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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, Wkg*J3O  
    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, |{7e#ww]  
    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? Q\o$**+{  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? u>,lf\Fgz  
    g,n-s+  
    om".j  
    DOF?(:8Y  
    [j:}=:feQ  
    function z = zernfun(n,m,r,theta,nflag) VMxYZkMNd_  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ){O1&|z-  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N i!SW?\  
    %   and angular frequency M, evaluated at positions (R,THETA) on the FylWbQU9  
    %   unit circle.  N is a vector of positive integers (including 0), and I;kf #nvao  
    %   M is a vector with the same number of elements as N.  Each element pAJ=f}",]E  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) M>?aa6@0  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, k_*XJ<S!Y  
    %   and THETA is a vector of angles.  R and THETA must have the same ~:/%/-^  
    %   length.  The output Z is a matrix with one column for every (N,M) ilDJwZg#  
    %   pair, and one row for every (R,THETA) pair. ->&BcPLn  
    % Xzx[C_G  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Yl)eh(\&J  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), TnN^2:cU  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral (j8GiJ]{L,  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Ud>`@2  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized (MgL"8TS  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. tk`: CT *  
    % F-$Z,Q]S  
    %   The Zernike functions are an orthogonal basis on the unit circle. xZ^ywa_  
    %   They are used in disciplines such as astronomy, optics, and ?vZWUWa  
    %   optometry to describe functions on a circular domain. 7XUhJN3n  
    % V~'k1P4  
    %   The following table lists the first 15 Zernike functions. -d|BO[4j  
    % ?-pxte8  
    %       n    m    Zernike function           Normalization 9"WRIHt'c  
    %       -------------------------------------------------- a);O3N/*I  
    %       0    0    1                                 1 " t5 +*  
    %       1    1    r * cos(theta)                    2 ]{K5zSK  
    %       1   -1    r * sin(theta)                    2 (g%JK3  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 8s QQK.N(  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ltNuLZ  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) McT\ R{/  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Rz`@N`U  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) J*}VV9H  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) v$t{o{3  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) m3U+ du  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Xy[}Gp  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?D1x;i9<  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) `[X6#` <  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) c*.G]nRc  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) SW3wMPy&s  
    %       -------------------------------------------------- &[NVP&9&U  
    % /t$rX3A  
    %   Example 1: P-[fHCg~  
    % L&:M8xiA~$  
    %       % Display the Zernike function Z(n=5,m=1)  &|/vM.  
    %       x = -1:0.01:1; !c\7  
    %       [X,Y] = meshgrid(x,x); &@=u+)^-{  
    %       [theta,r] = cart2pol(X,Y); PASuf.U$"  
    %       idx = r<=1; ?O!]8k`1$  
    %       z = nan(size(X)); W=~id"XtJ  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); L5R `w&Up  
    %       figure K1;z Mh  
    %       pcolor(x,x,z), shading interp La\Q'0  
    %       axis square, colorbar Mx^y>\X)v  
    %       title('Zernike function Z_5^1(r,\theta)') vkd *ER^  
    % Er`TryN|}  
    %   Example 2: XQK^$Iq]V  
    % $X`bm*  
    %       % Display the first 10 Zernike functions _i-\mR_~  
    %       x = -1:0.01:1; 1W*V2`0>  
    %       [X,Y] = meshgrid(x,x); Z/xV\Ggx  
    %       [theta,r] = cart2pol(X,Y); w-J"zC  
    %       idx = r<=1; a4%`"  
    %       z = nan(size(X)); 5 RW@_%C  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ex.+'m<g  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; -y%QRO(  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; v,n);  
    %       y = zernfun(n,m,r(idx),theta(idx)); }|AX_=a  
    %       figure('Units','normalized') 6e*%\2UA  
    %       for k = 1:10 % =y;L:S\p  
    %           z(idx) = y(:,k); (viWY  
    %           subplot(4,7,Nplot(k)) {!lNL[x  
    %           pcolor(x,x,z), shading interp  dFzYOG1  
    %           set(gca,'XTick',[],'YTick',[]) !zU/Hq{wcK  
    %           axis square HHZ`%  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) b~1iPaIh  
    %       end %z30=?VL  
    % u',b1 3g(  
    %   See also ZERNPOL, ZERNFUN2. %yeu"  
    \e_IFISC  
    @]*[c})/  
    %   Paul Fricker 11/13/2006 1bd$XnU  
    kPWBDpzN  
    $Tfm/=e  
    Qy/uB$q{A  
    L,#^&9bHa#  
    % Check and prepare the inputs: YDW|-HIF  
    % ----------------------------- NJk)z&M  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ;r3}g"D@  
        error('zernfun:NMvectors','N and M must be vectors.') (9E( Q*J5x  
    end lHcA j{6  
    >$Fp}?xX  
    f#1/}Hq/I  
    if length(n)~=length(m) [8.-(-/;  
        error('zernfun:NMlength','N and M must be the same length.') V- /YNRV  
    end XJc ,uj7  
    ,}KwP*:Z  
    5lGQ#r  
    n = n(:); <Kg2$lu(_`  
    m = m(:); >}CEN  
    if any(mod(n-m,2)) >[EBpYi  
        error('zernfun:NMmultiplesof2', ... Cpe#[mE  
              'All N and M must differ by multiples of 2 (including 0).') W8y$ Ve8m  
    end @' d6iYk_  
    7H4L-J3  
    +^Fp&K+^  
    if any(m>n) 7N| AA^I  
        error('zernfun:MlessthanN', ... &Bm&i.r  
              'Each M must be less than or equal to its corresponding N.') -;vT<G3  
    end Y KY2Cw  
    mf$Sa58  
    Oo1ecbY  
    if any( r>1 | r<0 ) g>_OuQ|c  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') P<vo;96JT  
    end ;I+H>$%jZ  
    0iV;g`%  
    \X5 3|Y;=  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 3-iD.IAUm@  
        error('zernfun:RTHvector','R and THETA must be vectors.') !j0_ cA  
    end TU%bOAKF\  
    (vnoP< 0  
    #~S>K3(  
    r = r(:); =HS4I.@c_5  
    theta = theta(:); \ADLMj`F|  
    length_r = length(r); T{tn.sT  
    if length_r~=length(theta) Q(e{~ ]*  
        error('zernfun:RTHlength', ... eIJ[0c b}  
              'The number of R- and THETA-values must be equal.') ioWo ]  
    end ^&NN]?  
    $it@>L8  
    ^&MK42,\  
    % Check normalization: *7Xzht&f  
    % -------------------- xG1?F_]  
    if nargin==5 && ischar(nflag) o0l7 4  
        isnorm = strcmpi(nflag,'norm'); o<rsAe  
        if ~isnorm n[P\*S  
            error('zernfun:normalization','Unrecognized normalization flag.') Im+ 7<3Z  
        end j`9Qzi1  
    else 7h`^N5H.q  
        isnorm = false; P$OUi!"  
    end m]P/if7  
    eF5;[v  
    _C$X04bU3V  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /tZ0 |B(  
    % Compute the Zernike Polynomials 8#l+{`$z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7]Rk+q2:  
    N 2Ssf$  
    'fn$'CeM(  
    % Determine the required powers of r: zSXA=   
    % ----------------------------------- iZ "y7s  
    m_abs = abs(m); }LQC.!  
    rpowers = []; Cfv]VQQE  
    for j = 1:length(n) |vz9Hs$@l  
        rpowers = [rpowers m_abs(j):2:n(j)]; AG>\aV"b  
    end X}W)3v  
    rpowers = unique(rpowers); O:YJ%;w  
    R5kH0{zM  
    )i-gs4[(QN  
    % Pre-compute the values of r raised to the required powers, (:7a&2/M  
    % and compile them in a matrix: :j,}{)5=  
    % ----------------------------- 9yL6W'B!  
    if rpowers(1)==0 > c:Zx!  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); RG}}Oh="v  
        rpowern = cat(2,rpowern{:}); D5L{T+}Oi%  
        rpowern = [ones(length_r,1) rpowern]; b 4OnZ;FI  
    else N}mh}  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); esI'"hVJ  
        rpowern = cat(2,rpowern{:}); ,Xtj;@~-  
    end AY88h$a  
    cz(G]{N  
    6 64q~_@B1  
    % Compute the values of the polynomials: X~L!e}Rz  
    % -------------------------------------- ) EXJ   
    y = zeros(length_r,length(n)); `0@z"D5c  
    for j = 1:length(n) q3+8]-9|5  
        s = 0:(n(j)-m_abs(j))/2;  KGT3|)QN  
        pows = n(j):-2:m_abs(j); q.T:0|  
        for k = length(s):-1:1 th 2<o5  
            p = (1-2*mod(s(k),2))* ... x0<^<D&Q  
                       prod(2:(n(j)-s(k)))/              ... X8R1a?  
                       prod(2:s(k))/                     ... ;;Tq$#vd  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 1-o V-K  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 0Oap39  
            idx = (pows(k)==rpowers); 1Es qQz*$u  
            y(:,j) = y(:,j) + p*rpowern(:,idx); n&d/?aJ7a\  
        end /b%Q[ Ck_  
         $[z<oN_Q  
        if isnorm YgimJsm  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); :1_mfX  
        end (Ilsk{aB;A  
    end vpLMhf`  
    % END: Compute the Zernike Polynomials doLNz4W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "DpKrVuG  
    nzuF]vo  
    #PFO]j!_b  
    % Compute the Zernike functions: MLS;SCl  
    % ------------------------------ AC4 l<:Yh  
    idx_pos = m>0; 0( //D;j  
    idx_neg = m<0; U^?= 0+  
    (U9a@ 1  
    3U;1D2"AE  
    z = y; e U;jP]FA  
    if any(idx_pos) Y /lN@  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); RxG^  
    end P%)b+H{$h  
    if any(idx_neg) <L&eh&4c  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); hW' HT  
    end i0ybJOa4  
    c<jB6|.=2  
     SFpQ#  
    % EOF zernfun GV6mzD@ <  
     
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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)  u\3ZIb  
    _yJ|`g]U3  
    DDE还是手动输入的呢? DaCblX  
    K0 QH?F  
    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