jssylttc |
2012-04-23 19:23 |
如何从zernike矩中提取出zernike系数啊
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, IADHe\. 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, Qz[^J 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? hEB5=~A_ 那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? xZ`z+) $$m0mK |)!f".` BFW b0;+ kAEq +{h function z = zernfun(n,m,r,theta,nflag) R#hy2kA %ZERNFUN Zernike functions of order N and frequency M on the unit circle. kC. !cPd % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N |qMG@ % and angular frequency M, evaluated at positions (R,THETA) on the 5c]:/9& % unit circle. N is a vector of positive integers (including 0), and *Mhirz%iD % M is a vector with the same number of elements as N. Each element ]Kq<U%x$ % k of M must be a positive integer, with possible values M(k) = -N(k) LXo$\~M8G8 % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, o^~ZXF} % and THETA is a vector of angles. R and THETA must have the same [cnuK % length. The output Z is a matrix with one column for every (N,M) T$lV+[7 % pair, and one row for every (R,THETA) pair. ?\8aT"o % gFp3=s0~ % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike wL8ji>" % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !-OZ/^l|O` % with delta(m,0) the Kronecker delta, is chosen so that the integral .JLJ(WM % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, id:,\iJ % and theta=0 to theta=2*pi) is unity. For the non-normalized 1k6asz^T % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. AT1cN1:4? % eP]y\S*P % The Zernike functions are an orthogonal basis on the unit circle. D@La-K*5 % They are used in disciplines such as astronomy, optics, and o5s6$\" % optometry to describe functions on a circular domain. ;=,-C;` % yDqwz[v b % The following table lists the first 15 Zernike functions. 72Bc0Wg
% u"qu!EY2 % n m Zernike function Normalization VF2,(f-* % -------------------------------------------------- vSnVq>-q& % 0 0 1 1 FXBmatBck % 1 1 r * cos(theta) 2 <CVX[R]U % 1 -1 r * sin(theta) 2 zU!{_Ao9 % 2 -2 r^2 * cos(2*theta) sqrt(6) *:H,-@ % 2 0 (2*r^2 - 1) sqrt(3) Z (6.e8fK % 2 2 r^2 * sin(2*theta) sqrt(6) s8,YQ5- % 3 -3 r^3 * cos(3*theta) sqrt(8) 9$$ Ijf % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) /^xv1F{ % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) i$] :Y`3h % 3 3 r^3 * sin(3*theta) sqrt(8) nZB~l= % 4 -4 r^4 * cos(4*theta) sqrt(10) Trs~KcsD % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) W~mo*EJ'^ % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) )(G<(eiD % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) @LI;q % 4 4 r^4 * sin(4*theta) sqrt(10) V5lUh#@TN& % -------------------------------------------------- h/tCve3Z % ,Sgo_bC/| % Example 1: [UXVL}tk % #|E#Rkw! % % Display the Zernike function Z(n=5,m=1) Sk53Lc % x = -1:0.01:1; %q|*}l % [X,Y] = meshgrid(x,x); 5b"=m9{g % [theta,r] = cart2pol(X,Y); =Lkn
% idx = r<=1; UUfM7gq % z = nan(size(X)); N-2#-poDe % z(idx) = zernfun(5,1,r(idx),theta(idx)); <2]h$53y! % figure E}4{{{r % pcolor(x,x,z), shading interp Mk#r_:[BS % axis square, colorbar }K 'A/]' % title('Zernike function Z_5^1(r,\theta)') ,5zY1C==Ut % Kc[^Pu % Example 2: (DvPdOT+3 % ^*l
dsc % % Display the first 10 Zernike functions o+hp#e % x = -1:0.01:1; dE8f?L' % [X,Y] = meshgrid(x,x); "*#f^/LS % [theta,r] = cart2pol(X,Y); (KC08 % idx = r<=1; g"sb0d9 % z = nan(size(X)); uH$hMg % n = [0 1 1 2 2 2 3 3 3 3]; Z]Xa:[ % m = [0 -1 1 -2 0 2 -3 -1 1 3]; (QIU 3EN % Nplot = [4 10 12 16 18 20 22 24 26 28]; ~Zsj@d % y = zernfun(n,m,r(idx),theta(idx)); XwEMF5[ % figure('Units','normalized') sRT5i9TQ % for k = 1:10 Po=:-Of: % z(idx) = y(:,k); Ydsnu % subplot(4,7,Nplot(k)) 4'D^>z!c % pcolor(x,x,z), shading interp 'KmM%tN % set(gca,'XTick',[],'YTick',[]) @{qcu\sZ % axis square x=>dmi3 % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) =?Ry,^=b % end w@2NXcmw % NUnwf
h % See also ZERNPOL, ZERNFUN2. vrmMEWPV 1sza\pR< /|U;_F Pmc % Paul Fricker 11/13/2006 @8J*vY =e A?HDY_u ~uY5~Qs9G @}UOm-M O~8jz % Check and prepare the inputs: ZWyf.VJ % -----------------------------
rY Puo if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) |7|'JTy error('zernfun:NMvectors','N and M must be vectors.') "=]'"'B: end (~\HizSl =Cf@!wZ^ w`boQ_Ir if length(n)~=length(m) y46sL~HRv error('zernfun:NMlength','N and M must be the same length.') I@N/Y{y# end U{EcV%C2 um PN=0u6 HHyN\ n = n(:); a$uDoi m = m(:); 1|
WDbk if any(mod(n-m,2)) T:'<:*pD error('zernfun:NMmultiplesof2', ... }:?_/$}; 'All N and M must differ by multiples of 2 (including 0).') O:V.;q2]U end HRahBTd(z :[?7,/w
_JpTHpqu if any(m>n) ve/.q^JeJ error('zernfun:MlessthanN', ... meB9:w[m 'Each M must be less than or equal to its corresponding N.') }rVLWt end toG- Dz& | |