切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9231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 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, oQV3  
    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, islHtX VE  
    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? I3r")}P  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 0<uLQVoR2n  
    w6h83m 3  
    \),f?f-m  
    $M0l (htR  
    >@cBDS<6R  
    function z = zernfun(n,m,r,theta,nflag) @6+_0^  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. N#RC;  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N X[$|I9  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 0QPY+6  
    %   unit circle.  N is a vector of positive integers (including 0), and -bdWG]w"  
    %   M is a vector with the same number of elements as N.  Each element &nVekE:!  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ntPj9#lf  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, /=AFle2(  
    %   and THETA is a vector of angles.  R and THETA must have the same -:'%YHxX  
    %   length.  The output Z is a matrix with one column for every (N,M) 6)ZaK  
    %   pair, and one row for every (R,THETA) pair. :*[mvF  
    % ._A4 :  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike h@1/  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), fJ _MuAv  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral cmU0=js.  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, T95FoA  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized \h s7>5O^K  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *>zOWocxD  
    % :6)!#q'g  
    %   The Zernike functions are an orthogonal basis on the unit circle. iR{@~JN=)  
    %   They are used in disciplines such as astronomy, optics, and |T"j7  
    %   optometry to describe functions on a circular domain. mC\<fo-u  
    % ;w]1H&mc*A  
    %   The following table lists the first 15 Zernike functions. EXH,+3fQp  
    % 33eOM(`D[  
    %       n    m    Zernike function           Normalization JgP%4)]LV  
    %       -------------------------------------------------- s m G?y~  
    %       0    0    1                                 1 {&D$U'ye  
    %       1    1    r * cos(theta)                    2 gN(kRhp  
    %       1   -1    r * sin(theta)                    2 N.]~%)K:{  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) lWJYT <kt  
    %       2    0    (2*r^2 - 1)                    sqrt(3) jgXr2JQ<  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) "d~<{(:N^  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) l& sEdEA  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) \SwqBw  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) @V1FBw9S!@  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ouoIbA9X  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Yh"9,Z&wiR  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) \}=W*xxB  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) }z\t}lven  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) P@5-3]m=  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) a5pM~.]  
    %       -------------------------------------------------- ;"9Ks.  
    % ~=HPqe8  
    %   Example 1: Zg4wd/y?  
    % J &=5h.G$  
    %       % Display the Zernike function Z(n=5,m=1) /J!hKK^k  
    %       x = -1:0.01:1; ^:f)XZ  
    %       [X,Y] = meshgrid(x,x); Iw#[K  
    %       [theta,r] = cart2pol(X,Y); PV=sqLM~  
    %       idx = r<=1; !&@t  
    %       z = nan(size(X)); gr.G']9lNq  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); UaQW<6+  
    %       figure 3y:),;|5  
    %       pcolor(x,x,z), shading interp Qch'C0u  
    %       axis square, colorbar X ) =-a  
    %       title('Zernike function Z_5^1(r,\theta)') 4{6,Sx  
    % ?=kH}'igq  
    %   Example 2: |Df`Aq(eYJ  
    % Y1qbu~!  
    %       % Display the first 10 Zernike functions E J6|y'  
    %       x = -1:0.01:1; k4:=y9`R}$  
    %       [X,Y] = meshgrid(x,x); a~-k} G5  
    %       [theta,r] = cart2pol(X,Y); }>YEtA  
    %       idx = r<=1; OH 88d:  
    %       z = nan(size(X)); wb]Z4/j#  
    %       n = [0  1  1  2  2  2  3  3  3  3]; EF 8rh  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; C'sA0O@O  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ;1TQr3w  
    %       y = zernfun(n,m,r(idx),theta(idx)); YsTF10  
    %       figure('Units','normalized') ~"R;p}5 "  
    %       for k = 1:10 # .OCoc  
    %           z(idx) = y(:,k); `<R^ZL,  
    %           subplot(4,7,Nplot(k)) ?i~mt'O  
    %           pcolor(x,x,z), shading interp =}SC .E\  
    %           set(gca,'XTick',[],'YTick',[]) x.(Sv]+[  
    %           axis square }~<9*M-P  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) \9:IL9~F  
    %       end D;DI8.4`N  
    % ;,B $lgF  
    %   See also ZERNPOL, ZERNFUN2. f+QDjJ?z  
    eGbjk~,f'  
    DpL|aRdbK  
    %   Paul Fricker 11/13/2006 :C#(yp  
    RUV:   
    +C=^,B!,  
    c 9zMI  
    rPJbbV",+^  
    % Check and prepare the inputs: =l}XKl->  
    % ----------------------------- )TkXdA?.  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <\\,L@  
        error('zernfun:NMvectors','N and M must be vectors.') 1jj.oa]  
    end p+{*&Hm5  
    XSo$;q\  
    3Ett9fBd  
    if length(n)~=length(m) zCpXF< _C  
        error('zernfun:NMlength','N and M must be the same length.') zT _[pa)O`  
    end 2?9gf,U  
    ]}N&I_mU  
    ADlLodG  
    n = n(:); P&8QKX3 j^  
    m = m(:); 83gp'W{|  
    if any(mod(n-m,2)) 3 *[YM7y  
        error('zernfun:NMmultiplesof2', ... 9]*hP](  
              'All N and M must differ by multiples of 2 (including 0).') 4.i< `'  
    end A>Oi9%OY:  
    c *]6>50  
    3h>5 6{P  
    if any(m>n) dfA4OZ&  
        error('zernfun:MlessthanN', ... VA@t8H,  
              'Each M must be less than or equal to its corresponding N.') -JB~yO?0  
    end ;kiL`K  
    jMB&(r  
    D= LLm$y  
    if any( r>1 | r<0 ) y6 _,U/9  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') j?cE0 hz  
    end bpWEF b'f  
    fp2.2 @[  
    x5)YZ~5  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 2K&5Kt/  
        error('zernfun:RTHvector','R and THETA must be vectors.') n{v[mqm^  
    end $bT<8:g  
    6 P6Pl&  
    lZ` CFZR0  
    r = r(:); W|aFEY  
    theta = theta(:); ?3{:[*  
    length_r = length(r); BK16~Wl  
    if length_r~=length(theta) Qt+;b  
        error('zernfun:RTHlength', ... yw9)^JU8"  
              'The number of R- and THETA-values must be equal.') uzpW0(_i3a  
    end F 7~T=X)1  
    @z!|HLD+  
    |Q)c{9sD  
    % Check normalization: "TboIABp:H  
    % --------------------  UX& ?^]  
    if nargin==5 && ischar(nflag) >SR! *3$5  
        isnorm = strcmpi(nflag,'norm'); OLS.0UEc  
        if ~isnorm O0VbKW0h3  
            error('zernfun:normalization','Unrecognized normalization flag.') =6N%;2`84  
        end N-O"y3W}  
    else /T_@rm  
        isnorm = false; +jPs0?}s  
    end  A/zZ%h  
    yRt>7'@X  
    ;$Q&2}L[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a*4l!-7  
    % Compute the Zernike Polynomials  *Fe  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  @{|vW  
    Z+*t=?L,,G  
    +hJ@w-u,G  
    % Determine the required powers of r: Pq@%MF]5  
    % ----------------------------------- )Zcw G(o0  
    m_abs = abs(m); 3{wmKo|_X  
    rpowers = []; cR0OJ'w  
    for j = 1:length(n) LR5X=&k  
        rpowers = [rpowers m_abs(j):2:n(j)]; >8*J ;(:W  
    end v7ShXX:  
    rpowers = unique(rpowers); ?IWLH-fkP  
    CSC sJE#4  
    0"*!0s ~  
    % Pre-compute the values of r raised to the required powers, VX!UT=;  
    % and compile them in a matrix: |NsrO8H   
    % ----------------------------- EO<{Bj=2  
    if rpowers(1)==0 0pYCh$TL1  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); &ZmHR^Flz  
        rpowern = cat(2,rpowern{:}); Q>,EYb>wI  
        rpowern = [ones(length_r,1) rpowern]; @w H+,]xE  
    else COF_a%  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); m3cO { 1I  
        rpowern = cat(2,rpowern{:}); ?-y!FD}m&  
    end LZ97nvK  
    eq|G\XJ  
    Q9UBxpDV:  
    % Compute the values of the polynomials: mr`Lxy9e  
    % -------------------------------------- f^tCD'Vmi  
    y = zeros(length_r,length(n)); w4S0aR:yL  
    for j = 1:length(n) 'g4t !__  
        s = 0:(n(j)-m_abs(j))/2; !zd]6YL$  
        pows = n(j):-2:m_abs(j); KNkVI K  
        for k = length(s):-1:1 4z_>CiA  
            p = (1-2*mod(s(k),2))* ... cZ!%#A z  
                       prod(2:(n(j)-s(k)))/              ... yEaim~  
                       prod(2:s(k))/                     ... t3~ZGOn  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... *bl*R';  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); S690Y]:h$v  
            idx = (pows(k)==rpowers); hU:M]O0uw  
            y(:,j) = y(:,j) + p*rpowern(:,idx); n^svRM]eQ  
        end [a\U8 w  
         y%y F34  
        if isnorm bJ[{[|yEd  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); J?yNZK$WqN  
        end 4`x.d  
    end _1s\ztDpw  
    % END: Compute the Zernike Polynomials kl0!*j  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q'07  
    x1gfo!BN  
    ?r~|B/ ]  
    % Compute the Zernike functions: }b// oe7  
    % ------------------------------ FC'v= *  
    idx_pos = m>0; W[5a'}OV  
    idx_neg = m<0; /M:R|91:_  
    uxGY/Zf  
    X+'z@xpj  
    z = y; !>~W5c^  
    if any(idx_pos) 2wHvHH!  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); VtNY~  
    end bNUb  
    if any(idx_neg) *rY@(|  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); g4NxNjM;  
    end P/BWFN1  
    QB<9Be@e  
    X[Lwx.Ly8  
    % EOF zernfun  _\H MF  
     
    分享到
    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
    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)  4>x$I9^Y!  
    y4\(ynk  
    DDE还是手动输入的呢? ;mT|0&o>#  
    .uh>S!X, ]  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    在线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)