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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, @ gjA8mL  
    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, c{})Z=  
    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? Z4D[nPm$  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? `~2I  
    J,t`il T  
    6rN.)dL.#N  
    9+I /bl4  
    Ypx"<CKP}  
    function z = zernfun(n,m,r,theta,nflag) .c\iKc#  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ]eo%eaA   
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N W]M Fq5.  
    %   and angular frequency M, evaluated at positions (R,THETA) on the r|Q/:UV?w  
    %   unit circle.  N is a vector of positive integers (including 0), and }KR"0G[f  
    %   M is a vector with the same number of elements as N.  Each element G/yYIs  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) D[3QQT7c  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, %ZGG6Xgw  
    %   and THETA is a vector of angles.  R and THETA must have the same B$_-1^L e  
    %   length.  The output Z is a matrix with one column for every (N,M) #?7g_  
    %   pair, and one row for every (R,THETA) pair. {EyWSf"  
    % NPLJ*uHH  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike z#/"5 l   
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), P$&l1Mp  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 'oF('uR  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, :d wP  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized %8?XOkH)  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. q)OCY}QA  
    % FA}y"I'W  
    %   The Zernike functions are an orthogonal basis on the unit circle. \-r"%@OkW  
    %   They are used in disciplines such as astronomy, optics, and @81N{tg-  
    %   optometry to describe functions on a circular domain. kp^q}iS  
    % =&WH9IKz  
    %   The following table lists the first 15 Zernike functions. /NQ PTr  
    % jm,cVo  
    %       n    m    Zernike function           Normalization ?7A>|p?"  
    %       -------------------------------------------------- aA'of>'ib|  
    %       0    0    1                                 1 LU+}iA)  
    %       1    1    r * cos(theta)                    2 YhL^kM@c  
    %       1   -1    r * sin(theta)                    2 q5\iQ2f{WV  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) T%SK";PAU$  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ^n*:zmD  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) >YR2h/S  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 'q1cc5(ueV  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) I8{ mkh  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) =LKf.@]#  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) W>&!~9H  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ^m-w@0^z  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =-/sB>-C  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) OuyO_DSI  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Hd_,`W@  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) qD,/Qu62  
    %       -------------------------------------------------- _,3%)sn-)  
    % XzPUll;ZU  
    %   Example 1: g1)ZjABV  
    % mnFmShu  
    %       % Display the Zernike function Z(n=5,m=1) >{>X.I~  
    %       x = -1:0.01:1; D+{& zo  
    %       [X,Y] = meshgrid(x,x); 9hjzOJPuga  
    %       [theta,r] = cart2pol(X,Y); s \0,@A   
    %       idx = r<=1; 2Mj_wc   
    %       z = nan(size(X)); .pIO<ZAFT  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); "%#CMCE|f  
    %       figure wxy@XN"/i+  
    %       pcolor(x,x,z), shading interp EF'8-*  
    %       axis square, colorbar $J#Z`%B^y  
    %       title('Zernike function Z_5^1(r,\theta)') HJt '@t=Ak  
    % AYfL}X<Ig  
    %   Example 2: jOm7:+H  
    % |qpFR)l  
    %       % Display the first 10 Zernike functions D/+l$aBz  
    %       x = -1:0.01:1; f( <O~D  
    %       [X,Y] = meshgrid(x,x); K?>sP%m)  
    %       [theta,r] = cart2pol(X,Y); co-1r/ -O  
    %       idx = r<=1; V,]Fh5f  
    %       z = nan(size(X)); \=Od1i  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 0bteI*L  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; S84S/y  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 3!`_Q%  
    %       y = zernfun(n,m,r(idx),theta(idx)); ~vcua@  
    %       figure('Units','normalized') z=Xh  
    %       for k = 1:10 G|TnvZ KX  
    %           z(idx) = y(:,k); S_38U  
    %           subplot(4,7,Nplot(k)) 3 6t^iV*3  
    %           pcolor(x,x,z), shading interp ?RS4oJz,5g  
    %           set(gca,'XTick',[],'YTick',[]) w!-MMT4y  
    %           axis square p$cb&NNh*H  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Gh 352  
    %       end v>_83P`  
    % ~RV"_8`V9  
    %   See also ZERNPOL, ZERNFUN2. z>)lp$  
    oWEzzMRz  
     /#zs  
    %   Paul Fricker 11/13/2006 Y$s4 *)%  
    dFmpx%+p  
    ,P=.x%  
    ?} lqu7S  
    ,.0B0Y-X  
    % Check and prepare the inputs: pl/ek0QX  
    % ----------------------------- tJA"BP3f  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) O`T_'.Lk  
        error('zernfun:NMvectors','N and M must be vectors.') |XV`A)=f  
    end t<"%m)J  
    4gZ)9ya   
    Rw hKW?r+  
    if length(n)~=length(m) 2w fkXS=~6  
        error('zernfun:NMlength','N and M must be the same length.') T8d=@8g,%  
    end _%#Uh#7P$  
    )TEod!]  
    bz.sWBugR  
    n = n(:); ).-#  
    m = m(:); =sF4H_B  
    if any(mod(n-m,2)) U2CC#,b!(  
        error('zernfun:NMmultiplesof2', ... q/ (h{cq  
              'All N and M must differ by multiples of 2 (including 0).') &MPlSIg  
    end (-`PO]e48  
    lgZ9*@d  
    *$Zy|&[Z  
    if any(m>n) _&S;*?K.  
        error('zernfun:MlessthanN', ... P)LOAe1'  
              'Each M must be less than or equal to its corresponding N.') umCmxm r&  
    end aU_l"+5>vq  
    t+\<i8  
    ~(B%E'  
    if any( r>1 | r<0 ) |;&I$'i  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') }$g"|;<ha  
    end #g'j0N  
    q$"?P  
    i:jns>E  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) [f}`reRlZ  
        error('zernfun:RTHvector','R and THETA must be vectors.') \S&OAe/b  
    end RxNLn/?d@  
    Cq'{ %  
    `g4N]<@z  
    r = r(:); %e)? Mem  
    theta = theta(:); Ya(3Z_f+VZ  
    length_r = length(r); &Pc.[k  
    if length_r~=length(theta) m/,80J8L+f  
        error('zernfun:RTHlength', ... +ej5C:El_}  
              'The number of R- and THETA-values must be equal.') h<8c{RuoZC  
    end J #jFX F\  
    ;mC|> wSZ  
    Y0J:c?,  
    % Check normalization: A.h0H]*Ma  
    % -------------------- btC6R>0   
    if nargin==5 && ischar(nflag) (!qfd Qq#  
        isnorm = strcmpi(nflag,'norm'); @Ae&1O;Zh  
        if ~isnorm 'Gamb+[  
            error('zernfun:normalization','Unrecognized normalization flag.') PZO.$'L|7  
        end Cl3L)  
    else t=|}?lN<  
        isnorm = false; Qvel#*-4  
    end L\5:od[EP  
    h:sf?X[  
    QpRk5NeLe  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q laoa)d#  
    % Compute the Zernike Polynomials 8&3& ^!I  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5.DmMG[T^=  
    _+H $Pa}?  
    f6nuh&!-  
    % Determine the required powers of r: hpYv*WH:  
    % ----------------------------------- 4mtO"'|  
    m_abs = abs(m); TBky+]p@  
    rpowers = []; .mcohfR  
    for j = 1:length(n) -$_FKny  
        rpowers = [rpowers m_abs(j):2:n(j)]; aof'shS8  
    end N9s.nu  
    rpowers = unique(rpowers);  Z'l!/l!  
    :RwURv+kT  
    PgHmOs  
    % Pre-compute the values of r raised to the required powers, 7=Pj}x)  
    % and compile them in a matrix: BUV4L5(  
    % ----------------------------- {d]B+'  
    if rpowers(1)==0 2J%L%6z8~  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 2v;&`04V<  
        rpowern = cat(2,rpowern{:}); XKDX*x G  
        rpowern = [ones(length_r,1) rpowern]; :(.:bf  
    else .726^2sx  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Nl/ fvJ`4  
        rpowern = cat(2,rpowern{:}); D>ou,  
    end )?$@cvf  
    cIa`pU,6A  
    %M/L/_d  
    % Compute the values of the polynomials: 'o*\ N%  
    % -------------------------------------- j]` hy"  
    y = zeros(length_r,length(n)); bv7xh*/  
    for j = 1:length(n) 'tcve2Tt  
        s = 0:(n(j)-m_abs(j))/2; m-+>h:1b|9  
        pows = n(j):-2:m_abs(j); VS_\bIC  
        for k = length(s):-1:1 ]YfG`0eK<  
            p = (1-2*mod(s(k),2))* ... b$_qG6)IJO  
                       prod(2:(n(j)-s(k)))/              ... j 9GKz1  
                       prod(2:s(k))/                     ... .*xO/pn  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 7GG`9!l]D  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 8 nqF i  
            idx = (pows(k)==rpowers); #3eI4KJ4+l  
            y(:,j) = y(:,j) + p*rpowern(:,idx); mG\9Qkom|  
        end ;]=@;? 9  
         [eBt Dc*w  
        if isnorm W(?J,8>  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); u,}>I%21  
        end 2PUB@B' +  
    end m=v.<+>  
    % END: Compute the Zernike Polynomials Pth4_]US  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +ZGH  
    mA_EvzXk\  
    < <Y]P+uU  
    % Compute the Zernike functions: 1vCp<D9<  
    % ------------------------------ RBg2iG$ 8|  
    idx_pos = m>0; ~m0=YAlk?  
    idx_neg = m<0; S4_ZG>\VT  
    *f{4 _ts  
    yB=R7E7  
    z = y; zf5%|7o  
    if any(idx_pos) O U9{Y9e  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); yd'cLZd<}  
    end 5p:2gsk  
    if any(idx_neg) YcR: _ac  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); rM6S%rS  
    end ;05lwP* r]  
    Z![#Uz.z  
    yp@cn(:~  
    % EOF zernfun .i*ja*   
     
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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)
    在线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 $M  
    6M8(KN^  
    DDE还是手动输入的呢? c;R .rV<  
    ZJ_P=  
    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