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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, ?VrZM  
    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, #H{<nVvg^  
    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? Fh9%5-t:J  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? :@jhe8'w  
    j/4N  
    7F;"=DarOE  
    C7[ge&  
    4!p ~Mr[E  
    function z = zernfun(n,m,r,theta,nflag) *vc=>AEc  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. )A:2y +  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N W{O:j  
    %   and angular frequency M, evaluated at positions (R,THETA) on the jIv%?8+%  
    %   unit circle.  N is a vector of positive integers (including 0), and [_hHZMTH  
    %   M is a vector with the same number of elements as N.  Each element .281;] =  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) >8_#L2@  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, py`RH )  
    %   and THETA is a vector of angles.  R and THETA must have the same `*cT79  
    %   length.  The output Z is a matrix with one column for every (N,M) s\i=-`  
    %   pair, and one row for every (R,THETA) pair. eZ5UR014  
    % !<H[h4g  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike A"x1MjuqLM  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Vo}3E]  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral lE:X~RO"~  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, nv1'iSEeOl  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized &f'\9lO  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. g <^Y^~+E  
    % L@2%a'  
    %   The Zernike functions are an orthogonal basis on the unit circle. u +q}9  
    %   They are used in disciplines such as astronomy, optics, and NsJt=~  
    %   optometry to describe functions on a circular domain. &o{I9MD  
    % Yr@_X  
    %   The following table lists the first 15 Zernike functions. =A={ Dpv[>  
    % N]R<EBq  
    %       n    m    Zernike function           Normalization IG0$OtG  
    %       -------------------------------------------------- drP2% u  
    %       0    0    1                                 1 ?z%@;&  
    %       1    1    r * cos(theta)                    2 *T"JO |  
    %       1   -1    r * sin(theta)                    2 ?Y+xuY/t  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) s:lar4>kM  
    %       2    0    (2*r^2 - 1)                    sqrt(3) %^[45e  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 0Ge*\Q  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 5QB] 2c^  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) *6^|i}  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 9+"D8J7  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Os^sOOSY  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ]UKKy2r.  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) qH!}oPeU'  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) `fh^[Q|4n0  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *vv <@+gA  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 6eE%x?#  
    %       -------------------------------------------------- {k] 2h4 &h  
    % X).UvPZ/  
    %   Example 1: i)f3\?,,  
    % mbxJS_P  
    %       % Display the Zernike function Z(n=5,m=1) o0$R|/>i  
    %       x = -1:0.01:1; #q`[(`Bx  
    %       [X,Y] = meshgrid(x,x); E*ybf'  
    %       [theta,r] = cart2pol(X,Y); *k==2figz  
    %       idx = r<=1; jcHs!   
    %       z = nan(size(X)); v1<gNb)`  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); fpf1^ TZ  
    %       figure )#k*K9[@  
    %       pcolor(x,x,z), shading interp J"QXu M  
    %       axis square, colorbar O%5cMz?eU  
    %       title('Zernike function Z_5^1(r,\theta)') B 3|zR  
    % 'ah|cMRn  
    %   Example 2: _ _cJ+%e  
    % N ?Jr8  
    %       % Display the first 10 Zernike functions Yao>F--?  
    %       x = -1:0.01:1; WsRG>w3"  
    %       [X,Y] = meshgrid(x,x); D}'g4Ag  
    %       [theta,r] = cart2pol(X,Y); )~xL_yW_X  
    %       idx = r<=1; H|;6K`O_  
    %       z = nan(size(X)); JbpKstc;  
    %       n = [0  1  1  2  2  2  3  3  3  3]; =2uE\6Fl,  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; !O F#4N  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28];  hh<5?1  
    %       y = zernfun(n,m,r(idx),theta(idx)); !t "uNlN  
    %       figure('Units','normalized') -B :Z(]3#\  
    %       for k = 1:10 (1JZuR<?c  
    %           z(idx) = y(:,k); j[NA3Vj1P  
    %           subplot(4,7,Nplot(k)) xal,j*  
    %           pcolor(x,x,z), shading interp ,OWdp<z  
    %           set(gca,'XTick',[],'YTick',[]) Hn)K;?H4  
    %           axis square d,[.=Jqv[  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) sj a;NL  
    %       end *}R5=r0  
    % ;e;lPM{+  
    %   See also ZERNPOL, ZERNFUN2. nR4L4tdS  
    XSt5s06TM  
    ya5a7  
    %   Paul Fricker 11/13/2006 vb/*ILS  
    BF8n: }9U  
    IrQ8t!  
    *V#v6r7<Y/  
    Fn$/ K  
    % Check and prepare the inputs: NHA 2 i  
    % ----------------------------- /{YUM~  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) zk5sAHQ  
        error('zernfun:NMvectors','N and M must be vectors.') cd{3JGg B  
    end 5~k-c Ua  
    Pc{D,/EpR  
    .vNfbYH(  
    if length(n)~=length(m) {YZ)IaqZ  
        error('zernfun:NMlength','N and M must be the same length.') SFoF]U09  
    end hKtOh  
    b0X*+q   
    :Q2\3  
    n = n(:); Z)'jn8?P  
    m = m(:); Tj*o[2mD  
    if any(mod(n-m,2)) ,'5P[-  
        error('zernfun:NMmultiplesof2', ... KIn^,d0H  
              'All N and M must differ by multiples of 2 (including 0).') 5FqUFzVqsl  
    end RI w6i?/I  
    )<G>]IP<  
    toPA@V  
    if any(m>n) XOa<R  
        error('zernfun:MlessthanN', ... 8F($RnP3  
              'Each M must be less than or equal to its corresponding N.') Iu|G*~\  
    end gJi11^PK  
    -`wGF#}y(=  
    *7oPM5J|v  
    if any( r>1 | r<0 ) i_g="^  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 9F0B-aZ  
    end 9bgKu6-X  
    [UNfft=K3P  
    ~c ;7me.  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) efMv1>{  
        error('zernfun:RTHvector','R and THETA must be vectors.') %r6LU<;1@  
    end %#Wg>6  
    vq$%Ug/B  
    ,iCd6M{  
    r = r(:); TC U |k ,  
    theta = theta(:); &k\7fvF  
    length_r = length(r); V5KAiG<d  
    if length_r~=length(theta) _jH1Mcq  
        error('zernfun:RTHlength', ... \|R`wFn^P  
              'The number of R- and THETA-values must be equal.') ]=9%fA  
    end @SPmb o  
    W#e:rz8=  
    6`NsX  
    % Check normalization: BdUhFN*  
    % -------------------- ig; ~ T  
    if nargin==5 && ischar(nflag) [rTV)JsTb  
        isnorm = strcmpi(nflag,'norm'); 8v1asFxs.  
        if ~isnorm cgYMo{R3  
            error('zernfun:normalization','Unrecognized normalization flag.') 0VoC|,$U  
        end ~FZLA}  
    else PNT.9 *d  
        isnorm = false; `]5XY8^kI  
    end 8(KsU,%d  
    ~/3cQN^  
    g%j z,|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% po=*%Zs*T  
    % Compute the Zernike Polynomials dyWWgC%A  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -2> L*"^  
    p: sn>Y  
    b_V)]>v+  
    % Determine the required powers of r: FD|R4 V*3  
    % ----------------------------------- LU?#{dZ  
    m_abs = abs(m); rorzxp{  
    rpowers = []; dq:M!F  
    for j = 1:length(n) ~l6e&J  
        rpowers = [rpowers m_abs(j):2:n(j)]; }E>2U/wpXY  
    end U{>!`RN  
    rpowers = unique(rpowers); )yJeh  
    2:pq|eiF  
    >z^T~@m7l  
    % Pre-compute the values of r raised to the required powers, ys+?+dY2  
    % and compile them in a matrix: l*'8B)vN2  
    % ----------------------------- pKEMp&geo  
    if rpowers(1)==0 q6j]j~JxB  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 2d.I3z:[  
        rpowern = cat(2,rpowern{:}); BC@"WlD  
        rpowern = [ones(length_r,1) rpowern]; d /t'N-m  
    else 3J'a  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); s ]QzNc  
        rpowern = cat(2,rpowern{:}); _rs#h)  
    end 0QC*Z (  
    Pv1psKu  
    -v jjcyTt  
    % Compute the values of the polynomials: ~PlwPvWo  
    % -------------------------------------- lq.0?(  
    y = zeros(length_r,length(n)); "g=ux^+X\  
    for j = 1:length(n) N`iK1n4 X  
        s = 0:(n(j)-m_abs(j))/2; \re.KB#R  
        pows = n(j):-2:m_abs(j); t9K.Jc0  
        for k = length(s):-1:1 1>1|>%  
            p = (1-2*mod(s(k),2))* ... Ccc6 ko_  
                       prod(2:(n(j)-s(k)))/              ... N_gjOE`x5  
                       prod(2:s(k))/                     ... ~MhPzu&B  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 3ZZJYf=  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 3\(s=- vh  
            idx = (pows(k)==rpowers); [MiD%FfcNH  
            y(:,j) = y(:,j) + p*rpowern(:,idx); TfZO0GL$  
        end B=Zo0 p^  
         !)\`U/.W  
        if isnorm >_F& oA#  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); |23 }~c,  
        end (nE$};c<b2  
    end uO^{+=;A =  
    % END: Compute the Zernike Polynomials jG.*tuf  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Sl$dXB@  
    ?QuFRl,ZJ  
    1:>RQPXcWv  
    % Compute the Zernike functions: *Lh0E/5  
    % ------------------------------ [j!0R'T  
    idx_pos = m>0; 9*2hBNp+  
    idx_neg = m<0; vfy- ;R(  
    V_ ]4UE  
    "M:arP5f  
    z = y; Me`"@{r|#  
    if any(idx_pos) 9J|YP}%  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Gg;#U`  
    end 9J%>2AA  
    if any(idx_neg) be764do  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); uY;/3 ?k&  
    end C8t+-p  
    4\$Ze0tv  
    5#fLGXP  
    % EOF zernfun [p7le8=  
     
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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)  A]+h<Y~}  
    u#M)i30j  
    DDE还是手动输入的呢? s4gNS eA  
    xs I/DW  
    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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