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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, \}jMC  
    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$cm:uks  
    这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? %@$UIO,(  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 3h:j.8Z  
    FpoH m%+  
    %!aU{E|@_  
    .sMs_ 5D  
    Z\&f"z?L  
    function z = zernfun(n,m,r,theta,nflag) >)><u4}  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. h2l;xt  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N X{9^$/XsJ  
    %   and angular frequency M, evaluated at positions (R,THETA) on the {#,<)wFV\  
    %   unit circle.  N is a vector of positive integers (including 0), and /{M<FVXK+|  
    %   M is a vector with the same number of elements as N.  Each element ! 'zd(kv<  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) c-LzluWi  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ?gH[la  
    %   and THETA is a vector of angles.  R and THETA must have the same hor7~u+  
    %   length.  The output Z is a matrix with one column for every (N,M) fF Q|dE;cF  
    %   pair, and one row for every (R,THETA) pair. pYr"3BwG  
    % qJ ey&_  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike &L o TO+  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), `lf_wB+I  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral kA :Y^2X'  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, SzULy >e  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 1.hWgWDP  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. #-{<d% qk  
    % xtV+Le%  
    %   The Zernike functions are an orthogonal basis on the unit circle. FX:`7c]:9  
    %   They are used in disciplines such as astronomy, optics, and UwN Vvo  
    %   optometry to describe functions on a circular domain. W4^L_p>Tm^  
    % i'tMpS3  
    %   The following table lists the first 15 Zernike functions. k"wQ9=HP7  
    % [W[{ 4 Xu  
    %       n    m    Zernike function           Normalization KK|w30\f  
    %       -------------------------------------------------- spK8^sh  
    %       0    0    1                                 1 Sp `l>BL  
    %       1    1    r * cos(theta)                    2 {X{R]  
    %       1   -1    r * sin(theta)                    2 st'T._  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) h my%X`%j  
    %       2    0    (2*r^2 - 1)                    sqrt(3) $8EEtr,!  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) P.~UU S  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) -D^I;[j_  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) sXLW';Fz  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ' jciX]g  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) _nGx[1G( 5  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) F72#vS j  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /:|vJ|dJ  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Im]@#X  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 8R~<$ xz  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) XF`2*:7  
    %       -------------------------------------------------- ,p2UshOmd  
    % \;;M")$  
    %   Example 1: 2+]5}'M  
    % !R{IEray  
    %       % Display the Zernike function Z(n=5,m=1) DE13x *2  
    %       x = -1:0.01:1; B|`?hw@g+  
    %       [X,Y] = meshgrid(x,x); ns[/M~_r  
    %       [theta,r] = cart2pol(X,Y); B-I4(w($  
    %       idx = r<=1; n Ja!&G&  
    %       z = nan(size(X)); 7?lz$.*Avp  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); S"bN9?;#u  
    %       figure vu0Ql1  
    %       pcolor(x,x,z), shading interp i4D(8;  
    %       axis square, colorbar *CN *G"  
    %       title('Zernike function Z_5^1(r,\theta)') 1(' wg!  
    % c[@_t.%)  
    %   Example 2: "M%R{pGA7  
    % #*A'<Zm  
    %       % Display the first 10 Zernike functions 79DNNj~  
    %       x = -1:0.01:1; VZ]iep  
    %       [X,Y] = meshgrid(x,x); Z[O hZ 9  
    %       [theta,r] = cart2pol(X,Y); HZrA}|:h  
    %       idx = r<=1; F`=p/IAJK  
    %       z = nan(size(X)); uYW4$6S 3  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Omd;  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 3Tr,waV  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ]2zM~  
    %       y = zernfun(n,m,r(idx),theta(idx)); A;cA|`b  
    %       figure('Units','normalized') }G4I9Py  
    %       for k = 1:10 KGt:  
    %           z(idx) = y(:,k); }i9:k kfq2  
    %           subplot(4,7,Nplot(k)) N2:Hdu :  
    %           pcolor(x,x,z), shading interp y_PA9#v7  
    %           set(gca,'XTick',[],'YTick',[]) cXXZ'y>FP  
    %           axis square G1|1Z5r  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ?XKX&ws  
    %       end ^[hAj>7_8$  
    % ^^q&VL  
    %   See also ZERNPOL, ZERNFUN2. @ZEBtM%.O  
    'O a3 6@  
    @&T' h}|:  
    %   Paul Fricker 11/13/2006 wd:Yy  
    nD i^s{  
    zC50 @S3|  
    , ['}9:f9  
    hcVu`Bn  
    % Check and prepare the inputs: 2V~E <K-  
    % ----------------------------- fY]"_P  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) # epP~J_f  
        error('zernfun:NMvectors','N and M must be vectors.') fW = N  
    end he|Q (?  
    %/dOV[/  
    3ynkf77cn  
    if length(n)~=length(m) K6{wM  
        error('zernfun:NMlength','N and M must be the same length.') ?NBae\6r  
    end 6R :hsC$  
    %9YY \a {  
    XPhP1 ^>\  
    n = n(:); Jm!,=} oP'  
    m = m(:); Kebr>t8^  
    if any(mod(n-m,2)) Q{~g<G  
        error('zernfun:NMmultiplesof2', ... 9]Jv >_W*  
              'All N and M must differ by multiples of 2 (including 0).') ?}`- ?JB1  
    end ^%!{qAp}Z  
    8K4^05*S   
    l8~(bq1  
    if any(m>n) >/ _#+,  
        error('zernfun:MlessthanN', ... (iKJ~bJ  
              'Each M must be less than or equal to its corresponding N.') xLed];2G  
    end S(@kdL  
    |GMo"[  
    iM!Ya!  
    if any( r>1 | r<0 ) ")KqPD6k  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') _DxHJl  
    end -k + jMH  
     hh4R  
    ?22U0UF  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) cr;:5D%_  
        error('zernfun:RTHvector','R and THETA must be vectors.') aEdA'>  
    end K/9Jx(I,qL  
    :]:)c8!6  
    x[mz`0  
    r = r(:); ;PaU"z+Je~  
    theta = theta(:); qu^g~"s  
    length_r = length(r); `h'+4  
    if length_r~=length(theta) RB4n>&Y  
        error('zernfun:RTHlength', ... ;6@sC[  
              'The number of R- and THETA-values must be equal.') brp3xgQ`]  
    end he(K   
    S ,F[74K  
    z5gVP8*z5  
    % Check normalization: Uha.8  
    % -------------------- 7:B/ ?E  
    if nargin==5 && ischar(nflag) ~!ooIwNNz  
        isnorm = strcmpi(nflag,'norm'); YE@yts  
        if ~isnorm \k5"&]I3  
            error('zernfun:normalization','Unrecognized normalization flag.') A6[FH\f  
        end n*"r!&Dg  
    else dC,C[7\  
        isnorm = false; NCh-BinK@  
    end N!ihj:,  
    eL~xS: VT  
    W,EIBgR(R5  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~AjPa}@ f  
    % Compute the Zernike Polynomials umns*U%T;  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GXxI=,L8F  
    x^@oY5}cr  
    QM8Ic,QFvo  
    % Determine the required powers of r: c2 NB@T9'v  
    % ----------------------------------- {C&U q#V  
    m_abs = abs(m); lrZ]c:%k  
    rpowers = []; XB7*S*"!  
    for j = 1:length(n) hZfj$|<  
        rpowers = [rpowers m_abs(j):2:n(j)]; g"748LY>=p  
    end \dCGu~bT  
    rpowers = unique(rpowers); vyDxX  
    keC'/\e  
    {@CQ (  
    % Pre-compute the values of r raised to the required powers, MrzD ah9UG  
    % and compile them in a matrix: |kK5:\H  
    % ----------------------------- sJKr%2nVV  
    if rpowers(1)==0 "a].v 8l!  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); tx7 zG.,  
        rpowern = cat(2,rpowern{:}); M?YNK]   
        rpowern = [ones(length_r,1) rpowern]; >%;i@"  
    else W:8MqVm34  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ]=t}8H  
        rpowern = cat(2,rpowern{:}); ,r*Kxy  
    end 27 XM&ZrZ  
    lIS`_H}  
    3F]Dh^IR9  
    % Compute the values of the polynomials: 8!|vp7/  
    % -------------------------------------- IQU1 JVk Z  
    y = zeros(length_r,length(n)); .O"a:^i  
    for j = 1:length(n) C IMI?  
        s = 0:(n(j)-m_abs(j))/2; ;&<N1  
        pows = n(j):-2:m_abs(j); W6T4Zsg  
        for k = length(s):-1:1 Jy/< {7j  
            p = (1-2*mod(s(k),2))* ... x?o#}:S  
                       prod(2:(n(j)-s(k)))/              ... iO?AY  
                       prod(2:s(k))/                     ... 7YD+zd:  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... o)XrC   
                       prod(2:((n(j)+m_abs(j))/2-s(k))); nE u:& 4  
            idx = (pows(k)==rpowers); qK7:[\T|?T  
            y(:,j) = y(:,j) + p*rpowern(:,idx); %d];h  
        end - (WH+  
         ('J@GTe@xj  
        if isnorm -_nQn  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); f$QkzWvr  
        end <&Xl b0  
    end _!1LV[x!s  
    % END: Compute the Zernike Polynomials 0F-{YQr>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,V,mz?d^9  
    ?Fx~_GT  
    lXTE#,XVf  
    % Compute the Zernike functions: %B\x %e ;P  
    % ------------------------------ Qu[QcB{ro-  
    idx_pos = m>0; .F8[;+  
    idx_neg = m<0; ^Zz^h@+  
    B?i#m^S  
    KGM__ZO.  
    z = y; 0zNbux_  
    if any(idx_pos) 2|^@=.4\  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); :.ZWYze  
    end ,B'=$PO%  
    if any(idx_neg) iH4LZ  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); H2BRI d  
    end #dae^UjM  
    #?w07/~L  
    [TOo 9W  
    % EOF zernfun NH|I>vyN  
     
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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)  0J'^<G TL  
    0x~+=GUN  
    DDE还是手动输入的呢?  it H  
    |u%;"N'p)  
    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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