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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, v3t<rv  
    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, DcM/p8da  
    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? \dE{[^.5  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? kjdIk9 Y  
    Fn4yx~0  
    T3"'`Sd9;  
    45< gO1  
    P0OMu/  
    function z = zernfun(n,m,r,theta,nflag) sMUpkU-  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. L ed{#+  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N T;{:a-8  
    %   and angular frequency M, evaluated at positions (R,THETA) on the n6Uf>5  
    %   unit circle.  N is a vector of positive integers (including 0), and [P ;fv  
    %   M is a vector with the same number of elements as N.  Each element }0@@_Y]CC  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) u(f;4`  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, QXL .4r%  
    %   and THETA is a vector of angles.  R and THETA must have the same i`];xNR'  
    %   length.  The output Z is a matrix with one column for every (N,M) ZPq.|6&  
    %   pair, and one row for every (R,THETA) pair. S>*i\OnI'  
    % ?@FqlWz,  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Lr6C@pI  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !^0vi3I  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral r%X M`;bQX  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, S<'_{uz  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized /iQh'rp  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. _!Tjb^  
    % ~EXCYUp4v  
    %   The Zernike functions are an orthogonal basis on the unit circle.  QV\a f  
    %   They are used in disciplines such as astronomy, optics, and ~ QohP`_  
    %   optometry to describe functions on a circular domain. P2C>IS  
    % S+wT}_BQ  
    %   The following table lists the first 15 Zernike functions. _JTK$ \  
    % E.ji;5  
    %       n    m    Zernike function           Normalization +c C. ZOS  
    %       -------------------------------------------------- WwtVuc|  
    %       0    0    1                                 1 ;PU'"MeB "  
    %       1    1    r * cos(theta)                    2 1-PlRQs.1  
    %       1   -1    r * sin(theta)                    2 #G`K<%{?f  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) >#l: ]T  
    %       2    0    (2*r^2 - 1)                    sqrt(3) `"yxmo*0  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) W+U0Y,N6  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) pYr+n9)^  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) PE/uB,Wl  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) d8+@K&z|  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) J=: \b  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ~OvbMWu  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0rI/$  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) fR{_P  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) UQ7]hX9  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) U/cj_}uX  
    %       -------------------------------------------------- }BL7P-km  
    % >b=."i  
    %   Example 1: g"kI1^[nj  
    % 3tJfh=r=1  
    %       % Display the Zernike function Z(n=5,m=1) oJ3(7Sz  
    %       x = -1:0.01:1; e?B}^Dk0i  
    %       [X,Y] = meshgrid(x,x); =2=rPZw9  
    %       [theta,r] = cart2pol(X,Y); 3"v>y]$U  
    %       idx = r<=1; >qr/1mW  
    %       z = nan(size(X)); w{k^O7~  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); y06**f)  
    %       figure qz3 Z'  
    %       pcolor(x,x,z), shading interp TecMQ0 KD  
    %       axis square, colorbar $ xHtI]T  
    %       title('Zernike function Z_5^1(r,\theta)') { gs$pBu  
    % qq<T~^  
    %   Example 2: Ml{ ]{n  
    % oaPWeM+  
    %       % Display the first 10 Zernike functions 4KR`  
    %       x = -1:0.01:1; $0 vT_  
    %       [X,Y] = meshgrid(x,x); oD\t4]?E  
    %       [theta,r] = cart2pol(X,Y); w $-q&  
    %       idx = r<=1; U$+,|\9  
    %       z = nan(size(X)); {I $iD  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ]d7A|)q  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; } S]!W\a  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; sP2Uj  
    %       y = zernfun(n,m,r(idx),theta(idx)); ){'<67dK  
    %       figure('Units','normalized') e`LkCy[_  
    %       for k = 1:10 o 7tUv"Rs  
    %           z(idx) = y(:,k); zaLPPm&f  
    %           subplot(4,7,Nplot(k)) YVgH[-`,  
    %           pcolor(x,x,z), shading interp BN%cX 2j  
    %           set(gca,'XTick',[],'YTick',[]) ~TS!5Wiv  
    %           axis square Qox/abC h  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) [TUs^%2@  
    %       end y?O-h1"3,  
    % vazA@|^8  
    %   See also ZERNPOL, ZERNFUN2. ISFNP&& K  
    c^pQitPv  
    "a~r'+'<  
    %   Paul Fricker 11/13/2006 $%"hhju  
    ob2_=hQnC  
    uYg Q?*Z  
    {J,"iJKop  
    (GpP=lSSeY  
    % Check and prepare the inputs: 0#8, (6  
    % ----------------------------- a)=|{QR>W  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) m;{HlDez  
        error('zernfun:NMvectors','N and M must be vectors.') rXMc0SPk  
    end se2Y:v  
    hE`d@  
    KU oAxA  
    if length(n)~=length(m) PI`Y%!P  
        error('zernfun:NMlength','N and M must be the same length.') '/6f2[%Y"  
    end .xmB8 R  
    U\qbr.<  
    $|J+  
    n = n(:); AA=rjB9  
    m = m(:); u pUJF`3  
    if any(mod(n-m,2)) 0uW)&>W  
        error('zernfun:NMmultiplesof2', ... '/ Hoq  
              'All N and M must differ by multiples of 2 (including 0).') Fv %@k{  
    end =>3,]hnep  
    I(7iD. ^:  
    wXqwb|2  
    if any(m>n) <X4f2z{T{@  
        error('zernfun:MlessthanN', ... xZ`vcS(  
              'Each M must be less than or equal to its corresponding N.') ip}%Y6Wj  
    end &-Wt!X 3  
    O|=?!|`o  
    j?]+~  
    if any( r>1 | r<0 ) 0n`Temb/  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Q$]1juqg  
    end <D)@;A  
    85[ 7lO)[  
    ;^0ok'P\~9  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) &K9RV4M5  
        error('zernfun:RTHvector','R and THETA must be vectors.') kv2o.q  
    end !]A/ID0K  
    V( 0Y   
    CPcUB4a%#  
    r = r(:); L / WRVc6  
    theta = theta(:); BxlhCu  
    length_r = length(r); \_R<Q?D+  
    if length_r~=length(theta) NopfL  
        error('zernfun:RTHlength', ... $yj*n;  
              'The number of R- and THETA-values must be equal.') AI{0;0  
    end 2~g-k 3  
    :R:@V#Y  
    is^R8a  
    % Check normalization: l$c/!V[3  
    % -------------------- UukY9n];]  
    if nargin==5 && ischar(nflag) t5K#nRd Z:  
        isnorm = strcmpi(nflag,'norm'); +`Nu0y!rj  
        if ~isnorm Z"w}`&TC$^  
            error('zernfun:normalization','Unrecognized normalization flag.') (,+#H]L  
        end |P|2E~[r  
    else t!J>853  
        isnorm = false; Fec4#}|  
    end <_eEpG}9  
    }{:}K<  
    [kr-gV  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @zi0:3`#0\  
    % Compute the Zernike Polynomials W  wj+\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1'TS!/ll];  
    b 1Wz  
    UCj+V@{  
    % Determine the required powers of r: u0oTqD?  
    % ----------------------------------- `}sFT:1&  
    m_abs = abs(m); b.[9Adi >  
    rpowers = []; _]Ob)RUVH  
    for j = 1:length(n) G@jx&#v  
        rpowers = [rpowers m_abs(j):2:n(j)]; 06.8m;{N  
    end OT|0_d?bD  
    rpowers = unique(rpowers); )*uotV  
    $/#[,1  
    +=|%9%  
    % Pre-compute the values of r raised to the required powers, AOcUr)  
    % and compile them in a matrix: Lp|n)29+du  
    % ----------------------------- oVbs^sbRH  
    if rpowers(1)==0 &1 yErGXC  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ..'"kX:5  
        rpowern = cat(2,rpowern{:}); T5T[$%]6  
        rpowern = [ones(length_r,1) rpowern]; :ntAU2)H  
    else Ow5 VBw(  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Rh ]XJM  
        rpowern = cat(2,rpowern{:}); ".#h$  
    end !m'Rp~t  
    ?LU>2!jN  
    UM21Cfqex  
    % Compute the values of the polynomials: OQ<;w  
    % -------------------------------------- KX cRm)  
    y = zeros(length_r,length(n)); bi@'m?XwJ  
    for j = 1:length(n) ObreDv^,  
        s = 0:(n(j)-m_abs(j))/2; yn(bW\  
        pows = n(j):-2:m_abs(j); +`B^D  
        for k = length(s):-1:1 ]uh/!\  
            p = (1-2*mod(s(k),2))* ... TEj"G7]1$A  
                       prod(2:(n(j)-s(k)))/              ... pTTM(Hrx  
                       prod(2:s(k))/                     ... mO]dP;,  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... K~3Y8ca  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); > MRuoJ  
            idx = (pows(k)==rpowers); H)dZ0n4T  
            y(:,j) = y(:,j) + p*rpowern(:,idx); (47la$CR  
        end }jWg&<5+z  
         uXUuA/O5-  
        if isnorm ,->5 sJ{U  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); w&VDe(:~  
        end >X"\+7bw  
    end .~rg#*]^  
    % END: Compute the Zernike Polynomials [fvjvN`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #RSUChe7w  
    H`q[!5~8  
    JlRNJ#h>  
    % Compute the Zernike functions: ~P~q'  
    % ------------------------------ H%Lln#  
    idx_pos = m>0; '`W6U]7>  
    idx_neg = m<0; c_.Fe'E  
    Clap3E|a  
    2 1+[9  
    z = y; W* v3B.  
    if any(idx_pos) V joVC$ZX  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); WW^+X~Y  
    end 7xG~4N<)]  
    if any(idx_neg) 9GTp};Kg  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); hK{<&T  
    end mZM7 4!4X  
    ,i;#e  
    yO7#n0q  
    % EOF zernfun 4)'U!jSb  
     
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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)  K5 BL4N  
    eNVuw:Q+  
    DDE还是手动输入的呢? GdVF;  
    7Zd g314  
    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