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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, mm[SBiFO\  
    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, ^# e~g/  
    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? C3VLV&wF  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Z$'I Bv  
    z`$J_CjY  
    ;(6P6@+o  
    |n6Eg9  
    /_\W+^fE  
    function z = zernfun(n,m,r,theta,nflag) N/~N7MwJj  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ^J x$t/t  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Ec]|p6a3  
    %   and angular frequency M, evaluated at positions (R,THETA) on the onte&Ed\  
    %   unit circle.  N is a vector of positive integers (including 0), and D>sYPrf  
    %   M is a vector with the same number of elements as N.  Each element RuAlB*  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ijUzC>O+q  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, RT*5d;l0  
    %   and THETA is a vector of angles.  R and THETA must have the same !.Zt[g}  
    %   length.  The output Z is a matrix with one column for every (N,M) w'ybbv{c  
    %   pair, and one row for every (R,THETA) pair. UUtbD&\  
    % ii4B?E  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike IA*KaX2S<  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ZR3nK0  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral d^V$Z6* ]  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ?tYpc_p#  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized gPEqjj  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ;-@=  
    % &35|16z%@  
    %   The Zernike functions are an orthogonal basis on the unit circle. 8_we: 9A  
    %   They are used in disciplines such as astronomy, optics, and {0yu   
    %   optometry to describe functions on a circular domain. \4bWWy  
    % :tGYs8UK  
    %   The following table lists the first 15 Zernike functions. g9mG`f  
    % ]tt} #  
    %       n    m    Zernike function           Normalization t})$lM  
    %       -------------------------------------------------- 30F!kP*E  
    %       0    0    1                                 1 5EeDHsvV9  
    %       1    1    r * cos(theta)                    2 +=3CL2{An  
    %       1   -1    r * sin(theta)                    2 n:#TOU1ix<  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) jqcz\n d  
    %       2    0    (2*r^2 - 1)                    sqrt(3) cFZCf8:zB  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) E{Vo'!LY  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) SUdm 0y  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) J|QiH<  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) faJM^u  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) {aj/HFLNY  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) z&+ zl6  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =@r--E  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) s#-eN)1R  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) TI9X.E?  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) SGcBmjP  
    %       -------------------------------------------------- R p.W,)i  
    % }ot"Sx\.  
    %   Example 1: y?z\L   
    % _p}xZD\?,  
    %       % Display the Zernike function Z(n=5,m=1) hR)2xz  
    %       x = -1:0.01:1; 6rDfQ`f\p  
    %       [X,Y] = meshgrid(x,x); 2WCLS{@'  
    %       [theta,r] = cart2pol(X,Y); e<=;i" |  
    %       idx = r<=1; cCdX0@hY  
    %       z = nan(size(X)); 4zc<GL3[  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); a/:XXy |  
    %       figure ~1(j&&kXet  
    %       pcolor(x,x,z), shading interp Ok H\^  
    %       axis square, colorbar <Gb %uny  
    %       title('Zernike function Z_5^1(r,\theta)') Omyt2`q  
    % r|R7- HI  
    %   Example 2: |:q/Dt@  
    % !,&yyx.  
    %       % Display the first 10 Zernike functions y!Cc?$]_Y  
    %       x = -1:0.01:1; ~!:0iFE&H  
    %       [X,Y] = meshgrid(x,x); `rFAZcEj%  
    %       [theta,r] = cart2pol(X,Y); hU {-a`  
    %       idx = r<=1; 8 %Sb+w07  
    %       z = nan(size(X)); xAdq+$><  
    %       n = [0  1  1  2  2  2  3  3  3  3]; mN}7H:,  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; B@K[3  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; k3hkk:W  
    %       y = zernfun(n,m,r(idx),theta(idx)); SS3-+<z  
    %       figure('Units','normalized') jPJAWXB4a  
    %       for k = 1:10 .b>TK  
    %           z(idx) = y(:,k); "^rNr_  
    %           subplot(4,7,Nplot(k)) H5xzD9K;/C  
    %           pcolor(x,x,z), shading interp b-<HXn_Fd  
    %           set(gca,'XTick',[],'YTick',[]) isK;mU?<  
    %           axis square P%>?[9!Nt  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]H[8Z|i""  
    %       end *Xr$/N  
    % rY}B-6qJn  
    %   See also ZERNPOL, ZERNFUN2. P:!)9/.2  
    oyeG$mpg  
    W |UtY`1  
    %   Paul Fricker 11/13/2006 0D [@u3W  
    AXW!]=?X  
    Q:o 7G|C  
    t1i(;|8|  
    o$J6 ~dn  
    % Check and prepare the inputs: GESXc $E8  
    % ----------------------------- f(Hu {c5yV  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Fb_S&!  
        error('zernfun:NMvectors','N and M must be vectors.') PZ OKrW  
    end v 81rfB5  
    WH$HI/%*m  
    ^Kq|ID AP  
    if length(n)~=length(m) ;e{5)@h$  
        error('zernfun:NMlength','N and M must be the same length.') ef]B9J~h  
    end fE25(wCz7  
    K0fv( !r{  
    ;u!?QSvb  
    n = n(:); ])T/sO#'  
    m = m(:); |4>:M\h  
    if any(mod(n-m,2)) 8T5k-HwE  
        error('zernfun:NMmultiplesof2', ... e!0OW7 kV  
              'All N and M must differ by multiples of 2 (including 0).') z +VV}:Q  
    end n[" 9|  
    _l&ucA  
    /1.rz{wpb  
    if any(m>n) OyVm(%Z   
        error('zernfun:MlessthanN', ... ZK dh%8C  
              'Each M must be less than or equal to its corresponding N.') ; B$ *)X9  
    end &3DK^|Lq  
    2C/%gcN >  
    >BoSw&T$Q  
    if any( r>1 | r<0 ) .Q\\dESn"  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') '2v f|CX  
    end 3H8Al  
    e}"wL g]  
    !nw [  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) <GZhH:  
        error('zernfun:RTHvector','R and THETA must be vectors.') (F&YdWe:  
    end m|4LbWz  
    A3yi?y{[*  
    |uUuFm  
    r = r(:); {LB`)Kuu  
    theta = theta(:); Zu#<  
    length_r = length(r); r+\/G{+=}  
    if length_r~=length(theta) a%J /0'(d  
        error('zernfun:RTHlength', ... nCaLdj?  
              'The number of R- and THETA-values must be equal.') }$aNOf%:  
    end 7),*3c')  
    c@OP5L>{  
    pxplWP,  
    % Check normalization: -!R l(if  
    % -------------------- r8v:|Q1"  
    if nargin==5 && ischar(nflag) e,Zv]Cym  
        isnorm = strcmpi(nflag,'norm'); MSYN1  
        if ~isnorm S0xIvzS  
            error('zernfun:normalization','Unrecognized normalization flag.') 0yQe5i}  
        end !5.v'K'  
    else - L`7+  
        isnorm = false; ^5x4q  
    end JQT4N[rEE  
    >hb- 5xC  
    @ ;J|xkJ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wE2x:Ge:  
    % Compute the Zernike Polynomials  -$R5  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% o* _g$  
    +]L)>$6  
    (xUFl@I!  
    % Determine the required powers of r: 0O; Z  
    % ----------------------------------- hht+bpHl  
    m_abs = abs(m); (`mOB6j  
    rpowers = []; Sf/W9Jw  
    for j = 1:length(n) cVg$dt  
        rpowers = [rpowers m_abs(j):2:n(j)]; W-XN4:,qI  
    end *1v_6<;2i<  
    rpowers = unique(rpowers); 8Mb$+^zU  
    R `Q?J[e  
    yu_gNro L  
    % Pre-compute the values of r raised to the required powers, Eq7gcDQ  
    % and compile them in a matrix: h@Dw'w  
    % ----------------------------- 1gAc,s2  
    if rpowers(1)==0 ._(z~3s  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); YiNo#M91  
        rpowern = cat(2,rpowern{:}); vGyppm[0  
        rpowern = [ones(length_r,1) rpowern]; Tvrc%L(]  
    else c}\ d5R_L  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %w@ig~vD'  
        rpowern = cat(2,rpowern{:}); 2dyxKK!\a  
    end %Fm`Y .l  
    hhj ,rcsi  
    )SD_}BY%k  
    % Compute the values of the polynomials: 8fEAYRGd  
    % -------------------------------------- W7]mfy^  
    y = zeros(length_r,length(n)); qIk )'!Vk  
    for j = 1:length(n) GiFf0c 9  
        s = 0:(n(j)-m_abs(j))/2; h%|9]5(=  
        pows = n(j):-2:m_abs(j); (ai72#nFtb  
        for k = length(s):-1:1 cnYYs d{  
            p = (1-2*mod(s(k),2))* ... K1 6s)S'  
                       prod(2:(n(j)-s(k)))/              ... rl41# 6  
                       prod(2:s(k))/                     ... ls]Elo8h1f  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ;pCG9  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 9XY|V<}  
            idx = (pows(k)==rpowers); [L)V(o)v  
            y(:,j) = y(:,j) + p*rpowern(:,idx); GZ.?MnG  
        end U(8I+xZ  
         "SDsISWd  
        if isnorm @]<DR*<  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1=- X<M75  
        end LsUFz_  
    end 2 /UI>@By  
    % END: Compute the Zernike Polynomials vLD:(qTi  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Hv+:fr"  
     !zF4 G,W  
    r)(i{:@r`  
    % Compute the Zernike functions: B0NN>)h  
    % ------------------------------ fCs\Q  
    idx_pos = m>0; [v~Uy$d\  
    idx_neg = m<0; ^JiaR)#r  
    EgCp:L{  
    mp muziH  
    z = y; +}`p"<'u  
    if any(idx_pos) ~ rQ4n9G  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); i:AjWC@]  
    end nqUH6(  
    if any(idx_neg) 'aLPTVM^  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); k-;.0!D^  
    end AW]("pt  
    0<6rU  
    t=AE7  
    % EOF zernfun k?z [hZg0  
     
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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)  | 2c!t$O@v  
    %lq7; emtp  
    DDE还是手动输入的呢? 1:~m)"?I_^  
    /`]|_>'  
    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