切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9010阅读
    • 5回复

    [讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

    上一主题 下一主题
    离线jssylttc
     
    发帖
    25
    光币
    13
    光券
    0
    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, 9=wt9` ?  
    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, !J@!P?0. C  
    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? !f^'-  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? pf'-(W+  
    t:?8I9d  
    bw\a\/Dw  
    },@1i<Bb  
    &!E+l<.RF  
    function z = zernfun(n,m,r,theta,nflag) =5QP'Qt{O  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. sMhUVc4  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N TDtS^(2A7K  
    %   and angular frequency M, evaluated at positions (R,THETA) on the N-g=_86C"  
    %   unit circle.  N is a vector of positive integers (including 0), and q\fZ Q  
    %   M is a vector with the same number of elements as N.  Each element ;E{k+vkqy  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) hb_J. Q  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, @! gJOy  
    %   and THETA is a vector of angles.  R and THETA must have the same ZI8*PX%2  
    %   length.  The output Z is a matrix with one column for every (N,M) r6#It$NU  
    %   pair, and one row for every (R,THETA) pair. Q#} 0pq  
    % ,(  ?q  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike QlmZ4fT[r  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), t|ih{0  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral |_7AN!7j  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, H]XY  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized :"pA0oB  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 9ne13 qVm+  
    % O DLRzk(  
    %   The Zernike functions are an orthogonal basis on the unit circle.  3~mi  
    %   They are used in disciplines such as astronomy, optics, and {d%% nK~  
    %   optometry to describe functions on a circular domain. XYM 5'  
    % tf5h/:  
    %   The following table lists the first 15 Zernike functions. )zR(e>VX  
    % 0F495'*A  
    %       n    m    Zernike function           Normalization *C*'J7  
    %       -------------------------------------------------- rv\yS:2  
    %       0    0    1                                 1 2qF ?%  
    %       1    1    r * cos(theta)                    2 S-$N!G~!  
    %       1   -1    r * sin(theta)                    2 (pl|RmmDz  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) /2n-q_  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 0E5"}8  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 5ZXP$.  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) H:d@@/  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 8?> #  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) v%=@_`Ht  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Ig sK7wn  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) m@z.H;  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) _=wu>h&7  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Lcx)wof  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) w4m)lQM  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) "\x<Zg;  
    %       -------------------------------------------------- E,/<;  
    % >+ P5Zm(_  
    %   Example 1: / X #4  
    % FKX+ z  
    %       % Display the Zernike function Z(n=5,m=1) nF Mc'm  
    %       x = -1:0.01:1; ODbEL/  
    %       [X,Y] = meshgrid(x,x); kT jx.  
    %       [theta,r] = cart2pol(X,Y); 94>EA/+Ek  
    %       idx = r<=1; gtV^6(Y  
    %       z = nan(size(X)); w6RB|^  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 7j ]d{lD  
    %       figure V?.')?'V  
    %       pcolor(x,x,z), shading interp nkp,  
    %       axis square, colorbar 6dCS Gb  
    %       title('Zernike function Z_5^1(r,\theta)') #}8l9[Q|M  
    % )nK-39,G  
    %   Example 2: -/y]'_a  
    % cL]vJ`?Ih  
    %       % Display the first 10 Zernike functions Q||v U  
    %       x = -1:0.01:1; j>{Dbl:#2  
    %       [X,Y] = meshgrid(x,x); YPV@/n[N  
    %       [theta,r] = cart2pol(X,Y); Em%0C@C  
    %       idx = r<=1; &tAhRMa  
    %       z = nan(size(X)); %z0;77[1I  
    %       n = [0  1  1  2  2  2  3  3  3  3]; [dQL6k";b  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; &^v5 x"  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; kkyi`_ZKn  
    %       y = zernfun(n,m,r(idx),theta(idx)); \ r^#a  
    %       figure('Units','normalized') #GJ{@C3H8Q  
    %       for k = 1:10 *t)Y@=k3>  
    %           z(idx) = y(:,k); +PlA#DZu  
    %           subplot(4,7,Nplot(k)) j.?c~Fh  
    %           pcolor(x,x,z), shading interp '@ $L}C#OI  
    %           set(gca,'XTick',[],'YTick',[]) 1[; 7Ay  
    %           axis square V>$A\AWw  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Ap :mc:  
    %       end - kGwbV}  
    % MsaD@JY.y  
    %   See also ZERNPOL, ZERNFUN2. 7z_EX8^  
    Skb d'j  
    va`/Dp)M  
    %   Paul Fricker 11/13/2006 z!M8lpI M  
    A>?_\<Gp  
    7CK3t/3D  
    F&Bh\C)]  
    xF#'+Y  
    % Check and prepare the inputs: 4R(H@p%+r2  
    % ----------------------------- THVF(M4v  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) &}:]uC  
        error('zernfun:NMvectors','N and M must be vectors.') yGxAur=dE  
    end /S9(rI<'  
    T4M"s;::1  
    fj7\MTy  
    if length(n)~=length(m) =T?:b8yV  
        error('zernfun:NMlength','N and M must be the same length.') B2R^oL' }  
    end 5~pQ$-  
    ]g3RVA%\l  
    >!U oS  
    n = n(:); nT;Rwz$3  
    m = m(:); KBe\)Vs  
    if any(mod(n-m,2)) N<$dbqoT|  
        error('zernfun:NMmultiplesof2', ... ,:E*Mw:  
              'All N and M must differ by multiples of 2 (including 0).') <Lt%[dn  
    end /O^aFIxk  
    uZg[PS=@!X  
    Q[wTV3d  
    if any(m>n) Fx3CY W  
        error('zernfun:MlessthanN', ... U5iyvU=UG  
              'Each M must be less than or equal to its corresponding N.') tF/)DZ.to  
    end ,Vc>'4E-  
    e}PJN6"5  
    ]UMt  
    if any( r>1 | r<0 ) 6XFLWN-)  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 9sgyg3fv>5  
    end M3 TsalF  
    R [[ #r5q  
    mRNA,*  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) x}tg/` .=z  
        error('zernfun:RTHvector','R and THETA must be vectors.') Z]QpH<Z  
    end FJ/c(K  
    4X1!t   
    y37c&XYq  
    r = r(:); tQ@%3`  
    theta = theta(:); qDV t  
    length_r = length(r); P4VMGP  
    if length_r~=length(theta) B&M-em=  
        error('zernfun:RTHlength', ... r=J+  
              'The number of R- and THETA-values must be equal.') 5Y3L  
    end YAc~,N   
    ,(@JNtx  
    +wHrS}I#g  
    % Check normalization: WXj iKW(  
    % -------------------- v|7=IJ  
    if nargin==5 && ischar(nflag) Od,P,t9  
        isnorm = strcmpi(nflag,'norm'); 5fT"`FL?  
        if ~isnorm %aB RL6  
            error('zernfun:normalization','Unrecognized normalization flag.') 9*<=K  
        end YaT6vSz  
    else %0gcNk"=  
        isnorm = false; #$^vP/"$  
    end &Rp/y%9  
    dc+U #]tS  
    0DB8[#i%:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \,ko'4 8@  
    % Compute the Zernike Polynomials Bs!F |x(  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% E5+-N  
    l2*o@&.  
    XZ&cTjNB&  
    % Determine the required powers of r: "8#EA<lsS  
    % ----------------------------------- H5)8TR3La  
    m_abs = abs(m); k0(_0o  
    rpowers = []; Pe,:FIp,  
    for j = 1:length(n) /)-OK7x  
        rpowers = [rpowers m_abs(j):2:n(j)]; wR%F>[ 6.{  
    end us7t>EMmB  
    rpowers = unique(rpowers); GpZ}xY'|w,  
    u==`]\_@  
    49Q tfk  
    % Pre-compute the values of r raised to the required powers, Oj,v88=  
    % and compile them in a matrix: ?heg_ ~P  
    % ----------------------------- Q|7$SS6$  
    if rpowers(1)==0 >oGs0mej  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); _Oc(K "v  
        rpowern = cat(2,rpowern{:}); Pea2ENe3  
        rpowern = [ones(length_r,1) rpowern]; k E},>+W+  
    else =H_vRd  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); m5_  
        rpowern = cat(2,rpowern{:}); |\<L7|hb9  
    end 8t5o&8v  
    8-&c%h 1  
     X? l5}  
    % Compute the values of the polynomials: Rh,a4n?W  
    % -------------------------------------- *Tum(wWZ  
    y = zeros(length_r,length(n)); AeR*79x  
    for j = 1:length(n) o FS2*u  
        s = 0:(n(j)-m_abs(j))/2; 2/>u8j  
        pows = n(j):-2:m_abs(j); &~KAZ}xu  
        for k = length(s):-1:1 : =f!>_r+  
            p = (1-2*mod(s(k),2))* ... eD,'M  
                       prod(2:(n(j)-s(k)))/              ... _PPn =kuMa  
                       prod(2:s(k))/                     ... V~ q b2$  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... L6 IIk  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); WI6h G  
            idx = (pows(k)==rpowers); cfC}"As  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ( &!RX.i  
        end x+8%4]u`  
         Mc9JFzp  
        if isnorm <f9a%`d  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); .2{*>Dzi  
        end =oT4!OUf  
    end HJ+ Q7)  
    % END: Compute the Zernike Polynomials WYm<_1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \OW.?1d  
    H{4_,2h =m  
    ;Xl {m`E+  
    % Compute the Zernike functions: ,}:}"cl  
    % ------------------------------ JI[{n~bhGD  
    idx_pos = m>0; d<cqY<y VA  
    idx_neg = m<0; -A^o5s  
    odTa 2$O  
    Tvl"KVGm  
    z = y; SajasjE!^1  
    if any(idx_pos) /d*[za'0  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); )8`i%2i=  
    end f7b6!R;z_  
    if any(idx_neg) 6&;h+;h  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); V<ii  
    end mEg3.|  
    U'LPaf$O  
    jx#9  
    % EOF zernfun 69S*\'L  
     
    分享到
    离线phoenixzqy
    发帖
    4352
    光币
    5478
    光券
    1
    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
    发帖
    959
    光币
    1087
    光券
    1
    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
    发帖
    25
    光币
    13
    光券
    0
    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)  N36B*9m&p  
    NErvX/qK  
    DDE还是手动输入的呢? PS0/O k  
    .HRd6O;  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    离线jssylttc
    发帖
    25
    光币
    13
    光券
    0
    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    离线18257342135
    发帖
    51
    光币
    1518
    光券
    0
    只看该作者 5楼 发表于: 2016-12-13
    支持一下,慢慢研究