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    [讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

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    离线jssylttc
     
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    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, XM3N>OR.  
    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, BVxg=7%St  
    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? CjM+%l0MW  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? PIo/|1  
    (yE?)s  
    e#K =SV!H  
    v?rjQ'OP  
    9Y 1&SEsNX  
    function z = zernfun(n,m,r,theta,nflag) ^_JD 7-g  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ks7g*; 3{@  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N {*H&NI  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ;HM& ":7  
    %   unit circle.  N is a vector of positive integers (including 0), and B:5( sK  
    %   M is a vector with the same number of elements as N.  Each element g^(wZ$NH  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) !*.mcIQT  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, c&2ZjM  
    %   and THETA is a vector of angles.  R and THETA must have the same <CJua1l\  
    %   length.  The output Z is a matrix with one column for every (N,M) >z6 (fM`i  
    %   pair, and one row for every (R,THETA) pair. OA2<jrGB!  
    % m8H|cQ@Uu  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike `CW8Wj  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), PJN TIa  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral bp2l%A;  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 9@Yk8  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized XJsHy_6  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. -;'8#"{`^  
    % A1@tp/L=o  
    %   The Zernike functions are an orthogonal basis on the unit circle. 9 )u*IGj  
    %   They are used in disciplines such as astronomy, optics, and =?T\zLN=  
    %   optometry to describe functions on a circular domain.  vrdlI^  
    % .&.j?kb  
    %   The following table lists the first 15 Zernike functions. ?hvPPEJf  
    % KDgJ~T  
    %       n    m    Zernike function           Normalization /j./  
    %       -------------------------------------------------- Gvv~P3Dm  
    %       0    0    1                                 1 npg.*I/>  
    %       1    1    r * cos(theta)                    2 0 V*Di2  
    %       1   -1    r * sin(theta)                    2 ?8. $A2(Xw  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Adgh:'h  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ,Cj1S7GFR  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) XodA(73`i  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ( =0W[@k  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) R6] /g  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) =v=a:e  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ;oV dkp  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Ojq>4=Z\  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) WM NcPHcj  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) DCM ,|FE  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) EsXCi2]1  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) .MMFN }1O  
    %       -------------------------------------------------- !Sfy'v.  
    % x)l}d3   
    %   Example 1: 5b3Wt7  
    % _KC)f'Cx  
    %       % Display the Zernike function Z(n=5,m=1) qI\qpWS\  
    %       x = -1:0.01:1; $[5ihV$u  
    %       [X,Y] = meshgrid(x,x); Q.#@xaX'{`  
    %       [theta,r] = cart2pol(X,Y); {NXc<0a(  
    %       idx = r<=1; w-};\]I  
    %       z = nan(size(X));  y$7Fq'  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ;$l!mv 7  
    %       figure X|t?{.p  
    %       pcolor(x,x,z), shading interp e~=fo#*2?@  
    %       axis square, colorbar G+ PBV%gE[  
    %       title('Zernike function Z_5^1(r,\theta)') {o< 4 ^  
    % ~F^=7oq  
    %   Example 2: -}@3,G  
    % 048BQ  
    %       % Display the first 10 Zernike functions [>::@[  
    %       x = -1:0.01:1;  d_gm'  
    %       [X,Y] = meshgrid(x,x); pa Uh+"y>  
    %       [theta,r] = cart2pol(X,Y); 9d^o2Y o  
    %       idx = r<=1; kM|akG  
    %       z = nan(size(X)); DtG><g}[]  
    %       n = [0  1  1  2  2  2  3  3  3  3]; T!eeMsI  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; rc1EJ(c  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 0*YLFqN  
    %       y = zernfun(n,m,r(idx),theta(idx)); :q S=_!1  
    %       figure('Units','normalized') *5]fjh{  
    %       for k = 1:10 J/8aDr (+  
    %           z(idx) = y(:,k); S*Un$ngAh  
    %           subplot(4,7,Nplot(k)) qPuxYU  
    %           pcolor(x,x,z), shading interp ,,S5 8\x  
    %           set(gca,'XTick',[],'YTick',[]) K2>(C$Z  
    %           axis square sguE{!BO  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) FYR%>Em  
    %       end j!GJ$yd=-6  
    % RzQ1Wq  
    %   See also ZERNPOL, ZERNFUN2. YW9 [^  
    eG9tn{  
    Q]Q i  
    %   Paul Fricker 11/13/2006 Y*;Z(W.V#  
    BRYhL|d~.  
    u*Z>&]W_  
    j0^~="p%C  
    } *|_P  
    % Check and prepare the inputs: 'A .c*<_  
    % ----------------------------- Q ,30  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7kx)/Rw\B  
        error('zernfun:NMvectors','N and M must be vectors.') Enm#\(j  
    end EWNh:<F?  
    2MQgTFM9  
    4]A2Jl E  
    if length(n)~=length(m) )dgXS//Y  
        error('zernfun:NMlength','N and M must be the same length.') KRQKL`}}  
    end ^_#0\f  
    M 8a^yoZn  
    W_9-JM(r  
    n = n(:); \~d|MP}"F:  
    m = m(:); v~e@:7d i  
    if any(mod(n-m,2)) 5:/ zbt\C  
        error('zernfun:NMmultiplesof2', ... s$css{(ek  
              'All N and M must differ by multiples of 2 (including 0).') z(d@!Cd  
    end &$tBD@7  
    K@Q_q/(%;  
    )(~4fA5j)  
    if any(m>n) mv|eEz)r  
        error('zernfun:MlessthanN', ... /~NsHStn  
              'Each M must be less than or equal to its corresponding N.') rCi7q]_  
    end _ fha9`  
    l-xKfp`  
    J * $u  
    if any( r>1 | r<0 ) >Lp^QP1gU  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') zQM3n =y  
    end dqO!p6  
    \c3zK|^  
    5n! V^ !  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) p0~=   
        error('zernfun:RTHvector','R and THETA must be vectors.') NH$%g\GPs  
    end 0H,1"~,w]  
    /fbI4&SB!  
    JUXIE y^  
    r = r(:); n#t{3qzpD  
    theta = theta(:); MEMD8:['  
    length_r = length(r); U.is:&]E  
    if length_r~=length(theta) ] C_g: |q  
        error('zernfun:RTHlength', ... @-nCK Yj  
              'The number of R- and THETA-values must be equal.') ['ol]ZJ  
    end B?9K!c  
    x*& OvI/o  
    =8O057y  
    % Check normalization: &54fFyJF  
    % -------------------- lMz5))Rr  
    if nargin==5 && ischar(nflag) i*B@#;;F  
        isnorm = strcmpi(nflag,'norm'); 5_Yl!=  
        if ~isnorm __r]@hY   
            error('zernfun:normalization','Unrecognized normalization flag.') H((! BRl  
        end }ozlED`E  
    else vKN"o* q  
        isnorm = false; .}>d[},F  
    end . [DCL  
    ]Aap4+s  
    :Y/i%#*1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9 X}F{!p~1  
    % Compute the Zernike Polynomials qYv/" 1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s2A3.SN  
    d!kiWmw,  
    ;ZZ%(P=-  
    % Determine the required powers of r: <Z5ak4P  
    % ----------------------------------- yL/EIN  
    m_abs = abs(m); }YJ(|z""  
    rpowers = []; d2lOx|jt  
    for j = 1:length(n) M,@\*qlEJ  
        rpowers = [rpowers m_abs(j):2:n(j)]; WF\ hXO  
    end n B4)%  
    rpowers = unique(rpowers); S!Ue+jW  
    G0Zq:kJ  
    @/h_v#W  
    % Pre-compute the values of r raised to the required powers, Jcf'Zw"\  
    % and compile them in a matrix: 7uG@ hL36  
    % ----------------------------- %^s;{aN*!  
    if rpowers(1)==0 It'hmwu#  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); "+3p??h%Rq  
        rpowern = cat(2,rpowern{:}); 'U ',9  
        rpowern = [ones(length_r,1) rpowern]; nM:e<`r  
    else YSwAu,$jf  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); A5-y+   
        rpowern = cat(2,rpowern{:}); fy04/_,q  
    end xcdy/J&  
    =g4^tIYq  
    RG/M-  
    % Compute the values of the polynomials: d%_v eVIe  
    % -------------------------------------- 2|]$hjs  
    y = zeros(length_r,length(n)); *KNj5>6=  
    for j = 1:length(n) >m='#x0>Y  
        s = 0:(n(j)-m_abs(j))/2; Sx)b~*  
        pows = n(j):-2:m_abs(j); =H6"\`W  
        for k = length(s):-1:1 jqq96hP,  
            p = (1-2*mod(s(k),2))* ... tWR>I$O8F  
                       prod(2:(n(j)-s(k)))/              ... )\!_`ob  
                       prod(2:s(k))/                     ... 'Lu7cb^  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... $z,lq#zzl  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); .Tr!/mf_  
            idx = (pows(k)==rpowers); 'qcLK>E  
            y(:,j) = y(:,j) + p*rpowern(:,idx); gTWl];xja  
        end ceBu i8a |  
         K3*8JF7_F  
        if isnorm $;NxO0$  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); #<d f!)  
        end Yqz(@( %  
    end KdU!wsKfG  
    % END: Compute the Zernike Polynomials K{)N:|y%!$  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .),ql_sXr  
    HqNM31)  
    @wO"?w(  
    % Compute the Zernike functions: M mH[ 7R  
    % ------------------------------ m<L.H33'  
    idx_pos = m>0; 4mR{\ d  
    idx_neg = m<0; ufF$7@(+  
    WE\@ArY>  
    lc1?Vd$  
    z = y; D?;8bI%"  
    if any(idx_pos) S*;8z}5<\  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); )]x/MC:9r  
    end z5G<h  
    if any(idx_neg) l`c&nf6  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); t.wB\Kmt\  
    end sLiKcR8^  
    >7%Gd-;l  
    r.1/ * i  
    % EOF zernfun RbB y8ZVM  
     
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    离线phoenixzqy
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    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
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    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
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    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)  |&JCf =  
    2&*r1NXBE  
    DDE还是手动输入的呢? {d`e9^Z:  
    3+6s}u)  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    离线jssylttc
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    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    离线18257342135
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    只看该作者 5楼 发表于: 2016-12-13
    支持一下,慢慢研究