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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, A`KTm(  
    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, C-7.Sa  
    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? lF<(yF5  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Q1rwTg\  
    11u qs S2  
    ?@#<>7V  
    sXUM,h8$!+  
    S=Zjdbd  
    function z = zernfun(n,m,r,theta,nflag) UkUdpZ.[il  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. PHoW|K_e  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 8LL);"$  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Vy biuP  
    %   unit circle.  N is a vector of positive integers (including 0), and *KM CU m  
    %   M is a vector with the same number of elements as N.  Each element z y.Ok 49  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) x>Kem$z  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, [|3 %~s|Sv  
    %   and THETA is a vector of angles.  R and THETA must have the same @`3)?J[w  
    %   length.  The output Z is a matrix with one column for every (N,M) Y#G '[N>  
    %   pair, and one row for every (R,THETA) pair. CA3.fu3(p  
    % q+z,{K  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike zr,jaR;  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ,J[sg7v cv  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral QeK~A@|F&  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, X,p&S^  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Z7(hW,60  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 49CMRO,T  
    % r6A7}v  
    %   The Zernike functions are an orthogonal basis on the unit circle. kys?%Y1  
    %   They are used in disciplines such as astronomy, optics, and kn! J`"b  
    %   optometry to describe functions on a circular domain. 9QpKB c  
    % p7z#4 GW  
    %   The following table lists the first 15 Zernike functions. ]fR 3f  
    % )2a!EEHz  
    %       n    m    Zernike function           Normalization DQ,QyV  
    %       -------------------------------------------------- P<bA~%<7"[  
    %       0    0    1                                 1 twJck~l~n  
    %       1    1    r * cos(theta)                    2  9TeDLp  
    %       1   -1    r * sin(theta)                    2 P)T:6K  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 5~qr+la  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ]xuq2MU,l  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) CxO) d7c  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) XOxm<3gXn  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) wc;5tb#  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) <4Ak$ E %"  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) f6DPah#  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 3T_-_5[c  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) mCg5-E~;  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) LnBkd:>}  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f1JvP\I0Q  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) PoC24#vS  
    %       -------------------------------------------------- }ts?ZR^V,  
    % Rq;R{a  
    %   Example 1: p{.EFa>H  
    % %bddR;c  
    %       % Display the Zernike function Z(n=5,m=1) #ujcT%1G  
    %       x = -1:0.01:1; ,O2Uj3"  
    %       [X,Y] = meshgrid(x,x); aFhsRE?YC=  
    %       [theta,r] = cart2pol(X,Y); sO6+L #!  
    %       idx = r<=1; k%hif8y  
    %       z = nan(size(X)); D@mDhhK_  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); O^LzS&I*  
    %       figure keX0br7u_  
    %       pcolor(x,x,z), shading interp ak<?Eu9rV  
    %       axis square, colorbar 7^S&g.A  
    %       title('Zernike function Z_5^1(r,\theta)') K~[/n<ks  
    % SMnbI .0  
    %   Example 2: (!;4Y82#  
    % I5  
    %       % Display the first 10 Zernike functions x *(pr5k  
    %       x = -1:0.01:1; #B54p@.}  
    %       [X,Y] = meshgrid(x,x); 4/HyO\?z5  
    %       [theta,r] = cart2pol(X,Y); 7n %QP  
    %       idx = r<=1; (R.k.,z  
    %       z = nan(size(X)); a "8/y4Y  
    %       n = [0  1  1  2  2  2  3  3  3  3]; GK:*|jV  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3];  ~B/|#o2  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; R9{6$djq\:  
    %       y = zernfun(n,m,r(idx),theta(idx)); x_#yH3kJ  
    %       figure('Units','normalized') 16x M?P  
    %       for k = 1:10 >:8GU f*  
    %           z(idx) = y(:,k); :  wb\N'b  
    %           subplot(4,7,Nplot(k)) az7L0pp  
    %           pcolor(x,x,z), shading interp oU67<jq  
    %           set(gca,'XTick',[],'YTick',[]) DLf6D | "  
    %           axis square o:m:9dn  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) m/CA  
    %       end .{~ygHQ`f  
    % =TU"B-*  
    %   See also ZERNPOL, ZERNFUN2.  _8t{4C  
    <.~j:GbsE  
    tXwnK[~x  
    %   Paul Fricker 11/13/2006 }[? X%=  
    ) 3Eax_?Z  
    ."cC^og  
    g5_]^[up w  
    izOtt^#DZt  
    % Check and prepare the inputs: y1FS?hSD0  
    % ----------------------------- vA"yy"B+ V  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) (7&[!PS  
        error('zernfun:NMvectors','N and M must be vectors.') JoIffI?{(D  
    end iwrS>Sm  
    @>&UoH}2  
    Ig*!0(v5$  
    if length(n)~=length(m) [Nsv]Yz  
        error('zernfun:NMlength','N and M must be the same length.') #*XuU8q?  
    end ]#KZ W)M  
    J!~?}Fq/z  
    sYgpK92  
    n = n(:); (hs[B4nV  
    m = m(:); K%Jy?7 U  
    if any(mod(n-m,2)) 9Iy>oV  
        error('zernfun:NMmultiplesof2', ... |'Z6M];8t  
              'All N and M must differ by multiples of 2 (including 0).') e\tcP  
    end 44]/rP_m  
    u6$fF=  
    <Hig,(=`.  
    if any(m>n) WR%x4\,d#  
        error('zernfun:MlessthanN', ... rt^<=|Z  
              'Each M must be less than or equal to its corresponding N.') 9g|o17  
    end K9 :I8E<  
    vrLI`3n]  
    < Pg4>  
    if any( r>1 | r<0 ) ":tQYo]d  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') T\NvN&h-  
    end $x)C_WZj?  
    s: ~3|D][  
    now\-XrS  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) E0o=  
        error('zernfun:RTHvector','R and THETA must be vectors.') L?23Av0W  
    end Kp!sn,:  
    7?Q<kB=f  
    ~L<q9B( @  
    r = r(:); ]Wa.k  
    theta = theta(:); OjcxD5"v9  
    length_r = length(r); pA&CBXio  
    if length_r~=length(theta) A|Up >`QH  
        error('zernfun:RTHlength', ... _ )b:F=4j  
              'The number of R- and THETA-values must be equal.') k}(C.`.  
    end oQ{(7.e7)  
    nB[Aw7^|A  
    8*k#T\  
    % Check normalization: "u@)   
    % -------------------- }uz*6Z(S  
    if nargin==5 && ischar(nflag) \=P+]9  
        isnorm = strcmpi(nflag,'norm'); oj/,vO:QT  
        if ~isnorm 7Y"CeU-S  
            error('zernfun:normalization','Unrecognized normalization flag.') URz$hcI8  
        end 4 Z.G  
    else k z"F4?,  
        isnorm = false; B b_R~1 l  
    end ]2`PS<a2  
    +] s"*'V$  
    iaPrkMhd  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% qv>?xKSm  
    % Compute the Zernike Polynomials |gxT-ZM  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @ )owj^sA  
    |j3mI\ANF  
    7O84R^!|2  
    % Determine the required powers of r: v1*Lf/  
    % ----------------------------------- )u)]#z  
    m_abs = abs(m); bKRz=$P?  
    rpowers = []; }d?"i@[  
    for j = 1:length(n) !Bcd\]q  
        rpowers = [rpowers m_abs(j):2:n(j)]; }D02*s  
    end 3\j{*f$J  
    rpowers = unique(rpowers); ^vw? 4O  
    +n_`*@SE  
    KjFNb;mM  
    % Pre-compute the values of r raised to the required powers, aZ"9)RJe  
    % and compile them in a matrix: )L fXb9}  
    % ----------------------------- ~?T*D*  
    if rpowers(1)==0 @62QDlt;  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); g).k+  
        rpowern = cat(2,rpowern{:}); X2^`Znq9  
        rpowern = [ones(length_r,1) rpowern]; XMzL\Edo  
    else DlIy'@ .  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); RR R'azT  
        rpowern = cat(2,rpowern{:}); 8#b>4 Dx  
    end #!!Ea'3Iq  
    MDI[TNYG  
    )xwWig.  
    % Compute the values of the polynomials: I[E/)R{\  
    % -------------------------------------- Huzw>  
    y = zeros(length_r,length(n)); WB~ ^R<g  
    for j = 1:length(n)  0].*eM  
        s = 0:(n(j)-m_abs(j))/2; s"G;rcS}#  
        pows = n(j):-2:m_abs(j); KFd !wZ @e  
        for k = length(s):-1:1 0`y;[qAG[  
            p = (1-2*mod(s(k),2))* ... :wtr{,9rZ  
                       prod(2:(n(j)-s(k)))/              ... 'oNY4.[  
                       prod(2:s(k))/                     ... q):Ph&'r  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... X$z@ *3=  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); &aD ]_+b  
            idx = (pows(k)==rpowers); U6SgV 8  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ETQ.A< v  
        end BfQRw>dZ"{  
         E07g^y"}i  
        if isnorm Id-?her>B  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); <~ E'% 60;  
        end &Xw{%Rg  
    end >:7W.QLRU  
    % END: Compute the Zernike Polynomials 96M?tTa  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^3`CP4DT  
    U-+%e:v  
    } ti+tM*  
    % Compute the Zernike functions: DxX333vC  
    % ------------------------------ ;533;(d* o  
    idx_pos = m>0; ODE9@]a  
    idx_neg = m<0; k8]=5C?k  
    ~xz3- a/  
    eq>E<X#<  
    z = y; ]u~6fknm  
    if any(idx_pos) %*4Gx +b  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); %) A-zzj  
    end /y2upu*!  
    if any(idx_neg) '&~A  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); doJ\7c5uU  
    end Gp6|0:2,L~  
    =l%"Om*A  
    GUUVE@Z  
    % EOF zernfun >C|/%$kk:f  
     
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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)
    在线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)  Cm}ZeQ  
    }}~ ^!  
    DDE还是手动输入的呢? $i@5'[jA  
    e*vSGT$KgL  
    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