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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, IN/$b^Um  
    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%"5?)[z  
    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? [C EV&B  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? >8/Otg+h  
    - G>J  
    bqH [-mu6  
    B!mHO*g  
    j)/Vtf  
    function z = zernfun(n,m,r,theta,nflag) pmP~1=3  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. V(Pw|u" e  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N K\ Wzh;  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 5 Y&`ZJ  
    %   unit circle.  N is a vector of positive integers (including 0), and T"P}`mT  
    %   M is a vector with the same number of elements as N.  Each element 9X*Z\-  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) X_'tgP9  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, r ??_2>Q  
    %   and THETA is a vector of angles.  R and THETA must have the same O^\:J 2I(  
    %   length.  The output Z is a matrix with one column for every (N,M) /\Nc6Z/ L  
    %   pair, and one row for every (R,THETA) pair. Xc NL\fl1  
    % g5#LoGc  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike gH7  +#/  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), DSHvBFQ  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral n`^jNXE  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Xj]9/?B?  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized / ^)3V}  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. (P? |Bk [  
    % :sw5@JdJ  
    %   The Zernike functions are an orthogonal basis on the unit circle. *i*\ dl  
    %   They are used in disciplines such as astronomy, optics, and *JImP9SE  
    %   optometry to describe functions on a circular domain. 3]1 ! g6  
    % +E9G"Z65iP  
    %   The following table lists the first 15 Zernike functions. V^tD@N  
    % Oa:C'M b  
    %       n    m    Zernike function           Normalization  gwIR3u  
    %       -------------------------------------------------- ]?_~QE`  
    %       0    0    1                                 1 .}F 39TS2  
    %       1    1    r * cos(theta)                    2 \ o2oQ3  
    %       1   -1    r * sin(theta)                    2 nN$.^!;&  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) L[44D6Vg  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ~I N g9|  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) $|Ol?s  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) [BdRx`  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) o.Ww .F  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) fwUvFK1G  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) j+'ua=T3  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) M p <r`PM2  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) F ]X<q uuL  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) [3=Y 9P:  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ma\%uEgTD  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) q;R&valn  
    %       -------------------------------------------------- b`%u}^B {  
    % 'r=2f6G>cP  
    %   Example 1: Wk^{Tn/]  
    % {_W8Qm`.  
    %       % Display the Zernike function Z(n=5,m=1) :!Z|_y{b  
    %       x = -1:0.01:1; fph+ 05.%  
    %       [X,Y] = meshgrid(x,x); nv0D4 t  
    %       [theta,r] = cart2pol(X,Y); \aPH_sf,  
    %       idx = r<=1; Gfx !.[Y  
    %       z = nan(size(X)); bkR~>F]FAu  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); F%zMhX'AG  
    %       figure P;(@"gD8z5  
    %       pcolor(x,x,z), shading interp <9H3d7%  
    %       axis square, colorbar s8:epcL`A  
    %       title('Zernike function Z_5^1(r,\theta)') yU(}1ZID  
    % DNDzK iMk  
    %   Example 2: _Cf:\Xs m  
    % k"7ZA>5jk  
    %       % Display the first 10 Zernike functions c{`!$Z'k<  
    %       x = -1:0.01:1; kqZRg>1A  
    %       [X,Y] = meshgrid(x,x); UazK0{t<f  
    %       [theta,r] = cart2pol(X,Y); [e\IHakj  
    %       idx = r<=1; )Dms9:  
    %       z = nan(size(X)); @lM-+q(tl  
    %       n = [0  1  1  2  2  2  3  3  3  3];  \aof  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; /qKor;x  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; rVhfj~Ts  
    %       y = zernfun(n,m,r(idx),theta(idx)); `"'u mIz  
    %       figure('Units','normalized') [V /f{y~ {  
    %       for k = 1:10 ;L",K?6#  
    %           z(idx) = y(:,k); i \Yd_  
    %           subplot(4,7,Nplot(k)) +5-|6  
    %           pcolor(x,x,z), shading interp F~fN7<9R  
    %           set(gca,'XTick',[],'YTick',[]) S_*Gv O  
    %           axis square _nzTd\L88  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) l' Li!u  
    %       end  3bd`q $  
    % vx6lud0k}  
    %   See also ZERNPOL, ZERNFUN2. vnf2Z,f%  
    O)R}|  
    TqS s*as5  
    %   Paul Fricker 11/13/2006 08AD~^^  
    TSJeS`I  
    1foG*   
    7C Sn79E  
    C_ ;nlG6  
    % Check and prepare the inputs: Y1AZ%{^0a  
    % ----------------------------- hb0)<^xu  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) *E>R1bJ8  
        error('zernfun:NMvectors','N and M must be vectors.') y] 9/Xr/  
    end P1L+Vnfu  
    FwKY;^`!d  
    ZLVgK@l  
    if length(n)~=length(m) 1H%p|'FKA  
        error('zernfun:NMlength','N and M must be the same length.') ,[N(XstI  
    end Z9h4 pd  
    o3GZcH?  
    usKP9[T$  
    n = n(:); /EHO(d!<  
    m = m(:); st.{AEv@  
    if any(mod(n-m,2)) 9 M?UPE  
        error('zernfun:NMmultiplesof2', ... ~[aV\r?  
              'All N and M must differ by multiples of 2 (including 0).') x~m$(LT  
    end eC 2~&:$L  
    Gys-Im6>~@  
    7[L%j;)bw  
    if any(m>n) m'G=WO*%  
        error('zernfun:MlessthanN', ... uARkf'  
              'Each M must be less than or equal to its corresponding N.') fMHw=wJQ  
    end EHl~y=9  
    WcXNc`x  
    5KTFf6Uq  
    if any( r>1 | r<0 ) grI#'x  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') l7<VHz0b  
    end &_@M 6[-  
    ^G5fs'd  
    5&A' +]  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) "9X(.v0ze  
        error('zernfun:RTHvector','R and THETA must be vectors.') DP*$@5  
    end .;U?%t_7  
    5yJ~ q  
    I@Y k &aU  
    r = r(:); *Br }U  
    theta = theta(:); s/3sOb}sA  
    length_r = length(r); q)@;8Z=_c  
    if length_r~=length(theta) Gw6Od j  
        error('zernfun:RTHlength', ... 'UGgY3  
              'The number of R- and THETA-values must be equal.') wsR\qq  
    end -nD} k  
    =_6 Q26  
    9qzHy}A  
    % Check normalization: JvCy&xrE;  
    % -------------------- F7=\*U  
    if nargin==5 && ischar(nflag) tmeg=U7  
        isnorm = strcmpi(nflag,'norm'); !6#.%"{-  
        if ~isnorm 9Ns%<FRO@  
            error('zernfun:normalization','Unrecognized normalization flag.') zT!.5qd  
        end ?}uvpB1}  
    else *y+K{ fM1  
        isnorm = false; PVN`k, 4  
    end HFYe@2r  
    2]x,joB  
    f*m^x7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5yW}#W>  
    % Compute the Zernike Polynomials gId :IR  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,>kXn1 ,  
    c*(=Glzn  
    !0Ak)Q]e'  
    % Determine the required powers of r: gA 5DEit  
    % ----------------------------------- e-xT.RnQ  
    m_abs = abs(m); O|9Nl*rXz  
    rpowers = []; xkkG#n)  
    for j = 1:length(n) 96gaun J  
        rpowers = [rpowers m_abs(j):2:n(j)]; O!F"w !5@  
    end ^Y8G}Z|  
    rpowers = unique(rpowers); i!<(R$ Lo  
    a94 nB  
    ^R+CkF4l l  
    % Pre-compute the values of r raised to the required powers,  -l"8L;`  
    % and compile them in a matrix: (f* r  
    % ----------------------------- q#8z%/~k  
    if rpowers(1)==0 sI#h&V,9  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?Qpi(Czbpq  
        rpowern = cat(2,rpowern{:}); S!iDPl~  
        rpowern = [ones(length_r,1) rpowern]; {-|El}.M  
    else #TgP:t]p  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 5["n] i  
        rpowern = cat(2,rpowern{:}); N B8Yn\{B  
    end &k(tDP  
    y7z ,I  
    1bCS4fs^>  
    % Compute the values of the polynomials: \x_$Pu  
    % -------------------------------------- UyMlk  
    y = zeros(length_r,length(n)); K$$%j"s  
    for j = 1:length(n) ]go.IfH  
        s = 0:(n(j)-m_abs(j))/2; 'E\qqE[;  
        pows = n(j):-2:m_abs(j); tU8aPiUl  
        for k = length(s):-1:1 X(]J\?n'  
            p = (1-2*mod(s(k),2))* ... E_xk8X~  
                       prod(2:(n(j)-s(k)))/              ... xf qu=z8X  
                       prod(2:s(k))/                     ... s21)*d  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... GlT/JZ9  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); /DSy/p0%  
            idx = (pows(k)==rpowers); 7l'1  
            y(:,j) = y(:,j) + p*rpowern(:,idx); kPnuU!  
        end Z~"8C Kz  
         oTpoh]|[  
        if isnorm s%N6^}N  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); pTYV@5|  
        end ;s-fYS6(>{  
    end A&Q!W)=  
    % END: Compute the Zernike Polynomials S.owVMQ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r+MqjdXG  
    ZgzYXh2  
    }sfv zw_  
    % Compute the Zernike functions: DH4|lb}  
    % ------------------------------ m&Y?]nbq  
    idx_pos = m>0; &([Gc+"5E.  
    idx_neg = m<0; ( "J_< p  
    %S%0/  
    y$?O0S%F  
    z = y;  Z Mf,3  
    if any(idx_pos) NB&zBJ#  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); TyaK_XW  
    end & y7~  
    if any(idx_neg) JaJyH%+$!  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); PO0/C q)  
    end Sr6?^>A@t  
    CDFkH  
    Dr#V^"Dte  
    % EOF zernfun u$1^=  
     
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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)  +\`t@Ht#  
    t%>x}b"2T  
    DDE还是手动输入的呢? 4\EvJg@Z.  
    Ol. rjz9  
    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