切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9078阅读
    • 5回复

    [讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

    上一主题 下一主题
    离线jssylttc
     
    发帖
    25
    光币
    13
    光券
    0
    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, TE3*ktB{N  
    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, EPc!p>  
    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? UM<@t%|>  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? #nKRTb+{  
    X]qCS0GD'  
    5N\+@grp  
    Ba<ngG !  
    d!o.ASL{  
    function z = zernfun(n,m,r,theta,nflag) sp|q((z{  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. &]w#z=5SXi  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N  1Yud~[c  
    %   and angular frequency M, evaluated at positions (R,THETA) on the M~-h-tG  
    %   unit circle.  N is a vector of positive integers (including 0), and Sa Cx)8ul0  
    %   M is a vector with the same number of elements as N.  Each element d7E7f  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) hHpx?9O+!  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, &`\ep9  
    %   and THETA is a vector of angles.  R and THETA must have the same [q'eEN G  
    %   length.  The output Z is a matrix with one column for every (N,M) @8|Gh]\P  
    %   pair, and one row for every (R,THETA) pair. bZ/ hgqS  
    % ei@3,{~5  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Rfht\{N 7  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), f3 !n$lj  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral TM0b-W (H  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, `4LJ;KC(  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized P@C c]Z  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ,(P %z.P@  
    % N r<9u$d9=  
    %   The Zernike functions are an orthogonal basis on the unit circle. o5P&JBX<  
    %   They are used in disciplines such as astronomy, optics, and (v!mR+\x  
    %   optometry to describe functions on a circular domain. ZPl PN;J^1  
    % [UoqIU  
    %   The following table lists the first 15 Zernike functions. 0pD[7~^o  
    % okz]Qc>G  
    %       n    m    Zernike function           Normalization pajy#0 U  
    %       -------------------------------------------------- mbyih+amCr  
    %       0    0    1                                 1 y1iX!m~)  
    %       1    1    r * cos(theta)                    2 *<r%aeG$em  
    %       1   -1    r * sin(theta)                    2 usy,V"{  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) bo1I&I  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ^#;RLSv   
    %       2    2    r^2 * sin(2*theta)             sqrt(6) >60"p~t  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) )A"jVQjI%w  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) pw3 (t  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) SiV*WxQe  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ailG./I+  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ';6X!KY+]  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #&V5H{  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) t@)my[!  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .a,(pq Jg  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 9<l-NU9 _  
    %       -------------------------------------------------- =UNT.]  
    % T%kKVr  
    %   Example 1: KzG_ <<  
    % B9*Sfw%  
    %       % Display the Zernike function Z(n=5,m=1) Y%g "Y  
    %       x = -1:0.01:1; cz#_<8'N  
    %       [X,Y] = meshgrid(x,x); +*C^:^jA  
    %       [theta,r] = cart2pol(X,Y); e\r7BW\Y  
    %       idx = r<=1; UfKkgq#  
    %       z = nan(size(X)); A#35]V06  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 0wFh%/:  
    %       figure n+?-�  
    %       pcolor(x,x,z), shading interp E~RV1)  
    %       axis square, colorbar b =b :  
    %       title('Zernike function Z_5^1(r,\theta)') WYLX?x  
    % @+&'%1  
    %   Example 2: /PqUXF  
    % W`x)=y]Z  
    %       % Display the first 10 Zernike functions uoCGSXsi  
    %       x = -1:0.01:1; e!Br>^8l  
    %       [X,Y] = meshgrid(x,x); nLJBq)i  
    %       [theta,r] = cart2pol(X,Y); bnr|Y!T}Bi  
    %       idx = r<=1; 3]^'  
    %       z = nan(size(X)); @6b[GekZ<  
    %       n = [0  1  1  2  2  2  3  3  3  3]; *S4aF*Qk  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; gI{ =0  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ;Iq5|rzDn  
    %       y = zernfun(n,m,r(idx),theta(idx)); lsY `c"NW>  
    %       figure('Units','normalized') B\[-fq  
    %       for k = 1:10 -!TcQzHUs  
    %           z(idx) = y(:,k); JYV\oV{  
    %           subplot(4,7,Nplot(k)) v9rVpYc"  
    %           pcolor(x,x,z), shading interp 3ji:O T  
    %           set(gca,'XTick',[],'YTick',[]) x: ~d@  
    %           axis square Gw{+xz KJ  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) o/1JO_41  
    %       end h0Jl_f#Y  
    % N09KVz2Q  
    %   See also ZERNPOL, ZERNFUN2. g$w6kz_[  
    X NE+(Bt  
    8l23%iWxe  
    %   Paul Fricker 11/13/2006 QNArZ6UQ  
    1lcnRHO  
    kA^A mfba  
    J=*K"8Qr  
    =}R~0|^  
    % Check and prepare the inputs: $.:3$et@/  
    % ----------------------------- tD-gc ''H  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) VR4%v9[1  
        error('zernfun:NMvectors','N and M must be vectors.') tpYa?ZCM  
    end tjxvN 4l  
    dy:d=Z  
    /{X_ .fv<v  
    if length(n)~=length(m) w$>3pQ8d  
        error('zernfun:NMlength','N and M must be the same length.') H$tb;:  
    end KlU qoJ;"  
    Rla4L`X;  
    O]qPmEj  
    n = n(:); 0H[LS  
    m = m(:); U$'y_}V  
    if any(mod(n-m,2)) "}zda*z8  
        error('zernfun:NMmultiplesof2', ... z-@ -O  
              'All N and M must differ by multiples of 2 (including 0).') Df@/cT  
    end d(S}NH  
    #DUh(:E'`  
    V;93).-$  
    if any(m>n) % {Q-8w!  
        error('zernfun:MlessthanN', ... <&U!N'CE  
              'Each M must be less than or equal to its corresponding N.') J^ ={}  
    end *#2Rvt*Ox  
    @^? XaU  
    |Q!4GeQL[  
    if any( r>1 | r<0 ) ``9 GY  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') $bGD%9 z  
    end ow.j+ <M  
    254~:eB0  
    J#7y< s  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ~E*d G  
        error('zernfun:RTHvector','R and THETA must be vectors.') &p"(-  
    end I7mG/  
    xo WT*f  
    irMd jG  
    r = r(:); *IZf^-=Q  
    theta = theta(:); NHkL24ve  
    length_r = length(r); XnXb&@Y  
    if length_r~=length(theta) ut5yf$%  
        error('zernfun:RTHlength', ... }Bff,q  
              'The number of R- and THETA-values must be equal.') Z;b+>2oL  
    end <LA^%2jT  
    \+Y!ILOI  
     .mPg0  
    % Check normalization: \!H{Ks{#R.  
    % -------------------- rAXX}"l6s  
    if nargin==5 && ischar(nflag) Kx6y" {me|  
        isnorm = strcmpi(nflag,'norm'); 0YS?=oi  
        if ~isnorm Nl*i5 io  
            error('zernfun:normalization','Unrecognized normalization flag.') o6|-=FcvC  
        end I]uhi{\C  
    else >. LKct*5K  
        isnorm = false; C5n?0I9  
    end d 4O   
    N[k<@Q?*a  
    eb!_ie"D  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f\K#>u* Q  
    % Compute the Zernike Polynomials -x'e+zT  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% T[.[ g/`  
    6/2v  
    c> G@+  
    % Determine the required powers of r: =n0*{~r  
    % ----------------------------------- 'b[0ci:  
    m_abs = abs(m); fp&Got!pB  
    rpowers = []; `ROEV~  
    for j = 1:length(n) UK3a{O[ 5  
        rpowers = [rpowers m_abs(j):2:n(j)]; &" h]y?Q  
    end U9ZbVjqv@  
    rpowers = unique(rpowers); @{}rG8  
    ,Y!zORv<7  
    n Ga1a  
    % Pre-compute the values of r raised to the required powers, t26ij`V  
    % and compile them in a matrix: nl@E[yA9[  
    % ----------------------------- kuS/S\Z5K  
    if rpowers(1)==0 p4mY0Y]mP  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); FH\CK  
        rpowern = cat(2,rpowern{:}); BGqa-d  
        rpowern = [ones(length_r,1) rpowern]; y\[r(4h  
    else b5 Q NEi  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); nj2gs,k  
        rpowern = cat(2,rpowern{:}); ULl_\5s2  
    end iBvOJs  
    I; j3*lV_  
    Z?1OdoT-  
    % Compute the values of the polynomials: ?q Xs-  
    % -------------------------------------- m6U8)!)T  
    y = zeros(length_r,length(n)); ~A >o O-0K  
    for j = 1:length(n) L[C*@ uK  
        s = 0:(n(j)-m_abs(j))/2; ,")F[%v  
        pows = n(j):-2:m_abs(j); nW5K[/1D  
        for k = length(s):-1:1 <lo`q<q  
            p = (1-2*mod(s(k),2))* ... V0NVGRQ  
                       prod(2:(n(j)-s(k)))/              ... dVGbe07  
                       prod(2:s(k))/                     ... =_QkH!vI  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ~@fR[sg<  
                       prod(2:((n(j)+m_abs(j))/2-s(k)));  _^T}_  
            idx = (pows(k)==rpowers); n,nisS  
            y(:,j) = y(:,j) + p*rpowern(:,idx); _!:@w9  
        end D'L{wm  
         )w"0w(   
        if isnorm )iSy@*nY  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); t)I0lnbs  
        end kaFnw(xa  
    end ;|30QUYh  
    % END: Compute the Zernike Polynomials Z[} $n-V  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (~#G'Hd  
    cU+>|'f &  
    s*JE)  
    % Compute the Zernike functions: n`<U"$*  
    % ------------------------------ e@j8T gI)  
    idx_pos = m>0; X47Ol  
    idx_neg = m<0; Jsn <,4DO8  
    Uu5C%9^s  
    k/#>S*Ne  
    z = y; .B$h2#i1  
    if any(idx_pos) 8^X]z|2  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 5Y-2 #  
    end lzfDH =&  
    if any(idx_neg) G (\Ckf:  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); %fpsc _  
    end F= i!d,S  
    7) 0q--B  
    F5IZ"Itu(  
    % EOF zernfun (C`@a/q  
     
    分享到
    离线phoenixzqy
    发帖
    4352
    光币
    5478
    光券
    1
    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)
    在线sansummer
    发帖
    959
    光币
    1087
    光券
    1
    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
    发帖
    25
    光币
    13
    光券
    0
    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)  90(UgK&Y  
    /D8EI   
    DDE还是手动输入的呢? :i}@Br+R7L  
    UT~4Cfb  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    离线jssylttc
    发帖
    25
    光币
    13
    光券
    0
    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    离线18257342135
    发帖
    51
    光币
    1518
    光券
    0
    只看该作者 5楼 发表于: 2016-12-13
    支持一下,慢慢研究