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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, *b/` Ya4  
    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, 2Yn <2U/^R  
    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? pDIVZC  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? SB|Qa}62  
    48qV >Gwf  
    2Mmz%S'd  
    (Dl$kGn  
    )V6Hl@v  
    function z = zernfun(n,m,r,theta,nflag) !0@Yplj  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. >eB\(EP  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N S.m{eur!,E  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ^,8)iV0j_  
    %   unit circle.  N is a vector of positive integers (including 0), and *q".-u!D[  
    %   M is a vector with the same number of elements as N.  Each element | >htvDL  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) TDNQu_E  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, e<h~o!z a  
    %   and THETA is a vector of angles.  R and THETA must have the same J/GSceHF  
    %   length.  The output Z is a matrix with one column for every (N,M) WP+oFkw>  
    %   pair, and one row for every (R,THETA) pair. yXF?H"h(  
    % I@%t.%O Jp  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike L>%o[tS  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^1aAjYFn  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 2hkRd>)&5  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, A1#%`^W9  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized $!(pF  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. J}+6UlD  
    % tj4VWJK  
    %   The Zernike functions are an orthogonal basis on the unit circle. !Kj,9NX{U  
    %   They are used in disciplines such as astronomy, optics, and jeX^}]x|%  
    %   optometry to describe functions on a circular domain. pxf$ 1  
    % V<@ o<R  
    %   The following table lists the first 15 Zernike functions. 7C ,UDp|  
    % \\7ZWp\fN  
    %       n    m    Zernike function           Normalization /fT+^&  
    %       -------------------------------------------------- :1^R9yWA4  
    %       0    0    1                                 1 OJ zs Q  
    %       1    1    r * cos(theta)                    2 9 ;Ox;;w  
    %       1   -1    r * sin(theta)                    2 [4C:r!  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ( %xwl  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Mt5PaTjj  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) MP 2~;T}~  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) /)(#{i*  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Jesjtcy<*  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) rT5Ycm@  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) %V{7DA&C  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Qj6/[mUr~  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) $8[r9L!  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) e9[|!/./5  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )>-ibf`#?  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) <l9-;2L4  
    %       -------------------------------------------------- ;Uu(zhbj  
    % Yvjc1  
    %   Example 1: 5<j%EQN|D  
    % 7{qy7,Gp  
    %       % Display the Zernike function Z(n=5,m=1) .j>hI="b  
    %       x = -1:0.01:1; a5!Fv54  
    %       [X,Y] = meshgrid(x,x); x,S P'fcP  
    %       [theta,r] = cart2pol(X,Y); ) ^3avRsC  
    %       idx = r<=1; hQHnwr  
    %       z = nan(size(X)); _b.qkTWUB  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); <_Q:'cx'  
    %       figure A\#P*+k0  
    %       pcolor(x,x,z), shading interp ]U7KLUY>:  
    %       axis square, colorbar /3:q#2'v  
    %       title('Zernike function Z_5^1(r,\theta)') P*Tx14xe4  
    % 'hv k  
    %   Example 2: )}'U`'q  
    % pd8Nke  
    %       % Display the first 10 Zernike functions 9*=W-v  
    %       x = -1:0.01:1; -s$F&\5by  
    %       [X,Y] = meshgrid(x,x); /<8N\_wh  
    %       [theta,r] = cart2pol(X,Y); QZhj b  
    %       idx = r<=1; jDN ]3Y`  
    %       z = nan(size(X)); k{$ ao  
    %       n = [0  1  1  2  2  2  3  3  3  3]; aKJQm '9Ks  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 1`9xIm*9w  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ]mXLg:3B  
    %       y = zernfun(n,m,r(idx),theta(idx)); 9Q-*@6G  
    %       figure('Units','normalized') M7+h(\H]2  
    %       for k = 1:10 <rL/B k  
    %           z(idx) = y(:,k); j"@93D~  
    %           subplot(4,7,Nplot(k)) b-*3 2Y%  
    %           pcolor(x,x,z), shading interp dwv6;x  
    %           set(gca,'XTick',[],'YTick',[]) ;6{@^  
    %           axis square u=/CRjot  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _fP&&}  
    %       end ]a3iEA2 (  
    % mA@Me7m}  
    %   See also ZERNPOL, ZERNFUN2. (q7 Ry4-  
    ;/*6U  
    I1>N4R-j  
    %   Paul Fricker 11/13/2006 D.6,VY H  
    FSb Hn{@  
    Q\,o :ZU_  
    -}6xoF?  
    g@Qgxsyk>  
    % Check and prepare the inputs: [e4]"v`N  
    % ----------------------------- 3#45m+D  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) zb Z4|_  
        error('zernfun:NMvectors','N and M must be vectors.') *d',Vuv&[  
    end cl*PFQp9j  
    wgRs Z  
    @ (i!Y L  
    if length(n)~=length(m) FG!X"<he  
        error('zernfun:NMlength','N and M must be the same length.') K[7EOXLy  
    end ^p/Ob'!  
    ^@_m "^C  
    q;wLa#4)J  
    n = n(:); *79m^  
    m = m(:); S$^ RbI  
    if any(mod(n-m,2)) KB!|B.ChN(  
        error('zernfun:NMmultiplesof2', ... ]}6w#)]"  
              'All N and M must differ by multiples of 2 (including 0).') vHE^"l5v  
    end OLj\-w^  
    )I-fU4?  
    *VkgQ`c  
    if any(m>n) 7RvUH-S[  
        error('zernfun:MlessthanN', ... P0-Fc@&Y  
              'Each M must be less than or equal to its corresponding N.') U70]!EaT  
    end T4;T6 9j;,  
    ez9k4IO  
    a3 >zoN  
    if any( r>1 | r<0 ) sfVf@0g  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') pBC<u  
    end z>[tF5  
    /)rkiwp  
    2w$t wW-  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) U`x bPQ  
        error('zernfun:RTHvector','R and THETA must be vectors.') {3Vk p5%l  
    end **[Z^$)u(  
    (:+>#V)pZ  
    kV Rn`n0  
    r = r(:); ^-M^gYBR  
    theta = theta(:); p=QYc)3F  
    length_r = length(r); Ih[+K#t+E  
    if length_r~=length(theta) }p9F#gr  
        error('zernfun:RTHlength', ... OlQ,Ce  
              'The number of R- and THETA-values must be equal.') #DkD!dW(l  
    end ^SfS~G Q  
    1 Ee>S\9t  
    cDXsi#Raj  
    % Check normalization: @oG)LT  
    % -------------------- 9%iFV N'  
    if nargin==5 && ischar(nflag) cxYfZ4++m  
        isnorm = strcmpi(nflag,'norm'); !z zW2>  
        if ~isnorm s/1 #DM"  
            error('zernfun:normalization','Unrecognized normalization flag.') =qvZpB7ZZ  
        end bO/*2oau  
    else WnAd5#G  
        isnorm = false; - n6jG}01b  
    end 0D(cXzQP  
    G;oFTP>o  
    (a6?s{(  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b]]N{: I  
    % Compute the Zernike Polynomials C6& ( c  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7XyOB+aQO  
    cUDgM  
    $'[q4wo<  
    % Determine the required powers of r: ,c)g,J9  
    % ----------------------------------- u>Ki$xP1  
    m_abs = abs(m); _hCJ|Rrln  
    rpowers = []; Ca$c;  
    for j = 1:length(n) :a< hQ|p  
        rpowers = [rpowers m_abs(j):2:n(j)]; 1;W=!Fx  
    end e"+dTq8W  
    rpowers = unique(rpowers); [D'Gr*5~{  
    <2P7utdZ  
    H*W):j}8  
    % Pre-compute the values of r raised to the required powers, ?Cci:Lin  
    % and compile them in a matrix: c/u_KJFF-n  
    % ----------------------------- i.rU&yT%  
    if rpowers(1)==0 /b.oEGqZX  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); PtKTm\,JL0  
        rpowern = cat(2,rpowern{:}); O=jN&<rb  
        rpowern = [ones(length_r,1) rpowern]; ur2!#bU9  
    else '0+$ m=   
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); vg8O] YF  
        rpowern = cat(2,rpowern{:}); LBX%HGH  
    end KC&`x |  
    ^@}#me@  
    ~r`Wr`]_z  
    % Compute the values of the polynomials: BGjb`U#%3  
    % -------------------------------------- FUaNiAr[  
    y = zeros(length_r,length(n)); z*.v_Mx  
    for j = 1:length(n) a%~yol0wO7  
        s = 0:(n(j)-m_abs(j))/2; Z%v6xP.  
        pows = n(j):-2:m_abs(j); Gidkt;lj  
        for k = length(s):-1:1 nN ~GP"}  
            p = (1-2*mod(s(k),2))* ... U7%28#@  
                       prod(2:(n(j)-s(k)))/              ... d]M[C[TOX  
                       prod(2:s(k))/                     ... FWTx&Ip  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... If}lJ6jZ  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); KP~-$NR  
            idx = (pows(k)==rpowers); xtJAMo>g  
            y(:,j) = y(:,j) + p*rpowern(:,idx); }~*rx7p  
        end w6EI{  
         X7e/:._SAH  
        if isnorm hmGdjw t$  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); v'nHFC+p  
        end Uh+jt,RB`  
    end org*z!;.   
    % END: Compute the Zernike Polynomials PqhlXqX9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% aii'}c  
    *j<@yG2\gP  
    {Nq?#%vdT  
    % Compute the Zernike functions: YkbO&~.  
    % ------------------------------ &N{zkMf  
    idx_pos = m>0; D_aR\  
    idx_neg = m<0; #,P(isEZ"  
    9N}W(>  
    om7`w ]  
    z = y; MYTS3(  
    if any(idx_pos) U,3d) ]Zy&  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); sfC@*Y2XT  
    end d[U1.SNL  
    if any(idx_neg) 1b `G2?%  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); v>^jy8$  
    end )[DpK=[N^p  
    H^v{Vo  
    \DyKtrnm%  
    % EOF zernfun 6 ">oo-  
     
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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)  a"MTQFm'  
    "<7$2!  
    DDE还是手动输入的呢? UU*0dSWr  
    Qu!OV]Cc  
    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