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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, >n^780S|  
    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, /3&MUB*z&y  
    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? xHMFYt+0$G  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? |Ve,Y  
    PD&gC88  
    (zh[1[a  
    3{f g3?  
    |c^?tR<  
    function z = zernfun(n,m,r,theta,nflag) AJm$(3?/D  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. FWA?mde  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N !I.}[9N  
    %   and angular frequency M, evaluated at positions (R,THETA) on the LT"H -fTgs  
    %   unit circle.  N is a vector of positive integers (including 0), and GC:q6}  
    %   M is a vector with the same number of elements as N.  Each element g(Q1d-L4e  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) <Se9 aD  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, z$WLx  
    %   and THETA is a vector of angles.  R and THETA must have the same 7B)1U_L0H  
    %   length.  The output Z is a matrix with one column for every (N,M) r! HXhl  
    %   pair, and one row for every (R,THETA) pair. xJ2I@*DN  
    % G:p85k `  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike yOt#6Vw  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), R3;%eyu  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral H>A6VDu  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 4(8tr D6  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Z`u$#<ukX  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. f:-l}Zj  
    % .p,VZ9  
    %   The Zernike functions are an orthogonal basis on the unit circle. ),-gy~  
    %   They are used in disciplines such as astronomy, optics, and Lm=;Y6'`N  
    %   optometry to describe functions on a circular domain. @0 /qP<E  
    % |/vJ+aKq  
    %   The following table lists the first 15 Zernike functions. marZA'u%B1  
    % P6R_W  
    %       n    m    Zernike function           Normalization h='F,r5#2  
    %       -------------------------------------------------- (v%24bv  
    %       0    0    1                                 1 V*{rHp{=p  
    %       1    1    r * cos(theta)                    2 Yu>DgMW  
    %       1   -1    r * sin(theta)                    2 fj)) Hnt(|  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) :Ys~Lt54  
    %       2    0    (2*r^2 - 1)                    sqrt(3) kQ}n~Hn  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) {X&lgj  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) r]UF<*$  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) \?d3Pn5`  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) +)iMJ]>  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) :#pdyJQ_  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ANy*'/f  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) lOk8VlH<h  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) =i&,I{3  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Cq"KKuf  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ^w.hI5ua)  
    %       -------------------------------------------------- -g]Rs!w'  
    % <ZF|2  
    %   Example 1: #uw&u6*\q  
    % jk{(o09  
    %       % Display the Zernike function Z(n=5,m=1) R<Lf>p>_  
    %       x = -1:0.01:1; Z0jgUq`r  
    %       [X,Y] = meshgrid(x,x); 12KC4,C&1i  
    %       [theta,r] = cart2pol(X,Y); )&Oc7\J,  
    %       idx = r<=1; ^k;]"NR  
    %       z = nan(size(X)); IB/3=4n^|  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); t82'K@sq  
    %       figure o)/Pr7Qn  
    %       pcolor(x,x,z), shading interp NEIkG>\7q  
    %       axis square, colorbar 6(Pan%  
    %       title('Zernike function Z_5^1(r,\theta)') La;G S  
    % BVNW1<_:  
    %   Example 2: rtRbr_  
    % zKO7`.*  
    %       % Display the first 10 Zernike functions e "A"  
    %       x = -1:0.01:1; lUm(iYv;H  
    %       [X,Y] = meshgrid(x,x); &0Yg:{k$  
    %       [theta,r] = cart2pol(X,Y); ]R#:Bq!F  
    %       idx = r<=1; \=A A,Il  
    %       z = nan(size(X)); '7-Yo Q  
    %       n = [0  1  1  2  2  2  3  3  3  3]; #]kjyT0  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; HYmC3  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; W]9*dabem  
    %       y = zernfun(n,m,r(idx),theta(idx)); a>'ez0C  
    %       figure('Units','normalized') 50W+!'  
    %       for k = 1:10 LH8jT  
    %           z(idx) = y(:,k); d,V#5l-6  
    %           subplot(4,7,Nplot(k)) <+i(CGw  
    %           pcolor(x,x,z), shading interp L>1hiD&  
    %           set(gca,'XTick',[],'YTick',[]) i2~uhGJ  
    %           axis square amu;grH  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #) aLD0p  
    %       end 3'0Pl8  
    % d;Vy59}eY  
    %   See also ZERNPOL, ZERNFUN2. H\67Pd(Z6  
    `6D?te  
    Ymk?@mV4  
    %   Paul Fricker 11/13/2006 Ke\\B o,  
    ;]>kp^C#  
    GM%+yS}(P  
    hmO2s/~  
    mgq!)  
    % Check and prepare the inputs: B`~EA] d  
    % ----------------------------- W$rWg>4>  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 0 &zp  
        error('zernfun:NMvectors','N and M must be vectors.') GXtMX ha,  
    end K4c:k; V  
    'o >)E>  
    >cu%Cs=m  
    if length(n)~=length(m) #z*,CU#S9d  
        error('zernfun:NMlength','N and M must be the same length.') _ E;T"SC  
    end 9DxHdpOk  
    y_Y(Xx3  
    Z2% HQL2  
    n = n(:); Rh!UbEPjC  
    m = m(:); " O&93#8  
    if any(mod(n-m,2)) HN5m%R&`  
        error('zernfun:NMmultiplesof2', ... Kg[OUBv  
              'All N and M must differ by multiples of 2 (including 0).') { "y/;x/  
    end )h{&O ,s  
     [XfR`@  
    7L{1S v  
    if any(m>n) `fu_){  
        error('zernfun:MlessthanN', ... Gm=qn]c  
              'Each M must be less than or equal to its corresponding N.') *o6}>;  
    end ^X=Q{nB  
    WRh5v8Wz0  
    37#&:[w>  
    if any( r>1 | r<0 ) fE#(M+(<  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') QQ*sjK.(  
    end {%V(Dd[B6  
    ;O"?6d0  
    oxwbq=a6yV  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 9 BCW2@Kp  
        error('zernfun:RTHvector','R and THETA must be vectors.') XH%L]  
    end *LT~:Gs#  
    o>el"0rn.h  
    Y[ G_OoU  
    r = r(:); .Ro/ioq  
    theta = theta(:); :cT)M(o  
    length_r = length(r); 7FB?t<x  
    if length_r~=length(theta) jkAjYR.  
        error('zernfun:RTHlength', ... M&Uy42,MR  
              'The number of R- and THETA-values must be equal.') ?bTfQH vX  
    end U&!TA(Yr  
    54 lD+%E  
    C"hN2Z!CD|  
    % Check normalization: 615Ya<3f8  
    % -------------------- D31X {dJ  
    if nargin==5 && ischar(nflag) q!Du J  
        isnorm = strcmpi(nflag,'norm'); #8$?# dT  
        if ~isnorm ;Rrh$Ag  
            error('zernfun:normalization','Unrecognized normalization flag.') jUe@xi s<T  
        end %b6$N_M{H1  
    else X\}l" ]  
        isnorm = false; =o@;K~-  
    end Ss3p6%V/  
    oV|O`n  
    i Ha?b2=)  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% o+A7hBM^  
    % Compute the Zernike Polynomials Z%t_1t  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% OgQd yU  
    rTPgHK]?l  
    7CIN!vrC|1  
    % Determine the required powers of r: =|t-0'RsN  
    % ----------------------------------- &i#$ia r  
    m_abs = abs(m); |;ztK[(  
    rpowers = []; TCr4-"`r-{  
    for j = 1:length(n) T(J'p4  
        rpowers = [rpowers m_abs(j):2:n(j)]; Ln"wj O ,  
    end _&<n'fK[  
    rpowers = unique(rpowers); AIF ?>wgq  
    m%'nk"p9  
    Y.^L^ "%dF  
    % Pre-compute the values of r raised to the required powers, inh0p^  
    % and compile them in a matrix: _&gi4)q  
    % ----------------------------- ,OE&e* 1  
    if rpowers(1)==0 C$[d~1t6  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ? SFBUX(p  
        rpowern = cat(2,rpowern{:}); 1\}vU  
        rpowern = [ones(length_r,1) rpowern]; x|H`%Z  
    else J_Lmy7~xbD  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); q_MN  
        rpowern = cat(2,rpowern{:}); coP->&(@U#  
    end r\NqY.U&  
    A8{jEJ=)P  
    aZ#FKp^8H  
    % Compute the values of the polynomials: VB |?S|<  
    % -------------------------------------- /MZ<vnN7f  
    y = zeros(length_r,length(n)); I _nQTWcm  
    for j = 1:length(n) Llfl I   
        s = 0:(n(j)-m_abs(j))/2; !)s(Lv%]  
        pows = n(j):-2:m_abs(j); 2)}n"ibbT  
        for k = length(s):-1:1 L.n@;*  
            p = (1-2*mod(s(k),2))* ... "?"  :  
                       prod(2:(n(j)-s(k)))/              ... ]RVu[k8  
                       prod(2:s(k))/                     ...  H.5 6  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 'gwh:  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Lg:1zC  
            idx = (pows(k)==rpowers); bz*@[NQ  
            y(:,j) = y(:,j) + p*rpowern(:,idx); _@5Xmr  
        end 5Xq+lLW>  
         '+Dsmoy  
        if isnorm T(!1\TB  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Ly= .  
        end pF;.nt)  
    end qe]D4K8`Q3  
    % END: Compute the Zernike Polynomials /[R=-s ;  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0s n$QmW:  
    FFT)m^4p.  
    lrrTeE*  
    % Compute the Zernike functions: ,NO[Piok  
    % ------------------------------ YPK@BmAdE  
    idx_pos = m>0; 5'!fi]Z  
    idx_neg = m<0; z)Rkd0/X  
    Kz'GAm\  
    ak7%  
    z = y; K1 f1 T  
    if any(idx_pos) {`HbpM<=m]  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); kQ\GVI11?  
    end ib,`0=0= O  
    if any(idx_neg) qq)5)S  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); +17!v_4^  
    end 3.Fko<D4jD  
    F|%PiC,,qO  
    G|cjI*  
    % EOF zernfun ,xwiJfG; ]  
     
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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)  'h ?  
    ql%K+4@  
    DDE还是手动输入的呢? .NWsr*Tel  
    FoE}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