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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, s@fTj$h  
    我输入10阶的n、m,r,theta为38025*1向量,最后得到的z是29525*10阶的矩阵, 0q&'(-{s1  
    这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? hTwA%  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ^ :F.  
    l&}y/t4%  
    R6=$u{D  
    }W ^: cp  
    Wq^qpN)5Y  
    function z = zernfun(n,m,r,theta,nflag) pC_O:f>vJ  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 'TA UE{{  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ?-Vjha@BO  
    %   and angular frequency M, evaluated at positions (R,THETA) on the "]-Xmdk09  
    %   unit circle.  N is a vector of positive integers (including 0), and ~@kU3ZGJZ  
    %   M is a vector with the same number of elements as N.  Each element ~xoF6 CF  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) wfjnA~1h  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, N:9>dpP}O  
    %   and THETA is a vector of angles.  R and THETA must have the same #0Tq=:AE>  
    %   length.  The output Z is a matrix with one column for every (N,M) ZNx$r]4nF  
    %   pair, and one row for every (R,THETA) pair. ]~\sA  
    % 57 #6yXQ  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike F-*2LMe  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), WQHd[2Z#e  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Vrvic4  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, vp.ZK[/`  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized wM|" I^[  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. /6_|]ijc  
    % 2W$cFC  
    %   The Zernike functions are an orthogonal basis on the unit circle. Ka`=WeJ|  
    %   They are used in disciplines such as astronomy, optics, and *@TZ+{t  
    %   optometry to describe functions on a circular domain. Vi$-Bw$@  
    % v 36%Pj`  
    %   The following table lists the first 15 Zernike functions. mRZ :ie  
    % }Nb8}(6  
    %       n    m    Zernike function           Normalization n>'Kp T9|  
    %       -------------------------------------------------- @}:uu$OH  
    %       0    0    1                                 1 F0690v0mB[  
    %       1    1    r * cos(theta)                    2 `g,8-  
    %       1   -1    r * sin(theta)                    2 6eokCc"o  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) uWrQ&}@  
    %       2    0    (2*r^2 - 1)                    sqrt(3) )7:J[0ZiQ  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) \);4F=h}f  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) x=#VX\5k:  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) A7c/N=Cp^  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) XQ*eP?OS{  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) #A8@CA^d  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) F9*g=  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 3T&6opaF  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Xo*DvD  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) PpsIhMq@  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) qn,O40/]  
    %       -------------------------------------------------- MJ=)v]a  
    % tBct  
    %   Example 1: !*`-iQo&  
    % 7G)H.L)$m"  
    %       % Display the Zernike function Z(n=5,m=1) AL5Vu$V~n}  
    %       x = -1:0.01:1;  RD tU43  
    %       [X,Y] = meshgrid(x,x); |A8/FU2{  
    %       [theta,r] = cart2pol(X,Y); lHV[Ln`\x  
    %       idx = r<=1; 21(p|`X  
    %       z = nan(size(X)); 1[ ]&(Pa  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); #b7$TV  
    %       figure +,2Jzl'-  
    %       pcolor(x,x,z), shading interp +S))3 5N[  
    %       axis square, colorbar 0KD]j8^  
    %       title('Zernike function Z_5^1(r,\theta)') rcGb[=Bf  
    % .] `f,^v<c  
    %   Example 2: BI j=!!  
    % }(<%`G6N  
    %       % Display the first 10 Zernike functions 1EyL#;k  
    %       x = -1:0.01:1; !p1qJ [  
    %       [X,Y] = meshgrid(x,x); M4WiT<|]R  
    %       [theta,r] = cart2pol(X,Y); ,hVvve,j}  
    %       idx = r<=1; }za[E>z  
    %       z = nan(size(X)); ` \A(9u*  
    %       n = [0  1  1  2  2  2  3  3  3  3]; srV.)Ur  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; mYc.x  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Gy[O)PEEh  
    %       y = zernfun(n,m,r(idx),theta(idx)); oBUxKisW  
    %       figure('Units','normalized') 2r%lA\,h$  
    %       for k = 1:10 z]3 `*/B  
    %           z(idx) = y(:,k); [TCP-bU  
    %           subplot(4,7,Nplot(k)) Od?qz1  
    %           pcolor(x,x,z), shading interp =YG _z^'  
    %           set(gca,'XTick',[],'YTick',[]) 45&8weXO:'  
    %           axis square M_LXg%  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) >q7BVF6V |  
    %       end *6 U&Qy-M  
    % {s3z"OV  
    %   See also ZERNPOL, ZERNFUN2. *UW=Mdt  
    Z@gnsPN^r  
    Bb{!Yh].:A  
    %   Paul Fricker 11/13/2006 L^^4=ao0  
    -VZRujl  
    *MI*Rz?4  
    Tg3!Rq55  
    ""svDfy$  
    % Check and prepare the inputs:  M/z}p  
    % ----------------------------- 3gQPKBpc  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2 3KyCV5  
        error('zernfun:NMvectors','N and M must be vectors.') umLb+GbI4  
    end xug)aE  
    #Dfo#]k(  
    gw _$  
    if length(n)~=length(m) ZX~ _g@  
        error('zernfun:NMlength','N and M must be the same length.') 0Aa`p3.)  
    end H.G!A6bd  
    wY"o`o Z  
    IaFr&  
    n = n(:); h8jD }9^  
    m = m(:); wNE$6  
    if any(mod(n-m,2)) A-CUv[pM  
        error('zernfun:NMmultiplesof2', ... z<]bv7V  
              'All N and M must differ by multiples of 2 (including 0).') 9SMiJad<  
    end mKq"3 4F  
    4e9q`~ sO  
    ?*r!{3T ,u  
    if any(m>n) ;3C:%!CdA]  
        error('zernfun:MlessthanN', ... X5g[ :QKP7  
              'Each M must be less than or equal to its corresponding N.') z&4~x!-_  
    end W 4YE~  
    (Y(E%  
    dRvin[R8  
    if any( r>1 | r<0 ) r+<{S\ Q  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') rsa&Oo D>  
    end #t!}K_  
    .]Mn^2#j  
    xn}BB}s{t  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ep(g`e  
        error('zernfun:RTHvector','R and THETA must be vectors.') VF0dE  
    end !NKmx=I]  
    pJ, @Y>  
    wHsB,2H  
    r = r(:); !PUp>(  
    theta = theta(:); rn.\tDeA  
    length_r = length(r); p SN~DvR  
    if length_r~=length(theta) jJwkuh8R  
        error('zernfun:RTHlength', ... }1+%_|Y-E  
              'The number of R- and THETA-values must be equal.') ?TEK=mD#u  
    end @kD8^,(oH  
    9>,Qgp,w  
    '~-IV0v9  
    % Check normalization: %c^ m\ E  
    % -------------------- xk~Nmb}  
    if nargin==5 && ischar(nflag) n<V1|X  
        isnorm = strcmpi(nflag,'norm'); FquFRx  
        if ~isnorm 6&2LWaWMo$  
            error('zernfun:normalization','Unrecognized normalization flag.') "PpjoM ~  
        end bdc\  
    else 8V4V3^_xs  
        isnorm = false; VGH/X.NJ  
    end <xS=#  
    -.vDF?@G  
    F}ukZ DB  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B!aK  
    % Compute the Zernike Polynomials &:?e&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'zgvQMu  
    m[2'd  
    I^Qx/uTKw  
    % Determine the required powers of r: heD,& OX  
    % ----------------------------------- 0|)19LR  
    m_abs = abs(m); DOm-)zl{|x  
    rpowers = []; r!/0 j)  
    for j = 1:length(n) >mIg@knE  
        rpowers = [rpowers m_abs(j):2:n(j)]; !eD+GDgE]  
    end Nh)[r x  
    rpowers = unique(rpowers); w;`m- 9<Y  
    O25m k X  
    ! gp}U#Yv  
    % Pre-compute the values of r raised to the required powers, F>Y9o- o2  
    % and compile them in a matrix: J^H =i)A  
    % ----------------------------- kC^.4n om  
    if rpowers(1)==0 QXk"?yT`E  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); .`LgYW  
        rpowern = cat(2,rpowern{:}); C*wdtEGq  
        rpowern = [ones(length_r,1) rpowern]; U|fTb0fB  
    else Ge}$rLu]0  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); .1ddv4Hk  
        rpowern = cat(2,rpowern{:}); B/YcSEY;  
    end WL~`u  
    DNth4z  
    dm^H5D/A  
    % Compute the values of the polynomials: ,hE/II`-d'  
    % -------------------------------------- m<fA|9 F#  
    y = zeros(length_r,length(n)); <NQyP{p  
    for j = 1:length(n)  ?f2G?Y  
        s = 0:(n(j)-m_abs(j))/2; J@bW^>g*6u  
        pows = n(j):-2:m_abs(j); X!0kK8v  
        for k = length(s):-1:1 R# 6H'TVE  
            p = (1-2*mod(s(k),2))* ... _.f@Y`4d  
                       prod(2:(n(j)-s(k)))/              ... 41;)-(1  
                       prod(2:s(k))/                     ... |[w^eg  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 0^\/ERK  
                       prod(2:((n(j)+m_abs(j))/2-s(k)));  1KJZWZy  
            idx = (pows(k)==rpowers); dF2@q@\.+  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Y. TYc;  
        end G)+Ff5e0L[  
         dIK{MA  
        if isnorm H'Iq~Ft1  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); $HRed|*.C  
        end |9]PtgQv7  
    end MuSaK %  
    % END: Compute the Zernike Polynomials <$C<Ba?;?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% OWV/kz5'H  
    Qk7J[4  
    QeK{MF  
    % Compute the Zernike functions: h3t$>vs2F"  
    % ------------------------------ B "n`|;r5  
    idx_pos = m>0; &l!$Sw-u;  
    idx_neg = m<0; t,?,F4 j  
    ,|x\MHd?t_  
    #J~Xv:LgD  
    z = y; QE6El'S  
    if any(idx_pos) ,Qo}J@e(  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); C "9"{  
    end {jG.=}/Dk  
    if any(idx_neg) ruHrv"29  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); iwkJ~(5z  
    end g =x"cs/[  
    SEU\}Ni{  
    Xv*}1PZH  
    % EOF zernfun w7ZG oh(  
     
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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)  #FEa 5  
    l6V%"Lo/)  
    DDE还是手动输入的呢? <nj IXa{  
    `'kc|!%MUq  
    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