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 %oqC5O6 function z = zernfun(n,m,r,theta,nflag) P|4qbm4%O, %ZERNFUN Zernike functions of order N and frequency M on the unit circle. gN/6%,H} % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N lq\/E`fc` % and angular frequency M, evaluated at positions (R,THETA) on the OMVK\_oXo % unit circle. N is a vector of positive integers (including 0), and @XFy^? % M is a vector with the same number of elements as N. Each element DZ~qk+,I % k of M must be a positive integer, with possible values M(k) = -N(k) 7! "OF % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, [agp06 $D? % and THETA is a vector of angles. R and THETA must have the same JVeb$_0k % length. The output Z is a matrix with one column for every (N,M) x*2' I % pair, and one row for every (R,THETA) pair. COk;z.Kn % |]8Hh> % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike RkuPMs
Hw; % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), DKxzk~sOM % with delta(m,0) the Kronecker delta, is chosen so that the integral ^&6'FE % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ffqz
:6 % and theta=0 to theta=2*pi) is unity. For the non-normalized _MC\\u/C/ % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. KZ;Q7 1 % N++jI( % The Zernike functions are an orthogonal basis on the unit circle. }+Ne)B E % They are used in disciplines such as astronomy, optics, and bI?YNt, % optometry to describe functions on a circular domain. WbW@V_rr % Ot#O];3 % The following table lists the first 15 Zernike functions. =UW!
7OzC % T,eP&IN % n m Zernike function Normalization 4#^?-6 % -------------------------------------------------- amY\1quD| % 0 0 1 1 <Fa]k'<^) % 1 1 r * cos(theta) 2 Vx6/Rehj % 1 -1 r * sin(theta) 2 ``p()^zT % 2 -2 r^2 * cos(2*theta) sqrt(6) 42wa9UL<Ka % 2 0 (2*r^2 - 1) sqrt(3) Zw`vPvb! % 2 2 r^2 * sin(2*theta) sqrt(6) v2uyn % 3 -3 r^3 * cos(3*theta) sqrt(8) g:sn/Zug] % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) z]Dbca1a` % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) w[S!U<9/ % 3 3 r^3 * sin(3*theta) sqrt(8) xbv % 4 -4 r^4 * cos(4*theta) sqrt(10) - s,M+Q(< % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) a*Oc:$ % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 0[qU k(=}[ % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6cV -iDOH % 4 4 r^4 * sin(4*theta) sqrt(10) ~Yw`w2 % -------------------------------------------------- N5%zbfKM % RN3-:Zd_X % Example 1: D<
h+r? % :DlgNR`bq % % Display the Zernike function Z(n=5,m=1) 30fsVwE2 % x = -1:0.01:1; !F_BLHig % [X,Y] = meshgrid(x,x); 9$u'2TV % [theta,r] = cart2pol(X,Y); Gx]J6Z8 % idx = r<=1; i,Q{Z@, % z = nan(size(X)); jeM/8~^4- % z(idx) = zernfun(5,1,r(idx),theta(idx)); EGZF@#N % figure YGj3W.eH % pcolor(x,x,z), shading interp nf7l}^/UE % axis square, colorbar UE[5Bw?4X % title('Zernike function Z_5^1(r,\theta)') lo%:$2*'p % Xo{|m[, % Example 2: < c}cgD4 % vIi#M0@N % % Display the first 10 Zernike functions JToc("V % x = -1:0.01:1; J4-64t nZ % [X,Y] = meshgrid(x,x); $H9+>Z0( % [theta,r] = cart2pol(X,Y); KfO$bmwmx % idx = r<=1; %$)[qa3 % z = nan(size(X)); FOFZ/q % n = [0 1 1 2 2 2 3 3 3 3]; d&dp#)._8 % m = [0 -1 1 -2 0 2 -3 -1 1 3]; TfYXF`d % Nplot = [4 10 12 16 18 20 22 24 26 28]; 4|9c+^%^ % y = zernfun(n,m,r(idx),theta(idx)); ~e,D`Lv % figure('Units','normalized') 8KQ]3Z9p % for k = 1:10 9D2}heTN % z(idx) = y(:,k); 8e`'Ox_5a % subplot(4,7,Nplot(k)) Y 7a<3> % pcolor(x,x,z), shading interp |,&5.|E 7 % set(gca,'XTick',[],'YTick',[]) $R ' % axis square =lzRx%tm % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ZZ<uiN$ % end d Q5_=(9 % 3 9|4)1e % See also ZERNPOL, ZERNFUN2. meHnT9a^ ! f\q0Gnl % Paul Fricker 11/13/2006 "gcHcboU5$ 8JP{`) 0'giAA % Check and prepare the inputs: cH&-/|N % ----------------------------- G\y: O9( if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) pohA??t2: error('zernfun:NMvectors','N and M must be vectors.') ~VRt6C end n(|~z CLb~6LD if length(n)~=length(m) 1e 8J-Nkj error('zernfun:NMlength','N and M must be the same length.') G d".zsn end [7Yfv
Xp k* ayzg3F> n = n(:); %6\e_y% m = m(:); {Lex(( if any(mod(n-m,2)) JF%eC}[d error('zernfun:NMmultiplesof2', ... O>Vb7`z0< 'All N and M must differ by multiples of 2 (including 0).') U4J9bp| end 5AvbKT ZeUA e if any(m>n) \GL!x 7s1A error('zernfun:MlessthanN', ... p7UdZOi2 'Each M must be less than or equal to its corresponding N.') `CWI%V end %_rdO(
:u%Jrc(W if any( r>1 | r<0 ) dE<}X7J% error('zernfun:Rlessthan1','All R must be between 0 and 1.') ^
|k7g end X=i^[?C ?onaJ=mT if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 2yu\fu error('zernfun:RTHvector','R and THETA must be vectors.') %S G**7 end tOJK~%' rOt`5_2f r = r(:); 7oj
^(R, theta = theta(:); sD|P*ir length_r = length(r); #J1vN]g if length_r~=length(theta) <oweLRt error('zernfun:RTHlength', ... ~
.} 'The number of R- and THETA-values must be equal.') OF$0]V end 5pF4{Jd1 tE i-0J % Check normalization: 9~bl % -------------------- 0y>]68D if nargin==5 && ischar(nflag) hJrcy!P<a isnorm = strcmpi(nflag,'norm');
1o&]=( if ~isnorm RTPxAp+\5 error('zernfun:normalization','Unrecognized normalization flag.') *Nv!Kuk end ^6tcB* #A else HgHhc&- isnorm = false; sFd"VRAV~E end L/2{}l>D IO,ddVO %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {s=n "*Qp) % Compute the Zernike Polynomials o5!"dxR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AOT +4*)% mlIX>ss|7B % Determine the required powers of r: .T*K4m{b0 % ----------------------------------- N!7r~B
m_abs = abs(m); Who7{|M\' rpowers = []; X67.%>#3 for j = 1:length(n) wv.FL$f[@ rpowers = [rpowers m_abs(j):2:n(j)]; l>l)m-;O end 3kAmRU rpowers = unique(rpowers); m!{}Y]FZn 5|&:l8= % Pre-compute the values of r raised to the required powers, <ejWl%4 % and compile them in a matrix: S >E|A% % ----------------------------- JfJUOaL if rpowers(1)==0 4)'8fi rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @,Je*5$o" rpowern = cat(2,rpowern{:}); 6XVr-ef rpowern = [ones(length_r,1) rpowern]; 1!u}~E_ else r"yA=d'c rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )_*<uSl rpowern = cat(2,rpowern{:}); aE[>^~Lv} end 0_Gi1) Mx?{[zT" % Compute the values of the polynomials: 2C9wOO % -------------------------------------- qT`sPEs;V y = zeros(length_r,length(n)); B;SN}I for j = 1:length(n) $"P9I-\m s = 0:(n(j)-m_abs(j))/2; ,@+7(W pows = n(j):-2:m_abs(j); ]Lc:M'V# for k = length(s):-1:1 g+5{&YD p = (1-2*mod(s(k),2))* ... E)eRi"a46 prod(2:(n(j)-s(k)))/ ... <+MNv#1:w prod(2:s(k))/ ... wzX
1!? prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... D`nW9i7 prod(2:((n(j)+m_abs(j))/2-s(k))); Ri$wt.b idx = (pows(k)==rpowers); UjmBLXz@T y(:,j) = y(:,j) + p*rpowern(:,idx); kF:4[d end 6S-1Wc4 ]LF Y2w< if isnorm C4Pi6.wf y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); F_8nxQ- end n^8LF9r end i^c % END: Compute the Zernike Polynomials 5):2;h k %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +y!B`'J sOc<'):TK % Compute the Zernike functions: 8 *@knkJ % ------------------------------ V K/;ohTTP idx_pos = m>0; sb
3l4(8g
idx_neg = m<0; w(w%~;\kLP TH_Vw,) z = y; > QwZt if any(idx_pos)
kyQUaFG z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); p5<2t SD end (<ybst6+I if any(idx_neg) S~WsGLF s z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); VKtrSY}6T end iz'#K?PF_ 4:$?u}9[:[ % EOF zernfun
|
|