切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9403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, :A8r{`R'N  
    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, sTd@/>S?p  
    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? Ur+U#}  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? AGFA;X  
    %V=%ARP|  
    X(tx8~z  
    =K:[26  
    q|ce7HnK  
    function z = zernfun(n,m,r,theta,nflag) Sn[xI9}O  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. |Z>-<]p9g  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N fizW\f8ai  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Y*BmBRN  
    %   unit circle.  N is a vector of positive integers (including 0), and &h/r]KrZ  
    %   M is a vector with the same number of elements as N.  Each element d dgDq0N1j  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) uqcG3Pi  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, My>q%lF=fw  
    %   and THETA is a vector of angles.  R and THETA must have the same 48 -j  
    %   length.  The output Z is a matrix with one column for every (N,M) %1 )c{7  
    %   pair, and one row for every (R,THETA) pair. 43k'96[2d  
    % pEwo}NS*H  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 2{j$1EdI@-  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 45 ^ Z5t  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral vN(~}gOd\  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, >T;!Z5L1  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized y^H5iB[SPL  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Milp"L?B%  
    % !Q"L)%)'A  
    %   The Zernike functions are an orthogonal basis on the unit circle. -;gQy[U  
    %   They are used in disciplines such as astronomy, optics, and u0#KBXRo  
    %   optometry to describe functions on a circular domain. e\X[\ve  
    % p l^;'|=M  
    %   The following table lists the first 15 Zernike functions. `!cdxKLR  
    % &vmk!wAs  
    %       n    m    Zernike function           Normalization fuj9x;8X0  
    %       -------------------------------------------------- K{d3)lVYCS  
    %       0    0    1                                 1 ,esEh5=Ir  
    %       1    1    r * cos(theta)                    2 4P#jMox  
    %       1   -1    r * sin(theta)                    2 )bg|l?  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) lq.:/_m0  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 8`L]<Dm  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) M_!]9#:K7  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) HsYzIQLL  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) !y$##PZ  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ~j[?3E4L}  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 6Mk#) ebM  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) _ uOi:Ti  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (8 7wWhH  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) f&$Bjq  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) <%.%q  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 07SW$INb  
    %       -------------------------------------------------- ;R 6f9tu2  
    % U~=?I)Ni  
    %   Example 1: Vl+UC1M}B>  
    % HIw)HYF 2  
    %       % Display the Zernike function Z(n=5,m=1) `.;U)}Tn  
    %       x = -1:0.01:1; Z4G%Ve[  
    %       [X,Y] = meshgrid(x,x); SOG(&)b  
    %       [theta,r] = cart2pol(X,Y); eTjPztdJbx  
    %       idx = r<=1; Zsapu1HoL\  
    %       z = nan(size(X)); b$;oty9Y  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); D'[:35z  
    %       figure s2L]H  
    %       pcolor(x,x,z), shading interp 0fstEExw  
    %       axis square, colorbar =xkaF)AW&v  
    %       title('Zernike function Z_5^1(r,\theta)') o.r D  
    % &M5v EPR  
    %   Example 2: k-&<_ghT \  
    % #qVvh3#g  
    %       % Display the first 10 Zernike functions ,62~u'hR5  
    %       x = -1:0.01:1; 1VYH:uGuAU  
    %       [X,Y] = meshgrid(x,x); ]N}/L lq  
    %       [theta,r] = cart2pol(X,Y); [<i3l'V/[  
    %       idx = r<=1; }s?3   
    %       z = nan(size(X)); E[t[R<v,P!  
    %       n = [0  1  1  2  2  2  3  3  3  3]; :kcqf,7  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ;e_us!Sn  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ,(oolx"Xa  
    %       y = zernfun(n,m,r(idx),theta(idx)); QN;5+p[N  
    %       figure('Units','normalized') x]YzVJ=Y  
    %       for k = 1:10 O: I]v@  
    %           z(idx) = y(:,k); #<Y3*^~5d  
    %           subplot(4,7,Nplot(k)) Ruq;:5u  
    %           pcolor(x,x,z), shading interp , l!>+@  
    %           set(gca,'XTick',[],'YTick',[]) 5Kd"W,  
    %           axis square @G]*]rkKb  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) vy2<'V*y}  
    %       end W8`6O2  
    % B{0]v-w  
    %   See also ZERNPOL, ZERNFUN2. U}HSL5v  
    7 `~0j6FY  
    ^+%bh/2_W  
    %   Paul Fricker 11/13/2006 851BOkRal4  
    /)v+|%U  
    a(IE8:yU`  
    0-OKbw5%=b  
    [,st: Y  
    % Check and prepare the inputs: O_s /BoB@  
    % ----------------------------- Q7pCF,;  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) F+VNrt-  
        error('zernfun:NMvectors','N and M must be vectors.') ~:,}?9  
    end ga KZ4#  
    $C=XSuPNK  
    <x$nw'H9  
    if length(n)~=length(m) **-rPonM[  
        error('zernfun:NMlength','N and M must be the same length.') 4T52vM  
    end 3,Z;J5VL4!  
    o *U-.&  
    *eD[[HbKX  
    n = n(:); r]}6iF.  
    m = m(:); \+Qd=,!i(  
    if any(mod(n-m,2)) gCYe ^KJ  
        error('zernfun:NMmultiplesof2', ... VxOWv8}|  
              'All N and M must differ by multiples of 2 (including 0).') ekfa"X_  
    end hG`@#9|f  
    f@9XSZ<.71  
    5mVO9Q j  
    if any(m>n) >8{{H"$;(  
        error('zernfun:MlessthanN', ... }X])055S  
              'Each M must be less than or equal to its corresponding N.') 2T%sHp~qt  
    end H!FaI(YZl  
    f3vl=EA4|  
    b&,Z mDJh  
    if any( r>1 | r<0 ) N DI4EA~z  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') &RuTq6)r  
    end +MYrNR.p  
    IuFr:3(  
    RI<s mt.Ng  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 1foG*   
        error('zernfun:RTHvector','R and THETA must be vectors.') 7C Sn79E  
    end C_ ;nlG6  
    Y1AZ%{^0a  
    hb0)<^xu  
    r = r(:); *E>R1bJ8  
    theta = theta(:); SG~HzQ\%  
    length_r = length(r); @D["#pe,}  
    if length_r~=length(theta) bFG?mG:  
        error('zernfun:RTHlength', ... E!WlQr:b$  
              'The number of R- and THETA-values must be equal.') YVHf-uP  
    end L|D9+u L  
    F;/^5T3wI  
    u"T9w]Z\  
    % Check normalization: ?&qQOM~b-\  
    % -------------------- 1Xh@x  
    if nargin==5 && ischar(nflag) {&Rz>JK  
        isnorm = strcmpi(nflag,'norm'); A3HN Mz  
        if ~isnorm E>E^t=; [  
            error('zernfun:normalization','Unrecognized normalization flag.') toj5b;+4F  
        end dA2@PKK  
    else >X[:(m'  
        isnorm = false; 2S:B%cj9m  
    end G.N `  
    lHliMBSc  
    7c%dSs6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Dbx zqd  
    % Compute the Zernike Polynomials B4zuWCE@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Lbwfd=  
    rHybP6C<  
    gDw(_KC  
    % Determine the required powers of r: ,9F3~Ryt(  
    % ----------------------------------- V3|" v4  
    m_abs = abs(m); DqI"B  
    rpowers = []; -ciwIS9L  
    for j = 1:length(n) xVI"sBUu  
        rpowers = [rpowers m_abs(j):2:n(j)]; C>-}BeY!  
    end V%t_,AT  
    rpowers = unique(rpowers); +wHa)A0MW  
    F }F{/  
    "-5FUKI-  
    % Pre-compute the values of r raised to the required powers, c/F!cW{z^  
    % and compile them in a matrix: Qi qRx  
    % ----------------------------- P uQ  
    if rpowers(1)==0 {65Y Tt%  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); S,'ekWVD  
        rpowern = cat(2,rpowern{:}); " :[;}f;  
        rpowern = [ones(length_r,1) rpowern]; JvCy&xrE;  
    else F7=\*U  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); E+ XR[p  
        rpowern = cat(2,rpowern{:}); !6#.%"{-  
    end 9Ns%<FRO@  
    @.dM1DN)  
    ;<cCT!A  
    % Compute the values of the polynomials: ,#^2t_c/  
    % -------------------------------------- vZ6R>f  
    y = zeros(length_r,length(n)); uzp\<\d-t  
    for j = 1:length(n) =:TQ_>$Nc2  
        s = 0:(n(j)-m_abs(j))/2; f*m^x7  
        pows = n(j):-2:m_abs(j); 5yW}#W>  
        for k = length(s):-1:1 gId :IR  
            p = (1-2*mod(s(k),2))* ... ,>kXn1 ,  
                       prod(2:(n(j)-s(k)))/              ... ?<OyJ|;V  
                       prod(2:s(k))/                     ... D51O/.:U2  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Pc+,iK>  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); `sv]/8RN  
            idx = (pows(k)==rpowers); 8H3O6ro  
            y(:,j) = y(:,j) + p*rpowern(:,idx); @P=n{-pIW  
        end h9nh9a(2  
         A~s6~  
        if isnorm @te}Asv  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ,&@FToR  
        end y#O/Xw  
    end M%!j\}2A  
    % END: Compute the Zernike Polynomials O "Aeg|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,6A/| K-  
    %u Dd#+{  
    D [v225  
    % Compute the Zernike functions: !l9 #a{#6l  
    % ------------------------------ I'<sJs*p  
    idx_pos = m>0; xKT;1(Mk  
    idx_neg = m<0; k?Zcv*[)D+  
    =wl0  
    $ Fy)+<  
    z = y; COH.`Tv{*  
    if any(idx_pos) nXh<+7  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); u b@'(*  
    end Lk)TK/JM)  
    if any(idx_neg) 1@+&6UC  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); X`]>J5  
    end j{m{hVa  
    LH~ t5  
    eW_EWVH  
    % EOF zernfun (d[JMO^@8  
     
    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)  is=|rY9$  
    >4AwjS }H  
    DDE还是手动输入的呢? Yk#$-"c/a  
    G)y'exk  
    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