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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, ph>7?3;t  
    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, &UCsBqIY  
    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? ml|W~-6l  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? |t iUej  
    ~9)"!   
    ps .]N   
    #rO8Kf  
    &!aAO(g  
    function z = zernfun(n,m,r,theta,nflag) {j5e9pg1L|  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. `U#55k9^5  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ##Q/I|  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 1i:|3PA~  
    %   unit circle.  N is a vector of positive integers (including 0), and 2&c9q5.b  
    %   M is a vector with the same number of elements as N.  Each element uXDq~`S  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ]lw|pvtd  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Z[\ O=1E,  
    %   and THETA is a vector of angles.  R and THETA must have the same Hn>B!Bm*  
    %   length.  The output Z is a matrix with one column for every (N,M) kF;D BN  
    %   pair, and one row for every (R,THETA) pair. "8^5>EJWv  
    % / N) W2  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike fFj grK8  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), P`s  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral \<}&&SuH  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Ev7J+TmXM  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized -C(b,F%%  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. M?b6'd9f  
    % Le<w R  
    %   The Zernike functions are an orthogonal basis on the unit circle. 63`{.yZ*z  
    %   They are used in disciplines such as astronomy, optics, and o?1;<gs  
    %   optometry to describe functions on a circular domain. .s+aZwTMT  
    % 2C{H$ A,pW  
    %   The following table lists the first 15 Zernike functions. B+^(ktZp@  
    % 1+-_s  
    %       n    m    Zernike function           Normalization l]~n3IK"  
    %       -------------------------------------------------- K=!Bh*  
    %       0    0    1                                 1 qd"_Wu6aF=  
    %       1    1    r * cos(theta)                    2 dq[Mj5eC  
    %       1   -1    r * sin(theta)                    2 =@k%&* Y?  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) AU-n&uX  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 2z\zh[(w  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) [mEql,x3  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) kJW N.  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) x.8TRMk^  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) btdb%Q*  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) "#(T  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ;<G=M2  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) F(na{<g};  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) kP/M< X"  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6s0_#wZC  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 5M9 I,  
    %       -------------------------------------------------- 0b4R  
    % 22f`LoM  
    %   Example 1: [<'-yQ{l\  
    % 5@^ dgq  
    %       % Display the Zernike function Z(n=5,m=1) [D*UT#FM  
    %       x = -1:0.01:1; ~z"= G5|  
    %       [X,Y] = meshgrid(x,x); 7^w >Rj  
    %       [theta,r] = cart2pol(X,Y); JK.ZdY%  
    %       idx = r<=1; p~*UpU8u  
    %       z = nan(size(X)); ,t\* ZTt$  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); (' -JY  
    %       figure hKzSgYxP=t  
    %       pcolor(x,x,z), shading interp `7/Y@}n  
    %       axis square, colorbar H\XP\4#u  
    %       title('Zernike function Z_5^1(r,\theta)') 4)1s M=u  
    % &QhX1dT+  
    %   Example 2: i hh/sPi  
    % sZW^ !z  
    %       % Display the first 10 Zernike functions $H+VA@_  
    %       x = -1:0.01:1; 5uxBK"q  
    %       [X,Y] = meshgrid(x,x); =0;^(/1Mc  
    %       [theta,r] = cart2pol(X,Y); ?_I[,N?@41  
    %       idx = r<=1; 765p/**  
    %       z = nan(size(X)); 4.IU!.Uo  
    %       n = [0  1  1  2  2  2  3  3  3  3]; #> j.$2G>  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 6;|n]m\Vd  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; MNSbtT*^  
    %       y = zernfun(n,m,r(idx),theta(idx)); 2(/g}  
    %       figure('Units','normalized') 8T(e.I  
    %       for k = 1:10 LVJxn2x6  
    %           z(idx) = y(:,k); /="~gq@  
    %           subplot(4,7,Nplot(k)) E*jP87g  
    %           pcolor(x,x,z), shading interp JwJ7=P=c  
    %           set(gca,'XTick',[],'YTick',[]) To?W?s  
    %           axis square 3>Y 6)  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) V{<xf f  
    %       end ?(R]9.5S  
    % }<dRj  
    %   See also ZERNPOL, ZERNFUN2. q7"7U=W0  
    =+AS/Jq  
    92^w8Z.  
    %   Paul Fricker 11/13/2006 B, 9w0  
    ATR!7i\|  
    ij?  
    9;veuX#(  
    P3oI2\)*i  
    % Check and prepare the inputs: +"1NC\<*  
    % ----------------------------- 6oBfB8]:d  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) up'Tit  
        error('zernfun:NMvectors','N and M must be vectors.') %Q.&ZhB  
    end .jj$Kh q]  
    [o?* "c  
    e [8LmuIZ  
    if length(n)~=length(m) gCxAG  
        error('zernfun:NMlength','N and M must be the same length.') /tUy3myJ  
    end `\+@Fwfx  
    *V+j%^91}  
    Dq)j:f#QM  
    n = n(:); 7^g&)P  
    m = m(:); &B|D;|7H  
    if any(mod(n-m,2)) {c (!;U  
        error('zernfun:NMmultiplesof2', ... A,`8#-AX  
              'All N and M must differ by multiples of 2 (including 0).') {uHU]6d3qy  
    end $#]]K  
    7PkJ-JBA  
    Mb]rY>B4  
    if any(m>n) qM.bF&&Go  
        error('zernfun:MlessthanN', ... lv]hTH 4T  
              'Each M must be less than or equal to its corresponding N.') :hM/f  
    end 0C>%LJ8r  
    &-mX ,   
    !tp1:'KG  
    if any( r>1 | r<0 ) 8KRba4[  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 0lv %`,  
    end W16,Alf:  
    LU9A#  
    'z$Q rFW  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) HvVts\f  
        error('zernfun:RTHvector','R and THETA must be vectors.') 0A( +ZMd  
    end ;f"0~D2  
    $ >EYhLBa  
    X@f "-\  
    r = r(:); x l#LrvxI  
    theta = theta(:); D#o}cC.  
    length_r = length(r); 0q'w8]m  
    if length_r~=length(theta) SGe^ogO"v  
        error('zernfun:RTHlength', ... -UD\;D?$  
              'The number of R- and THETA-values must be equal.') YiPoYlD*n<  
    end 3.qTLga|}  
    [3!~PR]  
    4vwTs*eB `  
    % Check normalization: .<Zy|1 4  
    % -------------------- -*XCxU'  
    if nargin==5 && ischar(nflag) ]Ei0d8Uo  
        isnorm = strcmpi(nflag,'norm'); |Z*J/v'@p  
        if ~isnorm }|XtypbL  
            error('zernfun:normalization','Unrecognized normalization flag.') DrO2y  
        end +mp@b942*  
    else 9F*+YG!  
        isnorm = false; )'4k|@8|  
    end Mv6 -|O  
    TEaJG9RU>v  
    IzpZwx^3''  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1Tm^  
    % Compute the Zernike Polynomials Lg+G; W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <NuUW9+  
    oDU ;E  
    B}&xaY  
    % Determine the required powers of r: u6bXv(  
    % ----------------------------------- !H}vu]R  
    m_abs = abs(m); R@`y>XGNJ  
    rpowers = []; Q !(pE&  
    for j = 1:length(n) 7K5P8N ,  
        rpowers = [rpowers m_abs(j):2:n(j)]; )-`;1ca)s  
    end b%S62(qP  
    rpowers = unique(rpowers); 1hziXC0WY  
    'FS?a  
    :=[XW?L%x  
    % Pre-compute the values of r raised to the required powers, }~Af/  
    % and compile them in a matrix: &T}''  
    % ----------------------------- sn?]n~z  
    if rpowers(1)==0 WuZ/C_  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ''Cay0h  
        rpowern = cat(2,rpowern{:}); @!8ZPiW<  
        rpowern = [ones(length_r,1) rpowern]; ](^(=%  
    else ti<;7Yb  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); C,.Ee3T  
        rpowern = cat(2,rpowern{:}); !1G."fo  
    end ME=/|.}D<  
    oun;rMq  
    ?:L:EW8  
    % Compute the values of the polynomials: qvv2O1c"A  
    % -------------------------------------- = hN !;7G  
    y = zeros(length_r,length(n)); B0ndcB-  
    for j = 1:length(n) R?p00  
        s = 0:(n(j)-m_abs(j))/2; xQ'2BAEa  
        pows = n(j):-2:m_abs(j); P:N1#|g  
        for k = length(s):-1:1 HuV J\%.  
            p = (1-2*mod(s(k),2))* ... s$a09x  
                       prod(2:(n(j)-s(k)))/              ... U_{Ux 2  
                       prod(2:s(k))/                     ... MG{YrX)oi  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... "^1L'4'S  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); pm9%%M$  
            idx = (pows(k)==rpowers); V}zEK0n(6  
            y(:,j) = y(:,j) + p*rpowern(:,idx); jr3ti>,xV  
        end &c*^VL\  
         jr`Ess  
        if isnorm 6HlePTf8  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Usta0Ag  
        end b?j< BvQ  
    end %bdjBa}  
    % END: Compute the Zernike Polynomials ?Sb8@S&J  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0]jA<vLR  
    o#hjvg  
    d%0~c'D8a  
    % Compute the Zernike functions: vC5n[0  
    % ------------------------------ 5A4&+rdU  
    idx_pos = m>0; Y9`5G%  
    idx_neg = m<0; $/7pYl\n  
    pm6>_Kz  
    :Pv*, qHE  
    z = y; c-Pw]Ju  
    if any(idx_pos) c?%(Dp E  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Dxk+P!!K  
    end ykFJ%sw3X  
    if any(idx_neg) Z*FrB58  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); %b^OeWip  
    end 1NcCy! +  
    q@jq0D)g  
    U5 r7j  
    % EOF zernfun o^V(U~m]  
     
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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)  MP>n)!R[`  
    k8]O65t|  
    DDE还是手动输入的呢? Wn|&cG9  
    +1 eCvt:,  
    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