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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, rEj Ez+wu  
    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, _+<AxE9\  
    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? EV_u8?va  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? X\5EF7:S  
    Er|j\(jM  
     >1q:-^  
    X3l6b+p  
    ,<;.'r  
    function z = zernfun(n,m,r,theta,nflag) \cQ+9e)  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. {<Xl57w-Q  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N P%ZU+ET  
    %   and angular frequency M, evaluated at positions (R,THETA) on the RggO|s+0;  
    %   unit circle.  N is a vector of positive integers (including 0), and Zig3WiD&  
    %   M is a vector with the same number of elements as N.  Each element /KhY,G'Z  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) v>5TTL~?  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, !X1 KOG  
    %   and THETA is a vector of angles.  R and THETA must have the same Lt {&v ^y  
    %   length.  The output Z is a matrix with one column for every (N,M) CL5t6D9Qi  
    %   pair, and one row for every (R,THETA) pair. 5G=fJAG  
    % 9w-;d=(Q  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike c22L]Sxo  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), E :UJ"6  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral LHs^Xo18  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, |^O3~!JP(>  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized hYVy65Ea  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. zI_pP?4;.q  
    % MaP-   
    %   The Zernike functions are an orthogonal basis on the unit circle. 3# idXc  
    %   They are used in disciplines such as astronomy, optics, and jtPHk*>^wu  
    %   optometry to describe functions on a circular domain. rrl{3 ?  
    % @Z89cTO  
    %   The following table lists the first 15 Zernike functions. 9)'wgI#  
    % BWzo|isv  
    %       n    m    Zernike function           Normalization ! ;R}=  
    %       -------------------------------------------------- M2M&L,/O  
    %       0    0    1                                 1 6}:(m#+  
    %       1    1    r * cos(theta)                    2 *!,k`=.([#  
    %       1   -1    r * sin(theta)                    2  !~]'&9  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) .FvIT] k-  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Olr'n% }  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 8zpTCae^=7  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Yz>8 Nn'_  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) $~/x;z:  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Y~U WUF%aK  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) dbfI!4  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) kj`h{Wc[)  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) F ZfhiIf  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) vcSb:('  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) xgWVxX^)  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) LP} j0)n  
    %       -------------------------------------------------- r,ep{ p  
    % _j]vR  
    %   Example 1: =@.5J'!  
    % hD7Lgi-N)W  
    %       % Display the Zernike function Z(n=5,m=1) J!iK W  
    %       x = -1:0.01:1; V.w!]{xm  
    %       [X,Y] = meshgrid(x,x); 5,du2  
    %       [theta,r] = cart2pol(X,Y); lv& y<d;  
    %       idx = r<=1; |k)Nf+(}W  
    %       z = nan(size(X)); La si)e=$<  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); W 6CNMI]  
    %       figure O_Z   
    %       pcolor(x,x,z), shading interp q`@8  
    %       axis square, colorbar ExSy/^4f  
    %       title('Zernike function Z_5^1(r,\theta)') -7m7.>/M  
    % 2bTM0-  
    %   Example 2: 7/FF}d  
    % &DWSu`z  
    %       % Display the first 10 Zernike functions z_87 ;y;=  
    %       x = -1:0.01:1; ksQw|>K  
    %       [X,Y] = meshgrid(x,x); XI5q>cd\Sz  
    %       [theta,r] = cart2pol(X,Y); yu=(m~KX   
    %       idx = r<=1; I(+%`{Wv  
    %       z = nan(size(X)); Ml+O - 3T  
    %       n = [0  1  1  2  2  2  3  3  3  3]; bYy7Ul6]  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3];  -to3I  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; }z_7?dn/  
    %       y = zernfun(n,m,r(idx),theta(idx)); @;{iCVW  
    %       figure('Units','normalized') 3@mW/l>X  
    %       for k = 1:10 4z,n:>oH  
    %           z(idx) = y(:,k); nY_+V{F  
    %           subplot(4,7,Nplot(k)) \_|r>vQ  
    %           pcolor(x,x,z), shading interp [K`d?&  
    %           set(gca,'XTick',[],'YTick',[]) }E\u2]  
    %           axis square 01o,9_|FL  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) a`zw5  
    %       end E^t}p[s  
    % ~ tqDh(  
    %   See also ZERNPOL, ZERNFUN2. $~:|Vj5iZ\  
    <O]B'Wc [  
    C8 "FTH'  
    %   Paul Fricker 11/13/2006  X&.LX  
    41\V;yib  
    N"2P]Z r  
    ,%,.c^-  
    7)y +QU]  
    % Check and prepare the inputs: [2nPr^  
    % ----------------------------- ;Y`k-R:E6A  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) :tBZu%N/N  
        error('zernfun:NMvectors','N and M must be vectors.') /w:~!3Aj0+  
    end i UW.$1l  
    JAI;7  
    aT PmW]w6  
    if length(n)~=length(m) Iqb|.vLG  
        error('zernfun:NMlength','N and M must be the same length.') 3+iQct[  
    end rfhvdwwD  
    d# q8-  
    aKC3v R0  
    n = n(:); >A1;!kGE#  
    m = m(:); ^|=3sJ4[U  
    if any(mod(n-m,2)) S&;D  
        error('zernfun:NMmultiplesof2', ... l*n4d[0J  
              'All N and M must differ by multiples of 2 (including 0).') (Kaunp5_`  
    end W&Kjh|[1QZ  
    5gY9D!;:0D  
    VHTr;(]hk  
    if any(m>n) \k*h& :$  
        error('zernfun:MlessthanN', ... -gb'DN1BG  
              'Each M must be less than or equal to its corresponding N.') v6+<F;G3y>  
    end f`8mES'gc8  
    pn4~?Aua0/  
    gD/% l[  
    if any( r>1 | r<0 ) kS$m$ D  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') %Dm:|><V$b  
    end  g=x1}nm  
    2~2j?\AEd.  
    L=5Fvm  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) V2_I=]p_  
        error('zernfun:RTHvector','R and THETA must be vectors.') - WK  
    end {-)*.l=  
    -K%~2M<  
    z+%74O"c  
    r = r(:); U Zc%XZ`"V  
    theta = theta(:); 2q*aq%  
    length_r = length(r); z7um9g  
    if length_r~=length(theta) vP{;'R  
        error('zernfun:RTHlength', ... \t@4)+s/)  
              'The number of R- and THETA-values must be equal.') hZNA I  
    end lF.yQ  
    :_"%o=  
    "N*i!h  
    % Check normalization: c %.vI  
    % -------------------- ?tFsSU  
    if nargin==5 && ischar(nflag) "4e{Cq  
        isnorm = strcmpi(nflag,'norm'); {>R'IjFc  
        if ~isnorm 5WG:m'$$  
            error('zernfun:normalization','Unrecognized normalization flag.') +2S#3m?1  
        end _=;ltO  
    else uV+.(sjH  
        isnorm = false; YN 31Lo  
    end k?'<f  
    N"rZK/@}  
    7__?1n~{  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #Ez+1  
    % Compute the Zernike Polynomials u#`FkuE\}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zCdzxb_h"  
    ZP^7`q)6  
    2OQDG7#Kc  
    % Determine the required powers of r: Y]>Qu f.!  
    % ----------------------------------- za oC  
    m_abs = abs(m); ?sm@lDZ\  
    rpowers = []; e3b|z.^8  
    for j = 1:length(n) W^AY:#eX~Q  
        rpowers = [rpowers m_abs(j):2:n(j)]; +qzCy/_gd  
    end FkJX)  
    rpowers = unique(rpowers); K7N.gT*4  
    8  }(ul  
    K JX@?1"  
    % Pre-compute the values of r raised to the required powers, Z-B b,8  
    % and compile them in a matrix: y:3d`E4Xw  
    % ----------------------------- K?:wX(JYT  
    if rpowers(1)==0 aR~Od Ys  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Yab=p 9V;;  
        rpowern = cat(2,rpowern{:}); {-?8r>  
        rpowern = [ones(length_r,1) rpowern]; xRU ~h Q  
    else j1{\nP/  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); bxA1fA;  
        rpowern = cat(2,rpowern{:}); ,t=12R]>  
    end pRLs*/Bw  
    Jf{ M[ z  
    |U4t 8  
    % Compute the values of the polynomials: wu2C!gyBo  
    % -------------------------------------- bR;Zc  
    y = zeros(length_r,length(n)); Hz6yy*  
    for j = 1:length(n) ~8 w(M  
        s = 0:(n(j)-m_abs(j))/2; .6D9m.Q,  
        pows = n(j):-2:m_abs(j); , JUP   
        for k = length(s):-1:1 <7%4=  
            p = (1-2*mod(s(k),2))* ... tuiQk=[ c  
                       prod(2:(n(j)-s(k)))/              ... mC}!;`$8p  
                       prod(2:s(k))/                     ... N2x!RYW  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... =!cI@TI  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); + $x;FT&  
            idx = (pows(k)==rpowers); |rbl sL2?Z  
            y(:,j) = y(:,j) + p*rpowern(:,idx); %g<J"/  
        end L!]~ J?)  
         2!4.L&Ki  
        if isnorm BLvI[b|3gn  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); X;?Z_3I:5  
        end fx783  
    end Mn=5yU  
    % END: Compute the Zernike Polynomials S"z cSkF  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% WZ<kk T  
    qJ|n73yn  
    P6E=*^^m(  
    % Compute the Zernike functions: 3oCw(Ff  
    % ------------------------------ QF fKEMN  
    idx_pos = m>0; M5Twulz/w  
    idx_neg = m<0; 6!3Jr  
    MK<VjpP0(  
    .u_k?.8|  
    z = y; >Lo!8Hen  
    if any(idx_pos) G{cTQH|  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); weOzs]uc  
    end z]YP  
    if any(idx_neg) Gkr^uXNg#  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Q l$t  
    end s\`Vr;R:|  
    4P>tGO&*x  
    u%7a&1c  
    % EOF zernfun 2 8j=q-9Z  
     
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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)  UuT[UB=x5  
    /1li^</|p`  
    DDE还是手动输入的呢? 9jPb-I-   
    b$R>GQ?#  
    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