niuhelen |
2011-03-12 23:00 |
非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 n#.~XNbxv function z = zernfun(n,m,r,theta,nflag) Uiz#QGt %ZERNFUN Zernike functions of order N and frequency M on the unit circle. "c'K8,+? % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N dM^1O-K: % and angular frequency M, evaluated at positions (R,THETA) on the ruf*-&Kr7 % unit circle. N is a vector of positive integers (including 0), and gPA),
NrN % M is a vector with the same number of elements as N. Each element Z:e|~# % k of M must be a positive integer, with possible values M(k) = -N(k) 8'nxc#& % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, s/"l ?d % and THETA is a vector of angles. R and THETA must have the same piy_9nk % length. The output Z is a matrix with one column for every (N,M) |Nfi y % pair, and one row for every (R,THETA) pair. ar!`8" % 9pPohR*#V % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ~Wox"h}( % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ~\C.Nm % with delta(m,0) the Kronecker delta, is chosen so that the integral /2&:sHWW % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, {dk%j~w8 % and theta=0 to theta=2*pi) is unity. For the non-normalized )q>mt/, % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. $
M8ZF(W %
AD=qB5: % The Zernike functions are an orthogonal basis on the unit circle. P%nN#Qm % They are used in disciplines such as astronomy, optics, and F^xhhz&e % optometry to describe functions on a circular domain. ([+u U! % w QnW2)9! % The following table lists the first 15 Zernike functions. .8I\=+Zi % c8"9Lv % n m Zernike function Normalization > 0kZ-M5 % -------------------------------------------------- }CoR$K % 0 0 1 1 1lf5xm. % 1 1 r * cos(theta) 2 5VXI/Lw# % 1 -1 r * sin(theta) 2 x9NcIa9 % 2 -2 r^2 * cos(2*theta) sqrt(6) OZ'=Xtbn % 2 0 (2*r^2 - 1) sqrt(3) 4)zHkN+ % 2 2 r^2 * sin(2*theta) sqrt(6) h\@\*Xz<v % 3 -3 r^3 * cos(3*theta) sqrt(8) &.ENcEic % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) "G+g(?N]j % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) K< ;I*cAX % 3 3 r^3 * sin(3*theta) sqrt(8) Xc!0'P0T % 4 -4 r^4 * cos(4*theta) sqrt(10) !MNnau%O % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0j--X?- % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Tw{}Ht_Qq % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) X,Q'Xe/ % 4 4 r^4 * sin(4*theta) sqrt(10) >&mNC\PA % -------------------------------------------------- Y<"BhE % YaWZOuxm % Example 1: Z/p>>SCak % 04u^Q % % Display the Zernike function Z(n=5,m=1) ";PW#VHC % x = -1:0.01:1; zlw+=NX % [X,Y] = meshgrid(x,x); ^Qx
qv % [theta,r] = cart2pol(X,Y); @5im*ubzM % idx = r<=1; zK5/0zMZ % z = nan(size(X)); KO ~_ % z(idx) = zernfun(5,1,r(idx),theta(idx)); >3v
j<v}m % figure sFvu@Wm'7W % pcolor(x,x,z), shading interp d5hYOhO[ % axis square, colorbar Tf|?j=f % title('Zernike function Z_5^1(r,\theta)') G(n
e8L8 % dE`a1H% % Example 2: O:'ENoQ:& % =F B[<% % % Display the first 10 Zernike functions s\CZ os& % x = -1:0.01:1; ./iC % [X,Y] = meshgrid(x,x); Vg>\@ C.s % [theta,r] = cart2pol(X,Y); ;AjY-w % idx = r<=1; )yOdRRP % z = nan(size(X)); e?V7<7$ % n = [0 1 1 2 2 2 3 3 3 3]; T@S\:P % m = [0 -1 1 -2 0 2 -3 -1 1 3]; b!h*I>` % Nplot = [4 10 12 16 18 20 22 24 26 28]; 'UW(0 PXw % y = zernfun(n,m,r(idx),theta(idx)); hI^Hqv % figure('Units','normalized') ;aY.CgX % for k = 1:10 _gGI&0(VM % z(idx) = y(:,k); *i=+["A % subplot(4,7,Nplot(k)) Q7~9~ % pcolor(x,x,z), shading interp
[b=l'e/ % set(gca,'XTick',[],'YTick',[]) ;`{PA
!> % axis square I|`/#BYbW % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) -Tzp;o % end xYg G % l$1z%|I % See also ZERNPOL, ZERNFUN2. j.b7<Vr4; QXQ'QEG % Paul Fricker 11/13/2006 sM4Qu./ ib3u: U:a-Wi+ % Check and prepare the inputs: {DI`HB[ % ----------------------------- "<e<0:: if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Ez= Q{g error('zernfun:NMvectors','N and M must be vectors.') iPD5
KsAOA end 9L"Z
~CUL s y ]k if length(n)~=length(m) N`G*
h^YQ error('zernfun:NMlength','N and M must be the same length.') :3uCW1 end n O^m M<A jtDF% n = n(:); WeqE9@V m = m(:); 7jj.maK if any(mod(n-m,2)) ({R-JkW:; error('zernfun:NMmultiplesof2', ... 5`!Bj0Uf 'All N and M must differ by multiples of 2 (including 0).') HB>&}z0 end HP$GI ')bas#=uP if any(m>n) c"pI+Q error('zernfun:MlessthanN', ... (.CEEWj%{ 'Each M must be less than or equal to its corresponding N.') J$]-)`[G& end fAW( x344}\ if any( r>1 | r<0 ) :jJ;&t^^ error('zernfun:Rlessthan1','All R must be between 0 and 1.') -w[j`}([P9 end !mM`+XH n]15 ~GO. if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) oCCTRLb02 error('zernfun:RTHvector','R and THETA must be vectors.') .nN>Ipv end Wk0"U
V jHU5>Gt-} r = r(:); N=JZtf/i theta = theta(:); [SJ)4e|) length_r = length(r); E`"<t:RzF if length_r~=length(theta) ~36)3W[4 error('zernfun:RTHlength', ... 6>fQe8Y 'The number of R- and THETA-values must be equal.') \V1geSoE end tK|jh by:"aDGK. % Check normalization: 65 P*Gu? % -------------------- >Q`\|m}x)Q if nargin==5 && ischar(nflag) dN8@ 0AMSf isnorm = strcmpi(nflag,'norm'); |4slG if ~isnorm b3zxiq
x error('zernfun:normalization','Unrecognized normalization flag.') -$x5[6bN end rryC^Vma else 3eg)O34 isnorm = false; [110[i^ end "%mu~&Ga ; qQ* p %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% VbwB<nQl % Compute the Zernike Polynomials Fm| h3.`V %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8iB}gHe9 $*KM%M6 % Determine the required powers of r: (>LJv |wn % ----------------------------------- ^L,Uz:[J m_abs = abs(m); vi4lmkyh^ rpowers = []; A#&,S4Wi| for j = 1:length(n) S260h,(, rpowers = [rpowers m_abs(j):2:n(j)]; `veq/! end si!jB%^ rpowers = unique(rpowers); f3p)Q<H>`( 2i4&*&A % Pre-compute the values of r raised to the required powers, jQV.U~25Q % and compile them in a matrix: ~8j4IO( % ----------------------------- %VSjMZ if rpowers(1)==0 ~+HZQv3Y rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 6>hW.aq} rpowern = cat(2,rpowern{:}); >k)zd- rpowern = [ones(length_r,1) rpowern]; <Rno; else a%IJ8t+mn rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )J"*[[e rpowern = cat(2,rpowern{:}); 4D4Y.g_x end QMI6l'"s pE~>k: % Compute the values of the polynomials: ZJod=^T % -------------------------------------- &|LP>'H; y = zeros(length_r,length(n)); J/{!_M- for j = 1:length(n) )[Bl3+' s = 0:(n(j)-m_abs(j))/2; 4(hHp6}b pows = n(j):-2:m_abs(j); 5LF#w_x for k = length(s):-1:1 \nKpJ9! p = (1-2*mod(s(k),2))* ... hE9UWa.Q> prod(2:(n(j)-s(k)))/ ... ,~TV/l< prod(2:s(k))/ ... ^T.icSxP prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... xc&&UKd prod(2:((n(j)+m_abs(j))/2-s(k))); (c'kZ9& idx = (pows(k)==rpowers); v=Y)
A ? y(:,j) = y(:,j) + p*rpowern(:,idx); Xh[02iL- end mT3'kUZ}] Z2t
r?] if isnorm W,53|9b@ y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); kuZs30^ end v<Ozr:lL end ;LhNz ()b % END: Compute the Zernike Polynomials U~}cib5W5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DEFh&n y?}R,5k % Compute the Zernike functions: sT;:V
% ------------------------------ SBbPO5^]( idx_pos = m>0; br[n5 idx_neg = m<0; 0\X'a}8Bu 'y?
HF@NJ z = y; pn._u`xMV if any(idx_pos) o(|fapK. z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); >+3tOv3: end %ylpn7I\6 if any(idx_neg) g:&V9 | |