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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, M;-PrJdyt  
    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, &r do Mc;  
    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? Wv8?G~>  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 2old})CLJ  
    PFu{OJg&  
    Ja"?Pb  
    )pbsvR_  
    f;x0Ho5C2  
    function z = zernfun(n,m,r,theta,nflag) mA@FJK_  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. #Ipi3  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N `zw XfY,%  
    %   and angular frequency M, evaluated at positions (R,THETA) on the `1{Y9JdQ  
    %   unit circle.  N is a vector of positive integers (including 0), and ~l+2Z4nV  
    %   M is a vector with the same number of elements as N.  Each element f; w\k7 #  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) m %]1~b}"  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, j4k\5~yzS  
    %   and THETA is a vector of angles.  R and THETA must have the same u%!/-&?wF  
    %   length.  The output Z is a matrix with one column for every (N,M) ;G.5.q[A  
    %   pair, and one row for every (R,THETA) pair. |Bz1u|uc  
    % X6 *4IE  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ~t^ Umx"Ew  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), JlR$"GU  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral %D1 |0v8}  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, & %A&&XT9  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized cD6S;PSg  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. /o OZ>B%1s  
    % l0 =[MXM4  
    %   The Zernike functions are an orthogonal basis on the unit circle. }C4wED.  
    %   They are used in disciplines such as astronomy, optics, and U}@xMt8@l  
    %   optometry to describe functions on a circular domain. J?{@pA  
    % iR?}^|]  
    %   The following table lists the first 15 Zernike functions. 5(>SFxz"t  
    % ~(nc<M[  
    %       n    m    Zernike function           Normalization VKV :U60  
    %       -------------------------------------------------- `6$|d,m5  
    %       0    0    1                                 1 <aztbq?  
    %       1    1    r * cos(theta)                    2 ;3x*pjLG:Q  
    %       1   -1    r * sin(theta)                    2 aD]! eP/)  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ZtyDip'x  
    %       2    0    (2*r^2 - 1)                    sqrt(3) E75/EQ5p]p  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 0vETg'r  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 3xg9D.A  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) n,U?]mr  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ] # VHx  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) DA1?M'N  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) sYjhQN=Y*  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) w4(L@1  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 2ah%,o  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /~M H]Gh  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) jp_|pC'  
    %       -------------------------------------------------- JIhEkY  
    % ]{oZn5F  
    %   Example 1: (+c1.h  
    % [\AOr`7  
    %       % Display the Zernike function Z(n=5,m=1) 6<EGH*GQ$  
    %       x = -1:0.01:1; AdVc1v&>  
    %       [X,Y] = meshgrid(x,x); l+[:Cni  
    %       [theta,r] = cart2pol(X,Y); ~w a6S?  
    %       idx = r<=1; ,DZvBS  
    %       z = nan(size(X)); 1W\E`)Z}]  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); /a'1 W/^2  
    %       figure J$Z=`=] t+  
    %       pcolor(x,x,z), shading interp 3/>7b (  
    %       axis square, colorbar #l ZK_N|1x  
    %       title('Zernike function Z_5^1(r,\theta)') 4;fuS_(X  
    % B*N1)J\5  
    %   Example 2: jMgXIK\  
    % Hs*["zFc  
    %       % Display the first 10 Zernike functions 3V<@ Vkf5  
    %       x = -1:0.01:1; Keozn*fzI  
    %       [X,Y] = meshgrid(x,x); L8 L1_  
    %       [theta,r] = cart2pol(X,Y); =hkYQq`Q  
    %       idx = r<=1; oQ 2$z8  
    %       z = nan(size(X)); "|h%Uy?XY  
    %       n = [0  1  1  2  2  2  3  3  3  3]; MfP)Pk5  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ZUHRATT-  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; tO&ffZP8$  
    %       y = zernfun(n,m,r(idx),theta(idx)); 1h&`mqY)L.  
    %       figure('Units','normalized') e"ehH#i  
    %       for k = 1:10 27EK +$  
    %           z(idx) = y(:,k); N7?B"p/  
    %           subplot(4,7,Nplot(k)) X_]rtG  
    %           pcolor(x,x,z), shading interp VG);om7`PD  
    %           set(gca,'XTick',[],'YTick',[]) GC{M"q|_  
    %           axis square SVZocTt  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) /' + >/  
    %       end dE7S[O  
    % q`VL i  
    %   See also ZERNPOL, ZERNFUN2. c2y,zq|H  
    Ax;=Zh<DAv  
    l~6K}g?  
    %   Paul Fricker 11/13/2006 c-sjYJXKM*  
    U[@y 8yN6M  
    Y()" 2CCV  
    1^!SuAA@  
    -QrC>3xZR  
    % Check and prepare the inputs: p49]{2GXb  
    % ----------------------------- QO2cTk m  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Rff F:,b  
        error('zernfun:NMvectors','N and M must be vectors.') Z!)~?<gcq:  
    end rm iOeS`:  
    u^1#9bAW8  
    }yz>(Pq  
    if length(n)~=length(m) aQCu3T  
        error('zernfun:NMlength','N and M must be the same length.') DxJ;C09xNa  
    end Z0F~?  
    0zaK&]oY0  
    V!W.P  
    n = n(:); x HRSzYn$  
    m = m(:); E>!=~ 7.  
    if any(mod(n-m,2)) F5h/>  
        error('zernfun:NMmultiplesof2', ... 4:`D3  
              'All N and M must differ by multiples of 2 (including 0).') 5 4gr'qvr  
    end fw%`[( hK  
    ]~({;;3o-  
    , NSf  
    if any(m>n) ZK5nN9`  
        error('zernfun:MlessthanN', ... @5Xo2}o-Q  
              'Each M must be less than or equal to its corresponding N.') \N,ox(f?gW  
    end l~c[}wv  
    C=: <[_m`  
    &X=7b@r  
    if any( r>1 | r<0 ) }LzBo\  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 0 j.K?]f)h  
    end (_T{Z>C/J  
    Yj %]|E-  
    jD: N)((  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) #b/qR^2qW  
        error('zernfun:RTHvector','R and THETA must be vectors.') :xd;=;q5  
    end y&/IJst&aq  
    |#oS7oV(  
    d1b] +AG4  
    r = r(:); c{z$^)A/  
    theta = theta(:); ekM? ' 9ez  
    length_r = length(r); Cp8=8N(Xb  
    if length_r~=length(theta) [q <'ty  
        error('zernfun:RTHlength', ... JU 9GJ"  
              'The number of R- and THETA-values must be equal.') Dw-d`8*  
    end l/eF P  
    +r:g}iR  
    }9~^}99}  
    % Check normalization: p_FM 2K7!  
    % -------------------- x9_mlZ  
    if nargin==5 && ischar(nflag) &m5zd$6  
        isnorm = strcmpi(nflag,'norm'); @:lM|2:  
        if ~isnorm |Splbs k  
            error('zernfun:normalization','Unrecognized normalization flag.') g3R(,IH  
        end S;|:ci<[=  
    else (3#PKfY+  
        isnorm = false; ^h(wi`i  
    end !l:GrT8J  
    e+ xQ\LH  
    $|K d<wv  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l$42MRi/  
    % Compute the Zernike Polynomials Dl,QCZeM  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %y1!'R:ZW  
    d*(aue=  
    K,b M9>}  
    % Determine the required powers of r: YeH!v, >  
    % ----------------------------------- @u~S!(7.Wi  
    m_abs = abs(m); 2*#|t: (c  
    rpowers = []; @Nu2 :~JO  
    for j = 1:length(n) _z\/{  
        rpowers = [rpowers m_abs(j):2:n(j)]; m'4f'tbN  
    end PwY/VGT  
    rpowers = unique(rpowers); 9}573M  
    {SoI;o_>  
    $=aO*i  
    % Pre-compute the values of r raised to the required powers, Y\|#Lu>B  
    % and compile them in a matrix: BZR{}Aj4pa  
    % ----------------------------- @^{Hq6_`  
    if rpowers(1)==0 FG?Mc'r&  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); b 2gng}  
        rpowern = cat(2,rpowern{:}); ."Ms7=  
        rpowern = [ones(length_r,1) rpowern]; iD^,O)b  
    else _|k$[^ln^  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); RObnu*  
        rpowern = cat(2,rpowern{:}); .@1+}0  
    end \kADh?phV  
    TpjiKM  
    Z6!Up1  
    % Compute the values of the polynomials: Z!p\=M,%  
    % -------------------------------------- RLF&-[mr3  
    y = zeros(length_r,length(n)); "oP^2|${  
    for j = 1:length(n) tbrU>KCBD  
        s = 0:(n(j)-m_abs(j))/2; ) SV.|  
        pows = n(j):-2:m_abs(j); bO~y=Pa \  
        for k = length(s):-1:1 -,bFGTvYQ  
            p = (1-2*mod(s(k),2))* ... [Nyt0l "z  
                       prod(2:(n(j)-s(k)))/              ... ^-o{3Q(w  
                       prod(2:s(k))/                     ... aSR-.r  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Na\ZV|;*tu  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); b@CB +8 $  
            idx = (pows(k)==rpowers); /dnwN7Gf  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 9A .RD`fg  
        end SV7;B?e%Y  
         AtT7~cVe  
        if isnorm ]5%0EE64  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ^r}c&@  
        end STKL  
    end Zxk~X}K\P  
    % END: Compute the Zernike Polynomials 0<M-asI?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @"w4R6l+*  
    `oRyw6Sko  
    ep>!jMhJa  
    % Compute the Zernike functions: "ra$x2|=}  
    % ------------------------------ 7h' C"rH  
    idx_pos = m>0; ,H7X_KbFD4  
    idx_neg = m<0; C{)1#<`  
    ?hoOSur+  
    [8V;Q  
    z = y; Cq5.gkS<  
    if any(idx_pos) ULx:2jz  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 'nmGHorp  
    end 0uy'Py@2<  
    if any(idx_neg) B|`?hw@g+  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); unDW2#GX  
    end "2%z;!U1  
    (leX` SN0u  
    %h. zkocM  
    % EOF zernfun so))J`ca)  
     
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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)  B*_K}5UO  
    '( I0VJJ   
    DDE还是手动输入的呢? -] wEk%j  
    Z*M{  
    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