| 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 7v]9) W=y function z = zernfun(n,m,r,theta,nflag) %2,'x %ZERNFUN Zernike functions of order N and frequency M on the unit circle. !cNw8"SIU % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N B:rzM:BQ % and angular frequency M, evaluated at positions (R,THETA) on the J>N^ FR9 % unit circle. N is a vector of positive integers (including 0), and w
21g& % M is a vector with the same number of elements as N. Each element dh K<5E % k of M must be a positive integer, with possible values M(k) = -N(k) %Fp1c K % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, /ei(Q'pc[ % and THETA is a vector of angles. R and THETA must have the same T0v{qQ % length. The output Z is a matrix with one column for every (N,M) \$W\[s4I % pair, and one row for every (R,THETA) pair. OKV/=]GS % /vNHb_- % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike xua
E\*m % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), bvF-F$n%F % with delta(m,0) the Kronecker delta, is chosen so that the integral sg%Ptp % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, e+O502] % and theta=0 to theta=2*pi) is unity. For the non-normalized y134m % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 4 zhg# % ^?R8>97_? % The Zernike functions are an orthogonal basis on the unit circle. ^u-;VoK % They are used in disciplines such as astronomy, optics, and -=4{X
R3 % optometry to describe functions on a circular domain. <_3OiU=w % 5ggsOqH % The following table lists the first 15 Zernike functions. %_.
fEFy07 % ?.Lq`~T` % n m Zernike function Normalization RxO!h8 % -------------------------------------------------- 7u<C&Z/ % 0 0 1 1 s`I]>e % 1 1 r * cos(theta) 2 |m F=X* % 1 -1 r * sin(theta) 2 6H^=\ % 2 -2 r^2 * cos(2*theta) sqrt(6) d7P'c!@+ % 2 0 (2*r^2 - 1) sqrt(3) VI k]`)# % 2 2 r^2 * sin(2*theta) sqrt(6) /Y0oA3am % 3 -3 r^3 * cos(3*theta) sqrt(8) YckLz01jh % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) r^T+I3 % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) UH`cWV Lpr % 3 3 r^3 * sin(3*theta) sqrt(8) H: ]'r5sw % 4 -4 r^4 * cos(4*theta) sqrt(10) <%"o-xZq7C % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) t~M0_TnXlP % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) o]TKL'gW % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) C Xh>'K % 4 4 r^4 * sin(4*theta) sqrt(10) Nin7AOO % -------------------------------------------------- f,'^"Me$c % M,dp; % Example 1: EI8KK o * % l5FKw;=K}: % % Display the Zernike function Z(n=5,m=1) s(pNg?R % x = -1:0.01:1; N?v}\ PU % [X,Y] = meshgrid(x,x); MuF{STE>-> % [theta,r] = cart2pol(X,Y); Xk`' m[ % idx = r<=1; tvcM<
e20 % z = nan(size(X)); Mz: "p. % z(idx) = zernfun(5,1,r(idx),theta(idx)); l#& \,T % figure dmPAPCm%y % pcolor(x,x,z), shading interp #n.XOet<\ % axis square, colorbar GQ6~Si2 % title('Zernike function Z_5^1(r,\theta)') $ Gs|Z$( % iC 4rzgq % Example 2: Bmv5yc+; % NeR1}W % % Display the first 10 Zernike functions 'Esz#@R % x = -1:0.01:1; ( 9(NP_s % [X,Y] = meshgrid(x,x); _rz7)%Y'#$ % [theta,r] = cart2pol(X,Y); {sF;R.P&r % idx = r<=1; Np@RK1} % z = nan(size(X)); qo7jrY5G % n = [0 1 1 2 2 2 3 3 3 3]; e'2w-^7 % m = [0 -1 1 -2 0 2 -3 -1 1 3]; <lP5}F87 % Nplot = [4 10 12 16 18 20 22 24 26 28]; l0lvca=; % y = zernfun(n,m,r(idx),theta(idx)); hVW1l&s % figure('Units','normalized') Sz-TarTF % for k = 1:10 l*b0uF % z(idx) = y(:,k); ;N^4R$Q. % subplot(4,7,Nplot(k)) -u~AY#* % pcolor(x,x,z), shading interp .5!Q( % set(gca,'XTick',[],'YTick',[]) juEH$7N! % axis square 1AQ3< % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 'o|=_0-7W % end 4[rX\?^e % <])kO`+G % See also ZERNPOL, ZERNFUN2. wit
x.>&|Ej % Paul Fricker 11/13/2006 -IS$1 ^zKP5nzL z-m:l; % Check and prepare the inputs: eA-$TSWh % ----------------------------- 8Ud.}<
Zi if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +HEL ^ error('zernfun:NMvectors','N and M must be vectors.') mV.26D<c end s]Z++Lh<{ VLC=>w\, if length(n)~=length(m) q3ebps9^ error('zernfun:NMlength','N and M must be the same length.') l} W">
yQ0 end p~z\&&0U0
vu3zZMl n = n(:); BHR(B]EI m = m(:); =xr2-K)e if any(mod(n-m,2)) |`O210B@ error('zernfun:NMmultiplesof2', ... eKe[]/}e9 'All N and M must differ by multiples of 2 (including 0).') gH/(4h end 0}-MWbG $.O(K4S if any(m>n) {CQI*\O error('zernfun:MlessthanN', ... Q#pgl 'Each M must be less than or equal to its corresponding N.') n!L}4Nmp end bq z*90 !_?#f| if any( r>1 | r<0 ) e/zz.cd){ error('zernfun:Rlessthan1','All R must be between 0 and 1.') (S8hr,%n end %Vhj<gN @gi / 1 cq if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) &X)^G# error('zernfun:RTHvector','R and THETA must be vectors.') &G_XgQsg{ end 7upN:7D- iz2;xa* r = r(:);
LDdgI theta = theta(:); ;M5]XCPk length_r = length(r); 7o9[cq w if length_r~=length(theta) wj\kx\+ error('zernfun:RTHlength', ... \iAs 'The number of R- and THETA-values must be equal.') MZ_dI"J, end 35Fs/Gf-n H'jo3d~+ % Check normalization: CPJ%<+4%b % -------------------- vgN%vw pL if nargin==5 && ischar(nflag) 4#ZZwa]y isnorm = strcmpi(nflag,'norm'); 90">l^HX= if ~isnorm s$xm error('zernfun:normalization','Unrecognized normalization flag.') 4B@Ir)^(* end Nx<fj=VJ else ,R=)^Gh{ isnorm = false; bEb+oRI end dQI6.$? zRgl`zREr %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% du&9mOrr % Compute the Zernike Polynomials gqDSHFm: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BCt>P?,UO 14yzGhA % Determine the required powers of r: c> ":g~w % ----------------------------------- $`_xP1bUT m_abs = abs(m); ,Ofou8C6 rpowers = []; p;)@R$* for j = 1:length(n) h 2C9p2. rpowers = [rpowers m_abs(j):2:n(j)]; -Hg,:re2 end URMxCL^" rpowers = unique(rpowers); [ip}f4K Y"E*#1/ % Pre-compute the values of r raised to the required powers, 6eW9+5oL % and compile them in a matrix: Ns.{$'ll % ----------------------------- mf\@vI if rpowers(1)==0 59k-,lyU, rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); <'vtnz rpowern = cat(2,rpowern{:}); 0|FQIhVuY rpowern = [ones(length_r,1) rpowern]; 6bUcrw/#
p else +{cCKRm rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); sLW e \o rpowern = cat(2,rpowern{:}); DhT8Kh{ end RT"JAJTi/ Q=#Wk$1. % Compute the values of the polynomials: +kT
o$_Wkz % -------------------------------------- aV G4Df y = zeros(length_r,length(n)); x_#'6H\1ga for j = 1:length(n) J pKCux s = 0:(n(j)-m_abs(j))/2; zJG=9C? pows = n(j):-2:m_abs(j); xi=Qxgx0I for k = length(s):-1:1 >RXDuCVi p = (1-2*mod(s(k),2))* ... XO}v8nWV prod(2:(n(j)-s(k)))/ ... &\<?7Qj3U| prod(2:s(k))/ ... $rH}2 prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... =p&uQ6.i+ prod(2:((n(j)+m_abs(j))/2-s(k))); WR}<^ax idx = (pows(k)==rpowers); /qweozW_+ y(:,j) = y(:,j) + p*rpowern(:,idx); [+b&)jN*2 end :6W* ;<o k9iB-=X?4s if isnorm t8t+wi! y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ^Dys#^ end 7z3YzQ=Kg end JmbWEX| % END: Compute the Zernike Polynomials Kj*$'(' %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |. LE` K"VRHIhfg % Compute the Zernike functions: ;a`I8F j % ------------------------------ Mgg m~|9) idx_pos = m>0; pxHJX2 idx_neg = m<0; vp`s< ;CA h2vD*W z = y; `D0Hu!; if any(idx_pos) K7]QgfpSZ z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); }&LLo end Kl w9 if any(idx_neg) +D|E8sz8 z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); N@\`DO end 1IWP~G $ cYKVhf % EOF zernfun
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