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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 正序阅读 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, #mkr]K8A4  
    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, 6{7O  
    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? &g& &-=7)  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? cC}s5`  
    ]NFDE-Jz]  
    LG<lZ9+y  
    YSa:"A  
    *?K` T^LS  
    function z = zernfun(n,m,r,theta,nflag) W^=89I4]  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. $}KYpSV  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 4uftx1o   
    %   and angular frequency M, evaluated at positions (R,THETA) on the t91CxZQ^s  
    %   unit circle.  N is a vector of positive integers (including 0), and `=KrV#/758  
    %   M is a vector with the same number of elements as N.  Each element  v$tS 2N2  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) HqF8:z?v  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, A+F-r_]}db  
    %   and THETA is a vector of angles.  R and THETA must have the same ~ml\|  
    %   length.  The output Z is a matrix with one column for every (N,M)  gA[M  
    %   pair, and one row for every (R,THETA) pair. ]#:xl}'LS  
    % _-!6@^+  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike E,6E-9  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), l&|{uk  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 2~`dV_  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, <=7)t.  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized @H_LPn  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ;XtDz  
    % rSJ}qRXwU  
    %   The Zernike functions are an orthogonal basis on the unit circle. P)\f\yb  
    %   They are used in disciplines such as astronomy, optics, and Xj@Kt|&`k  
    %   optometry to describe functions on a circular domain. F Q k;  
    % H~~(v52wD  
    %   The following table lists the first 15 Zernike functions. [KE4wz+s{  
    % jU#%@d6!#  
    %       n    m    Zernike function           Normalization ;< ][upn  
    %       -------------------------------------------------- .N'UnKz  
    %       0    0    1                                 1 fZ376Z:S$  
    %       1    1    r * cos(theta)                    2 <Q kfvK]Q  
    %       1   -1    r * sin(theta)                    2 [`b{eLCFX]  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) C=b5[, UCB  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Qdn:4yk  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ?#[K&$}  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) f7W=x6Z4  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) *7vPU:Q[  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Y$Ke{6 4  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 0=5i\*5 p  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 'q-h kN  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) FD-)nv2:  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) wS^-o  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) nSC>x:jY5/  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) .iYgRW=T  
    %       -------------------------------------------------- vJ'ho  
    % }rQ*!2Y?  
    %   Example 1: 37[C^R!1c  
    % 0IdD   
    %       % Display the Zernike function Z(n=5,m=1) WE"'3u^k  
    %       x = -1:0.01:1; y5ExEXa  
    %       [X,Y] = meshgrid(x,x); <f*0 XJ#  
    %       [theta,r] = cart2pol(X,Y); jl@8pO$  
    %       idx = r<=1; z?aD Oh  
    %       z = nan(size(X)); }*t~&l0  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); zKutx6=aj  
    %       figure Ii8jY_  
    %       pcolor(x,x,z), shading interp o MAK[$k;  
    %       axis square, colorbar fI|1@e1  
    %       title('Zernike function Z_5^1(r,\theta)') k$2Y)  
    % \[&]kPcDl  
    %   Example 2: Ygl!fC 4b  
    % F)IP~BE-k  
    %       % Display the first 10 Zernike functions 9e5UTJ  
    %       x = -1:0.01:1; 3 /e !7  
    %       [X,Y] = meshgrid(x,x); YH>n{o;- ?  
    %       [theta,r] = cart2pol(X,Y); <f6Oj`{f4  
    %       idx = r<=1; cjW]Nw  
    %       z = nan(size(X)); Pm_=   
    %       n = [0  1  1  2  2  2  3  3  3  3]; WDZi @9X_  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; HHYcFoJwYN  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Pla EI p  
    %       y = zernfun(n,m,r(idx),theta(idx)); GND[f}  
    %       figure('Units','normalized') @RP|?Xc{?  
    %       for k = 1:10 dB5DJ:$W$  
    %           z(idx) = y(:,k); T,fz/5w  
    %           subplot(4,7,Nplot(k)) 'n no)kQ"  
    %           pcolor(x,x,z), shading interp ^:j$p,0e*S  
    %           set(gca,'XTick',[],'YTick',[]) GM/1u fZH  
    %           axis square [ZbK)L+_  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) I}WJ0}R  
    %       end +=_Pl7?  
    % ;D1IhDC  
    %   See also ZERNPOL, ZERNFUN2. 8{YxUD  
    -{h[W bf  
    9PAp*`J@kr  
    %   Paul Fricker 11/13/2006 UQ)}i7v  
    OOCeZ3yF(  
    \abl|;fj  
    A M2M87{t  
    4=Ey\Px  
    % Check and prepare the inputs: >`:+d'Jv0  
    % ----------------------------- ||V:',#,W  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7yp*I[1Qf>  
        error('zernfun:NMvectors','N and M must be vectors.') ^XM;D/Gp~  
    end TRZ^$<AG  
    IoA;q)  
    lu_Gr=#O  
    if length(n)~=length(m) U6Ak"  
        error('zernfun:NMlength','N and M must be the same length.') y#+o*(=fRE  
    end g8Z14'Ke  
    (=j!P*  
    3_$eQ`AAA  
    n = n(:); lI 1lP 1  
    m = m(:); P `"7m-  
    if any(mod(n-m,2)) 8;8}Oq  
        error('zernfun:NMmultiplesof2', ... |BW,pT  
              'All N and M must differ by multiples of 2 (including 0).') 9K|lU:,  
    end *-_Np u6  
    &}O!l'  
    .KN]a"]  
    if any(m>n) )^TQedF  
        error('zernfun:MlessthanN', ... s /M~RB!w  
              'Each M must be less than or equal to its corresponding N.') ^v-'=1ub?  
    end TXcKuo=  
    YW<2:1A|  
    __j8jEV  
    if any( r>1 | r<0 ) ~-d.3A $u  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Ao T7sy7  
    end aB^G  
    _GqE'VX  
    M>@R=f  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) U[l%oLra  
        error('zernfun:RTHvector','R and THETA must be vectors.') (, "E9.  
    end Oq6n.:8g"  
    &J$5+"/;X  
    "ux]kfoT  
    r = r(:); |@.<} /  
    theta = theta(:); $0T"YC%  
    length_r = length(r); &F_rg,q&_  
    if length_r~=length(theta) 7-I>5 3@  
        error('zernfun:RTHlength', ... I})t  
              'The number of R- and THETA-values must be equal.') ,rQ)TT  
    end z :v, Vu  
    vv/,Rgv  
    f)Q]{cb6  
    % Check normalization: +E|ouFI  
    % -------------------- &Fjilx'k  
    if nargin==5 && ischar(nflag) /T)n5X  
        isnorm = strcmpi(nflag,'norm'); '*u;:[73  
        if ~isnorm ~+C?][T  
            error('zernfun:normalization','Unrecognized normalization flag.') V(LFH9.Mp  
        end MdZgS#`  
    else o '/C$E4W  
        isnorm = false; $3[\:+  
    end PMs_K"-K  
    uz3pc;0LPY  
    '-33iG  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8Ql'(5|T  
    % Compute the Zernike Polynomials 4UjE*Aq  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R2THL  
    _`i%9Ad.4  
    Fx5d@WNa>  
    % Determine the required powers of r: Cfo 8gX*  
    % ----------------------------------- %aBJ+V F  
    m_abs = abs(m); ggc?J<Dv  
    rpowers = [];  x9"4vp  
    for j = 1:length(n) ;+34g6  
        rpowers = [rpowers m_abs(j):2:n(j)]; _/~ ,a  
    end 9,f<Nb(\  
    rpowers = unique(rpowers); 'QojSq   
    Y{vwOs  
    Q4Fq=kTE  
    % Pre-compute the values of r raised to the required powers, 1]Q 2qs  
    % and compile them in a matrix: Du:p!nO  
    % ----------------------------- 5}bZs` C  
    if rpowers(1)==0 ?%/u/*9rj  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ywynx<Wg  
        rpowern = cat(2,rpowern{:}); ~vSAnjeR  
        rpowern = [ones(length_r,1) rpowern]; V!77YFen %  
    else F] ?@X  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); aq+IC@O  
        rpowern = cat(2,rpowern{:}); yISQYvSN  
    end E? eWv)//  
    D`:d'ow~KQ  
    h^+C)6(58n  
    % Compute the values of the polynomials: "lz[zFnO  
    % -------------------------------------- ``|RO[+2  
    y = zeros(length_r,length(n)); o.3YM.B#  
    for j = 1:length(n) S=H_9io  
        s = 0:(n(j)-m_abs(j))/2; 15KV} ){  
        pows = n(j):-2:m_abs(j); 'nWs0iH.  
        for k = length(s):-1:1 'K`Rbhy  
            p = (1-2*mod(s(k),2))* ... *Ht*)l?  
                       prod(2:(n(j)-s(k)))/              ... J4v0O="  
                       prod(2:s(k))/                     ... Th^(f@.w  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... KU|BT .o8  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Zfy~mv$  
            idx = (pows(k)==rpowers); MziZN^(  
            y(:,j) = y(:,j) + p*rpowern(:,idx); G/z\^Q  
        end y (nsyA  
         MuoctW  
        if isnorm 1%spzkE 3P  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); F|?+>c1}  
        end &^7uv0M<y  
    end WVWS7N\  
    % END: Compute the Zernike Polynomials ihiuSF<NaQ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xshAr J&A  
    }#OqU# q|  
    'ZC}9=_g  
    % Compute the Zernike functions: b-BM"~N'  
    % ------------------------------ |ck ZyDA  
    idx_pos = m>0; ,9Z2cgXwJ  
    idx_neg = m<0; q11QAx4p  
    yS)- &t!;  
    f `y" a@  
    z = y; j!x<QNNX  
    if any(idx_pos) =@JS88+  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 3WCqKXJ7  
    end R+{^@M&  
    if any(idx_neg) Zj1ZU[BEcL  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); V~T`&  
    end -VWCD,c  
    22GnbA7O  
    df4sOqU  
    % EOF zernfun eu}Fd@GO  
     
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    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
    离线jssylttc
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    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    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)  R|iEvt  
    [|4}~UV  
    DDE还是手动输入的呢? f'&GFL=c  
    0<g<GQ(E  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    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。
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