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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, \LXNdE2B  
    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, obGSc)?j  
    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? [-a /]  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? `r LMMYD=  
    izA3INT  
    Zwl?*t\D  
    <h0ptCB  
    roQIP%h!  
    function z = zernfun(n,m,r,theta,nflag) ] "_'o~  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. |[ofc!/  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N :6{HFMf"  
    %   and angular frequency M, evaluated at positions (R,THETA) on the aS 2 Y6  
    %   unit circle.  N is a vector of positive integers (including 0), and )BDi2: u  
    %   M is a vector with the same number of elements as N.  Each element 7G2N&v>  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) _95tgJy  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, GV/FK{v5  
    %   and THETA is a vector of angles.  R and THETA must have the same I`1=VC]^8  
    %   length.  The output Z is a matrix with one column for every (N,M) ](pD<FfS]'  
    %   pair, and one row for every (R,THETA) pair. ~o$=(EC  
    % ['j,S<Bu~  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike y0^FTSQ|  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), I}x*AM 7+  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Ho|n\7$  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, "m5ZZG#R`  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ]T`qPIf;yJ  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. hG]20n2  
    % 4mg&H0 !  
    %   The Zernike functions are an orthogonal basis on the unit circle. '@bA_F(  
    %   They are used in disciplines such as astronomy, optics, and 2{\Y<%.  
    %   optometry to describe functions on a circular domain. 2(|V1]6D?  
    % [g_@<?zg  
    %   The following table lists the first 15 Zernike functions. g!UM8I-$  
    % c$;enAf@  
    %       n    m    Zernike function           Normalization - Zh+5;8g  
    %       -------------------------------------------------- ap!<8N  
    %       0    0    1                                 1 d=XhOC$  
    %       1    1    r * cos(theta)                    2 6dp~19T^  
    %       1   -1    r * sin(theta)                    2 6(=:j"w0  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ~x+w@4)a>  
    %       2    0    (2*r^2 - 1)                    sqrt(3) `P~RG.HO  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) dewu@  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ]]4E)j8  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) B~IOM  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) fA^O  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) R<)uvW_@  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) `JCC-\9T_  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }PJ:9<G y  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) I/l]Yv!  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) tKs0]8tc  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) S3m+(N"&  
    %       -------------------------------------------------- $- L)>"  
    % K!X8KPo  
    %   Example 1: KpL82  
    % 5+r#]^eQY-  
    %       % Display the Zernike function Z(n=5,m=1) &nYmVwi?"Q  
    %       x = -1:0.01:1; &wfM:a/c  
    %       [X,Y] = meshgrid(x,x); STMcMm3  
    %       [theta,r] = cart2pol(X,Y); {+MMqJCa  
    %       idx = r<=1; :?TV6M  
    %       z = nan(size(X)); ~zx-'sc?  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); o0q{:An_Z  
    %       figure +qdK]RR}  
    %       pcolor(x,x,z), shading interp ]pt @  
    %       axis square, colorbar MX34qJ9k  
    %       title('Zernike function Z_5^1(r,\theta)') 03xQ%"TU<  
    % Kh>^;`h  
    %   Example 2: %`~8j H@  
    % <8Ad\MU  
    %       % Display the first 10 Zernike functions bm^ou#]|  
    %       x = -1:0.01:1; "6ZatRUd  
    %       [X,Y] = meshgrid(x,x); cX2b:  
    %       [theta,r] = cart2pol(X,Y); 0Z\fK>yw  
    %       idx = r<=1; f%af.cR*  
    %       z = nan(size(X)); 3yQ(,k#  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ,SBL~JJ  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 0y(d|;':  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; G100L}d"N  
    %       y = zernfun(n,m,r(idx),theta(idx)); !tVV +vT#  
    %       figure('Units','normalized') ~ rRIWfhb  
    %       for k = 1:10 z')'8155  
    %           z(idx) = y(:,k); 22GtTENd1h  
    %           subplot(4,7,Nplot(k)) ,J[sg7v cv  
    %           pcolor(x,x,z), shading interp qdOS=7]W  
    %           set(gca,'XTick',[],'YTick',[]) sU>*S$X8  
    %           axis square yF*JzE 7,  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) tY7u\Y;^  
    %       end vi'K|[!?  
    % ]}9EBf  
    %   See also ZERNPOL, ZERNFUN2. ve$P=ZuM  
    ? in&/ZrB  
    =I?p(MqW  
    %   Paul Fricker 11/13/2006 6>l-jTM  
    ]fR 3f  
    )2a!EEHz  
    DQ,QyV  
    #xO`k1W.  
    % Check and prepare the inputs: (T@ov~ @  
    % ----------------------------- YpiSH(70`  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) !nu#r$K(  
        error('zernfun:NMvectors','N and M must be vectors.') Dv$xP)./  
    end `/"z.~8  
    @sVBG']p  
    h7g9:10  
    if length(n)~=length(m) <#c2Hg%jh  
        error('zernfun:NMlength','N and M must be the same length.') Z*JZ Ubo-Q  
    end o;"!#Z 1SJ  
    @x)z" )>  
    1 @/+ c  
    n = n(:); > vgqf>)kk  
    m = m(:); |/q*Fg[f  
    if any(mod(n-m,2)) qoEOM%dAqV  
        error('zernfun:NMmultiplesof2', ... !OiP<8 ,H  
              'All N and M must differ by multiples of 2 (including 0).') L,R9jMx?_  
    end e Q0bx&  
    0ya_[\  
    RVD=CX  
    if any(m>n) 62.{8Uj  
        error('zernfun:MlessthanN', ... *G=n${'  
              'Each M must be less than or equal to its corresponding N.') wTOB'  
    end eM8u ;i  
    pnf3YuB  
    WC`<N4g|  
    if any( r>1 | r<0 ) iK)w3S}k1y  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') F7mzBrz  
    end ?Hq`*I?b9  
    :kgwKuhL  
    JBuorc  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) y1P?A]v  
        error('zernfun:RTHvector','R and THETA must be vectors.') B [03,zVf  
    end ?vvjwys@  
    x *(pr5k  
    #B54p@.}  
    r = r(:); WWD\EDnS  
    theta = theta(:); eGZId v1  
    length_r = length(r); w)hJ0k  
    if length_r~=length(theta) +-5CM0*&  
        error('zernfun:RTHlength', ... @UD6qA  
              'The number of R- and THETA-values must be equal.') yBeSvsm  
    end R\6#J0&Y-  
    9erTb?@S  
    #t9&X8:U  
    % Check normalization: +>{{91mN  
    % -------------------- { R&F_51)V  
    if nargin==5 && ischar(nflag) 1#XMUbFc  
        isnorm = strcmpi(nflag,'norm'); F)!B%4  
        if ~isnorm k4eV*e8  
            error('zernfun:normalization','Unrecognized normalization flag.') h}.0Ne  
        end b5KX`r  
    else ,>e)8  
        isnorm = false; S__+S7]Nr  
    end *|MPYxJ<  
    ]l`?"X|^  
    J/=b1{d"n  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Gh|q[s*k  
    % Compute the Zernike Polynomials 9CW .xX8  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I9TOBn|6   
    gy,B+~p  
    dfO84Z} 5  
    % Determine the required powers of r: %5$yz|:  
    % ----------------------------------- *=)%T(^  
    m_abs = abs(m); q>f1V3  
    rpowers = []; a'W-&j  
    for j = 1:length(n) enE8T3   
        rpowers = [rpowers m_abs(j):2:n(j)]; m8#+w0p)  
    end Lw1~$rZg  
    rpowers = unique(rpowers); bv-s}UP0  
    OV^) N  
    O~Pb u[C  
    % Pre-compute the values of r raised to the required powers, xLX:>64'o>  
    % and compile them in a matrix: ~O&3OL:L  
    % ----------------------------- +Z#lf  
    if rpowers(1)==0 L-",.U*;  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); $D<LND=o=  
        rpowern = cat(2,rpowern{:}); %Gh!h4Pv  
        rpowern = [ones(length_r,1) rpowern]; (khjP ,  
    else c2-NXSjsW  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ){ArZjG>  
        rpowern = cat(2,rpowern{:}); pd/{yX M  
    end U_B"B;ng+  
    fMP$o3;  
    [r<lAS{ .  
    % Compute the values of the polynomials: BbnY9"  
    % -------------------------------------- 2:Zb'Mj  
    y = zeros(length_r,length(n)); 5$`ihO?  
    for j = 1:length(n) xOp8[6Ga'  
        s = 0:(n(j)-m_abs(j))/2; BMgiXdv.B  
        pows = n(j):-2:m_abs(j); XN'x`%!*3#  
        for k = length(s):-1:1 ix [aS  
            p = (1-2*mod(s(k),2))* ... [2WJ>2r}6  
                       prod(2:(n(j)-s(k)))/              ... IhhB^E|  
                       prod(2:s(k))/                     ... T&j_7Q\;vI  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... $i7iv  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); M\ B A+  
            idx = (pows(k)==rpowers); &>XIK8*  
            y(:,j) = y(:,j) + p*rpowern(:,idx); [yJcM [p\  
        end i*_T\_=  
         f4@>7K]9TA  
        if isnorm /n"Ib )M  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); KD11<&4_x  
        end }YfM <  
    end -NGY+1  
    % END: Compute the Zernike Polynomials 3){ /u$iH.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /\q1,}M  
    ]X ,f  
    {=pRU_-^  
    % Compute the Zernike functions: xxLD8?@e7  
    % ------------------------------ w)2X0ev"  
    idx_pos = m>0; (&npr96f  
    idx_neg = m<0; 2^'|[*$k1@  
    _l<e>zj  
    KP(RK4F  
    z = y; ROw9l!YF  
    if any(idx_pos) *G"L]Nq#  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); mI_ ?hl?Pv  
    end #T &z`  
    if any(idx_neg) 'Y Bz?l9  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); h&|q>M3  
    end j-e/nZR@  
    Z/n\Ak sE  
    r+r-[z D(  
    % EOF zernfun sN]O]qYXJ  
     
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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)  q!l[^t|;  
    z;x1p)(xt  
    DDE还是手动输入的呢? "],amJ  
    +bnz%/v  
    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