  
              'Each M must be less than or equal to its corresponding N.') ?z.?(xZ 6  
    end oks;G([  
    [`@M!G.  
    w x]?D%l  
    if any( r>1 | r<0 ) E4% -*n  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') RHFRN&RU$  
    end `^[k8Z(  
    M[`[+5v  
    4G,FJjE`p  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) a]r+np]vTy  
        error('zernfun:RTHvector','R and THETA must be vectors.') "kP,v&n  
    end $bG*f*w  
    1L!;lP2  
    Po)U!5Tm  
    r = r(:); 7Vy_Cec1  
    theta = theta(:); DT`HS/~fH  
    length_r = length(r); _|u}^MLO  
    if length_r~=length(theta) 3/+kjY/  
        error('zernfun:RTHlength', ... bh@CtnO  
              'The number of R- and THETA-values must be equal.') Yk|6?e{+)  
    end b,^ "-r  
    =_`q;Tu=  
    jQ V[zcM  
    % Check normalization: n}UJ - \$  
    % -------------------- xfeED^?  
    if nargin==5 && ischar(nflag) VZt%cq  
        isnorm = strcmpi(nflag,'norm'); mS'Ad<  
        if ~isnorm ^UKAD'_#%O  
            error('zernfun:normalization','Unrecognized normalization flag.') [P+kQBL pL  
        end !\7 M7  
    else 3JGrJ!x  
        isnorm = false; ',R%Q0Q  
    end &)OI!^ (  
    G[ U5R?/  
    a=}1`Q  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >T]9.`xhK  
    % Compute the Zernike Polynomials h,$CJdDY]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GKFRZWXdT  
    P*!`AWn  
    ~2k.x*$  
    % Determine the required powers of r: i?!9%U!z4  
    % ----------------------------------- r<ww%2HTS  
    m_abs = abs(m); > '=QBW  
    rpowers = []; cjL)M=pIS  
    for j = 1:length(n) HX2u{2$  
        rpowers = [rpowers m_abs(j):2:n(j)]; {Phq39g  
    end LG:k}z/T  
    rpowers = unique(rpowers); hZ1enej)  
    /''=V.-N  
    )!-gT  
    % Pre-compute the values of r raised to the required powers, 2L&c91=wE  
    % and compile them in a matrix: Z|C,HF+m.  
    % ----------------------------- /[_aK0U3  
    if rpowers(1)==0 e#/&A5#Ya  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); sY!JB7!j  
        rpowern = cat(2,rpowern{:}); 9HJYrzf{%  
        rpowern = [ones(length_r,1) rpowern]; 5@5 *}[M  
    else B6dU6"  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); `+n0a@BVB  
        rpowern = cat(2,rpowern{:}); b3%x&H<j  
    end /C8}5)  
    ?TpjU*Cxy  
    }OEL] 5  
    % Compute the values of the polynomials: )'m;a_r`  
    % -------------------------------------- 0 8)f  
    y = zeros(length_r,length(n)); o:{Sws(=  
    for j = 1:length(n) bRu 9*4t  
        s = 0:(n(j)-m_abs(j))/2; #; !@Pf  
        pows = n(j):-2:m_abs(j); w=XIpWl  
        for k = length(s):-1:1 %JBLp xnq  
            p = (1-2*mod(s(k),2))* ... '/<\X{l8  
                       prod(2:(n(j)-s(k)))/              ... ^>E>\uz0v  
                       prod(2:s(k))/                     ... v'?Smd1v /  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... -]^JaQw  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); n5C,Z!)z  
            idx = (pows(k)==rpowers); UdrgUqq)  
            y(:,j) = y(:,j) + p*rpowern(:,idx); kS_#8 I  
        end cRs.@U\{R\  
         :lXY% [!6P  
        if isnorm ]AA|BeL?|  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); zd%f5L('  
        end [ifw}(  
    end CtMqE+j^  
    % END: Compute the Zernike Polynomials BlpyE[h T  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vQCb?+X&  
    :PBFFLe  
    bK6^<,~  
    % Compute the Zernike functions: }kt%dDU  
    % ------------------------------ 2n3&uvf'TL  
    idx_pos = m>0; j-BNHX  
    idx_neg = m<0; L E\rc A  
    SAq .W"ri  
    ynw(wSH=  
    z = y; 2z{B  
    if any(idx_pos) ?u#s?$Y?  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); YT?Lt!cl=  
    end Jd/d\P  
    if any(idx_neg) YD[AgToo0  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Ml c_w19C9  
    end Ze>R@rK  
    LT$t%V0?.e  
    gd * b0(  
    % EOF zernfun &S xF"pYV  
     
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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)  D@ sMCR  
    K!(WcoA&2i  
    DDE还是手动输入的呢? h*KDZ+{)  
    )CoFRqz<h  
    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