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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 正序阅读 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, !;Hi9,<#7g  
    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, 4cZig\mE;  
    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? &G63ReW7 @  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? P(iZGOKUs=  
    "p]Fq,  
    _<Hx1l~  
    X( Q*(_  
    K&1o!<|  
    function z = zernfun(n,m,r,theta,nflag) &qR1fbw"  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. iV+'p->/  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N +Smt8O<N  
    %   and angular frequency M, evaluated at positions (R,THETA) on the nT7{`aaQl  
    %   unit circle.  N is a vector of positive integers (including 0), and ?t;>]Wo;  
    %   M is a vector with the same number of elements as N.  Each element "F_o%!l  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 4a'O#;h o  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, si`{>e~`6P  
    %   and THETA is a vector of angles.  R and THETA must have the same X`xI~&t_  
    %   length.  The output Z is a matrix with one column for every (N,M) 2 uuI_9 "^  
    %   pair, and one row for every (R,THETA) pair. oL?[9aww  
    % [h"#Gwb=;  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike TTNgnP  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 1-z*'Ghys  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral lo}[o0X  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, _W@SCV)yH  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized *7 L*:g  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 44s K2  
    % ,p(4OZz5,  
    %   The Zernike functions are an orthogonal basis on the unit circle. w8~J5XS  
    %   They are used in disciplines such as astronomy, optics, and $`nKq4Y   
    %   optometry to describe functions on a circular domain. y&y(<  
    % sy^k:y?  
    %   The following table lists the first 15 Zernike functions. XTIRY4{ d  
    % W@S'mxk#*  
    %       n    m    Zernike function           Normalization  84PD`A  
    %       -------------------------------------------------- 7Pt*V@DHS  
    %       0    0    1                                 1 kBPFk t2  
    %       1    1    r * cos(theta)                    2 U3ygFW%  
    %       1   -1    r * sin(theta)                    2 pB @l+ n^  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) %9_wDfw~  
    %       2    0    (2*r^2 - 1)                    sqrt(3) >.R6\>N%  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 4SG22$7W  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) !U02>X   
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) |pIA9/~Z  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ":,HY)z  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ^V^In-[!y:  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) WY@x2bBi  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -25#Vh  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) >40B Fxc  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Z_%}pe39B  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ] =Js5  
    %       -------------------------------------------------- qA- ya6  
    % Q*TxjE7K  
    %   Example 1: #vO3*-hs  
    % Q9K+k*?{N  
    %       % Display the Zernike function Z(n=5,m=1) Z2chv,SqCJ  
    %       x = -1:0.01:1; )k&pp^q\  
    %       [X,Y] = meshgrid(x,x); 1B3,lYBM  
    %       [theta,r] = cart2pol(X,Y); Rl4r 9  
    %       idx = r<=1; `R@24 )  
    %       z = nan(size(X)); Ow\9vf6H  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); F"a^`E&  
    %       figure 0w >DU^+  
    %       pcolor(x,x,z), shading interp (l 2 2p  
    %       axis square, colorbar <$liWAGX\  
    %       title('Zernike function Z_5^1(r,\theta)') 6'C!Au  
    % S(A0),  
    %   Example 2: zIbl[[M&  
    % ;{|a~e?Y  
    %       % Display the first 10 Zernike functions Q6S[sTKR  
    %       x = -1:0.01:1; X7kJWX  
    %       [X,Y] = meshgrid(x,x); IidZ -Il  
    %       [theta,r] = cart2pol(X,Y); \h^bOxh  
    %       idx = r<=1; a@@!Eg A  
    %       z = nan(size(X)); y? [*qnPj  
    %       n = [0  1  1  2  2  2  3  3  3  3]; }\u~He%  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; C!w@Naj  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; bcpH|}[F)  
    %       y = zernfun(n,m,r(idx),theta(idx)); tYfhKJzGC  
    %       figure('Units','normalized') NrvS/ cI!t  
    %       for k = 1:10 w8%yX$<  
    %           z(idx) = y(:,k); m@JU).NKCS  
    %           subplot(4,7,Nplot(k)) 1elx~5v1.=  
    %           pcolor(x,x,z), shading interp _}]o~  
    %           set(gca,'XTick',[],'YTick',[]) >ge-yK 1  
    %           axis square Tu_dkif'  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 'D(Hqdr;:  
    %       end 7kn=j6I  
    % \Y9=d E}  
    %   See also ZERNPOL, ZERNFUN2. 9[N' HpQ3  
    SU# S'  
    @n(=#Q3  
    %   Paul Fricker 11/13/2006 1jmhh !,  
    [v-?MS  
    IJ, ,aCj4g  
    r"fu{4aX  
    :yT~.AK}>1  
    % Check and prepare the inputs: 9r1pdG_C@  
    % ----------------------------- BHj]w*Ov  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~Y)Au?d(a  
        error('zernfun:NMvectors','N and M must be vectors.') pq5)Ug  
    end ](_(1  
    j<deTK;.  
    aic6,>\!'  
    if length(n)~=length(m) O8u"Y0$*w  
        error('zernfun:NMlength','N and M must be the same length.') Tf@t.4\  
    end @YwaOc_%  
    |r-<t  
    8gC(N3/E"  
    n = n(:); XQ(`8Jl&^  
    m = m(:); Rl5}W\&  
    if any(mod(n-m,2)) uy\YJ.WMQ  
        error('zernfun:NMmultiplesof2', ... n]Dq  
              'All N and M must differ by multiples of 2 (including 0).') *7*g! km  
    end LO"HwN43h  
    iI&SI#; _  
    >4EcV1y  
    if any(m>n) [RpFC4W  
        error('zernfun:MlessthanN', ... U}A+jJ  
              'Each M must be less than or equal to its corresponding N.') cjN4U [  
    end N[pk@M\vX  
    OD1ns  
    6l_8Q w*5I  
    if any( r>1 | r<0 ) BC$In!  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') q=nMZVVlF(  
    end L0&!Qct  
    !Rb7q{@>  
    Kv#daAU  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) j|aT`UH03  
        error('zernfun:RTHvector','R and THETA must be vectors.')  Mx r#  
    end jilO%  "  
    rkD4}jV  
    t*}<v@,  
    r = r(:); [2\`Wh:%P  
    theta = theta(:); T@Q<oNU  
    length_r = length(r); G,"$Erx  
    if length_r~=length(theta) a|s=d  
        error('zernfun:RTHlength', ... |u}sX5/q  
              'The number of R- and THETA-values must be equal.') *<0g/AL  
    end Z#J{tXZc  
    b&_p"8)_  
    I(7gmCV  
    % Check normalization: mmjB1 L  
    % -------------------- U_8I$v-~  
    if nargin==5 && ischar(nflag) 3p4bOT5  
        isnorm = strcmpi(nflag,'norm'); j_H T  
        if ~isnorm }E1Eq  
            error('zernfun:normalization','Unrecognized normalization flag.') v'@LuF'e8  
        end 9Akwr}  
    else =:0(&NCRq  
        isnorm = false; [cW  
    end ^X;>?_Bk  
    h=U 4  
    *xjIl<`pK  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% JWdG?[$  
    % Compute the Zernike Polynomials 5g5pzww  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AN1bfF:C  
    h n ]6he  
    U&/S  
    % Determine the required powers of r: $?GO|.59  
    % ----------------------------------- }N|/b"j9  
    m_abs = abs(m); )I$Mh@F  
    rpowers = []; X'F$K!o*,:  
    for j = 1:length(n) {vH8X(m  
        rpowers = [rpowers m_abs(j):2:n(j)]; "nefRz%j+  
    end )/pPY  
    rpowers = unique(rpowers); }wb;ulN)  
    DtN6.9H2`  
    E<4}mSn)  
    % Pre-compute the values of r raised to the required powers, X5yhS  
    % and compile them in a matrix: |S}*M<0  
    % ----------------------------- OlJj|?z $  
    if rpowers(1)==0 S\rfR N  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 24Tw1'mW  
        rpowern = cat(2,rpowern{:}); E,$uN w']  
        rpowern = [ones(length_r,1) rpowern]; fh 3 6  
    else W!^=)Qs  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); l`]!)j|+  
        rpowern = cat(2,rpowern{:}); qs b4@jt+  
    end &ivIv[LV  
    n 3]y$wK  
    .KSGma6]  
    % Compute the values of the polynomials: [P,nW/H  
    % -------------------------------------- cA\W|A)  
    y = zeros(length_r,length(n)); Dw[Q,SE   
    for j = 1:length(n) 1mV0AE538  
        s = 0:(n(j)-m_abs(j))/2; `ouzeu9}  
        pows = n(j):-2:m_abs(j); &40]sxm  
        for k = length(s):-1:1 NeEV !V8  
            p = (1-2*mod(s(k),2))* ... Ye6O!,R  
                       prod(2:(n(j)-s(k)))/              ... "F}Ip&]hAG  
                       prod(2:s(k))/                     ... FHC7\#p/9Z  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... qQ'@yTVN  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); <i6MbCB  
            idx = (pows(k)==rpowers); eH8.O  
            y(:,j) = y(:,j) + p*rpowern(:,idx); k}.nH"AQ  
        end u2Obb`p S  
         q}i87a;m  
        if isnorm 4\3t5n  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 7KIQ)E'kG|  
        end Uy:.m  
    end FM)*>ax{  
    % END: Compute the Zernike Polynomials 2cl~Va=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g-}sVvM  
    9R[','x  
    ;p ('cwU%  
    % Compute the Zernike functions: AlxS?f2w  
    % ------------------------------ v],DBw9  
    idx_pos = m>0; 4C cb!?  
    idx_neg = m<0; ?OyW|jL  
    TbVL71c  
    eV0S:mit  
    z = y; +GS=zNw#  
    if any(idx_pos) xn8B|axB  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); R2`g?5v  
    end S/;Y4o  
    if any(idx_neg) 1n"X?K5;A  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Se8y-AL6x>  
    end 6%#'X  
    B_2>Yt"  
    L#Y;a 5b  
    % EOF zernfun 9(WC#-,  
     
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    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
    离线jssylttc
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    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    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)  P^m 6di  
    &~A*(+S  
    DDE还是手动输入的呢? ~w9 =Fd6  
    xp8f  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    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。
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