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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, }klE0<W|5\  
    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, cAYa=}~<  
    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? /j`i/Ha1  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? E {I)LdAqK  
    7YoofI  
    d&O'r[S  
    qn5y D!1  
    s@/B*r9  
    function z = zernfun(n,m,r,theta,nflag) ,w,ENU0~f  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. &8pCHGmV)  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N l~`txe  
    %   and angular frequency M, evaluated at positions (R,THETA) on the BERn _5gb  
    %   unit circle.  N is a vector of positive integers (including 0), and H(  
    %   M is a vector with the same number of elements as N.  Each element w:~nw;.T  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) n ;Ql=4  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ORUWsl Mt  
    %   and THETA is a vector of angles.  R and THETA must have the same =>gyc;{2K<  
    %   length.  The output Z is a matrix with one column for every (N,M)  EGp~Vo-  
    %   pair, and one row for every (R,THETA) pair. Fr1;)WV  
    % {JCSR2BB  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike )pkhir06t  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )->-~E}p9  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Km|9Too  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, s :-8 Z\,  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 2hjre3"?  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. jx^|2  
    % /vFxVBX  
    %   The Zernike functions are an orthogonal basis on the unit circle. QO1A976o  
    %   They are used in disciplines such as astronomy, optics, and Dme(Knly  
    %   optometry to describe functions on a circular domain. 4d{"S02h  
    % L8,H9T#e  
    %   The following table lists the first 15 Zernike functions. GC5#1+fQ  
    % eXskwV+7  
    %       n    m    Zernike function           Normalization +G3nn!g l4  
    %       -------------------------------------------------- TFiuz; *|  
    %       0    0    1                                 1 w>H%[\Qs  
    %       1    1    r * cos(theta)                    2 =)"NE>  
    %       1   -1    r * sin(theta)                    2 |r)>bY7  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) pIU#c&%<9  
    %       2    0    (2*r^2 - 1)                    sqrt(3) sRo<4U0M;l  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) rw}5nv  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) =]5DYRhX]  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) !`O_VV`/@  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ihpz}g  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) .N-'; %8  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) #cSw"A  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) <3],C)Zwc  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ?<>,XyY  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) S*2L4Uj`|  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 7gZVg@   
    %       -------------------------------------------------- _D7HQ  
    % SoXX}<~E4  
    %   Example 1: `JY>v io  
    % Mc#O+'](f  
    %       % Display the Zernike function Z(n=5,m=1) tF;& x g  
    %       x = -1:0.01:1; @4 Os?_gJ\  
    %       [X,Y] = meshgrid(x,x); "tg\yem  
    %       [theta,r] = cart2pol(X,Y); 82Z[eo  
    %       idx = r<=1; Y*5@|Q  
    %       z = nan(size(X)); R%]9y]HQ  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); %z!d4J75  
    %       figure ^w&5@3d  
    %       pcolor(x,x,z), shading interp PJSDY1T  
    %       axis square, colorbar e GqvnNv  
    %       title('Zernike function Z_5^1(r,\theta)') #(26t _a  
    % )\I? EU8  
    %   Example 2: @gu77^='  
    % XEgx#F ;F  
    %       % Display the first 10 Zernike functions dc\u$'F@S  
    %       x = -1:0.01:1; =Nv= Q mO  
    %       [X,Y] = meshgrid(x,x); >H=Q$gI  
    %       [theta,r] = cart2pol(X,Y); "t%1@b*u  
    %       idx = r<=1; 5b{yA~ty  
    %       z = nan(size(X)); =?`y(k4a  
    %       n = [0  1  1  2  2  2  3  3  3  3]; c9ov;Bw6S  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 5u u2 _B_L  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; yG4LQE  
    %       y = zernfun(n,m,r(idx),theta(idx)); !e#I4,fn  
    %       figure('Units','normalized') P98X[0&  
    %       for k = 1:10 D<D k1  
    %           z(idx) = y(:,k); $@:>7Y"  
    %           subplot(4,7,Nplot(k)) 0,L$x*Nj5  
    %           pcolor(x,x,z), shading interp WV !kA_  
    %           set(gca,'XTick',[],'YTick',[]) J?n)FgxS  
    %           axis square \{+nXn  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 5>4A}hSe  
    %       end . ;ea]_Z  
    % BhE~k?$9  
    %   See also ZERNPOL, ZERNFUN2. J.1ln = Y  
    ~D`oP/6  
    b0z{"  
    %   Paul Fricker 11/13/2006 e2Kpx8kWj  
    Z 9 q{r s  
    $E9daUt8"J  
    utm+\/  
    0@mX4.!  
    % Check and prepare the inputs: 0P%|)Ae  
    % ----------------------------- G4iLCcjY  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) q:~`7I  
        error('zernfun:NMvectors','N and M must be vectors.') 5S-o 2a  
    end ]RrP !|^  
    :9rhv{6Wp  
    /Y\E68_Fh  
    if length(n)~=length(m) {GH`V}Ob  
        error('zernfun:NMlength','N and M must be the same length.') Zm8 u:  
    end jO3u]5}.6  
    &"j).Ogm4  
    B,m$ur#$  
    n = n(:); @<w9fzi  
    m = m(:); EBL,E:_)  
    if any(mod(n-m,2)) <{z3p:\  
        error('zernfun:NMmultiplesof2', ... D'sboOY  
              'All N and M must differ by multiples of 2 (including 0).') 4pTu P /  
    end 1~xn[acy  
    m|cWX"#g  
    .jGsO0  
    if any(m>n) hZ\W ?r  
        error('zernfun:MlessthanN', ... L};;o+5uJD  
              'Each M must be less than or equal to its corresponding N.') .L(j@I t  
    end #+ lq7HJ1  
    <1 1Tqb  
    fe9& V2Uu  
    if any( r>1 | r<0 ) v`ZusHJ1d  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') |`t!aG8  
    end W!4V: (T  
    /&!d  
    +_XbHjhN/  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Sp$x%p0  
        error('zernfun:RTHvector','R and THETA must be vectors.') m[Ac'la  
    end :mtw}H 'F8  
    % x*Ec[l  
    DEwtP  
    r = r(:); vyx\N{  
    theta = theta(:); 53+rpU_  
    length_r = length(r); ]E8<;t)#  
    if length_r~=length(theta) $E_vCB _  
        error('zernfun:RTHlength', ... lbuW*)  
              'The number of R- and THETA-values must be equal.') IweK!,:>dN  
    end ):\{n8~  
    _kY[8e5  
    =&b$W/l)0  
    % Check normalization: z9kX`M+  
    % -------------------- Gx*0$4xJ3  
    if nargin==5 && ischar(nflag) 8 W<)c  
        isnorm = strcmpi(nflag,'norm'); 2=,Sz1`t  
        if ~isnorm M^JZ]W(  
            error('zernfun:normalization','Unrecognized normalization flag.') 2"Uk}Yz|  
        end 7 KdM>1!  
    else [dF=1E>W_J  
        isnorm = false; NUnc"@  
    end Z a1|fB  
    #`L}.  
    _NqT8C4C  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5eSTT#[+R  
    % Compute the Zernike Polynomials ._8cJf.ae  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;pyJ O_R[  
    Oi[9b  
    @[kM1:G-F{  
    % Determine the required powers of r: ]j$p_s>  
    % -----------------------------------  aC }1]7  
    m_abs = abs(m); DfzUGX  
    rpowers = []; -GWzMBS S  
    for j = 1:length(n) `FB?cPR  
        rpowers = [rpowers m_abs(j):2:n(j)]; MH8%-UV  
    end HN~4-6[q  
    rpowers = unique(rpowers); )"Br,uIv:/  
    8EEQV}4  
    3jeV4|  
    % Pre-compute the values of r raised to the required powers, g2>u]3&W  
    % and compile them in a matrix: o3=S<|V  
    % ----------------------------- n@,eZ!  
    if rpowers(1)==0 <07W&`Dw  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); =yhfL2`aw  
        rpowern = cat(2,rpowern{:}); V >uW|6  
        rpowern = [ones(length_r,1) rpowern]; 4-rI4A<  
    else K}/`YDu  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); +Y]*>afG  
        rpowern = cat(2,rpowern{:}); V;]VwsZ"  
    end e27CbA{_w  
    uvv-lAbjw  
    C^=gZ 6m  
    % Compute the values of the polynomials: $5CY<,f  
    % -------------------------------------- %c/"A8{eb  
    y = zeros(length_r,length(n)); y* Q-4_%,  
    for j = 1:length(n) 9.#R?YP$  
        s = 0:(n(j)-m_abs(j))/2; R/cq00g  
        pows = n(j):-2:m_abs(j); (0m$W<  
        for k = length(s):-1:1 zYF&Dv/u/  
            p = (1-2*mod(s(k),2))* ... m9w ; a  
                       prod(2:(n(j)-s(k)))/              ... SA n=9MG  
                       prod(2:s(k))/                     ... |A/_Qe|s2  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... [#6Esy8|  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); xWb?i6)z&  
            idx = (pows(k)==rpowers); UZ3Aq12U}a  
            y(:,j) = y(:,j) + p*rpowern(:,idx); RW[<e   
        end 78~V/L;@S2  
         pz}hh^]t  
        if isnorm f> [;|r@K  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ZLX`[   
        end xQ 3u  
    end mf[79:90^  
    % END: Compute the Zernike Polynomials ~EkGG .  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uOqDJM'RM  
    pcTXTy 28  
    7t Kft  
    % Compute the Zernike functions: <G?85*Nv_  
    % ------------------------------ aMg f6veM  
    idx_pos = m>0; G6mM6(Sr  
    idx_neg = m<0; >,vW  
    @@mW+16  
    ?ML<o>OKg  
    z = y; ~cj:AIF  
    if any(idx_pos) MJpTr5Vs  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); |RXC;zt9s  
    end ]!o,S{a&  
    if any(idx_neg) U I|@5:J  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); p:   
    end Cy'W!qH  
    @$} \S  
    MtTHKp   
    % EOF zernfun `9{C/qB  
     
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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)  /8R1$7  
    (reD  
    DDE还是手动输入的呢? |n/id(R+  
    ~ME=!;<_  
    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