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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, R] L|&{   
    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, Ju4={^#  
    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?  K6d9[;F  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? F$T@OT6  
    $)t ]av  
    Uax[Zh[Cg  
    Au(zvgP  
    dP}=cZ~  
    function z = zernfun(n,m,r,theta,nflag) \q(DlqTqs  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. bq{":[a  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N _7Z|=)  
    %   and angular frequency M, evaluated at positions (R,THETA) on the /2Q@M>  
    %   unit circle.  N is a vector of positive integers (including 0), and !c,=%4Pb  
    %   M is a vector with the same number of elements as N.  Each element d#6'dKV$  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) {U/a h2*  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ?$ T! =e"  
    %   and THETA is a vector of angles.  R and THETA must have the same 6fV%[.RR  
    %   length.  The output Z is a matrix with one column for every (N,M) |d =1|C%,  
    %   pair, and one row for every (R,THETA) pair. qP@d)XRQ  
    % Q[+&n*  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike w],+lN;  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), B@XnHh5y  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral "e4;xU-  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, wn2+4> |~p  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized M5DQ{d<r  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. R+,eXjz"  
    % p!5= 1$  
    %   The Zernike functions are an orthogonal basis on the unit circle. 5=]q+&y\H  
    %   They are used in disciplines such as astronomy, optics, and lYEMrr!KQw  
    %   optometry to describe functions on a circular domain. k/[*Wz$W  
    % $=?1>zvF  
    %   The following table lists the first 15 Zernike functions. qOOF]L9r%u  
    % I!'PvIyO  
    %       n    m    Zernike function           Normalization w;@DcX$]  
    %       -------------------------------------------------- T4MB~5,i  
    %       0    0    1                                 1 g%z'#E 97  
    %       1    1    r * cos(theta)                    2 ]r++YIg!j  
    %       1   -1    r * sin(theta)                    2 hwgLJY?  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) sDNV_} h  
    %       2    0    (2*r^2 - 1)                    sqrt(3) IRy!8A=X  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) L,G{ t^j  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) \z'A6@  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 44;ZX$HL  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) xe ng`!  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) zzmZ`Ya  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 'wh2787  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)  Z|zyO-  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) pe(31%(h  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Mle@.IIT  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Fh  t$7V  
    %       -------------------------------------------------- =fA* b  
    %  -)  
    %   Example 1: * ]uo/g  
    % K5X,J/n  
    %       % Display the Zernike function Z(n=5,m=1) NR3]MGBKv  
    %       x = -1:0.01:1; S<), ,(  
    %       [X,Y] = meshgrid(x,x); $gKMVgD"  
    %       [theta,r] = cart2pol(X,Y); #H]b Xr  
    %       idx = r<=1; d V+%x"[:  
    %       z = nan(size(X)); 1O" Mo  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); #XSs.i{  
    %       figure s-^B)0T!  
    %       pcolor(x,x,z), shading interp HzADz%~  
    %       axis square, colorbar 7PE3>cD  
    %       title('Zernike function Z_5^1(r,\theta)') q:Lw!'Z h  
    % :5kgJu  
    %   Example 2: ;uw`6 KJ  
    % o)w8 ]H /  
    %       % Display the first 10 Zernike functions > Y7nq\  
    %       x = -1:0.01:1; 8S;]]*cD~  
    %       [X,Y] = meshgrid(x,x); &=bWXNU.  
    %       [theta,r] = cart2pol(X,Y); f n]rMH4>  
    %       idx = r<=1; Z.9 ?u;  
    %       z = nan(size(X)); t{)Z$ )'  
    %       n = [0  1  1  2  2  2  3  3  3  3]; w7n6@"q  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; j9)WInYc:  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; %3v:c|r  
    %       y = zernfun(n,m,r(idx),theta(idx)); $|0_[~0-n  
    %       figure('Units','normalized') G01J1Ll}  
    %       for k = 1:10 Vp3r  
    %           z(idx) = y(:,k); f"^G\  
    %           subplot(4,7,Nplot(k)) i={ :6K?^  
    %           pcolor(x,x,z), shading interp .}KY*y  
    %           set(gca,'XTick',[],'YTick',[]) -w8c;5X  
    %           axis square uc6;%=%+  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) RRYm.dMIw  
    %       end 8<z]rLQw?%  
    % REd"}zDI  
    %   See also ZERNPOL, ZERNFUN2. q2qbbQ6H  
    4[@`j{  
    fC!]MhA"i  
    %   Paul Fricker 11/13/2006 <28L\pdG`  
    o+U]=q*|)$  
    u_0&`zq  
    yc|j]?  
    mDn*v( f  
    % Check and prepare the inputs: ts2;?`~  
    % ----------------------------- BIx Z4Ft  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) $@8$_g|Wz  
        error('zernfun:NMvectors','N and M must be vectors.') FScE3~R  
    end YHoj^=/b  
    lYZ5FacqC  
    ,^dyS]!d$  
    if length(n)~=length(m) x)'4u6;d  
        error('zernfun:NMlength','N and M must be the same length.') 6mH0|:CsY  
    end \k6Ho?PL  
    D@[Mk"f  
    n}8J-/(|+  
    n = n(:); y1DP`Ro  
    m = m(:); {~=Edf  
    if any(mod(n-m,2)) 3U#z {%  
        error('zernfun:NMmultiplesof2', ... D;@*  
              'All N and M must differ by multiples of 2 (including 0).') K]mR9$/  
    end :*GLLjS;  
    J \iyc,M<M  
    3?Ckk{)&  
    if any(m>n) ~T<yp  
        error('zernfun:MlessthanN', ... 'qRK6}"T  
              'Each M must be less than or equal to its corresponding N.') bv&A)h"S  
    end EYc, "'  
    OLAw Rha  
    AF5$U8jf  
    if any( r>1 | r<0 ) ZVo%ssVt  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') zo*YPDEm"  
    end JX_hLy@`  
    P 19nF[A  
    p"9a`/  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 1(V>8}zn  
        error('zernfun:RTHvector','R and THETA must be vectors.') esCm`?qCP  
    end L%,tc~)A  
    LQVa,'  
    I>k >^  
    r = r(:); 4@6!E^  
    theta = theta(:); U1?*vwfKZ  
    length_r = length(r); 'I|A*rO  
    if length_r~=length(theta) l#P)9$%  
        error('zernfun:RTHlength', ... R:+2}kS5e{  
              'The number of R- and THETA-values must be equal.') 2mVcT3  
    end 74*1|S <  
    Vl;GQe  
    [zp v3Uw  
    % Check normalization: v J*IUy  
    % -------------------- +QNFu){G  
    if nargin==5 && ischar(nflag) .k5 TQt  
        isnorm = strcmpi(nflag,'norm'); ns_5|*'  
        if ~isnorm Jd_w:H.  
            error('zernfun:normalization','Unrecognized normalization flag.') >Lo 0,b$  
        end /s.O3x._'  
    else ..yuEA  
        isnorm = false; *@'4 A :A  
    end S4]}/Imn)  
    @DgJxY|  
    J{$+\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X+;F5b9z  
    % Compute the Zernike Polynomials ne nYP0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b#h?O}  
    FoM4QO  
    '0_Z:\ laU  
    % Determine the required powers of r: uG1 1~uAt  
    % ----------------------------------- Sfc0 ~1  
    m_abs = abs(m); aaq{9Y#  
    rpowers = []; .uzg2Kd_  
    for j = 1:length(n) D8P<mIu}Y  
        rpowers = [rpowers m_abs(j):2:n(j)]; &0*l=!:G^  
    end '0g1v7Gx  
    rpowers = unique(rpowers); %V-\|cw   
    [Af&K22M(X  
    q0Fq7rWP  
    % Pre-compute the values of r raised to the required powers, ]@OGp:Hz  
    % and compile them in a matrix: O[Xl*9P  
    % ----------------------------- usiv`.  
    if rpowers(1)==0 0;z-I"N  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); BcaMeb-Z  
        rpowern = cat(2,rpowern{:}); uG2(NwOL  
        rpowern = [ones(length_r,1) rpowern]; $ wGDk  
    else Xv 3u}nPMq  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ?Dro)fH1  
        rpowern = cat(2,rpowern{:}); %&KJtKe  
    end z'a#lA.$}  
    }B2H)dG^K  
    4fBgmL  
    % Compute the values of the polynomials: v(^{ P  
    % -------------------------------------- QjETu  
    y = zeros(length_r,length(n)); w)-@?jN  
    for j = 1:length(n) X1U7$/t  
        s = 0:(n(j)-m_abs(j))/2; 6GCwc1g  
        pows = n(j):-2:m_abs(j); BQVpp,]  
        for k = length(s):-1:1 }OO(uC2  
            p = (1-2*mod(s(k),2))* ... &T?>Kx  
                       prod(2:(n(j)-s(k)))/              ... ]T\K-;i  
                       prod(2:s(k))/                     ... \a+F/I$hwa  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... LLv~yS O  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); <mlQn?u  
            idx = (pows(k)==rpowers); |M|'S~z  
            y(:,j) = y(:,j) + p*rpowern(:,idx); MfUG@  
        end N#{d_v^H?d  
         /km^IH  
        if isnorm TkhbnO g6  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); BMU}NZA  
        end \7Hzj0hSi  
    end E> Ukxi1  
    % END: Compute the Zernike Polynomials 21GjRPs\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% quc?]rb  
    [0CoQ5:d?&  
    KquHc-fzqr  
    % Compute the Zernike functions: uy\< t  
    % ------------------------------ 8r /]Q  
    idx_pos = m>0; L>$yslH; b  
    idx_neg = m<0; =zXpeo&|m  
    FT73P0!8.  
    +U&aK dQs  
    z = y; uRG0} >]|U  
    if any(idx_pos) (:E_m|00;  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); e:{v.C0ez  
    end Vnuz! 6.  
    if any(idx_neg)  Py\xN  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); STu!v5XY}-  
    end ,(Fo%.j  
    a`(6hL3IT  
    @& #df  
    % EOF zernfun $s.:wc^  
     
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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)  c 6}d{B[  
    b=:AFs{  
    DDE还是手动输入的呢? !~04^(  
    Fj"g CBaR  
    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