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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, #v<+G=r*O  
    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, 4j{ }{  
    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? Hs`  '](  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? e76)z; '  
    bL%)k61G_v  
    1x)%9u}  
    _!D$Aj  
    t~M_NEPxV  
    function z = zernfun(n,m,r,theta,nflag) HB^azHr  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. u,q#-d0g;  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N T@XiG:b7  
    %   and angular frequency M, evaluated at positions (R,THETA) on the .?TVBbc%5  
    %   unit circle.  N is a vector of positive integers (including 0), and cR} =3|t  
    %   M is a vector with the same number of elements as N.  Each element x@)u:0  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) fE iEy%o  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, S7*:eo  
    %   and THETA is a vector of angles.  R and THETA must have the same vY koh/(/u  
    %   length.  The output Z is a matrix with one column for every (N,M) a{=~#u8  
    %   pair, and one row for every (R,THETA) pair. #wfR$Cd  
    % zrM|8Cu  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike J)_ 42Z  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), NgKNT}JDv  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral dX*PR3I-3  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, sj~'.Zs%  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized M9nYt~vHX  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 'u~use"  
    % .u&g2Y  
    %   The Zernike functions are an orthogonal basis on the unit circle. g=wnly  
    %   They are used in disciplines such as astronomy, optics, and +X%yF{^m(  
    %   optometry to describe functions on a circular domain. D]REZuHOI  
    % .*{LPfD|  
    %   The following table lists the first 15 Zernike functions. M'sJ5;^5  
    % z#b6 aP  
    %       n    m    Zernike function           Normalization H^~!t{\  
    %       -------------------------------------------------- xb\lbS{ f  
    %       0    0    1                                 1 n&^Rs )%v  
    %       1    1    r * cos(theta)                    2 L`BLkDm  
    %       1   -1    r * sin(theta)                    2 V Puzu|  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) $= gv  
    %       2    0    (2*r^2 - 1)                    sqrt(3) {^F_b% a4z  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) Cb<\  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) }j x{Cw  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ]v#Q\Q8>  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 8in8_/x  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 4I$#R  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) H4U;~)i  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >*&[bW'}?  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) w$_ooQ(_;Q  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) MWB?V?qPSC  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ugz1R+f_4{  
    %       -------------------------------------------------- d{  Z  
    % H3JWf MlW  
    %   Example 1: iPao54Z  
    % lxbZM9A2  
    %       % Display the Zernike function Z(n=5,m=1) TA*49Qp  
    %       x = -1:0.01:1; };|'8'5  
    %       [X,Y] = meshgrid(x,x); D*b> l_  
    %       [theta,r] = cart2pol(X,Y); .[7m4iJf  
    %       idx = r<=1; `y4+OXZ^  
    %       z = nan(size(X)); {az8*MR=X  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); `#~@f!';  
    %       figure !HFwQGP.Y  
    %       pcolor(x,x,z), shading interp 4&tY5m>  
    %       axis square, colorbar ]0o78(/w2  
    %       title('Zernike function Z_5^1(r,\theta)') [e (-  
    % gxF3gM  
    %   Example 2: a83o (9  
    % @E1N9S?>  
    %       % Display the first 10 Zernike functions R]dc(D  
    %       x = -1:0.01:1; ]>!]X*\9  
    %       [X,Y] = meshgrid(x,x); k5^'b#v  
    %       [theta,r] = cart2pol(X,Y); F$.M2*9  
    %       idx = r<=1; 7l?-2I'c  
    %       z = nan(size(X)); W /IyF){  
    %       n = [0  1  1  2  2  2  3  3  3  3]; "p<f#s}  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 3N?uY2  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; mIOx)`$  
    %       y = zernfun(n,m,r(idx),theta(idx)); K}6}Opr,Tt  
    %       figure('Units','normalized') p0b&CrALx  
    %       for k = 1:10 qk+:p]2  
    %           z(idx) = y(:,k); ?P}7AF A(W  
    %           subplot(4,7,Nplot(k)) UJO+7h'  
    %           pcolor(x,x,z), shading interp ?=6zgb"9-  
    %           set(gca,'XTick',[],'YTick',[]) Oa{M9d,l  
    %           axis square XBBsdldZ  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @D%VV=N~[  
    %       end o|*,<5t  
    % )x]/b=m  
    %   See also ZERNPOL, ZERNFUN2. o)w'w34FCT  
    =*t)@bn  
    g=b 'T-  
    %   Paul Fricker 11/13/2006 VF;%Z  
    ee6Zm+.B  
    5<9}{X+@o  
    ugOcK Gf  
    By%aTuV$  
    % Check and prepare the inputs: N"T+. r  
    % ----------------------------- ^,,|ED\M{m  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) *PD7H9m  
        error('zernfun:NMvectors','N and M must be vectors.') Xq9%{'9  
    end hX8;G!/  
    .7{,u1N'  
    ~|riFp=J  
    if length(n)~=length(m) a&9+<  
        error('zernfun:NMlength','N and M must be the same length.') *r=6bpi  
    end )%P!<|s:5  
    b16\2%Ea1  
    K-sJnQ23'  
    n = n(:); ?z p$Wz;k  
    m = m(:);  T=9+  
    if any(mod(n-m,2)) (FP- K  
        error('zernfun:NMmultiplesof2', ... L -<!,CASW  
              'All N and M must differ by multiples of 2 (including 0).') rqSeh/<iD  
    end K%)u zP  
    1ih|b8)Dn  
    [/\}:#MLe  
    if any(m>n) ("ql//SL  
        error('zernfun:MlessthanN', ... KftZ ^mk+p  
              'Each M must be less than or equal to its corresponding N.') rM Un ~  
    end #Mrof9  
    (6!W8x7  
    B/AS|i] sM  
    if any( r>1 | r<0 ) `|+!H.3  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') sBt,y _LW  
    end [Q6PFdQ_JT  
    v:|_!+g:  
    22(7rUkI  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) \9jEpE^Ju(  
        error('zernfun:RTHvector','R and THETA must be vectors.') 7 wH9w  
    end "`s{fy~mV  
    w`x4i fZ0q  
    BRyrdt*_e  
    r = r(:); V9bn  
    theta = theta(:); D.su^m_1  
    length_r = length(r); nF`_3U8e  
    if length_r~=length(theta) ,Y  ./9F  
        error('zernfun:RTHlength', ... @T)kqT  
              'The number of R- and THETA-values must be equal.') ~x4]^XS  
    end C/_Z9LL?F  
    8Q4yllv4  
    a'r8J~:jy  
    % Check normalization: 4c0 =\v  
    % -------------------- ,%6!8vX  
    if nargin==5 && ischar(nflag) $MhfGMk!'  
        isnorm = strcmpi(nflag,'norm'); N3"O#C  
        if ~isnorm ?g+uJf  
            error('zernfun:normalization','Unrecognized normalization flag.') > &tmdE  
        end '(fQtQ%  
    else )jm!bR`  
        isnorm = false; *5m4 j=-  
    end Pg4go10|  
    |q!O~<H@  
     OXDEU.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;#)sV2F\&  
    % Compute the Zernike Polynomials V96:+r  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q|h#J}\  
    Tg/?v3M88  
    I'[gGK4 F  
    % Determine the required powers of r: M$,4B  
    % ----------------------------------- >W>3w  
    m_abs = abs(m); ` "Lk@  
    rpowers = []; Z@(m.&ZRx  
    for j = 1:length(n) zpgRK4p,I"  
        rpowers = [rpowers m_abs(j):2:n(j)]; efN5(9*9R  
    end uidoz f2}  
    rpowers = unique(rpowers); wjy<{I  
    vb.}SG>  
    f0M5^  
    % Pre-compute the values of r raised to the required powers, BMi5F?Q'G  
    % and compile them in a matrix: !KC4[;Y  
    % ----------------------------- y?OK#,j  
    if rpowers(1)==0 T\v~"pMu*0  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?LSwJ @#  
        rpowern = cat(2,rpowern{:});  hik.c3  
        rpowern = [ones(length_r,1) rpowern]; zoibinm}Eg  
    else E\1e8Wyh  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); FeL!%z  
        rpowern = cat(2,rpowern{:}); ,eSII2,r4  
    end F81Kxcs  
    %_ (Xn  
    /JjSx/  
    % Compute the values of the polynomials: PjE%_M<  
    % -------------------------------------- )6b`1o!7  
    y = zeros(length_r,length(n)); ?+_Y!*J2b  
    for j = 1:length(n) thLx!t  
        s = 0:(n(j)-m_abs(j))/2; pN=>q <]L  
        pows = n(j):-2:m_abs(j); fD<0V  
        for k = length(s):-1:1 VV-%AS6;  
            p = (1-2*mod(s(k),2))* ... \ v2-}jU(  
                       prod(2:(n(j)-s(k)))/              ... NjFlV(XT}  
                       prod(2:s(k))/                     ... blx"WVqo  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ?Gx-q+H  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); *JArR1J  
            idx = (pows(k)==rpowers); kF-7OX0)  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ^V0I!&7lx  
        end sjy/[.4-  
         R+# g_"1@p  
        if isnorm ]u|5ZCv0  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); * `3+x  
        end @Zzg^1Ilpu  
    end +8}8b_bgH  
    % END: Compute the Zernike Polynomials bQ`2ll*(  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &SMM<^P.  
    *#.Ku(C+  
    II]-mb  
    % Compute the Zernike functions: /_Fi4wZ  
    % ------------------------------ wBCBZs$H  
    idx_pos = m>0; <YAs0  
    idx_neg = m<0; 8;i'dF:)  
    af_b G;  
    PG{"GiZz=  
    z = y; QE6L_\l  
    if any(idx_pos) R[W'LRh~:1  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); :DJLkMP  
    end lm8<0*;,  
    if any(idx_neg) ts &sr  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); !DBaC%TGC  
    end wV q4DE  
    H<Zs2DP`  
    :M$8<03>F  
    % EOF zernfun R:y u  
     
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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)  c}g:vh  
    }lt]]094,  
    DDE还是手动输入的呢? &-6 D'@  
    N0G-/  
    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