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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, Hl*vS  
    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, %>_[b,  
    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? jX53 owZ  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 0D-`>_  
    !^J;S%MB:K  
    j!;LN)s@?  
    )7q$PcY  
    7Z-j'pq  
    function z = zernfun(n,m,r,theta,nflag) 7]{g^g.9-  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 9hp&HL)BOa  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Uqr>8|t?  
    %   and angular frequency M, evaluated at positions (R,THETA) on the yzK;  
    %   unit circle.  N is a vector of positive integers (including 0), and ">uN={Iy  
    %   M is a vector with the same number of elements as N.  Each element /-=fWtA  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 8&<:(mAP  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, gesbt  
    %   and THETA is a vector of angles.  R and THETA must have the same ~ELY$G.xl  
    %   length.  The output Z is a matrix with one column for every (N,M) =u~nLL  
    %   pair, and one row for every (R,THETA) pair. %&ejO= r  
    % X -pbSq~5  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike j50vPV8m  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), dj gk7  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral  tm1 =  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, r924!zdbR  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized =C\Tl-$\f  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. F^ q{[Z  
    % HB07 n4 |  
    %   The Zernike functions are an orthogonal basis on the unit circle. 'g v0;L  
    %   They are used in disciplines such as astronomy, optics, and X1o",,N^M  
    %   optometry to describe functions on a circular domain. ;p`1Y<d-O  
    % m*0YMS>Y |  
    %   The following table lists the first 15 Zernike functions. dab]>% M  
    % |}"YUk^  
    %       n    m    Zernike function           Normalization @!ChPl  
    %       -------------------------------------------------- &OR(]Wt0  
    %       0    0    1                                 1 I8H3*DE  
    %       1    1    r * cos(theta)                    2 K7}.#*% ~  
    %       1   -1    r * sin(theta)                    2 0cG'37[  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) rYUIFPN  
    %       2    0    (2*r^2 - 1)                    sqrt(3) hA=uoe\  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) jP@ @<dt  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 2D\ pt  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ZR>BK,  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Q3@zUjq_Q  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) SX4*804a_  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) "ubp`7%67  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7Sdo*z  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Z)!8a$M~  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :X>Wd+lY:_  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) n,I3\l9  
    %       -------------------------------------------------- ly0R'4j \  
    % oEIpv;:_  
    %   Example 1: fk*(8@u>  
    % fQ+whGB  
    %       % Display the Zernike function Z(n=5,m=1) *d._H1zT  
    %       x = -1:0.01:1; Hv6h7-  
    %       [X,Y] = meshgrid(x,x); dX(JV' 18A  
    %       [theta,r] = cart2pol(X,Y); j^G=9r[,  
    %       idx = r<=1; \w9}O2lL  
    %       z = nan(size(X)); Q%e<0t7  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 'g#%>  
    %       figure OB>Hiy   
    %       pcolor(x,x,z), shading interp S^O9}<2g  
    %       axis square, colorbar `}X3f#eO&  
    %       title('Zernike function Z_5^1(r,\theta)') |)x7qy`  
    % qxZIH  
    %   Example 2: "*vrrY  
    % 9a`Lr B  
    %       % Display the first 10 Zernike functions $6"sRI6u  
    %       x = -1:0.01:1; m8n)sw,,  
    %       [X,Y] = meshgrid(x,x); 7x)Pt@c  
    %       [theta,r] = cart2pol(X,Y); Okq,p=D6  
    %       idx = r<=1; =v2 |QuS$  
    %       z = nan(size(X)); ^PG"  
    %       n = [0  1  1  2  2  2  3  3  3  3]; +!lDAkW0  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; k 9i W1  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; =<]`'15"V  
    %       y = zernfun(n,m,r(idx),theta(idx)); <4r8H-(%  
    %       figure('Units','normalized') fCt|8,-H  
    %       for k = 1:10 v h,(]t  
    %           z(idx) = y(:,k); D4%J!L<P  
    %           subplot(4,7,Nplot(k)) ;"dX]":  
    %           pcolor(x,x,z), shading interp o78u>Oy  
    %           set(gca,'XTick',[],'YTick',[]) Q)75?mn  
    %           axis square i>M%)HN  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) =\Q< TY  
    %       end *_D/_Rp7  
    % H cmW  
    %   See also ZERNPOL, ZERNFUN2. uq5?t  
    EN m%(G$  
    AVT % AS  
    %   Paul Fricker 11/13/2006 -K|1w'E  
    JFv70rBe  
    ~J\qkQ  
    ? AfThJc  
    s8-RXEPb  
    % Check and prepare the inputs: o3 0C\  
    % ----------------------------- Q68~D.V%r  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) M9)4ihK  
        error('zernfun:NMvectors','N and M must be vectors.') yr\ClIU  
    end B=A!hXNa  
    TdFU,  
    ^0]0ss;##R  
    if length(n)~=length(m) pg{VKrT`  
        error('zernfun:NMlength','N and M must be the same length.') l";Yw]:^  
    end Q4XlYgIV2A  
    TV`1&ta  
    \$9C1@B@  
    n = n(:); yaz6?,)  
    m = m(:); Pe`mZCd^  
    if any(mod(n-m,2)) m6R/,  
        error('zernfun:NMmultiplesof2', ... i1evB9FZ1z  
              'All N and M must differ by multiples of 2 (including 0).') UPtj@gtcY  
    end  h,/Aq  
    UL[,A+X8D  
    SkuR~!  
    if any(m>n) L{/% "2>  
        error('zernfun:MlessthanN', ... !wp1Df[  
              'Each M must be less than or equal to its corresponding N.') f*%kHfaXgN  
    end BX/3{5Y>{  
    dN5{W0_  
    h$5[04.Q  
    if any( r>1 | r<0 ) IiE6i43  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') (d4btcg  
    end  kN=&"  
    EE 9w^.3a  
    cWW?@ _  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) )<5k+O~  
        error('zernfun:RTHvector','R and THETA must be vectors.') I>?oVY6M@u  
    end HH*y$  
    J~%43!X\K  
    9#9 UzKX#  
    r = r(:); : UeK0  
    theta = theta(:); }=X: F1S  
    length_r = length(r); oC`F1!SfOO  
    if length_r~=length(theta) $w(RJ/  
        error('zernfun:RTHlength', ... NP;W=A F  
              'The number of R- and THETA-values must be equal.') n])#<0  
    end :k\#=u(  
    *2 Pr1U  
    biHacm  
    % Check normalization: <0d2{RQ;  
    % -------------------- i q`}c |c  
    if nargin==5 && ischar(nflag) _(-jk4 L  
        isnorm = strcmpi(nflag,'norm'); a&>NuMDI  
        if ~isnorm {+9RJmZg  
            error('zernfun:normalization','Unrecognized normalization flag.') z Rna=h!  
        end d,GOP_N8I  
    else \%<M[r=  
        isnorm = false; vAX(3  
    end f<8Hvumw  
    \Th<7WbR6#  
    3(c-o0M  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'xH^ksb"  
    % Compute the Zernike Polynomials ]k!Xb  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^+x?@$rq  
    Et3I(X3  
    Cd*h4Q]S  
    % Determine the required powers of r: c)#P}Ai  
    % ----------------------------------- $Ivjcs:  
    m_abs = abs(m); vH+g*A0S<  
    rpowers = []; {KgA V  
    for j = 1:length(n) w(@r-2D"  
        rpowers = [rpowers m_abs(j):2:n(j)]; coAXYn  
    end =zFROB\  
    rpowers = unique(rpowers); n#+EG3  
    N,TV?Q5l7  
    ! JA;0[;l=  
    % Pre-compute the values of r raised to the required powers, nL 5tHz:e  
    % and compile them in a matrix: c`<2&ke  
    % ----------------------------- Na91K4r#  
    if rpowers(1)==0 )9H5'Wh#  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 9[/0  
        rpowern = cat(2,rpowern{:}); ?I=1T.  
        rpowern = [ones(length_r,1) rpowern]; $e+sqgU  
    else +Kk1[fh-  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Vg'R=+Wb  
        rpowern = cat(2,rpowern{:}); LwB1~fF  
    end iTHwH{!  
    ~A>fB2.pM  
    necY/&Ld-  
    % Compute the values of the polynomials: `/sNX<mp  
    % -------------------------------------- HJ&P[zV^  
    y = zeros(length_r,length(n)); i >3`V6  
    for j = 1:length(n) -m@c{&r  
        s = 0:(n(j)-m_abs(j))/2; c~hH 7/v  
        pows = n(j):-2:m_abs(j); FW-I|kK.  
        for k = length(s):-1:1 `N\ ^JAGW  
            p = (1-2*mod(s(k),2))* ... P}4&J ^  
                       prod(2:(n(j)-s(k)))/              ... ^xHKoOTj[  
                       prod(2:s(k))/                     ... ZxvH1qx8  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... l\Ozy  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ( eKgc  
            idx = (pows(k)==rpowers); JX0M3|I=  
            y(:,j) = y(:,j) + p*rpowern(:,idx); /V,xSK9.&  
        end NQqw|3  
         %"`p&aE:  
        if isnorm 8Qg{@#Wr  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ]r;rAOWVV  
        end B_d\eD  
    end =7V4{|ESfy  
    % END: Compute the Zernike Polynomials kgo#JY-4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CE3l_[c  
    8C{&i5kj\E  
    m%L!eR  
    % Compute the Zernike functions: hJM& rM7  
    % ------------------------------ 5az%yS  
    idx_pos = m>0; q=t!COS  
    idx_neg = m<0; kQ>2W5o-d-  
    <%^/uS  
    U =J5lo  
    z = y; Mqr]e#"o  
    if any(idx_pos) P3Ql[ 2  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); G`l\R:Q  
    end 1"y !wsM%  
    if any(idx_neg) -`' |z+V  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); "5N4 of 8  
    end jV2H61d  
    4r$#-  
    Xy(QK2|  
    % EOF zernfun 0$|VkMq(  
     
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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)  A f?&VD4K  
    LzXIqj'H7T  
    DDE还是手动输入的呢? .f!'> _  
    'PMzm/;8st  
    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