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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, w7(jSPB  
    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, ckglDhC  
    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? o 0-3[W'x<  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? U2lDTRt  
    ?qdZ]M4e  
    \-Oq/g{j  
    */T.]^  
    MPexc5_  
    function z = zernfun(n,m,r,theta,nflag) \Y>!vh X  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. [K*>W[n  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N $@ous4&  
    %   and angular frequency M, evaluated at positions (R,THETA) on the e=8z,.Xk  
    %   unit circle.  N is a vector of positive integers (including 0), and QJsud{ada  
    %   M is a vector with the same number of elements as N.  Each element U{)|z-n  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) /]_a\x5Ss  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, J Uf{;nt  
    %   and THETA is a vector of angles.  R and THETA must have the same Q>G lA  
    %   length.  The output Z is a matrix with one column for every (N,M) |JR;E$  
    %   pair, and one row for every (R,THETA) pair. 7%%FYHMO:  
    % UC u4S >  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike nB8JdM2h{  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), T|/B}srm  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral na%DF@Rt#  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, |t1ij'N  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ?HsQ417.H  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. viLK\>>  
    % U1.w%b,  
    %   The Zernike functions are an orthogonal basis on the unit circle. "!fvEE  
    %   They are used in disciplines such as astronomy, optics, and 4!I;U>b b  
    %   optometry to describe functions on a circular domain. *Dz<Pi^  
    % |?kZfr&9q  
    %   The following table lists the first 15 Zernike functions. tH}$j  
    % 7jf%-X  
    %       n    m    Zernike function           Normalization M_ GN3  
    %       -------------------------------------------------- P ]prrKZe,  
    %       0    0    1                                 1 ssWSY(j]  
    %       1    1    r * cos(theta)                    2 jP{W|9@ (  
    %       1   -1    r * sin(theta)                    2 `H^?jX>7  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) _(TYR*  
    %       2    0    (2*r^2 - 1)                    sqrt(3) t$*V*gK{  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ^T{ww=/v  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 1z#0CX}Y/H  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) TqZ&X| G  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) [h3y8O  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 3Mw2;.rk  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) cc$L56q  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :'t+*{ff  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) bSKe@4C  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) G OzV#  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) =$^<@-;  
    %       -------------------------------------------------- 6{qI  
    % >o#^)LN  
    %   Example 1: Wl4T}j  
    % 2f=7`1RCD  
    %       % Display the Zernike function Z(n=5,m=1) IIrXI8'}  
    %       x = -1:0.01:1; }+" N '  
    %       [X,Y] = meshgrid(x,x); nj@l5[  
    %       [theta,r] = cart2pol(X,Y); ?9?eA^X%  
    %       idx = r<=1; R24ZjbKL  
    %       z = nan(size(X));  =Y0>b4  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); >`@c9 m  
    %       figure S]P80|!|  
    %       pcolor(x,x,z), shading interp VgoN=S  
    %       axis square, colorbar :Hn*|+'  
    %       title('Zernike function Z_5^1(r,\theta)') }EW@/; kC  
    % "]"!"#aMv  
    %   Example 2: N?7vcN+-t)  
    % p-6(>,+E[  
    %       % Display the first 10 Zernike functions ]Q%|69H}B  
    %       x = -1:0.01:1; UB4M=R|  
    %       [X,Y] = meshgrid(x,x); T9c=As_EM  
    %       [theta,r] = cart2pol(X,Y); 9aE.jpN  
    %       idx = r<=1; )/:r $n7  
    %       z = nan(size(X)); f\Fk+)e@  
    %       n = [0  1  1  2  2  2  3  3  3  3]; -d|VXD5N  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; upJ|`,G{  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; W/U_:^[-  
    %       y = zernfun(n,m,r(idx),theta(idx)); bhI yq4N  
    %       figure('Units','normalized') 5:=ECtKi  
    %       for k = 1:10 o`!7 ~n  
    %           z(idx) = y(:,k); XO=UKk+EK  
    %           subplot(4,7,Nplot(k)) _QhB0/C  
    %           pcolor(x,x,z), shading interp @k~_ w#  
    %           set(gca,'XTick',[],'YTick',[]) GmA5E  
    %           axis square LPOZA`  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) *@C4~Zo  
    %       end WLb *\  
    % NWGSUUa  
    %   See also ZERNPOL, ZERNFUN2. =t+{ )d.w  
    )ny,vcU]  
    CkJU5D  
    %   Paul Fricker 11/13/2006 NW$C1(oT  
    %/^k r ZD  
    D-/aS5wM  
    6Wos6_  
    &.\|w  
    % Check and prepare the inputs: ,r,~1oV<"  
    % ----------------------------- X@ljZ  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 'm;M+:l 6  
        error('zernfun:NMvectors','N and M must be vectors.') owA0I'|V-A  
    end ~vCfMV[F  
    +Rtz`V1d  
    %Ps DS  
    if length(n)~=length(m) 0P5!fXs*  
        error('zernfun:NMlength','N and M must be the same length.') #$vef  
    end U2tsHm.O  
    ")q{>tV  
    pvz*(u  
    n = n(:); .>(?c92  
    m = m(:); '.@'^80iQ  
    if any(mod(n-m,2)) u% ^Lu.l_c  
        error('zernfun:NMmultiplesof2', ... J.`z;0]op  
              'All N and M must differ by multiples of 2 (including 0).') jU#/yM "Y  
    end O1o.^i$-M  
    &wZ ggp  
    Kb_R "b3v  
    if any(m>n) !U,^+"l'GP  
        error('zernfun:MlessthanN', ... 8e'0AI_>  
              'Each M must be less than or equal to its corresponding N.') =jik33QV<  
    end m_%1I J  
    2u[:3K-@,  
    nP9@yI*7  
    if any( r>1 | r<0 ) cx]O#b6B.  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') dYg}qad5:  
    end a0"gt"q A  
    M y:9  
    N*PF&MyB  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) imx/hz!  
        error('zernfun:RTHvector','R and THETA must be vectors.') lR!Sdd} -  
    end I#Q Tmg.  
    )shzJ9G  
    1}E`K#  
    r = r(:); o W)M&$oS  
    theta = theta(:); LzEAA{  
    length_r = length(r); Ozk^B{{o  
    if length_r~=length(theta) Yx_[vLm  
        error('zernfun:RTHlength', ... q8:Z.<%8  
              'The number of R- and THETA-values must be equal.') K_V44f1f  
    end r9Ux=W\  
    WgdL^PN(h  
    {IeW~S' &  
    % Check normalization: #vy:aq<bjE  
    % -------------------- &jgpeFiiC  
    if nargin==5 && ischar(nflag) @:@0}]%z9  
        isnorm = strcmpi(nflag,'norm'); *G^n<p$"  
        if ~isnorm l`2X'sw[/  
            error('zernfun:normalization','Unrecognized normalization flag.') eNlE]W,=  
        end Na^1dn  
    else Sf}>~z2  
        isnorm = false; {xw*H<"f<  
    end L~1u?-zu  
    gmfux b/  
    j.$#10*:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pti`q )  
    % Compute the Zernike Polynomials }DTpl?l  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9&A-o  
     M/5e4b  
    &nk6_{6 c  
    % Determine the required powers of r: 40pGu  
    % ----------------------------------- M}4%LjD  
    m_abs = abs(m); j380=? 7  
    rpowers = []; Y[gj2vNe4g  
    for j = 1:length(n) \5^#5_<  
        rpowers = [rpowers m_abs(j):2:n(j)]; %T*lcg  
    end pb`F_->uq  
    rpowers = unique(rpowers); m",wjoZe*  
    ^*C+^l&J!  
    ?H7*?HV  
    % Pre-compute the values of r raised to the required powers, rE"`q1b#  
    % and compile them in a matrix: c/ wzV  
    % ----------------------------- ]GYO`,  
    if rpowers(1)==0 &I.UEF2,  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); -6MgC9]  
        rpowern = cat(2,rpowern{:}); : j&M&+  
        rpowern = [ones(length_r,1) rpowern]; t=iSMe  
    else ]Ff"o7gT  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 59";{"sw  
        rpowern = cat(2,rpowern{:}); m~9Qx`fi`  
    end 58*s\*V` \  
    lhTjG,U=  
    Cz x U @  
    % Compute the values of the polynomials: St&xe_:^<  
    % -------------------------------------- hG cq>Cvf  
    y = zeros(length_r,length(n)); a +Q9kh  
    for j = 1:length(n) y3$i?}?A  
        s = 0:(n(j)-m_abs(j))/2; d$s1l  
        pows = n(j):-2:m_abs(j); 4VPL -":6  
        for k = length(s):-1:1 @L^2VVWk^  
            p = (1-2*mod(s(k),2))* ... \pZ,gF;y  
                       prod(2:(n(j)-s(k)))/              ... \4I1wdd|^  
                       prod(2:s(k))/                     ... ^~(vP:  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... x^}kG[s  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); !<]%V]5[_  
            idx = (pows(k)==rpowers); XF(I$Mxl6  
            y(:,j) = y(:,j) + p*rpowern(:,idx); km'3[}8o&  
        end tfj6#{M5  
         8qn1? Lb  
        if isnorm 0\%/:2   
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Kxz<f>`b/  
        end QRXsLdf$$  
    end elb|=J`M0  
    % END: Compute the Zernike Polynomials ,"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O^hWG ~o  
    B2VC:TG>  
    F{ J>=TC  
    % Compute the Zernike functions: D61CO-E(D  
    % ------------------------------ < i*v  
    idx_pos = m>0; r#*kx#"  
    idx_neg = m<0; j[gX"PdQ  
    "T@9]>6.f  
    E#r6e+e1Q%  
    z = y; *}Zd QJL  
    if any(idx_pos) v0|A N  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); rH8^Fl&jT  
    end d7qY(!&  
    if any(idx_neg) }N(-e$88  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); y.2_5&e/  
    end `C`CU?D  
    ZO{uG(u  
    vL@N21u  
    % EOF zernfun KZzOs9 s  
     
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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)  vBx*bZ  
    rCi7q]_  
    DDE还是手动输入的呢? _R<eWp  
    0b+OB pqN  
    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