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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, ;lldxS  
    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, &q+ %OPV  
    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? )xU70:X  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? (1R,   
    &fWZ%C7|jC  
    WA+v&* ]  
    *|cvx:GO  
    );C !:?  
    function z = zernfun(n,m,r,theta,nflag) MLJ8m  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 59LIK&w  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N #"jWPe,d  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Q"\[ICu!,  
    %   unit circle.  N is a vector of positive integers (including 0), and t}K?.To$  
    %   M is a vector with the same number of elements as N.  Each element SU1, +7"  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) HV>Wf"1  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, MTQdyTDHl  
    %   and THETA is a vector of angles.  R and THETA must have the same /[mCK3_  
    %   length.  The output Z is a matrix with one column for every (N,M) \J6T:jeS,  
    %   pair, and one row for every (R,THETA) pair. ky*-_  
    % dM)fr  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike H7WKnn@  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), RNPqW,B!0  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 5s0H4?S  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, n '&WIf3  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ?x:\RNB/  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. n1v%S"^  
    % 7XZ!UC;i  
    %   The Zernike functions are an orthogonal basis on the unit circle. |_-FQ~Hf F  
    %   They are used in disciplines such as astronomy, optics, and yjr!8L:m  
    %   optometry to describe functions on a circular domain. >_R5Li  
    % !j- 7,  
    %   The following table lists the first 15 Zernike functions. :R _(+EK1  
    % 0 {w?u%'  
    %       n    m    Zernike function           Normalization 1w35 H9\g  
    %       -------------------------------------------------- W}KtB1J  
    %       0    0    1                                 1 QkA79%;j  
    %       1    1    r * cos(theta)                    2 D:f0W v  
    %       1   -1    r * sin(theta)                    2 K'y;j~`-  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) )@Ly{cw   
    %       2    0    (2*r^2 - 1)                    sqrt(3) CFVe0!\  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) G|.>p<q   
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) $U<xrN>O  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) z"R-Sme  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) I#m5Tl|#  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) =6/0=a[  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) .aF+>#V=Q  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 8J Gt|,  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Cdc6<8  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Uq7 y4zJ  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) m"NZ;*d'  
    %       -------------------------------------------------- ><dSwwu  
    % OLlNCb#t  
    %   Example 1: <kt,aMw[*  
    % z6$W@-Vd  
    %       % Display the Zernike function Z(n=5,m=1) :FB#,AOa_  
    %       x = -1:0.01:1; ]7Tjt A.\q  
    %       [X,Y] = meshgrid(x,x); ](:aDHa  
    %       [theta,r] = cart2pol(X,Y); Uk5jZ|  
    %       idx = r<=1; j $a,93P5  
    %       z = nan(size(X)); 7$k[cL1  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); sd xl@  
    %       figure k?KKb /&b  
    %       pcolor(x,x,z), shading interp L@XhgQ  
    %       axis square, colorbar Yu`b[]W  
    %       title('Zernike function Z_5^1(r,\theta)') Rcfh*"k  
    % H"6Sj-<=  
    %   Example 2: QD-#sU]  
    % XzIhFX6  
    %       % Display the first 10 Zernike functions ggIz) </  
    %       x = -1:0.01:1; );'8*e'  
    %       [X,Y] = meshgrid(x,x); Tn8Z2iC  
    %       [theta,r] = cart2pol(X,Y); )=8MO-{  
    %       idx = r<=1; ]^uO3!+  
    %       z = nan(size(X)); l 2y_Nz-;  
    %       n = [0  1  1  2  2  2  3  3  3  3]; |MY6vRJ(  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; O|}97a^  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; k.NgE/;3  
    %       y = zernfun(n,m,r(idx),theta(idx)); IDyf9Zra?  
    %       figure('Units','normalized') "hdc B 0  
    %       for k = 1:10 18jI6$DY  
    %           z(idx) = y(:,k); >LRt,.hy6  
    %           subplot(4,7,Nplot(k)) :''^a  
    %           pcolor(x,x,z), shading interp m_wBRan  
    %           set(gca,'XTick',[],'YTick',[]) n(\5Z&  
    %           axis square E=+v1\t)]  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) <E[X-S%&  
    %       end *"2TT})   
    % sg RY`U.C  
    %   See also ZERNPOL, ZERNFUN2. b`)^Ao:  
    N&n2\Y  
    I@76ABu^  
    %   Paul Fricker 11/13/2006 (sSMH6iCif  
    )/A IfH  
    t>:2F,0K9  
    C(qqGK{  
    ~_OtbNj#  
    % Check and prepare the inputs: &_n~#Mex  
    % ----------------------------- <iDqt5)N  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 4RTuy+ M  
        error('zernfun:NMvectors','N and M must be vectors.') </(bwc~2  
    end of!Bz  
    cPZD#";f  
    v0&E!4q*'  
    if length(n)~=length(m) :f<3`x'  
        error('zernfun:NMlength','N and M must be the same length.') P]hS0,sE<(  
    end `],'fT|,S  
    8T6.Zhv  
    Op%}.9ed  
    n = n(:); {fW(e?8)  
    m = m(:); E(N?.i-%$  
    if any(mod(n-m,2)) !l-^JPb  
        error('zernfun:NMmultiplesof2', ... ?UuJk  
              'All N and M must differ by multiples of 2 (including 0).') 2YI#J.6]H  
    end 8:E)GhX  
    b.V\E Ok  
    jp?;8rS3  
    if any(m>n) T5(]/v,UT  
        error('zernfun:MlessthanN', ... R%B"Gtl)  
              'Each M must be less than or equal to its corresponding N.') %5.aC|^}  
    end XG2&_u&  
    Y?G\@ 6  
    B@XnHh5y  
    if any( r>1 | r<0 ) ,w&8 &wj  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') c@H:?s!0R  
    end KKpO<TO  
    Ct2m l  
    ) in hPd  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ODa+s>a`^  
        error('zernfun:RTHvector','R and THETA must be vectors.') wi]ya\(*yl  
    end 3{OY&   
    r;m_@*]  
    x|C[yu^c  
    r = r(:); qOOF]L9r%u  
    theta = theta(:); c 4Q{  
    length_r = length(r); a![x^@nF  
    if length_r~=length(theta) (3PkTQlE  
        error('zernfun:RTHlength', ... "f/91gIzm'  
              'The number of R- and THETA-values must be equal.') 6~g`B<(?  
    end ?M?S+@(  
    $ qOV#,@  
    '@OqWdaR  
    % Check normalization: 7u8HcHl  
    % -------------------- " o.V`Bj  
    if nargin==5 && ischar(nflag) 8/lv,m#  
        isnorm = strcmpi(nflag,'norm'); 9gFb=&1k  
        if ~isnorm LS1r}cl  
            error('zernfun:normalization','Unrecognized normalization flag.') iEd%8 F h  
        end 2p'ujAK  
    else {c5%.<O  
        isnorm = false; #m 2Ss  
    end i"|="O0v5  
    |KSd@   
    R7axm<PR=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ut"~I)S{LT  
    % Compute the Zernike Polynomials $r0~& $T&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "XQj ~L  
    dMkDNaH,  
    rzmd`)g  
    % Determine the required powers of r: a3}#lY):  
    % ----------------------------------- |M&i#g<A;  
    m_abs = abs(m); Vy*&po[   
    rpowers = []; 5:[<pY!s#  
    for j = 1:length(n) ki/xo^Y2<  
        rpowers = [rpowers m_abs(j):2:n(j)]; V/%tFd1  
    end 00s&<EM  
    rpowers = unique(rpowers); 2 de[ yz  
    #'"zyidu  
    GJlkEWs  
    % Pre-compute the values of r raised to the required powers, }~gBnq_DDU  
    % and compile them in a matrix: L0ZgxG3:g  
    % ----------------------------- ~~J xw ]  
    if rpowers(1)==0 rKZ1 c,y  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); WSA;p=_  
        rpowern = cat(2,rpowern{:}); \ ) H}  
        rpowern = [ones(length_r,1) rpowern]; T(UdV]~]"  
    else z=ItKoM*<  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9rhIDA(wc  
        rpowern = cat(2,rpowern{:}); c,WRgXL  
    end 3@u<Sa  
    {P'TtlEp  
    ;^QG>OP$  
    % Compute the values of the polynomials:  XL@Y!  
    % -------------------------------------- "YIrqk  
    y = zeros(length_r,length(n)); ?~G D^F  
    for j = 1:length(n) zk)9tm;i{  
        s = 0:(n(j)-m_abs(j))/2; dQhh,}  
        pows = n(j):-2:m_abs(j); hVvPI1[2  
        for k = length(s):-1:1 pz'l9Gp;@  
            p = (1-2*mod(s(k),2))* ... ;Dl< GW3<  
                       prod(2:(n(j)-s(k)))/              ... OC0dAxq  
                       prod(2:s(k))/                     ... FmU>q)  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... e_Cns&  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Dx<">4   
            idx = (pows(k)==rpowers);  VlGg?  
            y(:,j) = y(:,j) + p*rpowern(:,idx); hg8gB8Xq  
        end Z<j(ZVO  
         fC!]MhA"i  
        if isnorm <28L\pdG`  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); RI,Z&kXj2o  
        end JE~ci#|!  
    end OKDBzl  
    % END: Compute the Zernike Polynomials 3:q\]]]S  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% JryCL]  
    L%D:gy9o  
    5o2W[<%v  
    % Compute the Zernike functions: aBV{Xr~#(  
    % ------------------------------ d,"?tip/SX  
    idx_pos = m>0; 4J lB\8rc  
    idx_neg = m<0; vo'=d"zm  
    JXR_klx  
    ] i;xeo,  
    z = y; J{98x zb  
    if any(idx_pos) JaC =\\B  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); &p\fdR4e  
    end +-=o16*{ !  
    if any(idx_neg) fX)C8J^=G  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 3`9H  
    end XqD/~_z;  
    76i)m!  
    =ytB\e  
    % EOF zernfun I?sA)!8  
     
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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)  J<x?bIetj  
    $,ev <4I&  
    DDE还是手动输入的呢? }B2H)dG^K  
    8cn)ox|J[  
    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