切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9165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, z_|oCT!6  
    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, V4w=/e _  
    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? Q 9F)  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ?uLeFD  
    ~+F;q vq  
    D@ek9ARAq  
    G@ \Pi#1  
    "f.Z}AbP  
    function z = zernfun(n,m,r,theta,nflag) kma?v B  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. +C]&2zc.  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Av J4\  
    %   and angular frequency M, evaluated at positions (R,THETA) on the CB\{!  
    %   unit circle.  N is a vector of positive integers (including 0), and }ut]\]b  
    %   M is a vector with the same number of elements as N.  Each element 7*o*6,/  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) pL1i|O  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, : esg(  
    %   and THETA is a vector of angles.  R and THETA must have the same $^/0<i$   
    %   length.  The output Z is a matrix with one column for every (N,M) _u0$,Y?&|  
    %   pair, and one row for every (R,THETA) pair. Ka!I`Yf  
    % cR7wx 0Aj  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike El_Qk[X|A  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), yBpk$  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral X@N$Z{  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, s54nF\3V  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized +|cI:|H>  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. $m$;v<PSe  
    % U%<rn(xWXD  
    %   The Zernike functions are an orthogonal basis on the unit circle. XKOUQc4!R  
    %   They are used in disciplines such as astronomy, optics, and Njc%_&r  
    %   optometry to describe functions on a circular domain. IXLO>>`  
    % @exey  
    %   The following table lists the first 15 Zernike functions. &7mW9]  
    % ff? t[GS  
    %       n    m    Zernike function           Normalization TA18 gq  
    %       -------------------------------------------------- SUCU P<G  
    %       0    0    1                                 1 +!t}  
    %       1    1    r * cos(theta)                    2 }Nj97 R  
    %       1   -1    r * sin(theta)                    2 d;[u8t  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) /hWd/H]  
    %       2    0    (2*r^2 - 1)                    sqrt(3) < E|s\u  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) |iYg >  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) +]xFoH  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) BcWcdr+}9  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) F'P Qqb{  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) jjs&`Fy,  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) rt7<Q47QE  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) AbwbAm+  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) od<b!4k~s  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) MZv]s  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) @ T ;L$x  
    %       -------------------------------------------------- BbOu/i|  
    % j|fd-<ng  
    %   Example 1: CBT>"sYE1  
    % ^ZeJ[t&!#  
    %       % Display the Zernike function Z(n=5,m=1) km5~Gc}  
    %       x = -1:0.01:1; I+ l%Sn#\  
    %       [X,Y] = meshgrid(x,x); =s97Z-  
    %       [theta,r] = cart2pol(X,Y); 7Ey#u4Q  
    %       idx = r<=1; t G.(flW,  
    %       z = nan(size(X)); ,<,:8B  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); V3N0Og3  
    %       figure o5o^TW{  
    %       pcolor(x,x,z), shading interp ?9MVM~$  
    %       axis square, colorbar LE^G&<!  
    %       title('Zernike function Z_5^1(r,\theta)') OKOu`Hz@  
    % zJlQ_U-!  
    %   Example 2: L6P1L)  
    % mg:!4O$K  
    %       % Display the first 10 Zernike functions 4NR@u\S  
    %       x = -1:0.01:1; G* b2,9&F  
    %       [X,Y] = meshgrid(x,x); A~ (l{g  
    %       [theta,r] = cart2pol(X,Y); 34|a\b}  
    %       idx = r<=1; ,8G{]X)  
    %       z = nan(size(X)); SjEAuRDvUz  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Btt]R  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 9.O8/0w7LV  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Bvjl-$m!v  
    %       y = zernfun(n,m,r(idx),theta(idx)); ygZ  #y L  
    %       figure('Units','normalized') `\Ku]6J]5  
    %       for k = 1:10 hIv@i\`  
    %           z(idx) = y(:,k); =nUW'  
    %           subplot(4,7,Nplot(k)) vH %gdpxX  
    %           pcolor(x,x,z), shading interp )U<Y0bZA!  
    %           set(gca,'XTick',[],'YTick',[]) ~|Y>:M+0Z  
    %           axis square g+8hp@a  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 9a$56GnW1  
    %       end g&/p*c_  
    % .S\&L-{  
    %   See also ZERNPOL, ZERNFUN2. SF ]@|  
    +?D6T!)  
    lG<hlYckv  
    %   Paul Fricker 11/13/2006 [e e%c Xo  
    %WFu<^jm  
    ,38Eq`5&W  
    N6QVt f.  
    |dE -^"_  
    % Check and prepare the inputs: VzS&`d.h  
    % ----------------------------- "%2xR[NF  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 5 x2Ay=s  
        error('zernfun:NMvectors','N and M must be vectors.') ?wpB`  
    end &:*q_$]Oz  
    3*S{;p  
    donw(_=  
    if length(n)~=length(m) TB6m0qX(  
        error('zernfun:NMlength','N and M must be the same length.') X*oMFQgP  
    end s=I'e/"7  
    s$h] G[x  
    (kBP(2V  
    n = n(:); 9<CG s3\  
    m = m(:); _cDF{E+;  
    if any(mod(n-m,2)) 3+7^uR$/I4  
        error('zernfun:NMmultiplesof2', ... k5d\ w@G"~  
              'All N and M must differ by multiples of 2 (including 0).') N^?9ZO   
    end ^>4o$}  
    }_.:+H!@  
    wYsZM/lw  
    if any(m>n) ER&UBUu"  
        error('zernfun:MlessthanN', ... eKZ%2|+j!7  
              'Each M must be less than or equal to its corresponding N.') 7[v%GoE  
    end X+8B!F  
    U$&hZ_A  
    XhU@W}}  
    if any( r>1 | r<0 ) 7iC *Pr  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') [V# r7a  
    end ("7M b{  
    8U2dcx:G3  
    )QKf7 [:  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) &7?R+ZGo  
        error('zernfun:RTHvector','R and THETA must be vectors.') 79J-)e9  
    end }`_@'4:t  
    z T%U!jqI  
    d~tuk4F  
    r = r(:); '$|UwT`s  
    theta = theta(:); #>;FUZuJr  
    length_r = length(r); $e%2t^ i.g  
    if length_r~=length(theta) 1-SVCk -  
        error('zernfun:RTHlength', ... 8am`6;O:!  
              'The number of R- and THETA-values must be equal.') 0 5\dl  
    end @!=\R^#p  
    uw\1b.r'B  
    Y[ reD  
    % Check normalization: ZBD;a;wx  
    % -------------------- RH)EB<PV  
    if nargin==5 && ischar(nflag) VUU]Pu &  
        isnorm = strcmpi(nflag,'norm'); pI`?(5iK6|  
        if ~isnorm &UHPX?x  
            error('zernfun:normalization','Unrecognized normalization flag.') $ls[|N:y0l  
        end -OZ 5vH0  
    else _S0+;9fhY  
        isnorm = false; wVs.Vcwr  
    end i y YJR  
    {aKqXL[UP  
    xk~IN%\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yKagT$-  
    % Compute the Zernike Polynomials %bXx!x8(  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @=S}=cl  
    wHjLd$ +o  
    4] > ]-b  
    % Determine the required powers of r: )-rW&"{U  
    % ----------------------------------- =%)+%[wv  
    m_abs = abs(m); Uh}seB#mJj  
    rpowers = []; $V>98M>j  
    for j = 1:length(n) n#Dv2 E=6  
        rpowers = [rpowers m_abs(j):2:n(j)]; [a[/_Sf{  
    end K?x,T8<aW  
    rpowers = unique(rpowers); Id'RL2Kq*&  
    !4"sX+z9  
    Rn%N&1 Ef  
    % Pre-compute the values of r raised to the required powers, ]-o"}"3Ef  
    % and compile them in a matrix: I<b?vR 'F  
    % ----------------------------- >M!xiQX  
    if rpowers(1)==0 8$N8}q%  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); `L;eba  
        rpowern = cat(2,rpowern{:}); O^>jdl!TZ  
        rpowern = [ones(length_r,1) rpowern]; wle@v Cmr  
    else Gnm4gF!BI  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); WnFG{S{s  
        rpowern = cat(2,rpowern{:}); 6Z?j AXGSq  
    end K[\'"HyQ,X  
    :m=m}3/:  
    _;A $C(  
    % Compute the values of the polynomials: `pZs T ^G[  
    % -------------------------------------- /76 1o\Q  
    y = zeros(length_r,length(n)); V+-$ jOh  
    for j = 1:length(n) j Ib  
        s = 0:(n(j)-m_abs(j))/2; ~\nBjM2  
        pows = n(j):-2:m_abs(j); Vpfp}pL  
        for k = length(s):-1:1 kU5.iK'  
            p = (1-2*mod(s(k),2))* ... .N4  
                       prod(2:(n(j)-s(k)))/              ... 7DW]JK l  
                       prod(2:s(k))/                     ... pqM~l&  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... NY$uq+Z>  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); f"#m=_Xm  
            idx = (pows(k)==rpowers); 'J*<iA*W  
            y(:,j) = y(:,j) + p*rpowern(:,idx); SQsSa1  
        end bok.j  
         ?zJpD8e  
        if isnorm ~cAZB9Fa  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); &MR/6"/s  
        end G |*(8r()  
    end vqslirC  
    % END: Compute the Zernike Polynomials 5lKJll^2:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;T]d M fO  
    .o%^'m"=D[  
    z><5R|Gf  
    % Compute the Zernike functions: b/$km?R  
    % ------------------------------ a~h:qpg c  
    idx_pos = m>0; P27%xV-n>  
    idx_neg = m<0; rn@`yTw^  
    r,SnXjp@  
    :_<_[Y]1  
    z = y; &MmU  
    if any(idx_pos) =+5,B\~q@C  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); zxb/  
    end |as!Ui/J/  
    if any(idx_neg) 6[ qA`x#  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); TjWE_Bq]g  
    end @YvOoTyb  
    }F6<w{|  
    148V2H)  
    % EOF zernfun 5F+APz7  
     
    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)  tH=P6vY  
    u4z&!MT}  
    DDE还是手动输入的呢? =_j vk.  
    l1ZY1#%j  
    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