切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9368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, F%Xj'=  
    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, u%o2BLx  
    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? &jg..R  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? mbij& 0  
    (M>[D!Yt  
    ^an3&  
    0R#T3K}  
    ]TE,N$X  
    function z = zernfun(n,m,r,theta,nflag) 7U@;X~c  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. >~nc7j u  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 2feiD?0  
    %   and angular frequency M, evaluated at positions (R,THETA) on the *0*1.>Vg  
    %   unit circle.  N is a vector of positive integers (including 0), and k*?Axk#  
    %   M is a vector with the same number of elements as N.  Each element p$qpC$F  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) >+9f{FP 9  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, {JJq/[j  
    %   and THETA is a vector of angles.  R and THETA must have the same $aY*1UVq  
    %   length.  The output Z is a matrix with one column for every (N,M) I6jDRC0<  
    %   pair, and one row for every (R,THETA) pair. 5kRP Sfh  
    % 6 4fB$  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ogOUrJ}P  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), THFzC/~Q  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral mYE8]4  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, A9?h*/$  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized I3#h  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ;;*'<\lP.j  
    % qoifzEc`U  
    %   The Zernike functions are an orthogonal basis on the unit circle. ,h#U<CnP#  
    %   They are used in disciplines such as astronomy, optics, and f&n6;N  
    %   optometry to describe functions on a circular domain. b <1k$0J6  
    % Hq>"rrVhx  
    %   The following table lists the first 15 Zernike functions. b8>2Y'X  
    % 5bfd8C  
    %       n    m    Zernike function           Normalization ?HsQ417.H  
    %       -------------------------------------------------- viLK\>>  
    %       0    0    1                                 1 U1.w%b,  
    %       1    1    r * cos(theta)                    2 "!fvEE  
    %       1   -1    r * sin(theta)                    2 4!I;U>b b  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) {m<NPtp910  
    %       2    0    (2*r^2 - 1)                    sqrt(3) bnm3 cR:h"  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ZeL v!  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ';CL;A;  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) kOQq+_Y  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 7[b]%i  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) b{Qg$ZJeR  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) B?-~f^*,jG  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) _w'N&#  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) W=$cQ(x4Z  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) B(omD3jzN  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) _LOV&83O(  
    %       -------------------------------------------------- <+/:}S4w)  
    % "%,KZI  
    %   Example 1: [h3y8O  
    % 3Mw2;.rk  
    %       % Display the Zernike function Z(n=5,m=1) cc$L56q  
    %       x = -1:0.01:1; ^EG@tB $<  
    %       [X,Y] = meshgrid(x,x); /F3bZ3F  
    %       [theta,r] = cart2pol(X,Y); Bl >)GX\l  
    %       idx = r<=1; gmU0/z3&  
    %       z = nan(size(X)); 1>$}N?u:T  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); kJOSGrg  
    %       figure ?puZqVu5  
    %       pcolor(x,x,z), shading interp ~I_v {  
    %       axis square, colorbar V*|#j0}b  
    %       title('Zernike function Z_5^1(r,\theta)') 60A E~  
    % MmvMuX]#)  
    %   Example 2: e@GR[0~  
    % M<s Y_<z  
    %       % Display the first 10 Zernike functions Y XBU9T{r  
    %       x = -1:0.01:1; Za&.sg3RG  
    %       [X,Y] = meshgrid(x,x); B F,rZZL  
    %       [theta,r] = cart2pol(X,Y); +( *;F4>  
    %       idx = r<=1; v)TFpV6b{p  
    %       z = nan(size(X)); 2u> [[U1:  
    %       n = [0  1  1  2  2  2  3  3  3  3]; tSZd0G<A<o  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ,%L>TD'48s  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ,z*-93H1  
    %       y = zernfun(n,m,r(idx),theta(idx)); z ]d^%>Ef  
    %       figure('Units','normalized') oI!L2  
    %       for k = 1:10 Yy_o*Ozq  
    %           z(idx) = y(:,k); #4iiY6  
    %           subplot(4,7,Nplot(k)) *>ilT5q  
    %           pcolor(x,x,z), shading interp ?;//%c8,.  
    %           set(gca,'XTick',[],'YTick',[]) @t;WdbxB%  
    %           axis square w(y#{!%+  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) )RQX1("O  
    %       end N~w4|q!]  
    % gm-m_cB<  
    %   See also ZERNPOL, ZERNFUN2. [q MFLY$  
    -quWnn/  
    2K 8?S  
    %   Paul Fricker 11/13/2006 )bM #s">Y  
     F%}0q&  
    \mBH6GS  
    Sb9In_* 0  
    e>Z F? (a0  
    % Check and prepare the inputs: N1O& fMz  
    % ----------------------------- u_5O<UP5  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /f:)I.FUm  
        error('zernfun:NMvectors','N and M must be vectors.') JE`mB}8s/  
    end LGOeBEAMV^  
    ${/"u3a_  
    m^8KHa  
    if length(n)~=length(m) P p}N-me>_  
        error('zernfun:NMlength','N and M must be the same length.') Cw=wU/)  
    end PR&D67:Jy  
    Ul<'@A8  
    B Bub'  
    n = n(:); ATeXOe  
    m = m(:); }x[d]fcC  
    if any(mod(n-m,2)) s1[_Pk;!  
        error('zernfun:NMmultiplesof2', ... 4zF|}aiQ  
              'All N and M must differ by multiples of 2 (including 0).')  l*+"0  
    end ]Tje6i F  
    Se o3a6o  
    rQncW~  
    if any(m>n) $jd>=TU|  
        error('zernfun:MlessthanN', ... _t:l:x.;T  
              'Each M must be less than or equal to its corresponding N.') XZcT-w 7  
    end zEQ<Q\"1  
    C~2/ 5  
    >PsP y.  
    if any( r>1 | r<0 ) "4{_amgm&<  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') (okCZ-_Jn  
    end IZm_/  
    ,|: a7b]  
    brQkVt_)EE  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) I6]|dA3G  
        error('zernfun:RTHvector','R and THETA must be vectors.') }T?MWcG4  
    end m_%1I J  
    mErXdb|L  
    >.D0McQg  
    r = r(:); (1bz.N8z  
    theta = theta(:); ZKG S?z  
    length_r = length(r); %j].' ;  
    if length_r~=length(theta) pai>6p  
        error('zernfun:RTHlength', ... '~-Lxvf'  
              'The number of R- and THETA-values must be equal.') iL-I#"qT,  
    end 23/!k}G"  
    (% fl  
    o:\RJig<  
    % Check normalization: O<R6^0B42  
    % -------------------- x8a?I T.  
    if nargin==5 && ischar(nflag) n'/w(o$&  
        isnorm = strcmpi(nflag,'norm'); hT&,5zaWdv  
        if ~isnorm o6pnTu  
            error('zernfun:normalization','Unrecognized normalization flag.') AgsMk  
        end 9T47U; _)  
    else @jW_ r j:<  
        isnorm = false; 2Yx6.e<  
    end d[.kGytUt  
    (}Ql#q K  
    k_9tz}Z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [aF?1KxNMt  
    % Compute the Zernike Polynomials 8wz4KG3SK  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% rK*s/mX <  
    q+{-p?;;  
    , #yE#8  
    % Determine the required powers of r: Na^1dn  
    % ----------------------------------- Sf}>~z2  
    m_abs = abs(m); ]McLace&  
    rpowers = []; 9.| +KIRb  
    for j = 1:length(n) 3G9YpA_}X  
        rpowers = [rpowers m_abs(j):2:n(j)]; fGiN`j} j  
    end O)MKEMuA  
    rpowers = unique(rpowers); \ ?[#>L4  
    _=Y]ZX`j  
     6h N~<  
    % Pre-compute the values of r raised to the required powers, $Yt29AQ  
    % and compile them in a matrix: #Zpp*S55  
    % ----------------------------- 2}u hPW+  
    if rpowers(1)==0 zCD?5*7  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); a z 7Vy-  
        rpowern = cat(2,rpowern{:}); %T*lcg  
        rpowern = [ones(length_r,1) rpowern]; pb`F_->uq  
    else sk~rjH]-g$  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); nnmn@t(%r  
        rpowern = cat(2,rpowern{:}); uROt h_/  
    end Q>nq~#3?  
    3Ak,M-Jp  
    <@n/[ +3  
    % Compute the values of the polynomials: )2}{fFa%  
    % -------------------------------------- GzK{. xf  
    y = zeros(length_r,length(n)); o #{D;'  
    for j = 1:length(n) Wy%q9x]}  
        s = 0:(n(j)-m_abs(j))/2; )t{oyBT  
        pows = n(j):-2:m_abs(j); e*uaxh+7  
        for k = length(s):-1:1 SsDz>PP  
            p = (1-2*mod(s(k),2))* ... 58*s\*V` \  
                       prod(2:(n(j)-s(k)))/              ... lhTjG,U=  
                       prod(2:s(k))/                     ... Vg/{;uLAe  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... w[z^B&  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); hG cq>Cvf  
            idx = (pows(k)==rpowers); a +Q9kh  
            y(:,j) = y(:,j) + p*rpowern(:,idx); y3$i?}?A  
        end d$s1l  
         4VPL -":6  
        if isnorm Pze$QBNoRd  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); >#B%gxff  
        end D%umL/[]  
    end s z/7cLo  
    % END: Compute the Zernike Polynomials %y33evX/B  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &R/)#NAp  
    /hf}f=7kH  
    L,.Ae i9  
    % Compute the Zernike functions: 7]Y Le+Ds  
    % ------------------------------ m8H|cQ@Uu  
    idx_pos = m>0; p~I+ZYWF'  
    idx_neg = m<0; m/n_e g  
    XF(I$Mxl6  
    ^8aj\xe(  
    z = y; tfj6#{M5  
    if any(idx_pos) 8qn1? Lb  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 0\%/:2   
    end r_T\%  
    if any(idx_neg) xh[Mmq/R  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ?"PUw3V3lB  
    end ,"  
    O^hWG ~o  
    B2VC:TG>  
    % EOF zernfun F{ J>=TC  
     
    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)  ^%-$8sV  
    su/l'p'  
    DDE还是手动输入的呢? 'A .c*<_  
    Q ,30  
    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