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 V"=(I'X
function z = zernfun(n,m,r,theta,nflag) mEsOYIu{
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. +$R4'{9q
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 6rlafISvO
% and angular frequency M, evaluated at positions (R,THETA) on the ?h)Z ;,}
% unit circle. N is a vector of positive integers (including 0), and I_66q7U"0
% M is a vector with the same number of elements as N. Each element Zhb)n
% k of M must be a positive integer, with possible values M(k) = -N(k) W.b?MPy]
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, "bZ{W(h
% and THETA is a vector of angles. R and THETA must have the same J
WaI[n}
% length. The output Z is a matrix with one column for every (N,M) ,YzrqVY
% pair, and one row for every (R,THETA) pair. ]#>;C: L
% 8,Iil:w
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike w_o|k&~,
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), `BA wef
% with delta(m,0) the Kronecker delta, is chosen so that the integral &wc%mQV
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Xk=bb267
% and theta=0 to theta=2*pi) is unity. For the non-normalized JD1IL` ta;
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ;w,+x 7
% ,{=pFs2
% The Zernike functions are an orthogonal basis on the unit circle. B;f\H,/59
% They are used in disciplines such as astronomy, optics, and P9(]9np,,
% optometry to describe functions on a circular domain. e@[9WnxYe
% +RLHe]9&
% The following table lists the first 15 Zernike functions. $*EK
v'g[n
% xW92ZuzSH
% n m Zernike function Normalization ox9$aBjJ
% -------------------------------------------------- 'r_{T=
% 0 0 1 1
[7d>c
% 1 1 r * cos(theta) 2 ,m<t/@^]
% 1 -1 r * sin(theta) 2 a(>oQG8F
% 2 -2 r^2 * cos(2*theta) sqrt(6) 20glz(
% 2 0 (2*r^2 - 1) sqrt(3) V2-fJ!
% 2 2 r^2 * sin(2*theta) sqrt(6) !Yuu~|
% 3 -3 r^3 * cos(3*theta) sqrt(8) E#'JYz@
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Y"J'
'K
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) daamP$h9
% 3 3 r^3 * sin(3*theta) sqrt(8) \va'>?#o1
% 4 -4 r^4 * cos(4*theta) sqrt(10) ux-puG
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) C4vmgl&
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ]ADj9
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) d&mSoPf
% 4 4 r^4 * sin(4*theta) sqrt(10) dUAZDoLi
% -------------------------------------------------- #NU;$&
% o/Z
% Example 1: K/)*P4C-
% t+C9QXY
% % Display the Zernike function Z(n=5,m=1) |l5ol@2*
% x = -1:0.01:1; vFuf{ @P
% [X,Y] = meshgrid(x,x); lP$bxUNt
% [theta,r] = cart2pol(X,Y); 1CS[%)-c
% idx = r<=1; ?LE\pk
R
% z = nan(size(X)); 1eiV[z$?
% z(idx) = zernfun(5,1,r(idx),theta(idx)); XN+~g.0
% figure FdrH,
% pcolor(x,x,z), shading interp _^{!`*S
% axis square, colorbar Nr24Rv
% title('Zernike function Z_5^1(r,\theta)') C*O648yz[
% ;IklS*p]
% Example 2: &
w%%{lM
% px `o.%`'
% % Display the first 10 Zernike functions t^N
92$|
% x = -1:0.01:1; VK/@jrL+
% [X,Y] = meshgrid(x,x); $nX4!X
% [theta,r] = cart2pol(X,Y); !nX}\lw
% idx = r<=1; \1k(4MWd
% z = nan(size(X)); ;%u'w;sgq
% n = [0 1 1 2 2 2 3 3 3 3]; fb8"hO]s
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; N!O.=>8<
% Nplot = [4 10 12 16 18 20 22 24 26 28]; NI(fJ%U
% y = zernfun(n,m,r(idx),theta(idx)); cRt[{HE
% figure('Units','normalized') ,:RHhg
% for k = 1:10 JY$B%R4;]
% z(idx) = y(:,k); / {|<3CEe
% subplot(4,7,Nplot(k)) Ps<6 kQ(
% pcolor(x,x,z), shading interp L`$m<9w'
% set(gca,'XTick',[],'YTick',[]) ]
T`6Hz!
% axis square _oOEMQb
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) s06tCwPp
% end GtYtB2U
% Dm=d
% See also ZERNPOL, ZERNFUN2. }o>6 y>=
T(<
[k:`
% Paul Fricker 11/13/2006 #.
mc+n:I
{Pi+VuLE
] qT\z<}
% Check and prepare the inputs: jlhyn0
% ----------------------------- CYIp 3D'k
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) irqNnnMGEa
error('zernfun:NMvectors','N and M must be vectors.') j/I^\Ms
end ;QMRm<CLV
hqEnD
if length(n)~=length(m) l)JNNcej
error('zernfun:NMlength','N and M must be the same length.') &``dI,NC
end f-Yp`lnn.d
["5Z=4
n = n(:); v
};r
m = m(:); )s @}|`
if any(mod(n-m,2)) 6[q<%wA
error('zernfun:NMmultiplesof2', ... >]6inS9
'All N and M must differ by multiples of 2 (including 0).') aSu6SU
end BQ&G7V
`5VEGSP]
if any(m>n) wi{qN___
error('zernfun:MlessthanN', ... A6?+$ Hr
'Each M must be less than or equal to its corresponding N.') B/P E{ /
end J;?#Zt]`L
KY1(yni&8[
if any( r>1 | r<0 ) 5C03)Go3Z
error('zernfun:Rlessthan1','All R must be between 0 and 1.') H;#3S<
end %RlG~a
wHGiN9A+
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) F*&A=@/3
error('zernfun:RTHvector','R and THETA must be vectors.') ]p,svevo
end l4 @
/}%$fB
r = r(:); eB]cPo4gW
theta = theta(:); L|O'X4"&_
length_r = length(r); (A\X+S(
if length_r~=length(theta) ;0)|c}n+.5
error('zernfun:RTHlength', ... }|MPQy
'The number of R- and THETA-values must be equal.') x1g0_&F
end )qgcz<p?W
'\vmm>
% Check normalization: 'X()|{
% -------------------- \KBE+yj
if nargin==5 && ischar(nflag) --(e(tvf
isnorm = strcmpi(nflag,'norm'); ck=x_HB1
if ~isnorm 4# pn]
error('zernfun:normalization','Unrecognized normalization flag.') z#$>f*b
end i)vbmV
else B%~hVpm,eM
isnorm = false; x.=Np\#\G-
end =}bDT2Nb
9Ai e$=
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TFIP>$*_C
% Compute the Zernike Polynomials ~ULD{Ov'F
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (\CT
"u-
|4=Du-e
% Determine the required powers of r: sj"zgE)
% ----------------------------------- `k ^d)9
m_abs = abs(m); )#^5$5
rpowers = []; qDMVZb-(#
for j = 1:length(n) )<fa1Gz#^
rpowers = [rpowers m_abs(j):2:n(j)]; f!3$xu5
end S;!l"1[;
rpowers = unique(rpowers); \!+sL JP
.3yoDab
% Pre-compute the values of r raised to the required powers, /QA:`_</oh
% and compile them in a matrix: /< OoZf+[
% ----------------------------- ;y"=3-=vM"
if rpowers(1)==0 **oaR
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 8'niew
5d
rpowern = cat(2,rpowern{:}); mes/gqrJ1I
rpowern = [ones(length_r,1) rpowern]; ]c67zyX=%
else .u+ZrA#
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); x#_\b-
rpowern = cat(2,rpowern{:}); }bU1wIW9I
end rA=iBb3`
oS2L"#
% Compute the values of the polynomials: Ne 2tfiI`
% -------------------------------------- =vd9mb-
y = zeros(length_r,length(n)); 1E1oy(\V
for j = 1:length(n) ws^ 7J/8
s = 0:(n(j)-m_abs(j))/2; X&s@S5=r]
pows = n(j):-2:m_abs(j); *OX;ZQg0
for k = length(s):-1:1 JO|%Vpco
p = (1-2*mod(s(k),2))* ... /h.hFM/
prod(2:(n(j)-s(k)))/ ... E41ay:duAl
prod(2:s(k))/ ... iSiez'
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... l\Q--
prod(2:((n(j)+m_abs(j))/2-s(k))); nIckI!U#D
idx = (pows(k)==rpowers); K!L0|WH%!
y(:,j) = y(:,j) + p*rpowern(:,idx); |
Ns-l
(l
end ,aA%,C.0U
:1O49g3R
if isnorm `$fKS24u
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); PP]Z~ne0X
end [EdX6
end j'2:z#
% END: Compute the Zernike Polynomials ,V>7eQt?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1@$n)r`
+NM`y=@@
% Compute the Zernike functions: %^zGM^PD
% ------------------------------ B&bQvdp
idx_pos = m>0; j\/Rjn+:[
idx_neg = m<0; v`G [6Z
i_[nW
z = y; dTATJ)NH
if any(idx_pos) y)Y0SY1\j
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); l-~
o&n
end a7Xa3 vlpO
if any(idx_neg) Ub"6OT1tl
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); x/)o'#d$|l
end <k c9KE
kuQ+MQHs
% EOF zernfun