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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, , +J)`+pJx  
    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, 0$c(<+D  
    这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? gBh X=2%  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? yP# Y:s  
    )Jk$j  
    F"k`PF*b  
    9v`sSTlSd  
    YcX"Z~O6j=  
    function z = zernfun(n,m,r,theta,nflag) \ui'~n_t]  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. O2ktqAWx@  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N m4oj1h_4  
    %   and angular frequency M, evaluated at positions (R,THETA) on the -*KKrte  
    %   unit circle.  N is a vector of positive integers (including 0), and 1}Q9y`65  
    %   M is a vector with the same number of elements as N.  Each element =|aZNHqH  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ()Kaxcs?+  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, VFZ?<m  
    %   and THETA is a vector of angles.  R and THETA must have the same ,LxZbo!  
    %   length.  The output Z is a matrix with one column for every (N,M) g$#A'Du  
    %   pair, and one row for every (R,THETA) pair. LH_H yP_  
    % Cy uRj[;B  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike O.X;w<F/V  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )uOtQ0  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral >Rt:8uurAG  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, dR.?Kv(,E  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Mz(?_7  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. )'f=!'X  
    % ejyx[CF  
    %   The Zernike functions are an orthogonal basis on the unit circle. Hy\q{  
    %   They are used in disciplines such as astronomy, optics, and (nq""kO6'  
    %   optometry to describe functions on a circular domain. s<#BxN  
    % G \MeJSt*  
    %   The following table lists the first 15 Zernike functions. tjRw bnT"  
    % ElpZzGj+  
    %       n    m    Zernike function           Normalization %La7);SeY  
    %       -------------------------------------------------- %G 2g @2  
    %       0    0    1                                 1 $t^Td<  
    %       1    1    r * cos(theta)                    2 TA/hj>rV  
    %       1   -1    r * sin(theta)                    2 H $Az,-P  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) v{|y,h&]a  
    %       2    0    (2*r^2 - 1)                    sqrt(3) e#k rr  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 2HBey  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) z(Uz<*h8  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) @]#[TbNo  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) !y~nsy:&7x  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) `3ha~+Goo!  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) U4-RI]Cpf  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) KG(FA  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ;`pIq-=  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) YHom9& A  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) p<'pqf  
    %       -------------------------------------------------- 7K.],eo0  
    % 7J5jf231  
    %   Example 1: klAlS%  
    % qonStIP  
    %       % Display the Zernike function Z(n=5,m=1) o:ow"cOEf  
    %       x = -1:0.01:1; FIfLDT+Wh  
    %       [X,Y] = meshgrid(x,x); LlgFQfu8  
    %       [theta,r] = cart2pol(X,Y); W&cs&>F#  
    %       idx = r<=1; ZG1TR F "  
    %       z = nan(size(X)); !m~r0M7  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); (_FeX22+  
    %       figure $PRd'YdL/  
    %       pcolor(x,x,z), shading interp HU/4K7e`  
    %       axis square, colorbar hG~.Sc:G  
    %       title('Zernike function Z_5^1(r,\theta)') J5jI/P  
    % $Bc3| `K1v  
    %   Example 2: }z/%b<o_  
    % =to.Oa RR  
    %       % Display the first 10 Zernike functions @>$qb|j  
    %       x = -1:0.01:1; zmD7]?|  
    %       [X,Y] = meshgrid(x,x); q'y< UyT6  
    %       [theta,r] = cart2pol(X,Y); ucz~y! 4L{  
    %       idx = r<=1; NQuqM`LSQ  
    %       z = nan(size(X)); 4noy!h  
    %       n = [0  1  1  2  2  2  3  3  3  3]; >h~ik/|*  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; i9qIaG/  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; l?_Fy_fBt  
    %       y = zernfun(n,m,r(idx),theta(idx)); /%7&De6Xg  
    %       figure('Units','normalized') VuTTWBx  
    %       for k = 1:10 98 NFJ  
    %           z(idx) = y(:,k); ]G8"\J4 &  
    %           subplot(4,7,Nplot(k)) jHE^d<=O^  
    %           pcolor(x,x,z), shading interp AZik:C"Q  
    %           set(gca,'XTick',[],'YTick',[]) ~&<vAgy,  
    %           axis square Zw{?^6;cS  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 3rHn?  
    %       end 0Ba]Zo Z  
    % N8kNi4$mp=  
    %   See also ZERNPOL, ZERNFUN2. iyR"O1]  
    A\9LJ#E  
    =~W=}  
    %   Paul Fricker 11/13/2006 JJg;X :p  
    Ylu\]pr9|C  
    nTtEv~a_n  
    Ja&S_'P[  
    ` s+kYWg'Z  
    % Check and prepare the inputs: a @3s71  
    % ----------------------------- sz/^Ie-~  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Q1yXdw  
        error('zernfun:NMvectors','N and M must be vectors.') .)WEg|D0Ku  
    end mqsAYzG  
    $'eY-U8q  
    %#&njP  
    if length(n)~=length(m) -(lP8Y~gFY  
        error('zernfun:NMlength','N and M must be the same length.') x3U>5F@  
    end +03/A`PKrB  
    =w`uZ;l$Q  
    7p!ROl^  
    n = n(:); 0,@^<G8?  
    m = m(:); #l- 0$  
    if any(mod(n-m,2)) YjL'GmL<  
        error('zernfun:NMmultiplesof2', ... 2,g4yXws5  
              'All N and M must differ by multiples of 2 (including 0).') h*1T3U$  
    end W)T'?b'.  
    /uR/,R++  
    H=~7g3  
    if any(m>n) eGpKoq7a  
        error('zernfun:MlessthanN', ... \Z42EnJ  
              'Each M must be less than or equal to its corresponding N.') )'RaMo` 4  
    end [ "3s  
    IqepR >5t  
    #XqCz>Z  
    if any( r>1 | r<0 ) :IJ<Mmb  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') U~?mW,iRL  
    end +zLw%WD[l  
    =)g}$r &<  
    #%E^cGfY  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Q}<QE:-&E  
        error('zernfun:RTHvector','R and THETA must be vectors.') ?ILjt?X8  
    end 3pW4Ul@e  
    ]&D= *:c  
    b.?;I7r   
    r = r(:); jgPUR#)  
    theta = theta(:); r7?nHF  
    length_r = length(r); { 29aNm  
    if length_r~=length(theta)  |xg#Q`O  
        error('zernfun:RTHlength', ...  !=*8*?@  
              'The number of R- and THETA-values must be equal.') H%rNQxA2 +  
    end .b<W*4{j0H  
    _&s pMf  
    :WQlpLn  
    % Check normalization: _ gYj@ %  
    % -------------------- xHaz*w1|  
    if nargin==5 && ischar(nflag) L1g0Dd\Ox  
        isnorm = strcmpi(nflag,'norm'); cbm;45 L|  
        if ~isnorm ao.vB']T  
            error('zernfun:normalization','Unrecognized normalization flag.') P3 =#<Q.  
        end 'yA/sZ  
    else _$D!"z7i  
        isnorm = false; 3)?WSOsL :  
    end -gba&B+D"  
    ]sVWQj  
    (/]#G8  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h2Th)&Fb>  
    % Compute the Zernike Polynomials <`; {gX1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % C2Vga#  
    nIfAG^?|*  
    #wRhR>6  
    % Determine the required powers of r: x@bqPZ t  
    % ----------------------------------- _JNYvng m  
    m_abs = abs(m); #Cu$y8~as  
    rpowers = []; g:y4C6b  
    for j = 1:length(n) U\j g X  
        rpowers = [rpowers m_abs(j):2:n(j)]; )b2O!p  
    end m$v >r\*X  
    rpowers = unique(rpowers); 4`:POu&  
    2?Jw0Wq5D  
    a L+>XN  
    % Pre-compute the values of r raised to the required powers, x lqP%  
    % and compile them in a matrix: w4TQ4 Y  
    % ----------------------------- t[X^4bZd  
    if rpowers(1)==0 ty[p5%L1  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); &$_!S!Sa/  
        rpowern = cat(2,rpowern{:}); u SQ#Y^V_  
        rpowern = [ones(length_r,1) rpowern]; 7'i{JPm  
    else 2; ,8 u  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); BQg3+w:>  
        rpowern = cat(2,rpowern{:}); 6XU p$Pd(  
    end o}/|"(K  
    x`@`y7(  
    h| wdx(4  
    % Compute the values of the polynomials: Kn@#5MC rU  
    % -------------------------------------- I{[Z  
    y = zeros(length_r,length(n)); ^5TVm>F@3  
    for j = 1:length(n) dz +Dk6"R  
        s = 0:(n(j)-m_abs(j))/2; w"dKOdY  
        pows = n(j):-2:m_abs(j); 'plUs<A  
        for k = length(s):-1:1 URbB2 Bi  
            p = (1-2*mod(s(k),2))* ... Q{950$ )L  
                       prod(2:(n(j)-s(k)))/              ... $^{#hYq)o  
                       prod(2:s(k))/                     ... K#X/j'$^  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Q/0gd? U?  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); c};%VB  
            idx = (pows(k)==rpowers); mS![J69(  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 7/QK"0  
        end OM\1TD/-  
         AL3iNkEa  
        if isnorm FibZT1-k  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); _[Imwu}  
        end HSRO gBNI:  
    end pl1CPxSdO  
    % END: Compute the Zernike Polynomials ; xp-MK  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% W~D_+[P|_  
    YkB@fTTS  
    _\tv ${  
    % Compute the Zernike functions: w@cW`PlF  
    % ------------------------------ t4v'X}7q]  
    idx_pos = m>0; oU\7%gQ  
    idx_neg = m<0; $;q }j vo  
    !f52JQyh  
     w0=  
    z = y; ycc G>%>r  
    if any(idx_pos) ^ `Ozw^~  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 6Nn+7z<*&z  
    end j+ -r(lZ  
    if any(idx_neg) X`Q+,tx$  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); `}=R  
    end 2m yxwA5  
    4^2>K C_  
    9AB U^ig  
    % EOF zernfun o|z@h][(l(  
     
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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)
    在线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)  8`+=~S  
    ,byc!P  
    DDE还是手动输入的呢? k(H]ILL  
    ]" V_`i7Z  
    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