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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 正序阅读 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, `4V"s-T'  
    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, 0I 5&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? 1 {Jb"  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? DK6? E\<  
    =1Z;Ma<;  
    ` H XEZ|  
    Ly7!R$X  
    K"\MU  
    function z = zernfun(n,m,r,theta,nflag) &cu!Hx  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. oq$w4D0Z  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Km!nM$=k  
    %   and angular frequency M, evaluated at positions (R,THETA) on the M4KWN'  
    %   unit circle.  N is a vector of positive integers (including 0), and /syVGmS'M  
    %   M is a vector with the same number of elements as N.  Each element ka/XK[/'  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 'e@=^FC  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, }mZV L~|V  
    %   and THETA is a vector of angles.  R and THETA must have the same }H RK?.Vj:  
    %   length.  The output Z is a matrix with one column for every (N,M) J#Z5^)$  
    %   pair, and one row for every (R,THETA) pair. dlDki.  
    % JYm7@gx  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ]6&$|2H?Ni  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^aF8wbuZ  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral c #lPc>0xb  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, PB9/m-\H  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized c0ez/q1S  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. MT6/2d  
    % X}cZxlqc  
    %   The Zernike functions are an orthogonal basis on the unit circle. s+Q;pRZW{  
    %   They are used in disciplines such as astronomy, optics, and (K :]7  
    %   optometry to describe functions on a circular domain. g5S?nHS}  
    % HjA_g0u  
    %   The following table lists the first 15 Zernike functions. |0.Xl+7  
    % XIAeCU  
    %       n    m    Zernike function           Normalization LA%bq_> f  
    %       -------------------------------------------------- iiG f'@/  
    %       0    0    1                                 1 ,=BLnsg  
    %       1    1    r * cos(theta)                    2 y(a!YicA?  
    %       1   -1    r * sin(theta)                    2 >&S0#>wmyG  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) qAY%nA>jO  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ?La Ued'  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) -*a?<ES`  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) zt=0o| k  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) k?6z_vu  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) EJ84rSp  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) bAwl:l\`  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) DmqSQA  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) g{:<2xI5P  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) A],ooiq<  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) e3(/qMl  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) IQH[Q9%  
    %       -------------------------------------------------- } JiSmi6o  
    % JC#>Td  
    %   Example 1: 3c3OG.H$8  
    % $` VFdAe  
    %       % Display the Zernike function Z(n=5,m=1) 9GLb"6+PK  
    %       x = -1:0.01:1; <F=9*.@D   
    %       [X,Y] = meshgrid(x,x); A,gEM4  
    %       [theta,r] = cart2pol(X,Y); 0N:XIGFa  
    %       idx = r<=1; Wu1{[a|  
    %       z = nan(size(X)); MJ{%4S{K,p  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); a W%5~3  
    %       figure 5n lMrK  
    %       pcolor(x,x,z), shading interp [I;^^#'P  
    %       axis square, colorbar P\nC?!Q%c  
    %       title('Zernike function Z_5^1(r,\theta)') 58tVx'1y  
    % H%F>@(U  
    %   Example 2: EZDy+6b  
    % 3)=c]@N0  
    %       % Display the first 10 Zernike functions %G3(,Qz  
    %       x = -1:0.01:1; I5m][~6.?  
    %       [X,Y] = meshgrid(x,x); .dMVoG5  
    %       [theta,r] = cart2pol(X,Y); q'Wr[A40j  
    %       idx = r<=1; BB$oq'  
    %       z = nan(size(X)); .L6Zm U  
    %       n = [0  1  1  2  2  2  3  3  3  3]; bM,1f/^  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; M]]pTU((  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; gJ$K\[+  
    %       y = zernfun(n,m,r(idx),theta(idx)); (la[KqqCO  
    %       figure('Units','normalized') ;)AfB#:d  
    %       for k = 1:10 ZraT3  
    %           z(idx) = y(:,k); LwcIGhy  
    %           subplot(4,7,Nplot(k)) DL'iS  
    %           pcolor(x,x,z), shading interp A]0:8@k5  
    %           set(gca,'XTick',[],'YTick',[]) 3r+.N  
    %           axis square NB_ )ZEmF  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _nh[(F<hz  
    %       end 7R4z}2F2  
    % 3*UR3!Z9 *  
    %   See also ZERNPOL, ZERNFUN2. SMH<'F7i  
    8T)&`dM6P~  
    8I;XS14Q  
    %   Paul Fricker 11/13/2006 EZ;"'4;W  
    X1{[}!  
    (6l+lru[  
    nrm+z"7  
    NEt1[2X%  
    % Check and prepare the inputs: XQ%4L-rhN  
    % ----------------------------- %WTEv?I{Ga  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 5irwz4.4  
        error('zernfun:NMvectors','N and M must be vectors.') fA/m1bYxg  
    end s~I6SA&i  
    HB+|WW t>  
    YOr:sb   
    if length(n)~=length(m) 7/7Z`  
        error('zernfun:NMlength','N and M must be the same length.') NA3 \  
    end k3?rp`V1  
    P 3MhU;  
    0:0NXVYs&  
    n = n(:); 7 >iU1zy  
    m = m(:); jHN +5=l  
    if any(mod(n-m,2)) WQ yLf;!Lz  
        error('zernfun:NMmultiplesof2', ... -=s(l.?Hm5  
              'All N and M must differ by multiples of 2 (including 0).') 5DOBs f8Jo  
    end GdV1^`M6  
    m,C1J%{^  
    \`V;z~@iA  
    if any(m>n) wo_,Y0vfB  
        error('zernfun:MlessthanN', ... v>z tB,,9  
              'Each M must be less than or equal to its corresponding N.') ^7zu<lX  
    end 1f",}qe;  
    !Z VU,b>  
    xGTP;NT_H  
    if any( r>1 | r<0 ) kmzH'wktt  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Bqma\1cgb  
    end Zo1,1O  
    ]Q]W5WDe:  
    4DZ-bt'  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ]smkTo/  
        error('zernfun:RTHvector','R and THETA must be vectors.') uqz]J$  
    end R.=}@oPb  
    Eu"_MgD  
     hI9  
    r = r(:); rZ8`sIWQt  
    theta = theta(:); p<=$&*  
    length_r = length(r);  V#VN %{  
    if length_r~=length(theta) Q.K,%(^;a  
        error('zernfun:RTHlength', ... =zQN[  
              'The number of R- and THETA-values must be equal.') KYzv$oK  
    end y;/VB,4V  
     H"A7Zo  
    LL:_L<  
    % Check normalization: 6Gf?m;  
    % -------------------- 6@DF  
    if nargin==5 && ischar(nflag) .\>v0Du  
        isnorm = strcmpi(nflag,'norm'); mI74x3 [  
        if ~isnorm >/|q:b^2r  
            error('zernfun:normalization','Unrecognized normalization flag.') I`NjqyTW  
        end m2AnXY\  
    else p K0"%eA  
        isnorm = false; 9 (QJT}qC  
    end '7O3/GDK  
    lg^Z*&(  
    !47n[Zs  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6gc>X%d`K  
    % Compute the Zernike Polynomials Ub6jxib  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}P~P$q%  
    B}^w_C2  
    JtER_(.  
    % Determine the required powers of r: F'0O2KQ  
    % ----------------------------------- F$)[kP,wtO  
    m_abs = abs(m); p5G?N(l  
    rpowers = []; Jv^h\~*jH  
    for j = 1:length(n) (+0v<uR^D  
        rpowers = [rpowers m_abs(j):2:n(j)]; wmTb97o  
    end eA<0$Gs,h  
    rpowers = unique(rpowers); -B +4+&{T  
    )ut&@]  
    M3.do^ss  
    % Pre-compute the values of r raised to the required powers, s0vDHkf8  
    % and compile them in a matrix: E>K!Vrh-L  
    % ----------------------------- ov, hI>0!D  
    if rpowers(1)==0 q<M2,YrbAI  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); kGl~GOB a  
        rpowern = cat(2,rpowern{:}); >7 ="8  
        rpowern = [ones(length_r,1) rpowern]; 4t=G   
    else vam;4vyu  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 'dn]rV0(C  
        rpowern = cat(2,rpowern{:});  094o'k  
    end W)bLSL]`E  
    gw!vlwC&T  
    7<*yS310  
    % Compute the values of the polynomials: [@.!~E)P  
    % -------------------------------------- ~A\GT$  
    y = zeros(length_r,length(n)); 6e |*E`I  
    for j = 1:length(n) {z{bY\  
        s = 0:(n(j)-m_abs(j))/2; o4Om}]Ti  
        pows = n(j):-2:m_abs(j); p>huRp^w  
        for k = length(s):-1:1 :;9F>?VN>0  
            p = (1-2*mod(s(k),2))* ... I`!<9OTBj  
                       prod(2:(n(j)-s(k)))/              ... LcTP #  
                       prod(2:s(k))/                     ... )P sY($ &  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... <J`0  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); GB=X5<;  
            idx = (pows(k)==rpowers); a!v1M2>  
            y(:,j) = y(:,j) + p*rpowern(:,idx); @J/K-.r  
        end n"c[,k+R`U  
         ]Gsv0Xk1  
        if isnorm %iQD /iT5  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); {ttysQ-  
        end yd d7I&$  
    end JkbQyn  
    % END: Compute the Zernike Polynomials = %TWX[w  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nWw":K<@Q_  
    + R~'7*EI  
    ^'PWI{ O  
    % Compute the Zernike functions: m+]K;}.}R  
    % ------------------------------ V@g'#= {r  
    idx_pos = m>0; cQ R]le %(  
    idx_neg = m<0; VAHh~Q6 ;e  
    a .k.n<  
    :74y!  
    z = y; s Z].8.  
    if any(idx_pos) QTk}h_<u  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); m;GCc8  
    end k%WTJbuG<)  
    if any(idx_neg) I&x=;   
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Mh]Gw(?w  
    end inMA:x}cF1  
    |v 3T!  
    '-Vt|O_Q  
    % EOF zernfun m#| 9hMu  
     
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    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
    离线jssylttc
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    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    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)  gh]cXuph  
    AofKw  
    DDE还是手动输入的呢? n:?a$Ldgm  
    Woy m/[i  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    在线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。
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