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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, mmf}6ABYT  
    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, #YEOY#  
    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? 0vi)m y;!  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ~a5-xWEZ  
    KMU2Po qD  
    T?!D?YV  
    0\/cTNN  
    y,YK Mc  
    function z = zernfun(n,m,r,theta,nflag) bOvMXj/HV=  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ?H30  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N -JMlk:~  
    %   and angular frequency M, evaluated at positions (R,THETA) on the EKr#i}(x<  
    %   unit circle.  N is a vector of positive integers (including 0), and I4Y; 9Gg  
    %   M is a vector with the same number of elements as N.  Each element y?r:`n  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) CLn}BxgD  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, K4.GAGd  
    %   and THETA is a vector of angles.  R and THETA must have the same 5:T)hoF@  
    %   length.  The output Z is a matrix with one column for every (N,M) 7UVhyrl  
    %   pair, and one row for every (R,THETA) pair. AJ%x"  
    % "{1SDbwmMo  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike D on8xk  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), +DpiX&^h   
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral s\Zp/-Q  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, M~o\K'  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized vwc)d{ND  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ){_D  
    % I7Uj<a=(q  
    %   The Zernike functions are an orthogonal basis on the unit circle. "&@v[O)!xu  
    %   They are used in disciplines such as astronomy, optics, and _7^4sR8=  
    %   optometry to describe functions on a circular domain. 0/g 0=dW=  
    % 5VLJ:I?0O  
    %   The following table lists the first 15 Zernike functions. KcW]"K>p!  
    % Uiz#QGt  
    %       n    m    Zernike function           Normalization O=A(x m#  
    %       -------------------------------------------------- `H#G/zOr  
    %       0    0    1                                 1 HHZGu8tzt  
    %       1    1    r * cos(theta)                    2 #&oL iz=hZ  
    %       1   -1    r * sin(theta)                    2 P p]Ygt'u  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) !.^%*6f  
    %       2    0    (2*r^2 - 1)                    sqrt(3) PrZs@ Y  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) L'KgB=5K&i  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) QnJ(C]cW  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Fh3>y2 `/  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) /1!Wet}f  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) LY? `+/  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) |V>_l' /  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) B(z?IW&  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) LYV\|a{Y  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) <O]TM-h  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) > ]()#z  
    %       --------------------------------------------------  > h>  
    % dh; L!  
    %   Example 1: Js'#=  
    % u*:;O\6l  
    %       % Display the Zernike function Z(n=5,m=1) {dk%j~w8  
    %       x = -1:0.01:1; Px$4.b[{_Y  
    %       [X,Y] = meshgrid(x,x); r^2>60q'  
    %       [theta,r] = cart2pol(X,Y); p^yuz (  
    %       idx = r<=1; TSPFi0PP  
    %       z = nan(size(X)); ~|>q)4is6a  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); `1Cg)\&[e0  
    %       figure = ;!$Qw4  
    %       pcolor(x,x,z), shading interp {)c2#h  
    %       axis square, colorbar iFi6,V*PRt  
    %       title('Zernike function Z_5^1(r,\theta)') %~$P.Zh  
    % %`F &,!d  
    %   Example 2: o;#9$j7QP!  
    % B>!OW2q0D  
    %       % Display the first 10 Zernike functions *$4EXwt'  
    %       x = -1:0.01:1; H`XE5Hk)P%  
    %       [X,Y] = meshgrid(x,x); -76l*=|  
    %       [theta,r] = cart2pol(X,Y); p3N/"t&>  
    %       idx = r<=1; bV~z}V&  
    %       z = nan(size(X)); :hA=(iz  
    %       n = [0  1  1  2  2  2  3  3  3  3]; b_p/ 1W:  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; gFx2\QV  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; R54wNm @  
    %       y = zernfun(n,m,r(idx),theta(idx)); C@7<0w  
    %       figure('Units','normalized') ,$xV&w8f\"  
    %       for k = 1:10 -#e3aXe  
    %           z(idx) = y(:,k); Z^'i16  
    %           subplot(4,7,Nplot(k)) 82z\^a  
    %           pcolor(x,x,z), shading interp \TF='@u.  
    %           set(gca,'XTick',[],'YTick',[]) d8o<Q 9   
    %           axis square 2y t)"DnFk  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 8pEiU/V  
    %       end P m Zb!|  
    % s_j ?L  
    %   See also ZERNPOL, ZERNFUN2. ^/H9`z;  
    8^8fUN4<=  
    Ac'0  
    %   Paul Fricker 11/13/2006 Z/p>>SCak  
    }\s\fNSQ/  
    cKbjW  
    >*v P*H:P  
    &ml7368@  
    % Check and prepare the inputs: l4:5(1  
    % ----------------------------- 2^\67@9  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ZYi."^l  
        error('zernfun:NMvectors','N and M must be vectors.') tE~OWjL  
    end R'#1|eWCa  
    p#yq'kY  
    q3CcXYY  
    if length(n)~=length(m) 'DDlX3W-  
        error('zernfun:NMlength','N and M must be the same length.') #2XX[d%  
    end &T i:IC%M  
    WFYbmfmV  
    lh N2xg5x  
    n = n(:); ^E)*i#."4  
    m = m(:); \9Z1'W  
    if any(mod(n-m,2)) V5ySOgzw,  
        error('zernfun:NMmultiplesof2', ... 19r4J(pV  
              'All N and M must differ by multiples of 2 (including 0).') mw[  
    end ~g6`Cp`  
    &"h 9Awn2  
    O>h,u[0  
    if any(m>n) X*Qtbm,  
        error('zernfun:MlessthanN', ... 0pC}+ +  
              'Each M must be less than or equal to its corresponding N.') s"7$SxMT  
    end i xf~3Y8  
    cg]\R1Gm  
    7;w x,7CUq  
    if any( r>1 | r<0 ) +J`HI1  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') MPtn$@  
    end ['*{f(AI  
    ,"@Tm01os  
    8 BHtN  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) hJ>Kfm  
        error('zernfun:RTHvector','R and THETA must be vectors.') [b=l'e/  
    end ;`{PA !>  
    I|`/#BYbW  
    nQ$4W  
    r = r(:); ]z%X%wL  
    theta = theta(:); Zs(I]^w;d  
    length_r = length(r); } ^}fx [  
    if length_r~=length(theta) h0=Q.Yz6  
        error('zernfun:RTHlength', ... `ZC{<eVJ}=  
              'The number of R- and THETA-values must be equal.') 4GiHp7Y&A  
    end CSA.6uIT  
    o `]o(OP  
    BJ c'4>  
    % Check normalization: E!,+#%O>  
    % -------------------- e13{G @  
    if nargin==5 && ischar(nflag) &?#,rEw<x  
        isnorm = strcmpi(nflag,'norm'); #)qn$&.H  
        if ~isnorm o9j*Yz  
            error('zernfun:normalization','Unrecognized normalization flag.') 2i~tzo  
        end %X--`91|u  
    else {N \ri{|  
        isnorm = false; R.Plfm06Ue  
    end ;T9u$4 <  
    =u*\P!$  
    $RFy9(>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ae&i]K;  
    % Compute the Zernike Polynomials Y`O"+Jr  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3!&PI  
    yR`X3.:*]  
    d>RoH]K4  
    % Determine the required powers of r: ="k9 y  
    % ----------------------------------- (O$PJLI  
    m_abs = abs(m); P ,%IZ.  
    rpowers = []; 3y[uH'  
    for j = 1:length(n) zQ&k$l9  
        rpowers = [rpowers m_abs(j):2:n(j)]; P  -O& X  
    end ?$ft3p}  
    rpowers = unique(rpowers); 0`LR!X  
    8RA]h?$$J  
    8|Q=9mmWOh  
    % Pre-compute the values of r raised to the required powers, n!Ic.T3PA  
    % and compile them in a matrix: yFD3:;}  
    % ----------------------------- 5-=&4R\k  
    if rpowers(1)==0 4TP AD)C  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); rx$B(z(c  
        rpowern = cat(2,rpowern{:}); JGJy_.C  
        rpowern = [ones(length_r,1) rpowern]; W N5`zD$  
    else ![>j`i  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); fP:n=A{  
        rpowern = cat(2,rpowern{:}); lBYc(cr  
    end 'e/= !"T  
     d*Wg>8|  
    &D/@H1fBe  
    % Compute the values of the polynomials: 2j*+^&M/  
    % -------------------------------------- L"_l(<g  
    y = zeros(length_r,length(n)); _#jR6g TY  
    for j = 1:length(n) DCv=*=6w  
        s = 0:(n(j)-m_abs(j))/2; c2tf7fkH  
        pows = n(j):-2:m_abs(j); 9{A[n}  
        for k = length(s):-1:1 U= Gw(  
            p = (1-2*mod(s(k),2))* ... ']x`d  
                       prod(2:(n(j)-s(k)))/              ... ]]EOCGZ"  
                       prod(2:s(k))/                     ... hxXl0egI  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... =SY`Xkj[  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Wubvvm8U  
            idx = (pows(k)==rpowers); }.L\O]~{  
            y(:,j) = y(:,j) + p*rpowern(:,idx); %u)niY-g  
        end ; qQ* p  
         VbwB<nQl  
        if isnorm Fm|h3.`V  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); eB]R<a60  
        end T> !Y-e.q  
    end _#SCjFz  
    % END: Compute the Zernike Polynomials +s`HTf  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :c_>(~  
    fFSQLtm?E  
    h&k*i  
    % Compute the Zernike functions: (59u<F  
    % ------------------------------ n/&}|998?  
    idx_pos = m>0; vg.K-"yQW  
    idx_neg = m<0; mBQp#-1\  
    ?}n\&|+  
    5LkpfmR  
    z = y; .#4;em%7  
    if any(idx_pos) odm!}stus  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 5C G ,l  
    end JM& :dzyIP  
    if any(idx_neg) >k)zd-  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); I?z*.yA*  
    end /}ADV2sF  
    ]46-TuH  
    >$g+Gx\v4  
    % EOF zernfun /Cl=;^)  
     
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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)
    在线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)  `;OEdeAM  
    O:>9yZhV  
    DDE还是手动输入的呢? zV<vwIUrr  
    b#2$Pd:(  
    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