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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, n<47#-  
    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, )i@j``P  
    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? JD1IL` ta;  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ^gx`@^su  
    K^0cL%dB  
    ] ;X[xs  
    f S-(Kmh  
    ()L[l@m  
    function z = zernfun(n,m,r,theta,nflag) R$qp3I  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. YU! SdT$  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N % \OG#36  
    %   and angular frequency M, evaluated at positions (R,THETA) on the QR4!r@*=  
    %   unit circle.  N is a vector of positive integers (including 0), and ox9$aBjJ  
    %   M is a vector with the same number of elements as N.  Each element 'r_{T=  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) }T([gc7~  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, B?d^JWTZ  
    %   and THETA is a vector of angles.  R and THETA must have the same w6ZyMR,T  
    %   length.  The output Z is a matrix with one column for every (N,M) `uL^!-  
    %   pair, and one row for every (R,THETA) pair. ;{@ [ek6  
    % _?]E)i'RI  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 7q_B`$ata  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), zq ;YE  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral   -58  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3q7Z?1'o  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized AWkXW l}  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. aKi&2>c5>  
    % *fs'%"w-  
    %   The Zernike functions are an orthogonal basis on the unit circle. ry bs9:_}  
    %   They are used in disciplines such as astronomy, optics, and o/ Z  
    %   optometry to describe functions on a circular domain. K/)*P4C-  
    % t+C9QXY  
    %   The following table lists the first 15 Zernike functions. |l5ol @2*  
    % vFuf{ @P  
    %       n    m    Zernike function           Normalization qfF/X"#0  
    %       -------------------------------------------------- QoagyL  
    %       0    0    1                                 1 j*2Q{ik>J  
    %       1    1    r * cos(theta)                    2 ,+`1/  
    %       1   -1    r * sin(theta)                    2 [QC<u1/"K  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 5\hJ&  
    %       2    0    (2*r^2 - 1)                    sqrt(3) (J!FW(Ma|=  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) VRr_s:CWK  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) C*O648yz[  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ;IklS*p]  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) p'# (^  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 3]Jl\<0  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) y*i_Ec\h  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) k 4|*t}o7  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Vaj4p""\F  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Cso!VdCX  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 0#K?SuY.eN  
    %       -------------------------------------------------- cL/ 6p0S  
    % 3aMfZa<=  
    %   Example 1: +n#kpi'T  
    % mc{gcZIm  
    %       % Display the Zernike function Z(n=5,m=1) \_H-TbU8  
    %       x = -1:0.01:1; 0UV5}/2rP  
    %       [X,Y] = meshgrid(x,x); cY&SKV#  
    %       [theta,r] = cart2pol(X,Y); RPH]@  
    %       idx = r<=1; A\{dq:  
    %       z = nan(size(X)); G8Hj<3`  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); rgth2y]  
    %       figure tCkKJ)m  
    %       pcolor(x,x,z), shading interp if|j)h&  
    %       axis square, colorbar "S#}iYp  
    %       title('Zernike function Z_5^1(r,\theta)') [=Qv?am  
    % Y\CR*om!W  
    %   Example 2: 0I|IL]JL  
    % kzZdYiC  
    %       % Display the first 10 Zernike functions *{3&?pxx  
    %       x = -1:0.01:1; ;W ZA  
    %       [X,Y] = meshgrid(x,x); %O9kq  
    %       [theta,r] = cart2pol(X,Y); \\<waU''  
    %       idx = r<=1; TDvUiJm  
    %       z = nan(size(X)); o;.6Y `-fJ  
    %       n = [0  1  1  2  2  2  3  3  3  3]; >G4EiJS  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 'g3!SdaLF  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; [A@K)A$f  
    %       y = zernfun(n,m,r(idx),theta(idx)); hXxgKi%  
    %       figure('Units','normalized') |~QHCg<  
    %       for k = 1:10 UkO L7M  
    %           z(idx) = y(:,k); MjGeH>c  
    %           subplot(4,7,Nplot(k)) 4';~@IBf  
    %           pcolor(x,x,z), shading interp cP >MsUZWl  
    %           set(gca,'XTick',[],'YTick',[]) {|Ew]Wq  
    %           axis square Mi|PhDXMh  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) t:pgw[UJ  
    %       end K YSyz)M}  
    % z|';Y!kQ  
    %   See also ZERNPOL, ZERNFUN2. U g 'y  
    mkJC *45  
    pn},ovR;  
    %   Paul Fricker 11/13/2006 E=Z;T   
    y.KFz9Qv  
    egOZ.oV  
    XynDo^+ru  
    wHGiN9A+  
    % Check and prepare the inputs: fgmu*\x<  
    % ----------------------------- -ahSFBZlg  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 6EHYIN^D  
        error('zernfun:NMvectors','N and M must be vectors.') W(5et5DN,  
    end Eb9 eEa<W  
    &&(^;+  
    W u4` 3  
    if length(n)~=length(m) Ek0zFnb[Gx  
        error('zernfun:NMlength','N and M must be the same length.') ]u;Ma G=;  
    end vr/O%mDp  
    gBF2.{"^  
    s7x&x;-  
    n = n(:); hJuR,NP  
    m = m(:); i{#5=np H  
    if any(mod(n-m,2)) u@( z(P  
        error('zernfun:NMmultiplesof2', ... i_ha^mq3  
              'All N and M must differ by multiples of 2 (including 0).') =dVPx<l5  
    end .@#GNZe  
    Ro&s\T+d  
    xJ/<G$LNJ0  
    if any(m>n) r&^xg`i[z>  
        error('zernfun:MlessthanN', ... =}bDT2Nb  
              'Each M must be less than or equal to its corresponding N.') 9Ai e$=  
    end TFIP>$*_C  
    ~ULD{Ov'F  
    (\CT "u-  
    if any( r>1 | r<0 ) |4=Du-e  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') k0%*{IVPN  
    end `k^d)9  
    )# ^5$5  
    G5C=p:o{/  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) # :^aE|s  
        error('zernfun:RTHvector','R and THETA must be vectors.') 17-D\ +}  
    end aFCma2  
    SN${cs%  
    *bi!iz5F  
    r = r(:); oWJ0>)  
    theta = theta(:); 9 n(.v}  
    length_r = length(r); 0j =xWC  
    if length_r~=length(theta) $#b@b[h<w  
        error('zernfun:RTHlength', ... XXx]~m  
              'The number of R- and THETA-values must be equal.') =/ b2e\  
    end T#HW{3  
    {LwV&u(  
    l ~b  
    % Check normalization: 5_|Sm=  
    % -------------------- -y@# ^SrJ  
    if nargin==5 && ischar(nflag) ,*y\b|<j  
        isnorm = strcmpi(nflag,'norm'); 676r0`  
        if ~isnorm RDX$Wy$@L  
            error('zernfun:normalization','Unrecognized normalization flag.') n54}WGo>9  
        end OA_WjTwDs  
    else w1#1s|  
        isnorm = false; 3lkz:]SsE  
    end OoG Nij  
    u$vA9g4  
    m1d*Lt>F@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% HDV@d^]-  
    % Compute the Zernike Polynomials g>@T5&1q*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RotWMGNK  
    " R=,W{=  
    vr:5+wew  
    % Determine the required powers of r: )z:"P;b"Nl  
    % ----------------------------------- vtm?x,h  
    m_abs = abs(m); ,Q7W))j  
    rpowers = []; xcVF0%wVC  
    for j = 1:length(n) &8w MGahp  
        rpowers = [rpowers m_abs(j):2:n(j)]; \dB)G<_  
    end li4"|T&  
    rpowers = unique(rpowers); a 8(mU%  
    ` oPUf!  
    I(bxCiRV  
    % Pre-compute the values of r raised to the required powers, +\Zr\fOe|%  
    % and compile them in a matrix: Q5kf-~Jx+  
    % ----------------------------- SU8vz/\%y  
    if rpowers(1)==0 rV5QKz6'  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); eu^B  
        rpowern = cat(2,rpowern{:}); eeOE\  
        rpowern = [ones(length_r,1) rpowern]; eG\|E3Cb9  
    else -45xa$vv  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); n'i~1pM,?  
        rpowern = cat(2,rpowern{:}); 54^2=bp  
    end :!cNkJa  
    {&I3qk2(  
    dPVl\<L1  
    % Compute the values of the polynomials: JSCZX:5  
    % -------------------------------------- V\2&?#GZ  
    y = zeros(length_r,length(n)); hFiJHV  
    for j = 1:length(n) }O7!>T  
        s = 0:(n(j)-m_abs(j))/2; x2_?B[z  
        pows = n(j):-2:m_abs(j);  f^vz  
        for k = length(s):-1:1 v}>5!*  
            p = (1-2*mod(s(k),2))* ... l ;fO]{  
                       prod(2:(n(j)-s(k)))/              ... HW"';M%  
                       prod(2:s(k))/                     ... u A=x~-I  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... C7hJE -  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ;oT!\$Mu  
            idx = (pows(k)==rpowers); 5 `Mos  
            y(:,j) = y(:,j) + p*rpowern(:,idx); !#b8QER  
        end }zE Qrfl  
         an<loL W  
        if isnorm F?3zw4Vt~  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Ln3<r&&Jz  
        end Wh7}G   
    end 8s@k0T<O  
    % END: Compute the Zernike Polynomials 2Jl$/W 3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% IT5a/;J  
    !^h{7NmP[  
    k04CSzE"%  
    % Compute the Zernike functions: @/yQ4Gr  
    % ------------------------------ o;^k"bo6   
    idx_pos = m>0;  FTk`Mq  
    idx_neg = m<0; 920 o]Dh=t  
    'xn3g;5  
    \0'0)@uziQ  
    z = y; -Y:^<C^^&8  
    if any(idx_pos) -h|YS/$f  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 4Ww.CkRG  
    end ndB [f  
    if any(idx_neg) FKVf_Ncf%  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 4^>FN"Ve`B  
    end T=- $ok`G  
    2(%C  
    >AUj4d  
    % EOF zernfun !92zC._  
     
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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)  f L}3I(VK  
    <:t D m  
    DDE还是手动输入的呢? XC D&Im  
    r{Cbx#;  
    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