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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, 9a9<I  
    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, :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? +Gqh  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ;Xg6'yxJ  
    KT~J@];Fb  
    kU$P?RD  
    5<w"iqZ\?N  
    $j0] +vT  
    function z = zernfun(n,m,r,theta,nflag) X[_w#Hwp-  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. fqZqPcT0  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N S1(. AI~  
    %   and angular frequency M, evaluated at positions (R,THETA) on the (2(I|O#  
    %   unit circle.  N is a vector of positive integers (including 0), and v^2K=f[nE  
    %   M is a vector with the same number of elements as N.  Each element 9#{?*c6  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) *X+T>SKL  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, <use+C2  
    %   and THETA is a vector of angles.  R and THETA must have the same mV^+`GWvo  
    %   length.  The output Z is a matrix with one column for every (N,M)  Q<B=m6~  
    %   pair, and one row for every (R,THETA) pair. fT [JU1  
    % _;3xG0+  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ?&EPZqI  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), *i|O!h1St  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral N(q%|h<Z/=  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, :$."x '  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Ug*:o d  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 0^nnR7  
    % "^VKs_U8o  
    %   The Zernike functions are an orthogonal basis on the unit circle. EpSVHD:*  
    %   They are used in disciplines such as astronomy, optics, and Qc#<RbLL  
    %   optometry to describe functions on a circular domain. EL$l . v  
    % J5h;~l!y  
    %   The following table lists the first 15 Zernike functions. -'3~Y 2#  
    % o#gb+[  
    %       n    m    Zernike function           Normalization r7o63]  
    %       -------------------------------------------------- a<7Ui;^@  
    %       0    0    1                                 1 eE5U|y)_  
    %       1    1    r * cos(theta)                    2 hd1(q33  
    %       1   -1    r * sin(theta)                    2 "f/lm 2<  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) [}q6bXM*  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 4CVtXi_Y  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) :pj#t$:!  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ~K]5`(KV  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) +pp|Qgr 3  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) -:b0fKn  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) | YmQO#''  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) (@@t,\iF  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ' _Ij9{M  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ,0O9!^  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (b%&DyOt  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) p9rnhqH6  
    %       -------------------------------------------------- {jO:9O @  
    % 'D(|NYY  
    %   Example 1: !4TMgM  
    % '1{co/Y  
    %       % Display the Zernike function Z(n=5,m=1) xU+c?OLi  
    %       x = -1:0.01:1; 4%>iIPXi.(  
    %       [X,Y] = meshgrid(x,x); (4=NKtA^G  
    %       [theta,r] = cart2pol(X,Y); ua[ d  
    %       idx = r<=1; W m\HZ9PN  
    %       z = nan(size(X)); 19O /Q,9  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ee}&~%  
    %       figure ,pL%,>R5  
    %       pcolor(x,x,z), shading interp N@Pf\D  
    %       axis square, colorbar xD+n2:I{  
    %       title('Zernike function Z_5^1(r,\theta)') F33&A<(,  
    % sT:$:=  
    %   Example 2: ``KimeA~  
    % " UaUaSg#  
    %       % Display the first 10 Zernike functions 9nF;$ HB  
    %       x = -1:0.01:1; 7\I,;swo  
    %       [X,Y] = meshgrid(x,x);  %~Vgz(/  
    %       [theta,r] = cart2pol(X,Y); e<o{3*%p)  
    %       idx = r<=1; ?EQ]f34  
    %       z = nan(size(X)); VsEMF i=  
    %       n = [0  1  1  2  2  2  3  3  3  3]; <nDuN*|  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; NT+%u-  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; s8;/'?K  
    %       y = zernfun(n,m,r(idx),theta(idx)); @9S3u#vP  
    %       figure('Units','normalized') t Dn{;ED<  
    %       for k = 1:10 [~e{58}J|  
    %           z(idx) = y(:,k); r5y p jT^  
    %           subplot(4,7,Nplot(k)) 9>,$q"M}?  
    %           pcolor(x,x,z), shading interp ?/"Fwjau  
    %           set(gca,'XTick',[],'YTick',[]) C3 >X1nU  
    %           axis square T= Q"| S]V  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) &L6xagR7M  
    %       end i i&kfy  
    % &("HH"!  
    %   See also ZERNPOL, ZERNFUN2. %6Wv-:LY  
    2_i/ F)W  
     g=W1y  
    %   Paul Fricker 11/13/2006 vzDoF0Ts*p  
    aVTTpMY  
    oAaUXkQE  
    T^FeahA7;  
    ,pfHNK-u  
    % Check and prepare the inputs: 7;0$UYDU*  
    % ----------------------------- <X]'":  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ^f][;>c  
        error('zernfun:NMvectors','N and M must be vectors.') IJX75hE0g  
    end G-FeDP  
    MP"Pqt  
    {[B^~Y>Lr  
    if length(n)~=length(m) ?+6w8j%\  
        error('zernfun:NMlength','N and M must be the same length.') c*F'x-TH  
    end |ci1P[y  
    #a7 Wx}  
    , c3gW2E  
    n = n(:); /;P* ?  
    m = m(:); EPO*{bN7O  
    if any(mod(n-m,2)) }t.J;(ff:  
        error('zernfun:NMmultiplesof2', ... PeCU V6  
              'All N and M must differ by multiples of 2 (including 0).') bWp40&vx  
    end 4-ijuqjN  
    k)l*L1Y4:  
    >v1E;-ZA  
    if any(m>n) uh,~Cv XU]  
        error('zernfun:MlessthanN', ... 6k14xPj  
              'Each M must be less than or equal to its corresponding N.') dt -EY  
    end c;RB!`9"  
    !a %6nBo  
    M? 7CBqZ  
    if any( r>1 | r<0 ) Hz A+Oi  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 2R W^Nqc9  
    end f+A!w8E  
    HI+87f_Q  
    fg[]>:ZT.  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 'dTJE--@  
        error('zernfun:RTHvector','R and THETA must be vectors.') UD.&p'^ /{  
    end ,V$PV,G  
    lawjGI  
    6:PQkr  
    r = r(:); y]9PLch]vZ  
    theta = theta(:); z 'iAj  
    length_r = length(r); pS [nKcyj  
    if length_r~=length(theta) "0BuQ{CQ  
        error('zernfun:RTHlength', ... 2y_R05O0  
              'The number of R- and THETA-values must be equal.') zpPzXQv]/  
    end ZmT N  
    e G8Zn<:s  
    Nd b_|  
    % Check normalization: MqA%hlq  
    % -------------------- p xj}%LH  
    if nargin==5 && ischar(nflag) !%v=9muay  
        isnorm = strcmpi(nflag,'norm'); 8[2.HM$Y  
        if ~isnorm ]J`yh$a  
            error('zernfun:normalization','Unrecognized normalization flag.') 52RFB!Z[  
        end =aL=SC+  
    else DM*GvBdR  
        isnorm = false; kTCWyc  
    end C3m](%?   
    kaKV{;UM  
    P:`tL)W_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G/cE2nD  
    % Compute the Zernike Polynomials *ud"?{)Z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~c;D@.e\  
    u0 & aw  
    , [ogh  
    % Determine the required powers of r: T*m;G(  
    % ----------------------------------- ss8de9T"'  
    m_abs = abs(m); sE,Q:@H5  
    rpowers = []; }Y{aVn&C  
    for j = 1:length(n) \QpH~&QIS  
        rpowers = [rpowers m_abs(j):2:n(j)]; N sUFM  
    end NZj_7j|o9  
    rpowers = unique(rpowers); ue YBD]3'  
    C^dnkuA  
    HOEjLwH  
    % Pre-compute the values of r raised to the required powers, ch^tq",1>  
    % and compile them in a matrix: pONBF3H8  
    % ----------------------------- m{~p(sQL  
    if rpowers(1)==0 #<^ngoOj  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); >Ei-Spy>Xl  
        rpowern = cat(2,rpowern{:}); =|@%5&.P  
        rpowern = [ones(length_r,1) rpowern]; W ix/Az  
    else kO1.27D  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); /M Hml0u  
        rpowern = cat(2,rpowern{:}); f,e7;u z%  
    end Iy2KOv@a5  
    pO2Y'1*  
    d|nJp-%V  
    % Compute the values of the polynomials: 'Z<V(;W  
    % -------------------------------------- ?2;gmZd7  
    y = zeros(length_r,length(n)); !3E %u$-}  
    for j = 1:length(n) ;OE=;\  
        s = 0:(n(j)-m_abs(j))/2; ,$lOQ7R1(  
        pows = n(j):-2:m_abs(j); f/_RtOSw  
        for k = length(s):-1:1 `0]kRA8=  
            p = (1-2*mod(s(k),2))* ... L}>XH*  
                       prod(2:(n(j)-s(k)))/              ... \P3[_kbf1  
                       prod(2:s(k))/                     ... 0cd`. ZF  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... )^G&p[G  
                       prod(2:((n(j)+m_abs(j))/2-s(k)));  P[l?  
            idx = (pows(k)==rpowers); L`fDc  
            y(:,j) = y(:,j) + p*rpowern(:,idx); [c{/0*  
        end > @Ux8#  
         ` 0YI?$G1  
        if isnorm ws{2 0  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); yNw YP%"y  
        end (A6 -9g>  
    end <> jut  
    % END: Compute the Zernike Polynomials qre.^6x  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h{ &X`$  
    N*k`'T  
    FwdRM)1)  
    % Compute the Zernike functions: ( TQx3DGq  
    % ------------------------------ 8z?q4  
    idx_pos = m>0; $@[`/Uh   
    idx_neg = m<0; Anpx%NVo  
    ^>g7Kg"0  
    J9tQ@3{f  
    z = y; fgb%SIi?  
    if any(idx_pos) ]cz*k/*0  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); n1X.]|6'  
    end rv(Qz|K@  
    if any(idx_neg) gC}r$ZB(  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Y#'?3  
    end  CB<i  
    pa7Iz^i  
    |@}Yady@C  
    % EOF zernfun l7{Xy_66  
     
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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[ Pyxx_K  
    .M3]\I u  
    DDE还是手动输入的呢? PQ6.1}  
    [)K?e!c8  
    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