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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, (b//YyqN  
    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, _$c o Y  
    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? qX5>[qf-  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? CU\gx*=E  
    UWC4PWL,>C  
    1g{}O^ul  
    $M,<=.oT  
    I <D7 Jj  
    function z = zernfun(n,m,r,theta,nflag) G6zFQ\&f  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 6384$mT,S  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N {{Ox%Zm  
    %   and angular frequency M, evaluated at positions (R,THETA) on the fEXFnQ#  
    %   unit circle.  N is a vector of positive integers (including 0), and jDb\4QyC  
    %   M is a vector with the same number of elements as N.  Each element zgEN2d  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) >"b W'  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, wrgB =o  
    %   and THETA is a vector of angles.  R and THETA must have the same )~=8Ssu  
    %   length.  The output Z is a matrix with one column for every (N,M) \^" Vqx  
    %   pair, and one row for every (R,THETA) pair. G`O*AQ}[  
    % n]$rLm%^  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike s0;a j<J  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !'^l}K>  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ial{A6X  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1,  4bA^Gq  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized oio{@#DX`  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?SFQx \/  
    % SgewAng?@o  
    %   The Zernike functions are an orthogonal basis on the unit circle. er7(Wph  
    %   They are used in disciplines such as astronomy, optics, and GWuKDq  
    %   optometry to describe functions on a circular domain. AJEbiP  
    % O)vGIp?f't  
    %   The following table lists the first 15 Zernike functions. C}mhnU@  
    % !;|#=A9  
    %       n    m    Zernike function           Normalization hxMRmH[f:  
    %       -------------------------------------------------- Eej Lso#\  
    %       0    0    1                                 1 #W)m({}  
    %       1    1    r * cos(theta)                    2 B;(U ?gC  
    %       1   -1    r * sin(theta)                    2 C_Q3^mLx  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) S!u8JG1  
    %       2    0    (2*r^2 - 1)                    sqrt(3) a($7J6]M  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) r_$*euh@  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) W%>T{}4  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) V 9$T=[  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) u:|^L]{  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) _LwF:19Il  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) P1rjF:x[*  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) R;Dj70g  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) f EL 9J{  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) \DujF>:  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) g@!U^mr*3  
    %       -------------------------------------------------- /A,w{09G  
    % /g+-{+sx  
    %   Example 1: Xrb7.Y0d  
    % 63l& ihj  
    %       % Display the Zernike function Z(n=5,m=1) 85G-`T  
    %       x = -1:0.01:1; @z ",1^I  
    %       [X,Y] = meshgrid(x,x); !hq*WtIk  
    %       [theta,r] = cart2pol(X,Y); |E?r+]  
    %       idx = r<=1; N!~]D[D  
    %       z = nan(size(X)); SgxrU&::  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); dX/7n=  
    %       figure I m I$~q'  
    %       pcolor(x,x,z), shading interp ?HPAX  
    %       axis square, colorbar z7IJSj1gQI  
    %       title('Zernike function Z_5^1(r,\theta)') e&ysj:W5 "  
    % [yN+(^ i  
    %   Example 2: j8Z;}Ps  
    % @6~lZgXOV[  
    %       % Display the first 10 Zernike functions ]P wS3:x  
    %       x = -1:0.01:1; R&Nl!QTJj  
    %       [X,Y] = meshgrid(x,x); ow9a^|@a  
    %       [theta,r] = cart2pol(X,Y); y*^UGJC:  
    %       idx = r<=1; Ph""[0n%o  
    %       z = nan(size(X)); CBf[$[e  
    %       n = [0  1  1  2  2  2  3  3  3  3]; _N|%i J5  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Z S=H1  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Hj r'C?[  
    %       y = zernfun(n,m,r(idx),theta(idx)); R]%"YQ V  
    %       figure('Units','normalized') d*{Cv2A.  
    %       for k = 1:10 ?&wrz  
    %           z(idx) = y(:,k); oH6zlmqG"  
    %           subplot(4,7,Nplot(k)) qI7KWUR  
    %           pcolor(x,x,z), shading interp %dPk,Ylz  
    %           set(gca,'XTick',[],'YTick',[]) %Ve@DF8G  
    %           axis square o%~fJx:]y  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ?SgFD4<~P  
    %       end &6OY ^6<  
    % :a/rwZ[r  
    %   See also ZERNPOL, ZERNFUN2. {Ia1H  
    E<+ G5j  
    ^ 3 4Ng  
    %   Paul Fricker 11/13/2006 )-!)D  
    d lfjx  
    B,%6sa~I  
    p*lP9[7  
    8a 8a:d  
    % Check and prepare the inputs: ^yB]_*WJ  
    % ----------------------------- !Q|a R  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ;6PU  
        error('zernfun:NMvectors','N and M must be vectors.') trrNu  
    end Jkj7ty.J  
    }:faHLYT  
    qX\85dPn@}  
    if length(n)~=length(m) 3>VL>;75[  
        error('zernfun:NMlength','N and M must be the same length.') ]*| hd/j  
    end {2:baoG-  
    M5:.\0_  
    n+sv2Wv:  
    n = n(:); (LTu=1  
    m = m(:); m]U  
    if any(mod(n-m,2)) _@>*]g  
        error('zernfun:NMmultiplesof2', ... </_QldL_  
              'All N and M must differ by multiples of 2 (including 0).') ]>)shH=Yx  
    end ^V;r  
    o`Z3}  
    `uPO+2  
    if any(m>n) wwdmz;0S  
        error('zernfun:MlessthanN', ... ib(|}7Je  
              'Each M must be less than or equal to its corresponding N.') rR@]`@9  
    end [VXQ&  
    m33&obSP  
    iSf%N>y'K  
    if any( r>1 | r<0 ) W gyRK2#!  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') d>F7i~W  
    end X~VI}dJ  
    axC{azo|  
    Ld_uMe?Z  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) QmSj6pB>  
        error('zernfun:RTHvector','R and THETA must be vectors.') ;q-c[TZC  
    end sT1OAK\^  
    4qDO(YWf  
    46T(1_Xt~  
    r = r(:); Zex~ $r  
    theta = theta(:); <#BK(W~$  
    length_r = length(r); aK6dy\  
    if length_r~=length(theta) BDfMFH[1  
        error('zernfun:RTHlength', ... K3:z5j.X  
              'The number of R- and THETA-values must be equal.') .&b^6$dC  
    end tBzE(vW  
    _"Y7}A\9  
    `/m] K ~~  
    % Check normalization: -]KgLgJ  
    % -------------------- U*K4qJ6U  
    if nargin==5 && ischar(nflag) M)K!!Jqh  
        isnorm = strcmpi(nflag,'norm'); c(Y~5A{TXO  
        if ~isnorm )OQm,5F1  
            error('zernfun:normalization','Unrecognized normalization flag.') ][Tw^r&  
        end h/C{  
    else z<t2yh(DF  
        isnorm = false; DmgDhNXKq  
    end &0T7Uv-`  
    R $<{"b  
    +~F>:v?Rh  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1^NC=IS9z  
    % Compute the Zernike Polynomials ? XVE {N  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,O.iOT0=;  
    oAN,_1v)  
    .&]3wB~  
    % Determine the required powers of r: #QlxEs#%  
    % ----------------------------------- 2IHS)kkT|  
    m_abs = abs(m); _\dC<K *>  
    rpowers = []; [%LGiCU]  
    for j = 1:length(n) F ',1R"/}  
        rpowers = [rpowers m_abs(j):2:n(j)]; cyd_xB5K  
    end Ye|gW=FUR  
    rpowers = unique(rpowers); +-t&li%F  
    #('R`~  
    BuM #&]s  
    % Pre-compute the values of r raised to the required powers, ~^Al#@  
    % and compile them in a matrix: -|#/KKF  
    % ----------------------------- \s8h.xjU  
    if rpowers(1)==0 kQ\l7xd  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); g;R  
        rpowern = cat(2,rpowern{:}); OFv-bb*YZ  
        rpowern = [ones(length_r,1) rpowern];  !N\_D  
    else r!{i2I|  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); p{qA%D  
        rpowern = cat(2,rpowern{:}); #Z]Cq0=  
    end #l) o<Z  
    NV{= tAR  
    R ^@`]dX$  
    % Compute the values of the polynomials: XH0Vs.w  
    % -------------------------------------- uUBUUr  
    y = zeros(length_r,length(n)); XOS^&;  
    for j = 1:length(n) ]EN&EA"<  
        s = 0:(n(j)-m_abs(j))/2; RigS1A\2l  
        pows = n(j):-2:m_abs(j); "7(@I^'t6  
        for k = length(s):-1:1 B2BG*xa  
            p = (1-2*mod(s(k),2))* ... 'q/C: Yo  
                       prod(2:(n(j)-s(k)))/              ... 4u2_xbT  
                       prod(2:s(k))/                     ... @/01MBs;  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... [D?xd/G  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); A&KY7[<AC{  
            idx = (pows(k)==rpowers); Bd>ATc+580  
            y(:,j) = y(:,j) + p*rpowern(:,idx); f e6Op  
        end #\="^z6  
         iRW5*-66f  
        if isnorm H- WNu+  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); G'HLnx}Yi  
        end 02^\np  
    end rP6k}  
    % END: Compute the Zernike Polynomials Cx) N;x  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v+C D{Tc  
    ,pZz`B#  
    >9g^-~X;v  
    % Compute the Zernike functions: 4Im}!q5;:<  
    % ------------------------------ )i-`AJK-'v  
    idx_pos = m>0; ;%>X+/.y0  
    idx_neg = m<0; 0icB2Jm:D}  
    DAN"&&  
    :w4H$+j  
    z = y; "tK3h3/Xv  
    if any(idx_pos) f|!@H><  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); tJGPkeA  
    end k[1[Y{n.  
    if any(idx_neg) HqOnZ>D  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)');  -x/g+T-  
    end cwUor}<|  
    b]8\% =d  
    ws]d,]  
    % EOF zernfun 2NL|_W/  
     
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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,s .<TG  
    }tT*Ch?u  
    DDE还是手动输入的呢? OG}D;Ew  
    DV~1gr,\  
    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