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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, Wg20H23XW  
    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, J 5xMA-  
    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? h1l%\3ZH  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? J57; X=M  
    nLCaik_,m  
    <@Vf:`a!P>  
    nxNHf3   
    =3!o _  
    function z = zernfun(n,m,r,theta,nflag) .h\Py[h<^  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. D$@2H>.-  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N %k~ezn  
    %   and angular frequency M, evaluated at positions (R,THETA) on the xP/q[7>#Q  
    %   unit circle.  N is a vector of positive integers (including 0), and hRMya#%-  
    %   M is a vector with the same number of elements as N.  Each element aNA ]hl  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) e\O-5hp7  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, XMdCQ=  
    %   and THETA is a vector of angles.  R and THETA must have the same _GrifGU\  
    %   length.  The output Z is a matrix with one column for every (N,M) bwj{5-FU  
    %   pair, and one row for every (R,THETA) pair. #Ge_3^'  
    % FBbaLqgVF{  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike crN*eFeW  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 1z4_QZZ.NG  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 1vxQ`)a  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, j=Izwt>   
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized @$'pMg  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. &FuL {YL  
    % >239SyC-,  
    %   The Zernike functions are an orthogonal basis on the unit circle. n1PBpM9!  
    %   They are used in disciplines such as astronomy, optics, and A=IpP}7J  
    %   optometry to describe functions on a circular domain. v$w}UC%uf  
    % /sj*@HF=  
    %   The following table lists the first 15 Zernike functions. 5-y*]:g(  
    % +I3O/=)  
    %       n    m    Zernike function           Normalization ?c+$9  
    %       -------------------------------------------------- =,h'}(z_  
    %       0    0    1                                 1 4 Yv:\c  
    %       1    1    r * cos(theta)                    2 w%8y5v5  
    %       1   -1    r * sin(theta)                    2 @0]WMI9B"B  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ~KYzEqy  
    %       2    0    (2*r^2 - 1)                    sqrt(3) W]bgWKd  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) fGqX dlP  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) "j8)l4}  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) nj1o!+9>$  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) <oV[[wl  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ^+v1[U@  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) P/.<sr=2  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) t$wbwP  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) `-OzjbM  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^L)TfI_n  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) GBT|1c'i  
    %       -------------------------------------------------- `GdH ,:S>  
    % FO%pdLs,  
    %   Example 1: 'Grii,  
    % |R _rfJh  
    %       % Display the Zernike function Z(n=5,m=1) K@{0]6  
    %       x = -1:0.01:1; *OznZIn  
    %       [X,Y] = meshgrid(x,x); J?yasjjgP  
    %       [theta,r] = cart2pol(X,Y); Sk/@w[  
    %       idx = r<=1; 1[8^JVC>6  
    %       z = nan(size(X)); )#cZ& O  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); W/hzo*o'g  
    %       figure u}|v;:|j  
    %       pcolor(x,x,z), shading interp [DH4iG5  
    %       axis square, colorbar ;?tH8jf>  
    %       title('Zernike function Z_5^1(r,\theta)') 1*2ycfa  
    % <kPNe>-f  
    %   Example 2: U|V,&RlbR  
    % Tx!t3;Yz[  
    %       % Display the first 10 Zernike functions Mms|jF oQ  
    %       x = -1:0.01:1; Wc2&3p9 c  
    %       [X,Y] = meshgrid(x,x); c:u*-lYmK%  
    %       [theta,r] = cart2pol(X,Y); 6V%}2YE?X  
    %       idx = r<=1; 7Q9Hk(Z9  
    %       z = nan(size(X)); E+qLj|IU  
    %       n = [0  1  1  2  2  2  3  3  3  3]; \<*F#3U1  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; l<GN<[/.+  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; n]`]gLF\i  
    %       y = zernfun(n,m,r(idx),theta(idx)); G}g;<,g~  
    %       figure('Units','normalized') 'ia-h7QWS  
    %       for k = 1:10 GEF's#YWK  
    %           z(idx) = y(:,k); Eu'E;*- f  
    %           subplot(4,7,Nplot(k)) 3*~`z9-z  
    %           pcolor(x,x,z), shading interp #e-K It  
    %           set(gca,'XTick',[],'YTick',[]) O- QT+]  
    %           axis square ?'+]d;UO&  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) R/Bjc}J'  
    %       end 4KtD  k  
    % )'T].kWW  
    %   See also ZERNPOL, ZERNFUN2. 2Ax"X12{6  
      8sG?|u  
    ?Y3i-jY  
    %   Paul Fricker 11/13/2006 $q:l \  
    hmo4H3g!N  
    i/+^C($'f  
    :?\29j#*V  
    py:L-5  
    % Check and prepare the inputs: .1RQ}Ro,<  
    % ----------------------------- (m:Q'4Ep  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) jmwN1Se>  
        error('zernfun:NMvectors','N and M must be vectors.') SoM,o]s#y  
    end O}zHkcL  
    j0}wv~\  
    i@nRZ$K  
    if length(n)~=length(m) UTh2? Rh/  
        error('zernfun:NMlength','N and M must be the same length.') x5uz$g  
    end #%k_V+o3  
    2=fM\G  
    DdI%TU K,  
    n = n(:); f&4+-w.:V|  
    m = m(:); DSqA}r  
    if any(mod(n-m,2)) >^Wpc  
        error('zernfun:NMmultiplesof2', ... 'Xwv,  
              'All N and M must differ by multiples of 2 (including 0).') 0.x+ H9z  
    end Z,2?TT|p  
    pJ}U'*Z2  
    H!]&"V77  
    if any(m>n) lwX9:[Z  
        error('zernfun:MlessthanN', ... V'=;M[&  
              'Each M must be less than or equal to its corresponding N.') kE'p=dXx  
    end Z40k>t D  
    4)tY6ds)r|  
    en'[_43  
    if any( r>1 | r<0 ) 7V::P_aUY  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') #rX ^)2  
    end onSt%5{P%X  
    e8> X5  
    Zqm%qm:  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) `)y ;7%-  
        error('zernfun:RTHvector','R and THETA must be vectors.') RNw#s R  
    end W`9{RZ'  
    C;HEv q7  
    k{;?>=FH!  
    r = r(:); S;#:~?dU  
    theta = theta(:); I2CI9,0  
    length_r = length(r); %/w-.?bX  
    if length_r~=length(theta) )yb~ kbe  
        error('zernfun:RTHlength', ... _0rt.NRD  
              'The number of R- and THETA-values must be equal.') iu=Mq|t0  
    end J&~I4ko]  
    ASoBa&vX  
    faRQj:R8  
    % Check normalization: G`%rnu  
    % -------------------- 02;'"EmP$  
    if nargin==5 && ischar(nflag) _VdJFjY?zc  
        isnorm = strcmpi(nflag,'norm'); IrCl\HQN  
        if ~isnorm ,^c-}`!K  
            error('zernfun:normalization','Unrecognized normalization flag.') h )Y .jY  
        end ]@z!r2[  
    else jW8ad{  
        isnorm = false; V)~b+D  
    end {ObY1Y`ea  
    [rcM32  
    !O!:=wq  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s-Gd{=%/q  
    % Compute the Zernike Polynomials )fXw~  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #@BhGB`9Qt  
    83V\O_7j  
    n-M6~   
    % Determine the required powers of r: !1R?3rVQS  
    % ----------------------------------- <(Ar[Rp  
    m_abs = abs(m); { /!ryOA65  
    rpowers = []; I_8 n>\u  
    for j = 1:length(n) 3t9+YdNKU  
        rpowers = [rpowers m_abs(j):2:n(j)]; ,/p+#|>C=  
    end ^^ix4[1$Z  
    rpowers = unique(rpowers); Z<d=v3q  
    -2}ons(  
    Z:h'kgG&  
    % Pre-compute the values of r raised to the required powers, *^[6uaa  
    % and compile them in a matrix: Adiw@q1&  
    % ----------------------------- 6bj77CoB  
    if rpowers(1)==0 zwQ#Yvd  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); r\fkx>  
        rpowern = cat(2,rpowern{:}); ~P}ng{x4z  
        rpowern = [ones(length_r,1) rpowern]; |4/rVj"  
    else ~5|R`%  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); \anOOn@  
        rpowern = cat(2,rpowern{:}); w8q 2f-K-  
    end b (@GKH"W  
    ]:F !h2  
    5,4" CF$  
    % Compute the values of the polynomials: t;3).F  
    % -------------------------------------- EVqqOp1$v4  
    y = zeros(length_r,length(n)); DQu)?Rsk  
    for j = 1:length(n) X*7VDt=  
        s = 0:(n(j)-m_abs(j))/2; 7fWZ/;p  
        pows = n(j):-2:m_abs(j); vAG|Y'aO@%  
        for k = length(s):-1:1 'tMD=MH  
            p = (1-2*mod(s(k),2))* ... 'e<8j  
                       prod(2:(n(j)-s(k)))/              ... N6BOUU]  
                       prod(2:s(k))/                     ... yZ=O+H  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... {l/`m.Z  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); D5Rp<PBq,  
            idx = (pows(k)==rpowers); 0r$n  
            y(:,j) = y(:,j) + p*rpowern(:,idx); M).CyY;bm  
        end Zonn  
         j7I?K :op=  
        if isnorm j)ln"u0R^B  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); MR4k#{:w  
        end ^pY8'LF6  
    end 73u97oe>1  
    % END: Compute the Zernike Polynomials ryz NM3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ep5`&g]3  
    S Cn)j:gH;  
    _@)-#7  
    % Compute the Zernike functions: @R;k@b   
    % ------------------------------ p/LV^TQ  
    idx_pos = m>0; ^XYK }J  
    idx_neg = m<0; 2Kr8#_) 0  
    fO#nSB/ 8  
    ?c"i V  
    z = y; ?%ltoezf  
    if any(idx_pos) 58mpW`Q  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); +S#Xm4  
    end inq {" 6  
    if any(idx_neg) !H)!b#_  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); SuI^8^f=  
    end }{PG^Fc<P  
    >^,?0HP  
    ;U a48pSv  
    % EOF zernfun ]M"U 'Z  
     
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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)  !tx.2m*5  
    FlrYXau  
    DDE还是手动输入的呢? C}L2'l,  
    +Oo-8f*  
    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