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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, 2aM7zP[Z  
    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, T^1 Z_|A  
    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? 7pyzPc#_  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ai/|qYf  
    !,m  
    A#=TR_@:  
    3x0t[{l  
    sF{aG6u   
    function z = zernfun(n,m,r,theta,nflag) jb.H[n,\  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Oo|PZ_P  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 5.9<g>C  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Mqr_w!8d  
    %   unit circle.  N is a vector of positive integers (including 0), and u S1O-Q>  
    %   M is a vector with the same number of elements as N.  Each element "0An'7'm  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Wb-C0^dTn  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, sE pI)9  
    %   and THETA is a vector of angles.  R and THETA must have the same }4A] x`3  
    %   length.  The output Z is a matrix with one column for every (N,M) RRIh;HhX  
    %   pair, and one row for every (R,THETA) pair. } a9Ah:.7/  
    % 0ra'H/>Ly  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike aTuu",f  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), V\;Xa0  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral +i&<`ov  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, W,<q!<z\t  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized q!ZM Wg  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. o.{W_k/n  
    % vk92j?  
    %   The Zernike functions are an orthogonal basis on the unit circle. Ek_5% n  
    %   They are used in disciplines such as astronomy, optics, and l-+=Yk!X  
    %   optometry to describe functions on a circular domain. C`[<6>&y  
    % {o}U"b<+Ra  
    %   The following table lists the first 15 Zernike functions. p0Jr{hM  
    % O[Vet/^)  
    %       n    m    Zernike function           Normalization @NLcO}  
    %       -------------------------------------------------- 8s1nE_3  
    %       0    0    1                                 1 rAH!%~  
    %       1    1    r * cos(theta)                    2 C^J<qq &  
    %       1   -1    r * sin(theta)                    2 Jka>Er  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) VeYT[Us"  
    %       2    0    (2*r^2 - 1)                    sqrt(3) g+ c*VmY  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) nkW})LyB\  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) J}#gTG( '  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ?QOU9"@+B  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) yLnQ9BXB&  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) -s3`mc}*  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) }L\;W:0  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) VdlT+'HF  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) kxMvOB$  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)  LR97FG  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) z'YWomfZm  
    %       -------------------------------------------------- YM}a>o  
    % .-d'*$ yJ  
    %   Example 1: jn<?,UABD  
    % \P<aK$g  
    %       % Display the Zernike function Z(n=5,m=1) XO+BZB`F  
    %       x = -1:0.01:1; *~vB6V|1  
    %       [X,Y] = meshgrid(x,x); =;Gq:mHi  
    %       [theta,r] = cart2pol(X,Y); _~<sb,W  
    %       idx = r<=1; )1s5vNVa  
    %       z = nan(size(X)); ,md_eGF  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); , >LJpv  
    %       figure K./qu^+k  
    %       pcolor(x,x,z), shading interp PkvW6,lS  
    %       axis square, colorbar 7v5]% %E/  
    %       title('Zernike function Z_5^1(r,\theta)') my (@~'  
    % daE.y_9y  
    %   Example 2: 3s6obw$ki  
    % lvW T  
    %       % Display the first 10 Zernike functions ~gDYb#p  
    %       x = -1:0.01:1; cOVj @z  
    %       [X,Y] = meshgrid(x,x); g) Lf^  
    %       [theta,r] = cart2pol(X,Y); mY"7/dw<v  
    %       idx = r<=1; &<A,\ M  
    %       z = nan(size(X)); i2=- su  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 6{h\CU}"  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; /<rvaR  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 4V@%Y,:ee  
    %       y = zernfun(n,m,r(idx),theta(idx)); Pb5yz-?  
    %       figure('Units','normalized') 4^F[Gp?  
    %       for k = 1:10 eZ'8JU]  
    %           z(idx) = y(:,k); @j!,8JQEd  
    %           subplot(4,7,Nplot(k)) Y%KowgP\  
    %           pcolor(x,x,z), shading interp `Fd \dn  
    %           set(gca,'XTick',[],'YTick',[]) roADC?@r  
    %           axis square FM {f{2j  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) .5|[gBK  
    %       end 3=O [Q:8  
    % (i~UH04r>s  
    %   See also ZERNPOL, ZERNFUN2. tOIqX0dWd  
    x[0T$  
    uo"<}>iJ  
    %   Paul Fricker 11/13/2006 ] K$YtM^  
    )lG}B U.  
    P5Xp #pa  
    \|PiQy*_?  
    2j s/>L0  
    % Check and prepare the inputs: I9X \@ lTf  
    % ----------------------------- <V?2;Gy  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) *:%&z?<Fw  
        error('zernfun:NMvectors','N and M must be vectors.') S\GWMB!oF  
    end m{IlRf'  
    \s=r[0tj!  
    odhcD;^X1  
    if length(n)~=length(m) S;~eI8gQ"  
        error('zernfun:NMlength','N and M must be the same length.') m?e/MQr  
    end K#R]of~/  
    LU6R"c11  
    2F4<3k! &  
    n = n(:); 5CI {&E  
    m = m(:); 'uu*DgEr  
    if any(mod(n-m,2)) de:@/-|  
        error('zernfun:NMmultiplesof2', ... #V k?  
              'All N and M must differ by multiples of 2 (including 0).') &^`Wtd~g  
    end l2F#^=tp  
    pDS[ecx  
    g[} L ?  
    if any(m>n) GfONm6A  
        error('zernfun:MlessthanN', ... a 0SZw  
              'Each M must be less than or equal to its corresponding N.') W@R7CQE@  
    end UC`h o%OBF  
    \K$\-]N+  
    :8yebOs   
    if any( r>1 | r<0 ) ZF7n]LgSc&  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') F_@B ` ,  
    end x6cG'3&T  
    }qWnn>h9xv  
    U$y 9f  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) $}/tlA&e  
        error('zernfun:RTHvector','R and THETA must be vectors.') c.>f,vtcn  
    end o/-RGLzAo  
    O=%Ht-kOc  
    $0V+<  
    r = r(:); =M1}HF,7>l  
    theta = theta(:); P'KA-4!  
    length_r = length(r); @b(@`yz.a  
    if length_r~=length(theta) 1>*oN  
        error('zernfun:RTHlength', ... tddwnpnSw  
              'The number of R- and THETA-values must be equal.') "(=g7,I4  
    end & AK\Pw)  
    e66Ag}Sw|  
    K~:SLCv E%  
    % Check normalization: S)hDsf.I  
    % -------------------- d(^8#4  
    if nargin==5 && ischar(nflag) qc(e3x  
        isnorm = strcmpi(nflag,'norm'); YP,,vcut  
        if ~isnorm kqB# 9  
            error('zernfun:normalization','Unrecognized normalization flag.') kn:hxdZ  
        end b%l H=u  
    else DN%}OcpZ  
        isnorm = false; vA6`};|  
    end @ lB{!j&q  
    i ;B^I8  
    gdIk%m4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q 4Pv\YO  
    % Compute the Zernike Polynomials "rMfe>;FJ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l~$)>?ZD  
    |&K;*g|a  
    B  W*8  
    % Determine the required powers of r: 0[UI'2  
    % ----------------------------------- h[dJNawL  
    m_abs = abs(m); sqhMnDn[  
    rpowers = []; "E+;O,N-  
    for j = 1:length(n) dEYw_qJ2  
        rpowers = [rpowers m_abs(j):2:n(j)]; tQ@7cjq8bA  
    end ?=lb@U  
    rpowers = unique(rpowers); A{> w5T  
    ]s Euh~F  
    4L>8RiiQE;  
    % Pre-compute the values of r raised to the required powers, ;?q(8^A  
    % and compile them in a matrix:  8s22VL  
    % ----------------------------- ObM/~{rKx  
    if rpowers(1)==0 'A|c\sy  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 0d2RB^"i  
        rpowern = cat(2,rpowern{:}); OcUj_Zd  
        rpowern = [ones(length_r,1) rpowern]; E^J &?-  
    else -aBhN~  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); y)KIz  
        rpowern = cat(2,rpowern{:}); o|>=< l  
    end 0WzoI2Q  
    f\5w@nX  
    yLf9cS6=  
    % Compute the values of the polynomials:  IZrcn  
    % -------------------------------------- 4x ?NCD=k  
    y = zeros(length_r,length(n)); Kz b-a$  
    for j = 1:length(n) u$tst_y-  
        s = 0:(n(j)-m_abs(j))/2; uKzx >\}?1  
        pows = n(j):-2:m_abs(j); P, ZQ*Ju  
        for k = length(s):-1:1 uPl7u 1c  
            p = (1-2*mod(s(k),2))* ... 'T^MaLK  
                       prod(2:(n(j)-s(k)))/              ... F3V:B.C  
                       prod(2:s(k))/                     ... DI)"F OM6  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... l`~$cK!  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); gK~Z Ch  
            idx = (pows(k)==rpowers); . AA# G  
            y(:,j) = y(:,j) + p*rpowern(:,idx); P'iX?+*  
        end Q}Ah{H0C  
         M#Z^8(  
        if isnorm j)G%I y[`  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ;Mq'+4$  
        end !.%*Tp#k#  
    end o#"yFP1  
    % END: Compute the Zernike Polynomials (]sm9PO  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =P,mix|  
    (XR}U6^v]  
    -J!n7  
    % Compute the Zernike functions: >"UXY)  
    % ------------------------------ X*#\JF4$i  
    idx_pos = m>0; xN$V(ZX4  
    idx_neg = m<0; Q65M(x+oy  
    %{'[S0@Z  
    k6DJ(.n'%a  
    z = y; _!|$i  
    if any(idx_pos) {R(/Usg!=  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); )/f#~$ws  
    end [jN Vk3  
    if any(idx_neg) b i-Am/9  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ^xk4HF   
    end Ib2&L  
    | #a{1Z)  
    p\I3fI0i  
    % EOF zernfun !p ~.Y+  
     
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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)  VM3)L>x]/  
    > LU !Z  
    DDE还是手动输入的呢? 4 tt=u]:  
    {X\FS   
    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