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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, q*{"6"4(  
    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, BqZLqGO Ku  
    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? *B:{g>0  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ~ezCE4^&  
    cIM5;"gLP  
    (-dJ0!  
    fY%Sw7ql<  
    WtRy~5A2  
    function z = zernfun(n,m,r,theta,nflag) \TMRS(  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. R<UjhCvx.  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N [&3"kb  
    %   and angular frequency M, evaluated at positions (R,THETA) on the w5|@vB/pj  
    %   unit circle.  N is a vector of positive integers (including 0), and L6 _Sc-sU  
    %   M is a vector with the same number of elements as N.  Each element ;;nmF#  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) m(OBk;S~   
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ^* DKF  
    %   and THETA is a vector of angles.  R and THETA must have the same ui 2RTAb  
    %   length.  The output Z is a matrix with one column for every (N,M) UO:>^,(j  
    %   pair, and one row for every (R,THETA) pair. `SW`d<+L  
    % yAi4v[  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike =?*V3e3{  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), q6_1`Ew  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral J.?p?-"  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, :cynZab  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized @XIwp2A{+  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. yp*kMC,3  
    % m9e$ZZG$  
    %   The Zernike functions are an orthogonal basis on the unit circle. V;:A&  
    %   They are used in disciplines such as astronomy, optics, and HKxrBQr78  
    %   optometry to describe functions on a circular domain. J7cqnj  
    % uwQ4RYz  
    %   The following table lists the first 15 Zernike functions. fZ %ZV  
    % IB;y8e,  
    %       n    m    Zernike function           Normalization \'p7,F{:>5  
    %       -------------------------------------------------- 4P:vo$Cy  
    %       0    0    1                                 1 J| DWT+$#Z  
    %       1    1    r * cos(theta)                    2 lJYv2EZ  
    %       1   -1    r * sin(theta)                    2 +M.|D,wg2  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) dO8Z {wfs  
    %       2    0    (2*r^2 - 1)                    sqrt(3) X*Q7Yu  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 'Gt`3qG  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) V&}Z# 9Dx  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 9n%W-R.  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) }oU&J81  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Sv>aZ  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 1Gqtd^*;  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) QB@*/Le   
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) \Fe5<G'v  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) B"B  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) bFJn-g n  
    %       -------------------------------------------------- ^a}{u$<  
    % ?<}qx`+%Q  
    %   Example 1: q{5Vq_s\  
    % }}xR?+4A  
    %       % Display the Zernike function Z(n=5,m=1) hs*:!&E  
    %       x = -1:0.01:1; eo,]b1C2n  
    %       [X,Y] = meshgrid(x,x); ~g,QwaA[  
    %       [theta,r] = cart2pol(X,Y); ){(cRB$  
    %       idx = r<=1; pucHB<R@bL  
    %       z = nan(size(X)); dW5z0VuB$/  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); pKJ[e@E^  
    %       figure #,9|Hr%  
    %       pcolor(x,x,z), shading interp s`TBz8QO$  
    %       axis square, colorbar ujSzm=_P  
    %       title('Zernike function Z_5^1(r,\theta)') So 5{E 4[  
    % x-QP+M`Pu  
    %   Example 2: ZEMo`O  
    % j>:T)zhyY  
    %       % Display the first 10 Zernike functions 97 g-*K  
    %       x = -1:0.01:1; @kK=|(OB'  
    %       [X,Y] = meshgrid(x,x); @Uu\x~3y  
    %       [theta,r] = cart2pol(X,Y); E:tUbWVp  
    %       idx = r<=1; N1-LM9S  
    %       z = nan(size(X)); hPH7(f|c{g  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Eg:p_F*lr  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 2#[Y/p  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Z`!pU"O9l  
    %       y = zernfun(n,m,r(idx),theta(idx)); t2gjhn^p  
    %       figure('Units','normalized') h=tY 5]8  
    %       for k = 1:10 f_\-y&)+*  
    %           z(idx) = y(:,k); ) jvkwC  
    %           subplot(4,7,Nplot(k)) aD^MoB3  
    %           pcolor(x,x,z), shading interp (l,o UBRr  
    %           set(gca,'XTick',[],'YTick',[]) loB/w{r*x  
    %           axis square O^:h_L  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) r6)1Y`K=9  
    %       end 9..k/cH  
    % ~_&.A*Jh  
    %   See also ZERNPOL, ZERNFUN2. K|/a]I":  
    Rb0{t[IU  
    -a[{cu{  
    %   Paul Fricker 11/13/2006 O o:jP6r  
    *l^'v9  
    Y[DKj!v  
    Ss:,#|   
    ? `KOW  
    % Check and prepare the inputs: N'1~wxd  
    % ----------------------------- rifxr4c[X>  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) C"{on%  
        error('zernfun:NMvectors','N and M must be vectors.') g2]-Q.  
    end ?Sqm`)\>4  
    cn0Fz"d  
    @FV;5M:I  
    if length(n)~=length(m) m0"\3@kB  
        error('zernfun:NMlength','N and M must be the same length.') {;E/l(HNI  
    end -(.7/G'Vk>  
    12a #]E  
    [ lW "M  
    n = n(:); >N J$ac  
    m = m(:); {+:XVT_+  
    if any(mod(n-m,2)) g@k9w{_  
        error('zernfun:NMmultiplesof2', ... w!RH*S  
              'All N and M must differ by multiples of 2 (including 0).') \gkajY-?  
    end hl:eF:'hm  
    a AM UJk  
    q@9 i3*q;  
    if any(m>n) `[CJtd2\  
        error('zernfun:MlessthanN', ... }hYE6~pr  
              'Each M must be less than or equal to its corresponding N.') <T[N.mB  
    end +a-@ !J~:  
    HH?*"cKF~  
    to 6Q90(  
    if any( r>1 | r<0 ) DeA'D|  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') [R>   
    end ntV >m*^  
    CE,O m^  
    oDUMoX%4s  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) %ZyPK,("  
        error('zernfun:RTHvector','R and THETA must be vectors.') hH}/v0_jb  
    end S$52KOo  
    GuWBl$|+b  
    XB-|gPk  
    r = r(:); E{ s|#  
    theta = theta(:); QtQ^"d65  
    length_r = length(r); =bWq 3aP)P  
    if length_r~=length(theta) QJWES%m`  
        error('zernfun:RTHlength', ... Rk%M~D*-  
              'The number of R- and THETA-values must be equal.') dY<#a,eS  
    end ~iZF~PQ1_  
    %k #Nu  
    ;/ KF3 %  
    % Check normalization: X.l"f'`l  
    % -------------------- TSSt@xQ+  
    if nargin==5 && ischar(nflag) vw] D{OBv*  
        isnorm = strcmpi(nflag,'norm'); ,jsx]U/^  
        if ~isnorm JK"uj%  
            error('zernfun:normalization','Unrecognized normalization flag.') (B,t 1+%  
        end T1HiHvJ  
    else o%PoSZZ  
        isnorm = false; L"7` \4  
    end %@93^q[\2  
    j :Jdwf  
    P=[x!}.I  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j jT 2k  
    % Compute the Zernike Polynomials dG>Wu o  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% OCO,-(  
    [e@OHQM  
    +1jqCW  
    % Determine the required powers of r: h$ iyclX  
    % ----------------------------------- _8pkejg  
    m_abs = abs(m); TL{pc=eBo  
    rpowers = []; 1=5'R/k  
    for j = 1:length(n) s k6|_  
        rpowers = [rpowers m_abs(j):2:n(j)]; R 4QwWSBJ  
    end a 8hv.43  
    rpowers = unique(rpowers); rI66frbj  
    lEb R)B,  
    OGi4m |  
    % Pre-compute the values of r raised to the required powers, .Xz"NyW  
    % and compile them in a matrix: I u~aTgHX%  
    % ----------------------------- %802H%+  
    if rpowers(1)==0 zHc4e   
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); rn[}{1I33Q  
        rpowern = cat(2,rpowern{:}); ?Gv!d  
        rpowern = [ones(length_r,1) rpowern]; I cA\3j  
    else \]#;!6ge  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); j X!ftm2  
        rpowern = cat(2,rpowern{:}); %3#I:>si  
    end +fCyR  
    2td|8vDA  
    ="w8U'  
    % Compute the values of the polynomials: VmH_0IM^6  
    % -------------------------------------- CE7pg&dJ)i  
    y = zeros(length_r,length(n)); ^@LhUs>3  
    for j = 1:length(n) }Oh'YX#[  
        s = 0:(n(j)-m_abs(j))/2; 9c5G6n0  
        pows = n(j):-2:m_abs(j); =']};  
        for k = length(s):-1:1 > <  _Z  
            p = (1-2*mod(s(k),2))* ... 19w,'}CGk  
                       prod(2:(n(j)-s(k)))/              ... @uM3iO7&  
                       prod(2:s(k))/                     ... 7- 3N  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... >;&V~q:di  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); S}p&\w H  
            idx = (pows(k)==rpowers); -f;j1bQ  
            y(:,j) = y(:,j) + p*rpowern(:,idx); [F V=@NI  
        end )>X|o$2  
         # pjyhH@  
        if isnorm xBE RCO^  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); %f> |fs  
        end Up/u|A$0V  
    end cwWSNm|  
    % END: Compute the Zernike Polynomials 73s3-DS,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N7HbOLpM  
    Zc\h15+P  
    CMxjX  
    % Compute the Zernike functions: {cyo0-9nv  
    % ------------------------------ $L&9x3+?Kg  
    idx_pos = m>0; xX&>5 "  
    idx_neg = m<0; ^I0GZG  
    EO 9kE.g  
    <j1d~XU}  
    z = y; lfpt:5a9&  
    if any(idx_pos) _Dcc<-.  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); z Jo#3  
    end At[n<8_|  
    if any(idx_neg) %L\{kUam  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); B:A1W{l  
    end (|a$N.e&K  
    R!V5-0%  
    peTO-x^a-  
    % EOF zernfun U3|&Jee  
     
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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)  h`wMi}q'D  
    H&w:`JYDL3  
    DDE还是手动输入的呢? Q00R<hu@F  
    =jg#fdM -  
    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