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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, AJ)N?s-=  
    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, aIklAj)=  
    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? drh,=M\F  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? s|-g)  
    b%|6y  
    Wo<kKkx2  
    ZG1 {"J/z  
    \v p^[,SI  
    function z = zernfun(n,m,r,theta,nflag) )C%S`d<%,  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ANXN.V  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 0{sYD*gK]  
    %   and angular frequency M, evaluated at positions (R,THETA) on the uAv'%/  
    %   unit circle.  N is a vector of positive integers (including 0), and !sav~dB)  
    %   M is a vector with the same number of elements as N.  Each element >on' y+  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) V;1i/{  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, MFs W  
    %   and THETA is a vector of angles.  R and THETA must have the same a\Dw*h?b~  
    %   length.  The output Z is a matrix with one column for every (N,M) {#H'K*j{  
    %   pair, and one row for every (R,THETA) pair. tnFhL&  
    % !E9A=u{  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike c$~J7e6$  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), f}{Oj-:"CC  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral -ZBSkyMGy  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ?CZ*MMV  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Pc=:j(  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. "Sd2VSLg  
    % BnIZ+fg=  
    %   The Zernike functions are an orthogonal basis on the unit circle. `&>CK`%Xu  
    %   They are used in disciplines such as astronomy, optics, and m'5rzZP  
    %   optometry to describe functions on a circular domain. J3AS"+]  
    % 2jH&@g$cl;  
    %   The following table lists the first 15 Zernike functions. $jL+15^N0+  
    % 0A.9<&Lod  
    %       n    m    Zernike function           Normalization VMV~K7%0  
    %       -------------------------------------------------- bb"x^DtT  
    %       0    0    1                                 1 a~O](/+p;  
    %       1    1    r * cos(theta)                    2  y jY}o  
    %       1   -1    r * sin(theta)                    2 JURJN+)z  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) N="H 06t  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Rb_+C  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) I>45xVA  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) mY/x|)MmM  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) h/\/dp/tt  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) <!I^xo [  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) vAo|o *  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) c9axzg UA  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >}>cJh6  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Xsv^GmP+  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (vr v-4  
    %       4    4    r^4 * sin(4*theta)             sqrt(10)  hPgDK.R'  
    %       -------------------------------------------------- $_b^p=  
    % ~Is-^k)y  
    %   Example 1: * 2s(TW  
    % ^%2S,3*0  
    %       % Display the Zernike function Z(n=5,m=1) 6yPh0n  
    %       x = -1:0.01:1; i`HXBq!|w  
    %       [X,Y] = meshgrid(x,x); xgv&M:%D-  
    %       [theta,r] = cart2pol(X,Y); oM)4""|  
    %       idx = r<=1; yB,{:kq7D  
    %       z = nan(size(X)); IL N0/eH  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); rdQ'#}I x  
    %       figure Vh;P,no#  
    %       pcolor(x,x,z), shading interp O7GJg;>?  
    %       axis square, colorbar y|[YEY U)  
    %       title('Zernike function Z_5^1(r,\theta)') O5?3 nYHa  
    % %!QY:[   
    %   Example 2: </7_T<He.  
    % :h60  
    %       % Display the first 10 Zernike functions `]\:%+-  
    %       x = -1:0.01:1; #\r5Q>  
    %       [X,Y] = meshgrid(x,x); 0@*EwI  
    %       [theta,r] = cart2pol(X,Y); _ ^cFdP)8|  
    %       idx = r<=1; pG9qD2C f  
    %       z = nan(size(X));  R7-+@  
    %       n = [0  1  1  2  2  2  3  3  3  3]; #ysSfM6  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; g7nqe~`{  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Zi~-m]9U  
    %       y = zernfun(n,m,r(idx),theta(idx)); u[ 2B0a  
    %       figure('Units','normalized') k&8&D  
    %       for k = 1:10 3 tIno!|  
    %           z(idx) = y(:,k); b<?A  
    %           subplot(4,7,Nplot(k)) qLh[BR  
    %           pcolor(x,x,z), shading interp cp g+-Zf%  
    %           set(gca,'XTick',[],'YTick',[]) 4Hcds9y9  
    %           axis square IL2OVLX  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) &[iunJv:eq  
    %       end NamO5(1C  
    % (&t8.7O  
    %   See also ZERNPOL, ZERNFUN2. WjsE#9D!of  
    ;H:+w\?8f$  
    O9(6?n  
    %   Paul Fricker 11/13/2006 "=ogO/_Q"  
    <764|q  
    h|S6LgB  
    FR9*WI   
    '}eA2Q>BV  
    % Check and prepare the inputs: Q( \2(x\  
    % ----------------------------- Zn9ecN  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~*,e&I  
        error('zernfun:NMvectors','N and M must be vectors.') ss>p  
    end <fm0B3i?  
    H(k-jAO,  
    ?g\SF}2  
    if length(n)~=length(m) H[KTM'n  
        error('zernfun:NMlength','N and M must be the same length.') =ijVT_|u0  
    end (D 5.NB%@  
    Gv uX"J  
    / %:%la%  
    n = n(:); c (Gl3^  
    m = m(:); Jg\1(ix  
    if any(mod(n-m,2)) EM&;SQ;C9  
        error('zernfun:NMmultiplesof2', ... KJ&~z? X  
              'All N and M must differ by multiples of 2 (including 0).') jWL;ElM'  
    end ?}g#Mc  
    |w7D&p$  
    tQ > IJ  
    if any(m>n) ;YK{[$F  
        error('zernfun:MlessthanN', ... ehCZhi~  
              'Each M must be less than or equal to its corresponding N.') Hg}@2n)/  
    end +GqV9x 8  
    7 ,![oY[  
    (#"iZv,  
    if any( r>1 | r<0 ) jJfV_#'N'  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') M~/R1\'&j  
    end MH8Selnv  
    _x ;fTW0  
    b=-LQkcZhK  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) qIIl,!&}A  
        error('zernfun:RTHvector','R and THETA must be vectors.') hz8Z)xjJ V  
    end lh?TEQ  
    > l@ o\  
    D>~S-]  
    r = r(:); cA8"Ft{P)  
    theta = theta(:); qr~= S  
    length_r = length(r); ~>]/1JFz  
    if length_r~=length(theta) c[xH:$G?Y  
        error('zernfun:RTHlength', ... k}o*=s>M  
              'The number of R- and THETA-values must be equal.') d].(x)|st  
    end [8J/# !B  
    T)QT_ST.9  
    ~dO&e=6Hk  
    % Check normalization: *`HE$k!  
    % -------------------- F;&a=R!.  
    if nargin==5 && ischar(nflag) `?PpzDV7Y  
        isnorm = strcmpi(nflag,'norm'); 1*>lYd8 _  
        if ~isnorm Pd[&&!+gV  
            error('zernfun:normalization','Unrecognized normalization flag.') QNzx(IV@  
        end <&$:$_ah  
    else D`G ;kp  
        isnorm = false; cI Byv I-  
    end l"-F<^ U  
    IO4 8sV }  
    ct3^V M&/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9W[ ~c"Ku  
    % Compute the Zernike Polynomials ;1&7v  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% du:%{4  
    #el i_Cxe  
    vV?=r5j  
    % Determine the required powers of r:  !AGjiP$  
    % ----------------------------------- X~Yj#@  
    m_abs = abs(m); ,X2CV INb}  
    rpowers = []; %Z"I=;=nxI  
    for j = 1:length(n) dt efDsK  
        rpowers = [rpowers m_abs(j):2:n(j)]; dIUg e`O9  
    end MJ )aY2  
    rpowers = unique(rpowers); 9mT;> mE  
    /4R|QD  
    :]viLw\&g  
    % Pre-compute the values of r raised to the required powers, $ 4& )  
    % and compile them in a matrix: hu G]kv3F:  
    % ----------------------------- BZP~m=kq  
    if rpowers(1)==0 -PI_ *  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); =nmvG%.hd  
        rpowern = cat(2,rpowern{:}); i8tH0w/(M  
        rpowern = [ones(length_r,1) rpowern]; o$=D`B  
    else ?1f(@  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7|"gMw/  
        rpowern = cat(2,rpowern{:}); >c~ Fg s  
    end HZ#<+~J  
    Wn9b</ tf  
    5GP,J,J  
    % Compute the values of the polynomials: qOV6Kh)  
    % -------------------------------------- z8ox#+l  
    y = zeros(length_r,length(n)); zuR F6?un  
    for j = 1:length(n) #Zm%U_$<  
        s = 0:(n(j)-m_abs(j))/2; P7||d@VW,  
        pows = n(j):-2:m_abs(j); "2}E ARa  
        for k = length(s):-1:1 L b'HM-d  
            p = (1-2*mod(s(k),2))* ... ~;f,Ad`Q  
                       prod(2:(n(j)-s(k)))/              ... !]W}I  
                       prod(2:s(k))/                     ... Ier0F7]I  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... d0`5zd@S  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); RSNukg  
            idx = (pows(k)==rpowers); bOi`JJ^   
            y(:,j) = y(:,j) + p*rpowern(:,idx); &s|&cT  
        end Z"# /,?|3@  
         GTw3rD^wg  
        if isnorm "v"w ER?  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); bGl5=`  
        end y8$TU;  
    end j&G*$/lTO6  
    % END: Compute the Zernike Polynomials oM=Ltxv}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >lo,0oG  
    kT!Y~c  
    M-Az2x;6  
    % Compute the Zernike functions: )V}u}5  
    % ------------------------------ DL^}?Ve  
    idx_pos = m>0; L y!!+UM\  
    idx_neg = m<0; KT)A{i  
    H$ !78/f  
    ;+dB-g[  
    z = y; f$lf(brQ:  
    if any(idx_pos) f?iQ0wv)  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ;_yp@.,\T  
    end 9` /\|t|V  
    if any(idx_neg) t\hvhcbL  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); PQmgv&!DP  
    end z;dD }Fo  
    X]?qns7  
    vGK'U*gGD  
    % EOF zernfun (f^K\7HM  
     
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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)  $FEG0&  
    y?W8FL  
    DDE还是手动输入的呢? CMa~BOt#  
    ,mH2S/<}S  
    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