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 P6 mDwR function z = zernfun(n,m,r,theta,nflag) ]iiB|xT %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ev(E % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1.z !u%2 % and angular frequency M, evaluated at positions (R,THETA) on the f.U.( % unit circle. N is a vector of positive integers (including 0), and
l65Qk2<YC % M is a vector with the same number of elements as N. Each element xh!aB6m8R % k of M must be a positive integer, with possible values M(k) = -N(k) 4yR X{Bl| % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, S >\\n^SbT % and THETA is a vector of angles. R and THETA must have the same x/#.%Ga#T % length. The output Z is a matrix with one column for every (N,M) M0uC0\'#P % pair, and one row for every (R,THETA) pair. \'Ca%j % lK y4Nry9 % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike m\J"P'= % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), U,^jN|v % with delta(m,0) the Kronecker delta, is chosen so that the integral wEbO|S+K1 % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ]4&B*]j % and theta=0 to theta=2*pi) is unity. For the non-normalized coc:$Sr% % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ' ui`EL % % (I(k$g[> % The Zernike functions are an orthogonal basis on the unit circle. ~=pyA#VVJ" % They are used in disciplines such as astronomy, optics, and %J|xPp) % optometry to describe functions on a circular domain. +Ram%"Zwh % wHhIa3_v % The following table lists the first 15 Zernike functions. /)xQ# yfX % U\", !S~< % n m Zernike function Normalization ;i;;{j@$i % -------------------------------------------------- yg@}j % 0 0 1 1 M%FKg/ % 1 1 r * cos(theta) 2 x\6 i (k- % 1 -1 r * sin(theta) 2 m_>~e}2'A % 2 -2 r^2 * cos(2*theta) sqrt(6) 0'tm., % 2 0 (2*r^2 - 1) sqrt(3) 05vu{> % 2 2 r^2 * sin(2*theta) sqrt(6) m?Dk(DJ % 3 -3 r^3 * cos(3*theta) sqrt(8) \G &q[8F\ % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Qx !!
Ttd{ % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) V@1K % 3 3 r^3 * sin(3*theta) sqrt(8) oJ)v6"j % 4 -4 r^4 * cos(4*theta) sqrt(10) .aqP= % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Zl`sY5{1 % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) I'16- % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) uB uwE6 % 4 4 r^4 * sin(4*theta) sqrt(10) {_*$X % -------------------------------------------------- ZlsdO.G % l j*J|%~ % Example 1: d9uT*5f % Y@M
l}43 % % Display the Zernike function Z(n=5,m=1) {:d9q % x = -1:0.01:1; N&+DhKw % [X,Y] = meshgrid(x,x); e.^Y4( % [theta,r] = cart2pol(X,Y); nXF|AeAco % idx = r<=1; "t)|N
dZm % z = nan(size(X)); {V9}W< % z(idx) = zernfun(5,1,r(idx),theta(idx)); 9 k>=y n % figure wa4(tM2 % pcolor(x,x,z), shading interp /2PsC*y % axis square, colorbar SB` "%6 % title('Zernike function Z_5^1(r,\theta)') n`)wD~mk % s|=.L&" % Example 2: auT$-Ki8 % t#[u
X? % % Display the first 10 Zernike functions az ?2 % x = -1:0.01:1; iVGc\6+' % [X,Y] = meshgrid(x,x); 4FgY!k % [theta,r] = cart2pol(X,Y); p~THliwd % idx = r<=1; XZ8;Ow= % z = nan(size(X)); L]HYk}oD. % n = [0 1 1 2 2 2 3 3 3 3]; 0Ku%9wh- % m = [0 -1 1 -2 0 2 -3 -1 1 3]; Ev;ocb, % Nplot = [4 10 12 16 18 20 22 24 26 28]; ZM%z"hO9R % y = zernfun(n,m,r(idx),theta(idx)); R]{AJ"p % figure('Units','normalized') uua1_#a % for k = 1:10 B>&eciY % z(idx) = y(:,k); ku}I;k | % subplot(4,7,Nplot(k)) hq^@t6!C\m % pcolor(x,x,z), shading interp P>t[35/1 % set(gca,'XTick',[],'YTick',[])
S*1Km& % axis square )pXw 3Fo % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #D{//P|; % end ly]n2RK % ;hLne0|)} % See also ZERNPOL, ZERNFUN2. ~:%rg H k99ANW % Paul Fricker 11/13/2006 yxa~Rz/ &`'gO
9 mJ|7Jc % Check and prepare the inputs: rn?:utP % ----------------------------- o[!g,Gmoh if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) _8'F I_E3 error('zernfun:NMvectors','N and M must be vectors.') Q`}n;DV end E*Q><UU K>`7f]?H*e if length(n)~=length(m) #?z1cgCg error('zernfun:NMlength','N and M must be the same length.') ,e
~@ end jbqhNsTNK 1+^L,-k! n = n(:); :>[;XT< m = m(:); ?_F,HhQ if any(mod(n-m,2)) TvWhy`RQ error('zernfun:NMmultiplesof2', ... <Zc: 'All N and M must differ by multiples of 2 (including 0).') ?)cNe:KY end Ir*,fyl G1"=}Wt` if any(m>n) { [4Y(l1 error('zernfun:MlessthanN', ... 66%#$WH# 'Each M must be less than or equal to its corresponding N.')
U!-|.N, end ?6
"F.\O@ jz$)*Kdi* if any( r>1 | r<0 ) GGs3r;(t error('zernfun:Rlessthan1','All R must be between 0 and 1.') oWpy^=D_ end 8<t?o'9I k:w\4Oqd if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) r^?%N3 error('zernfun:RTHvector','R and THETA must be vectors.') OwQ 9y<v end E$FXs~a yV xR||e r = r(:); >=2nAv/( theta = theta(:); gTR:9E:B length_r = length(r); Wv"[,5
Z13 if length_r~=length(theta) PL8eM]XS error('zernfun:RTHlength', ... ,FJ9C3 'The number of R- and THETA-values must be equal.') (o\:rLZu end %rTXT
,h1r6&MEY % Check normalization: +MQf2|-- % -------------------- R9yK" if nargin==5 && ischar(nflag) P$@5&/] isnorm = strcmpi(nflag,'norm'); t9PS5O ; if ~isnorm 2D
MH@U2 error('zernfun:normalization','Unrecognized normalization flag.') Lvc*L6 end 1C=}4^Pu else f$k#\=2% isnorm = false; eR8qO"%2: end WZCX&ui | |