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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, cc3+ Wx_  
    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, cn ;2&  
    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? jdD`C`w|,  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? T,4REbm^  
    ]"vpCL  
    J s,.$t  
    ][T>052v  
    ; JHf0  
    function z = zernfun(n,m,r,theta,nflag) pmDFmES  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 04E#d.o '  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N OFlY"O S[  
    %   and angular frequency M, evaluated at positions (R,THETA) on the p:4oA<V  
    %   unit circle.  N is a vector of positive integers (including 0), and SM`n:{N(  
    %   M is a vector with the same number of elements as N.  Each element PkdL] !:  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ^<e(3S:  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, (Mt-2+"+  
    %   and THETA is a vector of angles.  R and THETA must have the same ]*AQT7PH  
    %   length.  The output Z is a matrix with one column for every (N,M) :AC(  \  
    %   pair, and one row for every (R,THETA) pair. vj<JjGP  
    % I6 Q{ Axy  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike nA#dXckoc  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), $C&E3 'O  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral hQbz}x  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ?xCWg.#l4V  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized <a%RKjQvT  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. O>2i)M-h9x  
    % ,y*|f0&"~  
    %   The Zernike functions are an orthogonal basis on the unit circle. Ne2eBmY}(  
    %   They are used in disciplines such as astronomy, optics, and -xU4s  
    %   optometry to describe functions on a circular domain. B P0*`TY  
    % fUS1`  
    %   The following table lists the first 15 Zernike functions. UJQGwTA W  
    % n]P,5  
    %       n    m    Zernike function           Normalization IdWFG?b3  
    %       -------------------------------------------------- q{+Pf/M5  
    %       0    0    1                                 1 #uH%J<U  
    %       1    1    r * cos(theta)                    2 a.s5>:Ct  
    %       1   -1    r * sin(theta)                    2 7 +kU8}  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) yK:b $S  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ABnJ{$=n#  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) &BJ"T  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) =$Sd2UD  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) R"qxT.P(  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) /gq VXDY+`  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) J0 x)NnWJ  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 3g5 n>8-  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) O3["5  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) i}r|Zo  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) @ZGD'+zd?  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) o",J{  
    %       -------------------------------------------------- rE$=~s  
    % o) ,1R:  
    %   Example 1: c R6:AGr  
    % NN@'79x  
    %       % Display the Zernike function Z(n=5,m=1) @PyZ u7'  
    %       x = -1:0.01:1;  F'9#dR?  
    %       [X,Y] = meshgrid(x,x); , LVZ  
    %       [theta,r] = cart2pol(X,Y); :c`Gh< u  
    %       idx = r<=1; RD0=\!w*5  
    %       z = nan(size(X)); =2.q=a|'  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); q!\4|KF~  
    %       figure MPD<MaW$  
    %       pcolor(x,x,z), shading interp ,\=,,1_  
    %       axis square, colorbar K+ @R [  
    %       title('Zernike function Z_5^1(r,\theta)') BDz 7$k]  
    % `ehcj G1nY  
    %   Example 2: wOs t).  
    % YGf<!  
    %       % Display the first 10 Zernike functions bOS; 1~~  
    %       x = -1:0.01:1; " TP^:Ln  
    %       [X,Y] = meshgrid(x,x); %{;1i  
    %       [theta,r] = cart2pol(X,Y); )@[##F2  
    %       idx = r<=1; `(o:;<&3  
    %       z = nan(size(X)); _%:$sAj  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ^n&_JQIXb  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 5v,_ Hgh  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; sA!$}W  
    %       y = zernfun(n,m,r(idx),theta(idx)); ~"nF$DB  
    %       figure('Units','normalized') K:(E"d;  
    %       for k = 1:10 OV,t|  
    %           z(idx) = y(:,k); )4e?-?bK!  
    %           subplot(4,7,Nplot(k)) <S68UN(Ke  
    %           pcolor(x,x,z), shading interp  jWqjGX`  
    %           set(gca,'XTick',[],'YTick',[]) kqQT^6S   
    %           axis square 6,a:s:$>}R  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])  aK33bn'j  
    %       end z^^)n  
    % Z]qbLxJV  
    %   See also ZERNPOL, ZERNFUN2. G[$g-NU+  
    :kQydCuK  
    f O,5 u;  
    %   Paul Fricker 11/13/2006 N`et]'_A}  
    ;9$71E  
    Xli$4 uL   
    zy(NJ  
    &OsO _F  
    % Check and prepare the inputs: WJj5dqatV  
    % ----------------------------- \45F;f_r6  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) i8->3uB  
        error('zernfun:NMvectors','N and M must be vectors.') dTZ$92<  
    end 6W[~@~D=  
    2mEvoWnJ  
    G4](!f!Kv  
    if length(n)~=length(m) B-UsMO  
        error('zernfun:NMlength','N and M must be the same length.') }\0ei(%H  
    end *WaqNMD[%  
    qsWy <yL+  
    LY;Fjb yU  
    n = n(:); ->L>`<7(  
    m = m(:); e2qSU[  
    if any(mod(n-m,2)) 'S%H"W\  
        error('zernfun:NMmultiplesof2', ... sm"s2Ci=}  
              'All N and M must differ by multiples of 2 (including 0).') je8 5G`{DC  
    end L Iz<fB  
    |p.|zH  
    &&g02>gE  
    if any(m>n) hjD%=Ri0Z  
        error('zernfun:MlessthanN', ... uH]oHh!}j  
              'Each M must be less than or equal to its corresponding N.') +}R#mco5K  
    end KX J7\}  
    Xz`0nU  
    \{ve6`7Rn  
    if any( r>1 | r<0 ) lHAWZyO  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 8qL.L(=\/  
    end iD*L<9  
    VwOcWKD  
    h:RP/ 0E  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) R,ZG?/#uM9  
        error('zernfun:RTHvector','R and THETA must be vectors.') T~L&c  
    end 6n^@Ps  
    9y&bKB2,  
    GZ^Qt*5 {  
    r = r(:); ?N^1v&Q  
    theta = theta(:); X|-[i hp;  
    length_r = length(r); :V1j*)  
    if length_r~=length(theta) ~7an j.  
        error('zernfun:RTHlength', ... *3)kr=x  
              'The number of R- and THETA-values must be equal.') Al=ByX@  
    end $,P:B%]  
    XBoq/kbw!  
    MU%7'J :_  
    % Check normalization: #q 4uS~  
    % -------------------- HuJc*op-6  
    if nargin==5 && ischar(nflag) $<yhEvv  
        isnorm = strcmpi(nflag,'norm'); P0pBR_:o  
        if ~isnorm WdH/^QvTP  
            error('zernfun:normalization','Unrecognized normalization flag.') `Qjs {H  
        end stUUez>  
    else @{W"mc+  
        isnorm = false; |Ve,Y  
    end oKb"Ky@s  
    cPv(VjS1;  
    tva=DS  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f7y.##WG  
    % Compute the Zernike Polynomials qV6WT&)T  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `nKN|6o#x  
    ,J8n}7aI  
    =3 Vug2*wd  
    % Determine the required powers of r: AgZ?Ry  
    % ----------------------------------- 9z..LD(  
    m_abs = abs(m); e[16 7uU  
    rpowers = []; <Se9 aD  
    for j = 1:length(n) z$WLx  
        rpowers = [rpowers m_abs(j):2:n(j)]; {`G d  
    end J>5rkR@/  
    rpowers = unique(rpowers); !|up"T I  
    a|"Uw `pX+  
    5dB62dqN  
    % Pre-compute the values of r raised to the required powers, 7[w<v(Rc  
    % and compile them in a matrix: s8)`wH ?  
    % ----------------------------- s M*ay,v;  
    if rpowers(1)==0 mf)+ 5On  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 1I{8 |  
        rpowern = cat(2,rpowern{:}); a eeor  
        rpowern = [ones(length_r,1) rpowern]; !1fZ7a  
    else 9 @xl{S-  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Q2D!Agq=D  
        rpowern = cat(2,rpowern{:}); HC/z3b;  
    end |/vJ+aKq  
    f um.G{}  
    T4HJy|  
    % Compute the values of the polynomials: a5GLbanF  
    % -------------------------------------- EG;E !0  
    y = zeros(length_r,length(n)); BqY_N8l&E  
    for j = 1:length(n) )+hV+rM jp  
        s = 0:(n(j)-m_abs(j))/2; P/girce0  
        pows = n(j):-2:m_abs(j); ZGDT 6,  
        for k = length(s):-1:1 Rh?bBAn8  
            p = (1-2*mod(s(k),2))* ... Ff%V1BH[  
                       prod(2:(n(j)-s(k)))/              ... EgU#r@7I  
                       prod(2:s(k))/                     ... u;gO+)wqv  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... i*Ldec^  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 4] uj+J  
            idx = (pows(k)==rpowers); uTxa5j  
            y(:,j) = y(:,j) + p*rpowern(:,idx); /rnI"ze`  
        end kB> ~Tb0  
         {VE h@yn  
        if isnorm Cq"KKuf  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ^w.hI5ua)  
        end -g]Rs!w'  
    end <ZF|2  
    % END: Compute the Zernike Polynomials #uw&u6*\q  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7l=;I%  
    LWN {  
    wOl?(w=|  
    % Compute the Zernike functions: a/,>fv9;$  
    % ------------------------------ `;E/\eG"  
    idx_pos = m>0; hd(FOKOP  
    idx_neg = m<0; .mt%8GM  
    2t-w0~O  
    ) ;\c{QF  
    z = y; |4Q*4s  
    if any(idx_pos) B6Vlc{c5SO  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); M  9t7y  
    end _XV%}Xb'  
    if any(idx_neg) [d6!  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); #Ub_m@@ 4  
    end t>I.1AS  
    o@Oz a  
    DPTk5o[  
    % EOF zernfun _`|1B$@x  
     
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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)  ]%\,.&=hT  
    fqNh\~kja  
    DDE还是手动输入的呢? H?$dnwR  
    %go2tv:|W  
    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