切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9488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, ~@D%qbN  
    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, X6,9D[Nw  
    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? =!^iiHF  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? /wE_eK.  
    s%oAsQ_y  
    \z9?rvT:  
    (NdgF+'=  
    >!1f`  
    function z = zernfun(n,m,r,theta,nflag) G)hH?_U#T  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. +ca296^  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N :dN35Y]a  
    %   and angular frequency M, evaluated at positions (R,THETA) on the \&5@yh  
    %   unit circle.  N is a vector of positive integers (including 0), and Wp}9%Mq~Jy  
    %   M is a vector with the same number of elements as N.  Each element >k}/$R+  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) UD2<!a'T  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Wk?|BR]O  
    %   and THETA is a vector of angles.  R and THETA must have the same e:LZs0  
    %   length.  The output Z is a matrix with one column for every (N,M) (QSWb>np  
    %   pair, and one row for every (R,THETA) pair. fVUBCu  
    % VaSNFl1_M  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike AvE^ F1  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), i*R:WTw#  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral &&1Y"dFs  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, H ?j-=Zka  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 'c0'P%[5A  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. I ~L Q1 _  
    % _(`X .D  
    %   The Zernike functions are an orthogonal basis on the unit circle. D?}m h1#  
    %   They are used in disciplines such as astronomy, optics, and s2?,'es  
    %   optometry to describe functions on a circular domain. +){a[@S@x  
    % 9]@J*A}=l  
    %   The following table lists the first 15 Zernike functions. ;"Y;l=9_  
    % K#UA M .  
    %       n    m    Zernike function           Normalization &]6K]sWJK{  
    %       -------------------------------------------------- p@8krOo`  
    %       0    0    1                                 1 #IaBl?}r^  
    %       1    1    r * cos(theta)                    2 N~5WA3xd  
    %       1   -1    r * sin(theta)                    2 ./nYXREO|  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) v?D kDnta  
    %       2    0    (2*r^2 - 1)                    sqrt(3) qH%L"J  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) SKSAriS~  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) `s83r hs`!  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ;D"P9b]9$  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 4 uy@ {  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 8U<.16+5Q  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) _jrA?pY  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) <]Pix )  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) wGzXp5 dl  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) qa$[L@h>  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) vg:J#M:  
    %       -------------------------------------------------- 3]`qnSYBv  
    % !qXq y}?w  
    %   Example 1: y:|.m@ j1  
    % 0Dm`Ek3A7x  
    %       % Display the Zernike function Z(n=5,m=1) QE#-A@c  
    %       x = -1:0.01:1; '5xuT _  
    %       [X,Y] = meshgrid(x,x); W|H4i;u  
    %       [theta,r] = cart2pol(X,Y); jO&f*rxN  
    %       idx = r<=1; bOxjm`B<  
    %       z = nan(size(X)); S?nNZW\6[  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); DtF![0w/  
    %       figure <[3lV)~t  
    %       pcolor(x,x,z), shading interp *M5$ h*;v  
    %       axis square, colorbar RM^?&PM85  
    %       title('Zernike function Z_5^1(r,\theta)') oj^5G ]_ <  
    % + <!)k?  
    %   Example 2: `! ,\kc1  
    % N}+B:l]Qy  
    %       % Display the first 10 Zernike functions SJ@8[n.x  
    %       x = -1:0.01:1; F@_Egi  
    %       [X,Y] = meshgrid(x,x); -Ty<9(~S  
    %       [theta,r] = cart2pol(X,Y); 4('0f:9z+  
    %       idx = r<=1; 9Nag%o{*S>  
    %       z = nan(size(X)); J"D&q  
    %       n = [0  1  1  2  2  2  3  3  3  3]; \}u7T[R=`  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 3d#9Wyxs  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; PK-}Ldj  
    %       y = zernfun(n,m,r(idx),theta(idx)); KF:]4`$  
    %       figure('Units','normalized') vbWJhj K0h  
    %       for k = 1:10 'TK$ndy;7}  
    %           z(idx) = y(:,k); t7*G91Hoq&  
    %           subplot(4,7,Nplot(k)) 2w x[D  
    %           pcolor(x,x,z), shading interp cy&  
    %           set(gca,'XTick',[],'YTick',[]) <nOuyGIZ  
    %           axis square zfP[1  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Lt;.Nw  
    %       end _cxm}*}\#  
    % =0PNHO\gl  
    %   See also ZERNPOL, ZERNFUN2. 2\nBqCxR  
    =#.8$oa^  
    f gK2.;>  
    %   Paul Fricker 11/13/2006  \]f5  
    Ersr\ZB  
    d739UhKC  
    qXP1Q3  
    w| -0@  
    % Check and prepare the inputs: EaM"=g  
    % ----------------------------- k Z+q  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 6:|!1Pg5  
        error('zernfun:NMvectors','N and M must be vectors.') FhY{;-W(T  
    end @sB}q 6>  
    Yn IM-  
    S|{Yvyp  
    if length(n)~=length(m) .*RB~c t  
        error('zernfun:NMlength','N and M must be the same length.') 0^<Skm27"  
    end r%Q8)nEo  
    jpYw#]Q  
    R (tiIo  
    n = n(:); r/N[7 *i  
    m = m(:); :Bx+WW&P.i  
    if any(mod(n-m,2)) t5ny"k!  
        error('zernfun:NMmultiplesof2', ... +X* F<6mZ  
              'All N and M must differ by multiples of 2 (including 0).') E(aX4^]g  
    end ;e#>n!<u  
    xE G+%Uk{  
    YiIddQ  
    if any(m>n) lgCHGv2@  
        error('zernfun:MlessthanN', ... ]/aRc=Gn  
              'Each M must be less than or equal to its corresponding N.') VL_)]LR*)  
    end e/]O<,*  
    >~`Y   
    Eonq'Re$  
    if any( r>1 | r<0 ) Ht`<XbQ>  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') <_BqpZ^`  
    end l]a^"4L4`o  
    L<f-Ed9|  
    `YFkY^T  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Qag|nLoT  
        error('zernfun:RTHvector','R and THETA must be vectors.') D:YN_J"kV  
    end tIi!* u  
    )^jQkfL  
    5z9r S<  
    r = r(:); ~&wXXVK3  
    theta = theta(:); jGk7=}nw  
    length_r = length(r); +?URVp  
    if length_r~=length(theta) FX7Cjo#=R  
        error('zernfun:RTHlength', ... 'sm[CNzS  
              'The number of R- and THETA-values must be equal.') S`pF7[%rp  
    end ax-=n(   
    !pd7@FwC  
    9O),/SH;:  
    % Check normalization:  4 "pS  
    % -------------------- 6obQ9L c  
    if nargin==5 && ischar(nflag) L]c 8d   
        isnorm = strcmpi(nflag,'norm'); Kwy1SyU  
        if ~isnorm *)j@G:  
            error('zernfun:normalization','Unrecognized normalization flag.') 4u3 \xR?w6  
        end v t^r1j  
    else ,3wI~ j=  
        isnorm = false; $?H]S]#|}.  
    end JiKImz  
    z{_mEE49  
    QDIsC  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #[no~&E  
    % Compute the Zernike Polynomials X?KGb{  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2B6^ ]pSk  
    21.YO]Et  
    Eem 2qKj  
    % Determine the required powers of r: 1k!D0f3qb  
    % ----------------------------------- rDpe_varA  
    m_abs = abs(m); UqD5 A~w  
    rpowers = []; cj$,ob&DX  
    for j = 1:length(n) ^OHZ767v  
        rpowers = [rpowers m_abs(j):2:n(j)]; LTg?5GwD\j  
    end "AT&!t[J  
    rpowers = unique(rpowers); Wl,%&H2S<  
    /DLr(  
    8&?^XcJ*x  
    % Pre-compute the values of r raised to the required powers, ,)Yao;Cvd  
    % and compile them in a matrix: 2;&mkc K'  
    % ----------------------------- c}YJqhk0J  
    if rpowers(1)==0 $`^H:Djr  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); \V._Z>]  
        rpowern = cat(2,rpowern{:}); -Bl/ 4p  
        rpowern = [ones(length_r,1) rpowern]; '*8  
    else jIKBgsiF/  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ^/G?QR  
        rpowern = cat(2,rpowern{:}); |c<XSX?ir  
    end G=vN;e_$_b  
    wG_4$kyj  
    w#W5}i&x  
    % Compute the values of the polynomials: l#b:^3  
    % -------------------------------------- ?A|zRj{  
    y = zeros(length_r,length(n)); H!p!sn  
    for j = 1:length(n) j6`6+W=S(  
        s = 0:(n(j)-m_abs(j))/2; #]"/{Z  
        pows = n(j):-2:m_abs(j); 7TP$  
        for k = length(s):-1:1 ;F|jG}M"  
            p = (1-2*mod(s(k),2))* ... $Xf~# uH  
                       prod(2:(n(j)-s(k)))/              ... X)I/%{  
                       prod(2:s(k))/                     ... P<8LAc$T  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... QT_Srw@  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); wbBE@RU>!  
            idx = (pows(k)==rpowers); TV? ^c?{5  
            y(:,j) = y(:,j) + p*rpowern(:,idx); OE6#YT  
        end  1U  
         ,Ie<'>hd  
        if isnorm 6s'[{Ov  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ~Hs=z$  
        end &.hoC Po$  
    end &/HoSj>HS  
    % END: Compute the Zernike Polynomials 'wa g |-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d"Bo8`_  
    <Uf|PFVj$  
    0xv\D0  
    % Compute the Zernike functions: .hxin [Y  
    % ------------------------------ NOV.Bs{ yL  
    idx_pos = m>0; "=FIFf  
    idx_neg = m<0; +5#x6[  
    }&mj.hGv  
    wI*Y{J  
    z = y; t`uc3ta"9  
    if any(idx_pos) iL+y(]  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); qv.n99?]  
    end +9TV:T  
    if any(idx_neg) g083J}08  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); OqtQA#uL  
    end (Bsw/wv  
     70{RDj6{  
    3zbXAR*  
    % EOF zernfun TWtC-wI;  
     
    分享到
    离线phoenixzqy
    发帖
    4352
    光币
    5479
    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
    960
    光币
    1088
    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)  F3Ap1-%z  
    -zTEL (r  
    DDE还是手动输入的呢? j %H`0  
    F3Dt7q  
    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